# Recontextualized Knowledge and Narrative Coalitions on Telegram

Tom Willaert1,2
1Brussels School of Governance, Vrije Universiteit Brussel, Belgium
2imec-SMIT, Vrije Universiteit Brussel, Belgium
<EMAIL_ADDRESS>
###### Abstract

A defining characteristic of conspiracy texts is that they negotiate power and
identity by recontextualizing prior knowledge. This dynamic has been shown to
intensify on social media, where knowledge sources can readily be integrated
into antagonistic narratives through hyperlinks. The objective of the present
chapter is to further our understanding of this dynamic by surfacing and
examining 1) how online conspiracy narratives recontextualize prior knowledge
by coupling it with heterogeneous antagonistic elements, and 2) how such
recontextualizing narratives operate as connectors around which diverse actors
might form narrative coalitions. To this end, the chapter offers an empirical
analysis of links to prior knowledge in public messaging channels from the
Pushshift Telegram dataset. Using transferable methods from the field of
bibliometrics, we find that politically extreme Telegram channels engage with
a variety of established knowledge sources, including scientific journals,
scientific repositories and other sources associated with the system of
scholarly communication. Channels engaging with shared knowledge sources
thereby form narrative coalitions ranging from scientific and technological
imaginaries to far-right extremist and antisemitic conspiracy theories. Our
analysis of these coalitions reveals (i) linguistic, political, and thematic
forces that shape conspiracy narratives, (ii) emerging ideological,
epistemological and ontological positions associated with online conspiracism,
and (iii) how references to shared knowledge contribute to the communicability
of conspiracy narratives.

Book chapter submitted to Advances in Sociolinguistics (Bloomsbury)

##  1 Introduction

If textual genres can be analysed in terms of intertextual relations to other
discourses Briggs Bauman (1992), then the genre of conspiracy theory may well
be defined by its recontextualizations of prior knowledge. By referencing
academic papers, monographs, websites, and other knowledge sources, conspiracy
texts appropriate the citation apparatus of “conventional scholarship” in
order to lay claim to evidence-based observations about the world (Barkun,
2013, p. 7). Such references thus constitute a key instrument by which
conspiracy texts negotiate power and identity, as they allow texts to assert
authority over the established system of knowledge production, subsequently
subverting it to lend credibility to antagonistic and other potentially
harmful narratives Milani (2020); Rothermel (2023). Previous research has
demonstrated that this recontextualization of knowledge is intensified on
social media Tuters . (2023), where the use of hyperlinks (which may directly
point to external sources) or hashtags (which may connect content with over-
arching themes) affords the creation of structural couplings between
heterogeneous forms of knowledge and discourse Costas . (2021); Mason (2019).

The objective of the present study is to further our understanding of how
references to prior knowledge relate to the formation of narratives and
narrative coalitions on social media. The chapter thus aims to surface and
examine two structural functions of online conspiracy narratives, namely: 1)
how conspiracy narratives recontextualize prior knowledge by coupling it with
seemingly heterogenous elements, and 2) how such recontextualizing narratives
operate as connectors around which diverse actors might form narrative
coalitions.

To this end, we conduct an empirical analysis of references to prior knowledge
on Telegram, a messaging platform that has become strongly associated with the
propagation of conspiracy theories and other forms of disinformation Peeters
Willaert (2022); Willaert . (2022); Simon . (2022). This can be attributed to
Telegram’s loose stance on content moderation, which has been shown to attract
extreme actors previously “deplatformed” from other social media Rogers
(2020). Telegram offers such actors the possibility of creating one-way,
public messaging channels through which they may broadcast content to large
audiences of followers. As messages can be forwarded freely between channels,
the platform removes much of the friction that might impede the circulation of
conspiracy narratives Willaert (2023).

Building on these observations, the present chapter specifically traces the
uses of hyperlinks to external knowledge in messages from the multilingual
Pushshift Telegram Dataset Baumgartner . (2020), which contains the contents
of 28,000 Telegram channels “snowballed” from primarily English-language seed
channels pertaining to right-wing extremist politics and cryptocurrencies, and
which covers a period between 2015 and 2019. By zooming in on the object of
the hyperlink, and specifically hyperlinks introduced in the textual body of
messages in public Telegram channels, we aim to uncover the dynamics through
which conspiracy narratives and narrative coalitions form on the platform. Our
approach is thereby informed by two theoretical assumptions about the nature
of conspiracy narratives and the communities in which they propagate.

First, we subscribe to the notion that on a conceptual level, conspiracy
narratives, just like political narratives more generally, may be
characterized as sense-making devices Bruner (1991) through which individuals
and groups “construct disparate facts in [their] worlds and weave them
together cognitively”, enabling them to “understand themselves as political
beings” Patterson Monroe (1998). Channelling Mark Fenster, we are thus
primarily concerned here with how conspiracy narratives operate as
“organization[s] of data” (Fenster, 2008, p. 120), in which real-world
knowledge and observations (as referenced through hyperlinks) become
associated with seemingly heterogeneous concepts to form coherent explanations
for the world’s workings. Drawing on insights from structuralist narratology,
conspiracy narratives might thus be analysed as devices that organize
representations of people, places and other real-world observations according
to underlying plot structures. Archetypical examples include Freytag’s
narrative arc Freytag (1895), Propp’s morphology of the folktale Propp (1968),
or Campbell’s hero’s journey Campbell (2008). Likewise, narratives may
organize their constituent parts according to “actantial” models such as
Greimas’ actantial narrative schema Greimas (1987); Herman (2005), which
comprises structural relations between entities (e.g. concepts, characters).
This actantial perspective has proven productive for the analysis of
conspiracy narratives in social media environments, as it lends itself to
operationalizations in terms of networks and network analysis Tangherlini .
(2020). Research following this approach has thereby demonstrated that
conspiracy narratives in participatory web environments are marked by dynamics
of intensified “narrative convergence” Tuters Willaert (2022); Willaert
(2023), in which disparate concepts are rapidly linked together into over-
arching narratives - a dynamic that elsewhere has been referred to as a
“conspiracy singularity” Klein (2020); Merlan (2020).

Secondly, in addition to organizing heterogeneous concepts on a structural
level, conspiracy narratives have been demonstrated to contribute to the
formation and organization of communities Bessi . (2015). This observation
recalls Maarten Hajer’s concept of “discourse coalitions”, wherein “different
actors from various backgrounds form specific coalitions around specific story
lines. Story lines are the medium through which actors try to impose their
view of reality on others, suggest certain social positions and practices, and
criticize alternative social arrangements.” (Hajer, 2002, p. 47). Following
this description, the objective of the present chapter is to identify what we
will refer to as the “narrative coalitions” that form around shared
emplotments of prior knowledge in social media. The chapter thus
(simultaneously) addresses the following empirical research questions:

  1. 1.

Which knowledge sources are referenced in conspiratorial Telegram channels and
which heterogeneous concepts are they associated with to form narratives?

  2. 2.

What are the latent narrative coalitions that emerge when we examine
associations between Telegram channels based on such shared references to
prior knowledge?

As recontextualized knowledge constitutes a core element of conspiracy
theories, we expect this empirical tracing of knowledge sources on Telegram to
yield more general insights into the thematic, linguistic, and political
forces that shape online conspiracy narratives.

##  2 Methodology

###  2.1 Bibliographic coupling analysis

In order to mine traces of the (ab)uses of knowledge sources from large
collections of Telegram messages, we turn to transferable methods form the
quantitative field of bibliometrics. Pioneered by scholars Eugene Garfield and
Derek John de Solla Price in the 1960s, bibliometrics is primarily concerned
with “gauging the perception of academic publications” on the basis of
citations (Ball, 2021, p. 2). The central principle underpinning its wide
range of methods and metrics, is that publications that are cited frequently
might be considered important to a field of research, whereas those cited less
frequently might be considered less relevant (Ball, 2021, p. 2). Today, the
field has become negatively associated with academic power structures and the
evaluation of scholars or institutions based on the quantification of
scientific output (Pendlebury, 2021, p.30). Yet we adhere here to a more
descriptive interpretation of bibliometrics, which is closely aligned with its
original purposes, and in which its methods are primarily used to map
“knowledge flows”, the “diffusion of ideas”, and the underlying “intellectual
structures” that mark a given field of research (Zhao Strotmann, 2015, p.2).
Rather than investigating references in traditional scholarly publications
such as books or journal articles, we focus on hyperlinks referencing
knowledge sources in conspiratorial Telegram channels. This transfer of
methods hinges on the aforementioned observation that conspiratorial texts and
discourse have been argued to mimic the scholarly apparatus of references and
citations (Barkun, 2013, p. 7).

At the center of our investigation is a bibliographic coupling frequency (BCF)
analysis, which quantifies the strength of association between two papers
based on the number of references they share. It is thereby assumed that
papers sharing references to the same knowledge sources might also be
intellectually similar to each other (Garfield, 1968, p. 541). This reasoning
extends to the present situation in the sense that we may argue that Telegram
channels sharing links to similar knowledge sources can be considered to
belong to the same “narrative coalition” (see Figure 1).

Figure 1: Illustration of bibliographic coupling of two Telegram Channels.
Both Telegram channel 1 and Telegram channel 2 contain at least on reference
to url1, url2 and url3. As such it can be argued that both channels are
connected with a bibliographic coupling frequency of 3.

As demonstrated by previous work, a BCF approach readily lends itself to
operationalizations in the context of large collections of data mined from
social media Yang Colavizza (2022). Our data processing pipeline is thus
geared towards preprocessing and visualizing our collection of Telegram
messages in ways that directly speak to our research questions: first, it
allows us to produce an overview of the different types of knowledge sources
referenced in the Telegram channels, and secondly, it offers a network view of
the communities (narrative coalitions) among channels that can be identified
by means of a BCF analysis.

###  2.2 Data processing pipeline

Figure 2: Overview of the main steps in our processing pipeline.

The main steps in our pipeline are illustrated in Figure 2. In a first step,
we iterate over each message included in the Pushshift Telegram Dataset
Baumgartner . (2020), which contains 317 million Telegram messages posted by
2.2 million users in 28,000 channels (covering both public broadcasting
channels and chat groups) between 2015 and 2019. From each message, we extract
the URLs that are included in the message text. We thus retain a dataset of
12,938,085 messages that contain at least one URL. If a URL is a shortened
url, we first expand it using the python urlExpander library Yin (2018). We
then filter out the second-level domain from each URL, that is: the domain
directly below the top-level domain, which might, among other things, refer to
a country code (e.g. “.be”) or an organization (e.g. “.org”). We subsequently
normalize the URL strings for further classification. The URL
https://www.journals.elsevier.com/fuel-and-energy-abstracts is thus converted
to “elsevier.com”.111We make an exception here for the Google subdomains
“books.google.com”, “sites.google.com”, “scholar.google.com”,
“docs.google.com”, “drive.google.com”, which are retained in full. Other
Google subdomains (e.g. “play.google.com”) are reduced to “google.com”.

Next, we classify each simplified URL according to whether it refers to an
established knowledge source or not. We approach this classification
inductively by referencing each simplified URL in the Telegram dataset against
the list of sources included in the OpenAlex knowledge graph Priem . (2022).
OpenAlex is a citation database that presents itself as a “fully open catalog
of the global research system” that be queried via an API
(https://docs.openalex.org/). The knowledge base offers an open alternative to
paywalled knowledge bases such as Scopus or Web of Science, and indexes
249,000 academic sources, including “journals, conferences, preprint
repositories, and institutional repositories” (https://docs.openalex.org/api-
entities/sources). These sources are identified and aggregated from other
projects including the Directory of Open Access Journals (DOAJ), Pubmed, the
Internet Archive and institutional repositories
(https://docs.openalex.org/additional-help/faq). In order to construct a
reference corpus for classifying the URLs retrieved from the Telegram dataset,
we compile a list of all source URLs for works contained in the knowledge
graph (https://docs.openalex.org/api-entities/sources). These URLs are then
filtered and cleaned following the same procedure as the URLs extracted from
the Telegram dataset, thus retaining a list of 18,494 domains of OpenAlex
sources.

In a next step, we encode each of the retrieved URLs from the Telegram dataset
according to whether it occurs in the OpenAlex knowledge graph or not,
focusing our analysis on Telegram’s public broadcasting channels. This results
in an enriched dataset that we aggregate on channel level, retaining for each
channel the counts of the URLs per type (OpenAlex knowledge object or not), as
well as metadata including the channel title and channel description. We then
proceed to submit the dataset of channels to a bibliographic coupling
analysis, in which we quantify associations between channels on the basis of
the number of references to knowledge sources they share between them. To this
end, we follow the method outlined in Zhao Strotmann 2015, pp. 31-39.

First, we construct a citation matrix, in which we keep track of the
relationship between channels (columns) and sources (rows) (see Figure 3). If
a source is referenced by a channel (i.e. if there is at least one message in
the channel that cites it), we set the corresponding row value for that source
to 1, if not, we set it to 0. In this step, we reduce the dataset by excluding
the five sources that are cited by the highest number of channels (i.e. those
with the highest row sum in our citation matrix). We omit these channels
because they connect large numbers of channels in our bibliographic coupling
graph graph, and as such they would obscure the more fine-grained intellectual
structures of the network. We provide an overview of these highly frequent
sources in the Findings section.

Figure 3: Illustration of a simple citation matrix. Each column represents a
Telegram channel, rows represent sources. If a channel refers to a source, we
set the value to 1, otherwise we set the value to 0.

Conceptually, the citation matrix, which maps connections between channel
names and knowledge sources, provides initial insights into structural
connections between knowledge sources and channels. Channel names can thereby
be considered important indicators of the “heterogeneous” narratives that
these sources are enveloped in.

Based on the citation matrix, we proceed to calculate the bibliographic
coupling frequency between each pair of Telegram channels in our dataset.
Following the example in Figure 3, we can do this by calculating the scalar
product of their corresponding vectors (columns) (Zhao Strotmann, 2015, p.
39). For example, the BCF of Telegram channel 1 and Telegram channel 2 is
calculated as V(1,1,1,0) x V(1,1,1,0) = 3. The BCF between Telegram channel 1
and Telegram channel 3 is calculated as V(1,1,1,0) x V(0,0,0,1) = 0.

We then construct a graph representation of connections between channels based
on the bibliographic coupling frequency. In this graph, which we will refer to
as a bibliographic coupling network, each node represents a channel, and each
edge between nodes represents the bibliographic coupling frequency (BCF)
between those nodes. We then use methods from network analysis to analyse the
structural properties of the bibliographic coupling network. Specifically, we
use the Louvain community detection algorithm to find non-overlapping
groupings of nodes in the graph Blondel . (2008). Conceptually, this approach
yields insights into possible narrative coalitions among channels, that is:
communities of conspiratorial or otherwise antagonistic channels that engage
with similar knowledge sources.

##  3 Findings

###  3.1 Heterogenous couplings

In the Pushshift Telegram dataset, we find 24,198 public broadcasting channels
that contain URLs. Out of these, we extract the top 10,000 channels by the
total number of references to sources that also occur in the OpenAlex
knowledge graph. This results in a citation matrix with 10,000 channels and
77,747 different cited sources (including OpenAlex and non-OpenAlex sources).

Figure 4: Overview of 50 most frequent domains in top 10,000 channels by
number of links to OpenAlex sources, segmented by: combined source types (a),
OpenAlex sources (b), non-OpenAlex sources (c)

As follows from Figure 4(a), we find that the Telegram channels in our dataset
refer to a heterogeneous set of websites. 7 of the 50 URLs that are cited by
the highest number channels thereby also occur in the OpenAlex knowledge
graph. As shown in Figure 4(c), the list of non-OpenAlex sources cited by most
channels first and foremostly contains what elsewhere has been referred to as
“Very Large Online Platforms and Search Engines” (VLOPS) European Commission
(2024). This includes references to Youtube, Instagram, Twitter (presently
named “X”), Facebook, and Wikipedia. Of the non-VLOPS, Telegram and its text
and editor and publisher Telegraph figure prominently in the list
(“telegram.org”, “telegra.ph”). Additionally, the list also contains
alternatives to these established platforms, such as the Russian social
networking site VKontake (“vk.com”), the Iranian video sharing platform Aparat
(“aparat.com”), and the search engine Yandex (“yandex.ru”). Likewise the list
contains references to established news websites (“bbc.com”), and state-owned
counterparts, such as Russia Today (“rt.com”).222From our overview of most
cited sources, it follows that both the OpenAlex and non-OpenAlex lists
contain references to news outlets. However, it seems to be the case that
OpenAlex prioritizes actual publications (e.g. “time.com”) over general news
websites (e.g. “bbc.com”). This could be explained on the grounds of the
citability of the former: academic publications might be more likely to
reference the contents of specialized news publications. On a methodological
note, it should be pointed out that the domains “goog.le” and “bit.ly” refer
to shortend URLs that could no longer be expanded. This might be attributed to
changes in the target domain (removal, migration to another domain) that might
have happened between the collection of this dataset (with the most recent
posts dating from 2019) and the moment of URL expansion (2024).

As shown in Figure 4(b), the OpenAlex sources cited by the highest numbers of
channels comprise a variety of sources that (by virtue of their inclusion in
the knowledge graph), have at some point become associated with the
established “apparatus” of scholarly communication. This includes file sharing
and archiving websites (“docs.google.com”, “drive.google.com”, “archive.org”),
and established news sources (“theguardian.com”, “nytimes.com”,
“washingtonpost.com”). We also find sources that are more central to
scholarship and scholarly communication, including references to scientific
journals and domains of scientific publishers (“nature.com”, “sciencemag.org”,
“springer.com”, “wiley.com”, “sciencedirect.com”), scientific repositories
(“arxiv.org”), as well as digital object identifiers (DOIs) typically
associated with scientific publications (“doi.org”). Further, we find
references to domains of higher education institutions (“stanford.edu”,
“mit.edu”) and governmental organizations (“nih.org”, “nasa.gov”). Finally, we
also find references to scientific magazines (“scientificamerican.com”) and
academic social networking websites (“researchgate.net”).

From this overview, it follows that Telegram channels snowballed from a seed
of English-language, right-wing extremist channels actively engage with a
variety of sources. One trend that emerges when we examine the non-OpenAlex
sources, is that the Telegram channels in the dataset interact quite
prominently with links referring to sources either based in or focused on
Russia. Some of these (viz. Russia Today) are, at the moment of writing,
banned in the European Union on the grounds of spreading disinformation Chee
(2022). While it might not come as a surprise that such sources figure in a
dataset that has been characterized as a key resource for studying computer-
mediated disinformation campaigns (Baumgartner ., 2020, p. 6), their presence
does foreground the rather heterogeneous nature of those links that refer to
more established knowledge sources. How exactly do these sources, which are
part of the OpenAlex knowledge graph, connect to the overall antagonistic
nature of our corpus? In order to examine the exact nature of these
“heterogeneous couplings” between prior knowledge and elements of
conspiratorial narratives, we proceed to identify the narrative coalitions
among channels that form around these sources.

###  3.2 Narrative coalitions

We analyse narrative coalitions based on a bibliographic coupling analysis of
channels referencing sources included in the OpenAlex knowledge graph. To this
end, we first filter our citation matrix by removing all non-OpenAlex sources,
and then remove the five OpenAlex sources that are cited by the highest number
of channels, notably: “docs.google.com” (cited by 1182 channels),
“wordpress.com” (cited by 785 channels), “theguardian.com” (cited by 704
channels), “nytimes” (cited by 626) channels, and drive.google.come (cited by
626 channels). Based on this filtered citation matrix, we construct a
bibliographic coupling matrix of 10,000 channels; (i.e. it contains 10.000
rows and 10.000 columns). We then construct a graph from the bibliographic
coupling matrix, where nodes represent channels, and weighted edges the BCF
between two channels. We only retain those edges that represent a
bibliographic coupling frequency of at least 2. This yields a bibliographic
coupling graph of 1382 nodes and 56,338 edges. We then inductively identify
narrative coalitions using the Louvain Community detection algorithm. Each
node in the graph is thereby coloured by the community to which it pertains.
The resulting bibliographic coupling network is shown in Figure 5.

As follows from this figure, we can identify 6 main communities in our
dataset, which we will refer to as narrative coalitions. We thereby disregard
three smaller communities, each of which contains less than 1% of the channels
in the dataset. A general visual inspection of the graph reveals three forces
that shape the narrative alliances that can be observed. First, coalitions are
marked by shared intellectual or thematic orientations, which may or may not
reflect real-world events. Second, narrative coalitions are marked by a shared
regional or linguistic dimension. Third, narrative coalitions can be
distinguished from each other on the basis of their political orientation.

While there is a rich diversity of channels in each of these coalitions,
whereby politically or thematically opposed channels may contend and connect
over the same sources, the following sections provide a broad characterization
and analysis of each of these coalitions based on visual inspections of the
graph and the associations evoked by the names of the channels that make up
the coalitions. As will follow, many of these channel names contain vernacular
elements that can be considered indicative of subcultural communities and
their idiosyncratic narratives Peeters . (2021). Our investigation thus
reveals narrative coalitions pertaining to (i) scientific and technological
imaginaries, (ii) far-right extremism and antisemitic conspiracy theories,
(iii) anarchist and Marxist discourse, (iv) anti-vaccination conspiracy
theories, (v) feminist and anti-feminist discourse, (vi) and national
politics.

Figure 5: Visualization of bibliographic coupling network of Telegram
channels. Nodes represent channels, weighted edges the bibliographic coupling
frequency (BCF) between two channels based on OpenAlex sources cited by both
channels. Only edges representing a BCF higher than or equal to 2 are
retained. Nodes are coloured according to Louvain communities (narrative
coalitions). Node positioning: ForceAtlas2 algorithm, Node size: node degree.

####  3.2.1 Science and technology

The largest narrative coalition in our dataset is broadly concerned with the
fields of science and technology. It thus deals explicitly with questions of
(scientific) knowledge, and shapes a series of idiosyncratic ontological and
epistemological positions, some of which might be situated at the fringes of
scientific inquiry.

This narrative coalition is primarily concerned with fleshing out ontological
and aesthetic positions that imagine humanity’s future position in light of
technological advances. Channels thus recontextualize knowledge sources to
shape concepts of transhumanism (“transhumanism_in_our_hearts”), cyberpunk
(“Subhuman Cyberpunk”) and solarpunk (“solarpunk”, “lunarpunk”), as well as
futurism (“Futurismo”), Orwellian dystopianism (“Pretty Orwellian Premium
Morbid Reality”) and dataism (“data scientology”).

On an epistemological level, the coalition is marked by traces of skepticism
(“skeptic salmon”) and rationalism (“the Indian Rationalist”). It also
contains explicit references to alternative forms of knowledge and modalities
for accessing knowledge, ranging from the neutral “copyleft” to refer to legal
frameworks for open access to knowledge, to more marked forms of knowledge,
such as “pagan” knowledge (“Pagan Knowledge”, “The Viking Programmer”).
Likewise, the narrative coalition contains references to the “redpilling” meme
(“Redpill Dump”). This right-wing metaphor finds its origins in the 1999
science fiction movie The Matrix, and refers to a process in which “taking the
red pill” leads to an “awakening” or an “epiphany about the rightness of white
nationalism” Hagen (2020).

The latter resonates with other overtly racist and subcultural vernacular
channels in this coalition, as illustrated by channel names such as “Racially
charged memes and posts”, and “Anon Politics” (which recalls elements of the
anonymous messaging board 4chan). Moreover, this coalition engages explicitly
with instances of the Qanon conspiracy theory (“Qlobal-change Italia”).
Closely associated with the “pizzagate” conspiracy theory, QAnon revolves
around the false narrative that established, wealthy elites form Satanic
cabals and child molesting rings that are conspiring against Donald Trump
Tuters Willaert (2022).

Interwoven with the abovementioned channels are channels that engage with a
variety of scientific disciplines and subjects, including biology (“Science of
Biology”, “astrobiology”), mathematics (“Mathematics Association”),
engineering (“Mechanical Engineering ( ut_redc)”), robotics (“Robotrends.ru”),
neuroscience (“neuroscienceru”, “The Brains”), pharmaceutics (“Small Pharma”,
“Development of Pharmacy Channel”), psychology (“International Scientific
Group on Positive Psychology”), the humanities (“philosophy bookshelf”,
“english book house”, “Linguistics Library”, “translation studies”,
“ecolinguistics”), spiritualism (“Spiritual Science Channel”), theology
(“theology bites”), and atheism and agnosticism (“Atheists & Agnostics”). The
narrative coalition likewise contains marked stances on health and diet (“case
against veganism”), and “based” reflections on education (“based education
hub”). Another right-wing slang term, the word “based” has become associated
with varying meanings revolving around the concepts of “not caring about what
other people think” and “being true to oneself” Hagen de Zeeuw (2023).

While an in-depth discussion of the complex semantic and vernacular dimensions
of this coalition, as well as the alternative knowledge it propagates are
beyond the scope of this chapter, our overview clearly illustrates how a
narrative coalition pertaining to scientific imaginaries can readily be marked
by extreme political and ideological positions. The latter is reflected even
more so in our second coalition.

####  3.2.2 Far-right extremism

The second major narrative coalition in the dataset consists of channels
explicitly associated with right-wing extremism, memes and elements pertaining
to the “deep vernacular web” de Zeeuw Tuters (2020). Of all coalitions, this
one engages most overtly with antisemitic conspiracy theories.

Interconnected channels in the dataset propagate the aforementioned
“Pizzagate” conspiracy and the antisemitic conspiracy theories of Jewish
cabals and ritualistic murder (e.g. “Pizzagate/Ritualistic Jewish Cabals”,
“Jewish Ritual Murder Abortion Satanism”). It contains explicit traces of
holocaust denial (“Shoahoax”) and antisemitic rhetoric (“Judenpresse
Monitor/Archive”, “K i K E s C e N t R a L”), the term “kikes” being a
disparaging and offensive term to refer to Jews. Illustrative of an overall
dynamic of contestation over knowledge, we find that these channels populate
the same coalition as channels associated with news outlets of the Jewish
community (e.g. ‘Jewish News TM”).

Further conspiracy theory channels in this coalition include overtly
transphobic conspiracy theories (“Gay Trans P**ophile Agenda”), as well as the
“white genocide” conspiracy theory (“White Genocide Immigration Anti-White
Agenda”), a white supremacist conspiracy theory that propagates unfounded
beliefs of hidden plans geared towards the extinction of whites. The coalition
furthermore contains channels that propagate conspiracy theories surrounding
the “New World Order” (NWO) and the Illuminati (“Anti-nwo”,“Antiilluminaten
TV”). As in the previous coalition, channels also explicitly refer to the
QAnon Conspiracy theory and its “where we go one we go all” slogan (“Qlobal-
change”, “Brexiteers MBGA WWG1WGA”). These channels co-occur with channels
marked by explicit evocations of Nazism and fascism (e.g. “Fascist Library”,
“FacistBook News & Media”, “THE CATHOLIC WIGNAT”, “NazBol Party Club”) as well
as radical politics (e.g. “Radical Agenda”). The coalition also includes
channels pertaining to right-wing and conspiracist organisations (“Infowars
chat”, “ProudBoys”). These channels are interconnected with channels evoking
elements of the “deep vernacular web”, typically associated with the language
that characterizes platforms such as 4chan (e.g. “CUCKISTAN”, “Chadistan”,
“Dank Memes Gang”, “4Chan /POL/HIS/INT”), as well as the practice of
shitposting (“Radical Shitposts”, “Vaguely Philosophical Shitposts”). Just
like the previous coalition, this cluster also contains explicit references to
the concept of “redpilling” (e.g. “Random Redpills Compilation”, “The Ultimate
Redpill”, “Your daily redpill”, “B R E A D P I L L E D”). If anything, this
cluster of channels thus further emphasizes the extreme nature of the
discourse and narratives that subsume prior knowledge on Telegram.

####  3.2.3 Marxism, anarchism, and technology

The third narrative coalition that forms around established sources of
knowledge is comprised of channels associated with left-wing politics, Marxism
and anarchism, in particular as these relate to technology. Channels in the
coalition thus refer to left-wing politics (e.g. “Free Left”, “Center-left
Gang AMERICA GOOD edition”, “LeftyGram”), as well as communism (“Meta-
Communism”, “National Bolshevik Ideology”) and anarchism (“ANARCHObois”,
“Anarquismo (canal)”, “MemoriaAnarquista”, “Anarcho-sheet”). In the same
narrative alliance, we find channels dealing with digital technologies (e.g.
“e m a c s”, “Linuxgram”, “Stuff and Docs”, “DOFH - DevOps from the hell”,
“Technical Writing 101”), some explicitly opposing technology (“Neo-Luddite’s
Cabin”). Interconnected with these Marxist, anarchist and technology-oriented
channels are channels pertaining to specific fandoms, such as furry culture,
Taylor Swift, and the Teenage Mutant Ninja Turtles (“Taylor Swift Updates”,
“Furry Senate”, “AltFurry Meme Factory”, “TMNTFans”). This cluster likewise
promotes a number of idiosyncratic positions (e.g. “Pretty Trippy Premium
Hippie Shit”, “Macchiagram”).

This cluster speaks to the narrative convergence between political discourse,
technological imaginaries, and popular culture. As will follow, the latter in
particular contributes to what we may refer to as the “communicability” of
conspiracy theory, disinformation and other antagonistic narratives.

####  3.2.4 German-language diagonalism and conspiracism

The fourth narrative coalition in our dataset is a more heterogeneous
coalition, situated at the intersections of previously discussed coalitions.
This hetorogeneity is exemplified by channel names contriving concepts such as
“Black Gay Space Communism”, which recalls the “Fully Automated Luxury Gay
Space Communism” meme ironically presenting a communist utopia. The most
distinguishing feature of this coalition however is a cluster of German-
language channels. These pertain to diagonalism or the “querdenken” movement
(“NRW stellt sich quer Info”), as well as conspiracism (“Die Wahrheit und nur
die Wahrheit. Was die offiziellen Medien nie zeigen.”, “Impfkritik”,
“Unabhängiger Infokanal”) and patriottic resistance (“Unabhängiger
Infokanal”). This cluster of channels is clearly held together by an
interaction with shared knowledge sources that cater specifically to the
German context, which highlights a prominent regional and linguistic dimension
of this type of online conspiracy discourse.

####  3.2.5 Spanish-language feminism, anarchism, left-wing politics

The fifth narrative coalition that forms around established knowledge sources
is marked by Spanish-language channels pertaining to feminism, anarchism, and
left-wing politics. This narrative coalition thus contains references to far-
left political parties (“Partido Pirata de Chile”, “Partido Popular”,
“Secretaría Rural PODEMOS”) and labour movements (“Movimiento laboral
sindical”). The coalition also contains references to anarchism
(“AnarcoCanal”), guerrilla warfare (“Guerrilla”), political counter
information (“Contrainfo”), and environmentalism and sustainability
(“BlogSOSTENIBLE - Ecología y más”). Strikingly, the coalition contains a
cluster of channels referring to feminism (“FemiNews”, “Comuneras Feministas”,
“CANAL CÍRCULOS FEMINISTAS PODEMITAS”, “Economía feminita”). As in the case of
the aforementioned coalitions, we again find traces of interactions with
technology and cybersecurity ( “Sombrero Blanco - Ciberseguridad”). Another
recurring dynamic that can again be observed, is that the same sources are
contended over by channels with diverging orientations. In the cluster dealing
with feminism, we for instance find examples of channels taking up anti-
feminist positions (e.g. “El Feminismo es cáncer”).

####  3.2.6 Channels pertaining to Brazil

The final narrative coalition that we identify in our dataset primarily
consists of channels with titles in Spanish and Portuguese, many of which
pertain to the context of Brazil. Within this thematically diverse cluster, we
see tensions play out that also mark the dataset at large, notably opposing
channels laying claim to the same sources of knowledge. As such, this
heterogeneous cluster at the same time contains references to conservative
politics (“CONSERVADORES”, “Canal Conservador”) as well as LGBTQ+ channels
(“LGBT+ News”), along with “politically incorrect” channels (“Politicamente
incorreto”). As was the case for some of the aforementioned channels, these
clusters are interlaced with references to technology and popular culture,
such as references to the movie Fantastic Beasts and Where to Find them
(“Animais fantásticos e onde habitam”).

##  4 Discussion

In light of our twofold research question, we find that politically extreme
channels in the Pushshift dataset engage with a variety of established
knowledge sources, including scientific journals, scientific repositories and
other sources associated with the system of scholarly communication. Channels
engaging with shared knowledge sources thereby form narrative coalitions on
scientific and technological imaginaries, far-right extremism and antisemitic
conspiracy theories, Marxist and anarchist discourse, anti-vaccination
conspiracy theories (in German), left-wing political discourse, and regional
politics (pertaining to Brazil).

These empirical observations about the use and abuse of knowledge contribute
to our understanding of conspiracy texts on social media in three ways. First,
our network visualizations surfaces the forces and dynamics of power and
identity in which pre-existing knowledge is (re)claimed by sometimes
contradictory channels. We have thereby seen that large narrative coalitions
form over shared knowledge, which are shaped and held together by linguistic
ties, references to over-arching themes (e.g. transhumanism), or a pre-
occupation with similar real-world events. It has furthermore been shown how
within each of these narrative coalitions, similar forces are at work, leading
to similar sources being discussed and pulled on by opposing channels (e.g.
the coincidence of Jewish news channels and antisemitic conspiracy channels).

Second, our examination of Telegram channels through the lens of knowledge
sources inductively reveals a variety of ideological, epistemological and
ontological positions associated with online conspiracism. This includes
confluences between science and spiritualism, as well as other idiosyncratic
positions that might recontextualize, refute or otherwise subvert established
(scientific) evidence.

Third, our analysis produces further empirical support for the notion of the
“communicability” of conspiracy theories (Milani, 2020, p. 13) by
demonstrating how narrative coalitions of conspiracism form around shared
intertextual links to pre-existing knowledge. These intertextual networks
might contribute to the impression that the conspiracy narratives under
discussion are both accurate and coherent. It can be further be observed that
these conspiracy theories are normalized through their connections - via
shared references - with the seemingly mundane and the banal Milani (2020).
Specifically, conspiracy channels are embedded within networks that also
contain channels devoted to memes, shitposts, popular culture, fandoms, and
technology.

##  5 Conclusion and avenues for future research

By way of conclusion, we highlight the methodological contribution of this
study, which may be extended to future work.

For one thing, the use of transferable methods from the field of bibliometrics
proves to be a productive pathway for research into online disinformation and
conspiracy theories. This approach, as illustrated here by the use of
bibliographic coupling networks, presumes that we can repurpose methods
initially designed to study the intellectual structures of established
scientific fields to uncover and study similar structures in the realm of
(antagonistic) social media. We have referred to such structures as “narrative
coalitions”. In the course of this investigation, it has become clear that
methods from bibliometrics can be used as heuristics to identify (networks of
channels related to) conspiracy theories. The Pushshift dataset under
consideration was for instance not originally constructed with the specific
aim of studying online conspiracy theories. However, as may follow from the
analysis above, tracing bibliographic couplings among channels citing pre-
existing knowledge sources proves to be an effective method for identifying
meaningful clusters of channels dealing with conspiracy theories (cf.
antisemitism) as well as to clusters of channels that recontextualize channels
in interesting ways. Our initial experiences here warrant further explorations
of how methods that were initially designed to identify deviations from
scientific norms in the field might be of interest to identify those phenomena
that are of interest to the research in online conspiracy theories and
disinformation.

A second methodological implication of our approach is that it highlights the
significance of archival data for the analysis of knowledge and narratives on
social media. As access to social media data for research purposes becomes
increasingly more restricted, we have demonstrated that valuable insights can
be obtained from re-examining historical secondary datasets such as the
Pushshift Telegram archive, thus uncovering previously unexplored phenomena
and dynamics. Further methodological gains can be expected from cross-
referencing such archives with open datasets for bibliometrics research, as we
have illustrated here by means of the OpenAlex knowledge graph. By
incorporating more fine-grained classifications of knowledge sources (e.g. by
domain, by accessibility method (open access or not), or by focusing on
scientific research that was later retracted, we expect future research to
yield even more elaborate distant readings of how scientific knowledge is
refracted and recontextualized in social media texts.

##  6 Data and software availability statement

The Pushshift Telegram dataset analysed in this study is available in its
entirety on Zenodo via https://zenodo.org/record/3607497. Instructions for
downloading data from the OpenAlex knowledge graph are available via
https://docs.openalex.org/. The Python scripts accompanying this chapter are
available on GitHub via https://github.com/willaertt.

##  7 Ethics and personal data

No user information was processed for the present research: only data from the
messages and channels files were used. For details on how these files were
processed and which fields were used in the analysis, we refer to the detailed
methodological description in the chapter.

##  8 Funding

This project has been funded by the European Union’s Horizon Europe programme
under grant agreement ID 101094752: Social Media for Democracy (SoMe4Dem) -
Understanding the Causal Mechanisms of Digital Citizenship. The funders had no
role in study design, data collection and analysis, decision to publish, or
preparation of the manuscript.

## References

  * Ball (2021) ball2021introductionBall, R.  202101\.  Introduction Introduction.  R. Ball (), Handbook bibliometrics Handbook bibliometrics ( 1–3).  De Gruyter.  https://www.degruyter.com/document/doi/10.1515/9783110646610-001/html 10.1515/9783110646610-001
  * Barkun (2013) barkun2013cultureBarkun, M.  2013\.  A culture of conspiracy: apocalyptic visions in contemporary America A culture of conspiracy: apocalyptic visions in contemporary America (Second Edition ) ( 15).  BerkeleyUniversity of California Press.
  * Baumgartner . (2020) baumgartner2020pushshiftBaumgartner, J., Zannettou, S., Squire, M. Blackburn, J.  202001\.  The Pushshift Telegram Dataset. The Pushshift Telegram Dataset.  arXiv.  http://arxiv.org/abs/2001.08438 arXiv:2001.08438 [cs]  10.48550/arXiv.2001.08438
  * Bessi . (2015) bessi2015scienceBessi, A., Coletto, M., Davidescu, GA., Scala, A., Caldarelli, G. Quattrociocchi, W.  201502\.  Science vs Conspiracy: Collective Narratives in the Age of Misinformation Science vs Conspiracy: Collective Narratives in the Age of Misinformation.  PLOS ONE102e0118093.  https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0118093 Publisher: Public Library of Science  10.1371/journal.pone.0118093
  * Blondel . (2008) blondel2008fastBlondel, VD., Guillaume, JL., Lambiotte, R. Lefebvre, E.  200810\.  Fast unfolding of communities in large networks Fast unfolding of communities in large networks.  Journal of Statistical Mechanics: Theory and Experiment200810P10008.  https://dx.doi.org/10.1088/1742-5468/2008/10/P10008 10.1088/1742-5468/2008/10/P10008
  * Briggs Bauman (1992) briggs1992genreBriggs, CL. Bauman, R.  1992\.  Genre, Intertextuality, and Social Power Genre, Intertextuality, and Social Power.  Journal of Linguistic Anthropology22131–172.  https://www.jstor.org/stable/43102167 Publisher: [American Anthropological Association, Wiley]
  * Bruner (1991) bruner1991narrativeBruner, J.  1991\.  The narrative construction of reality The narrative construction of reality.  Critical inquiry1811–21.
  * Campbell (2008) campbell2008heroCampbell, J.  2008\.  The hero with a thousand faces The hero with a thousand faces ( 17).  New World Library.
  * Chee (2022) chee2022russiaChee, FY.  202207\.  Russia Today loses fight against EU ban, Moscow warns of retaliation Russia Today loses fight against EU ban, Moscow warns of retaliation.  Reuters.  https://www.reuters.com/business/media-telecom/eu-court-backs-eu-ban-russia-today-2022-07-27/
  * Costas . (2021) costas2021heterogeneousCostas, R., de Rijcke, S. Marres, N.  2021\.  “Heterogeneous couplings”: Operationalizing network perspectives to study science-society interactions through social media metrics “Heterogeneous couplings”: Operationalizing network perspectives to study science-society interactions through social media metrics.  Journal of the Association for Information Science and Technology725595–610.  https://onlinelibrary.wiley.com/doi/abs/10.1002/asi.24427 _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/asi.24427  10.1002/asi.24427
  * de Zeeuw Tuters (2020) dezeeuw2020tehde Zeeuw, D. Tuters, M.  202007\.  Teh Internet Is Serious Business: On the Deep Vernacular Web and Its Discontents Teh Internet Is Serious Business: On the Deep Vernacular Web and Its Discontents.  Cultural Politics162214–232.  https://doi.org/10.1215/17432197-8233406 10.1215/17432197-8233406
  * European Commission (2024) ec2024dsaEuropean Commission.  202402\.  DSA: Very large online platforms and search engines | Shaping Europe’s digital future. DSA: Very large online platforms and search engines | Shaping Europe’s digital future.  https://digital-strategy.ec.europa.eu/en/policies/dsa-vlops.
  * Fenster (2008) fenster2008conspiracyFenster, M.  2008\.  Conspiracy theories: secrecy and power in American culture Conspiracy theories: secrecy and power in American culture (Rev. and updated ed ).  MinneapolisUniversity of Minnesota Press.  OCLC: ocn212908873
  * Freytag (1895) freytag1895techniqueFreytag, G.  1895\.  Technique of the drama: An exposition of dramatic composition and art Technique of the drama: An exposition of dramatic composition and art.  S. Griggs.
  * Garfield (1968) garfield1968worldGarfield, E.  1968\.  World brain or Memex? Mechanical and intellectual requirements for universal bibliographic control World brain or memex? mechanical and intellectual requirements for universal bibliographic control.  The foundations of access to knowledge: A symposium The foundations of access to knowledge: A symposium ( 169).
  * Greimas (1987) greimas1987actantsGreimas, A.  1987\.  Actants, Actors and Figures Actants, actors and figures.  PJ. Perron FH. Collins (), On meaning: selected writings in semiotic theory On meaning: selected writings in semiotic theory ( 106–120).  MinneapolisUniversity of Minnesota Press.
  * Hagen (2020) hagen2020reactionaryHagen, S.  202005\.  Reactionary Wokeness: How Redpilling Became a Thing on Reddit – OILab. Reactionary Wokeness: How Redpilling Became a Thing on Reddit – OILab.  https://oilab.eu/reactionary-wokeness-how-redpilling-became-a-thing-on-reddit/.
  * Hagen de Zeeuw (2023) hagen2023basedHagen, S. de Zeeuw, D.  202301\.  Based and confused: Tracing the political connotations of a memetic phrase across the web Based and confused: Tracing the political connotations of a memetic phrase across the web.  Big Data & Society10120539517231163175.  https://doi.org/10.1177/20539517231163175 Publisher: SAGE Publications Ltd  10.1177/20539517231163175
  * Hajer (2002) hajer2002discourseHajer, MA.  2002\.  Discourse coalitions and the institutionalization of practice: the case of acid rain in Great Britain Discourse coalitions and the institutionalization of practice: the case of acid rain in great britain.  The Argumentative Turn in Policy Analysis and Planning The argumentative turn in policy analysis and planning ( 51–84).  Routledge.
  * Herman (2005) herman2005actantHerman, D.  2005\.  Actant Actant.  D. Herman, M. Jahn ML. Ryan (), Routledge Encyclopedia of Narrative Theory Routledge encyclopedia of narrative theory ( 1–2).  London and New YorkRoutledge.
  * Klein (2020) klein2020greatKlein, N.  202012\.  The Great Reset Conspiracy Smoothie. The Great Reset Conspiracy Smoothie.  https://theintercept.com/2020/12/08/great-reset-conspiracy/.
  * Mason (2019) mason2019intertextualityMason, J.  2019\.  Intertextuality in Practice Intertextuality in Practice ( 33).  AmsterdamJohn Benjamins Publishing Company.  http://www.jbe-platform.com/content/books/9789027262318 10.1075/lal.33
  * Merlan (2020) merlan2020conspiracyMerlan, A.  2020\.  The Conspiracy Singularity Has Arrived. The Conspiracy Singularity Has Arrived.  https://www.vice.com/en/article/v7gz53/the-conspiracy-singularity-has-arrived.
  * Milani (2020) milani2020nogoMilani, TM.  202007\.  No-go zones in Sweden: The infectious communicability of evil No-go zones in Sweden: The infectious communicability of evil.  Language, Culture and Society217–36.  http://www.jbe-platform.com/content/journals/10.1075/lcs.19014.mil 10.1075/lcs.19014.mil
  * Patterson Monroe (1998) patterson1998narrativePatterson, M. Monroe, KR.  1998\.  Narrative in political science Narrative in political science.  Annual Review of Political Science11315–331.  https://doi.org/10.1146/annurev.polisci.1.1.315 10.1146/annurev.polisci.1.1.315
  * Peeters . (2021) peeters2021vernacularPeeters, S., Tuters, M., Willaert, T. de Zeeuw, D.  2021Aug.  On the Vernacular Language Games of an Antagonistic Online Subculture On the vernacular language games of an antagonistic online subculture.  Frontiers in Big Data4718368.  10.3389/fdata.2021.718368
  * Peeters Willaert (2022) peeters2022telegramPeeters, S. Willaert, T.  2022\.  Telegram and Digital Methods. Mapping Networked Conspiracy Theories through Platform Affordances Telegram and digital methods. mapping networked conspiracy theories through platform affordances.  M/C Journal25.  https://doi.org/10.5204/mcj.2878
  * Pendlebury (2021) pendlebury2021eugenePendlebury, DA.  202101\.  Eugene Garfield and the Institute for Scientific Information Eugene garfield and the institute for scientific information.  R. Ball (), Handbook bibliometrics Handbook bibliometrics ( 27–40).  De Gruyter.  https://www.degruyter.com/document/doi/10.1515/9783110646610-001/html 10.1515/9783110646610-001
  * Priem . (2022) priem2022openalexPriem, J., Piwowar, H. Orr, R.  202206\.  OpenAlex: A fully-open index of scholarly works, authors, venues, institutions, and concepts. OpenAlex: A fully-open index of scholarly works, authors, venues, institutions, and concepts.  arXiv.  http://arxiv.org/abs/2205.01833 arXiv:2205.01833 [cs]  10.48550/arXiv.2205.01833
  * Propp (1968) propp1968morphologyPropp, V.  1968\.  Morphology of the Folktale: Second Edition Morphology of the Folktale: Second Edition.  University of Texas Press.  https://books.google.be/books?id=3Md3u9UPgOEC
  * Rogers (2020) rogers2020deplatformingRogers, R.  2020\.  Deplatforming: Following extreme Internet celebrities to Telegram and alternative social media Deplatforming: Following extreme internet celebrities to telegram and alternative social media.  European Journal of Communication353213–229.  https://doi.org/10.1177/0267323120922066 10.1177/0267323120922066
  * Rothermel (2023) rothermel2023roleRothermel, AK.  202301\.  The role of evidence-based misogyny in antifeminist online communities of the ‘manosphere’ The role of evidence-based misogyny in antifeminist online communities of the ‘manosphere’.  Big Data & Society10120539517221145671.  https://doi.org/10.1177/20539517221145671 Publisher: SAGE Publications Ltd  10.1177/20539517221145671
  * Simon . (2022) simon2022linkedSimon, M., Welbers, K., Kroon, AC. Trilling, D.  2022\.  Linked in the dark: A network approach to understanding information flows within the Dutch Telegramsphere Linked in the dark: A network approach to understanding information flows within the Dutch Telegramsphere.  Information, Communication & Society001–25.  https://doi.org/10.1080/1369118X.2022.2133549 Publisher: Routledge _eprint: https://doi.org/10.1080/1369118X.2022.2133549  10.1080/1369118X.2022.2133549
  * Tangherlini . (2020) tangherlini2020automatedTangherlini, TR., Shahsavari, S., Shahbazi, B., Ebrahimzadeh, E. Roychowdhury, V.  202006\.  An Automated Pipeline for the Discovery of Conspiracy and Conspiracy Theory Narrative Frameworks: Bridgegate, Pizzagate and Storytelling on the Web An automated pipeline for the discovery of conspiracy and conspiracy theory narrative frameworks: Bridgegate, pizzagate and storytelling on the web.  PLOS ONE1561–39.  https://doi.org/10.1371/journal.pone.0233879 10.1371/journal.pone.0233879
  * Tuters Willaert (2022) tuters2022deepTuters, M. Willaert, T.  2022Aug.  Deep state phobia: Narrative convergence in coronavirus conspiracism on Instagram Deep state phobia: Narrative convergence in coronavirus conspiracism on instagram.  Convergence2841214–1238.  10.1177/13548565221118751
  * Tuters . (2023) tuters2023scienceTuters, M., Willaert, T. Meyer, T.  2023\.  How Science Gets Drawn Into Global Conspiracy Narratives How science gets drawn into global conspiracy narratives.  Issues in Science and Technology391232–36.  10.58875/POZR1536
  * Willaert (2023) willaert2023computationalWillaert, T.  202311\.  A computational analysis of Telegram’s narrative affordances A computational analysis of telegram’s narrative affordances.  PLOS ONE18111-23.  https://doi.org/10.1371/journal.pone.0293508 10.1371/journal.pone.0293508
  * Willaert . (2022) willaert2022disinformationWillaert, T., Peeters, S., Seijbel, J. Van Raemdonck, N.  2022Sep..  Disinformation Networks: A Quali-Quantitative Investigation of Antagonistic Dutch-speaking Telegram Channels Disinformation networks: A quali-quantitative investigation of antagonistic dutch-speaking telegram channels.  First Monday275.  https://firstmonday.org/ojs/index.php/fm/article/view/12533 10.5210/fm.v27i5.12533
  * Yang Colavizza (2022) yang2022mapYang, P. Colavizza, G.  202204\.  A Map of Science in Wikipedia A Map of Science in Wikipedia.  Companion Proceedings of the Web Conference 2022 Companion Proceedings of the Web Conference 2022 ( 1289–1300).  Virtual Event, Lyon FranceACM.  https://dl.acm.org/doi/10.1145/3487553.3524925 10.1145/3487553.3524925
  * Yin (2018) leon2018urlexpanderYin, L.  201808\.  SMAPPNYU/urlExpander: Initial release. Smappnyu/urlexpander: Initial release.  https://doi.org/10.5281/zenodo.1345144.  10.5281/zenodo.1345144
  * Zhao Strotmann (2015) zhao2015analysisZhao, D. Strotmann, A.  2015\.  Analysis and Visualization of Citation Networks Analysis and Visualization of Citation Networks.  ChamSpringer International Publishing.  https://link.springer.com/10.1007/978-3-031-02291-3 10.1007/978-3-031-02291-3


# The restricted minimum density power divergence estimator for non-
destructive one-shot device testing the under step-stress model with
exponential lifetimes

Narayanaswamy Balakrishnan, María Jaenada and Leandro Pardo

###### Abstract

One-shot devices data represent an extreme case of interval censoring. Some
kind of one-shot units do not get destroyed when tested, and so, survival
units can continue within the test providing extra information about their
lifetime. Moreover, one-shot devices may last for long times under normal
operating conditions, and so accelerated life tests (ALTs) may be used for
inference. ALTs relate the lifetime distribution of an unit with the stress
level at which it is tested via log-linear relationship. Then, mean lifetime
of the devices are reduced during the test by increasing the stress level and
inference results on increased stress levels can be easily extrapolated to
normal operating conditions. In particular, the step-stress ALT model
increases the stress level at pre-fixed times gradually during the life-
testing experiment, which may be specially advantageous for non-destructive
one-shot devices. However, when the number of units under test are few,
outlying data may greatly influence the parameter estimation. In this paper,
we develop robust restricted estimators based on the density power divergence
(DPD) under linearly restricted subspaces, for non-destructive one-shot
devices under the step-stress ALTs with exponential lifetime distributions. We
theoretically study the asymptotic and robustness properties of the restricted
estimators and we empirically illustrate such properties through a simulation
study.

Keywords : Accelerated lifetests; Exponential lifetime distributions; One-shot
devices; Restricted Minimum Density Power Divergence Estimator.

##  1 Introduction

One-shot devices, an extreme case of interval censoring, play an important
role in survival analysis. One-shot devices can be only tested at some
discrete inspections times, so we can only know if a test unit have failed or
not at certain fixed times. Real-life one-shot devices usually have large mean
lifetimes under normal operating conditions, and so accelerated life tests
(ALTs) plans may be useful to infer on their reliability. ALTs plans assume
that the mean lifetime of the devices is related to the stress level at which
units are tested, and therefore they accelerate the time to failure by
increasing the stress level.

Generally, one-shot devices are assumed to get destroyed when tested, and so
one-shot data are right and left censoring. However, the non-destructiveness
assumption may not be necessary in many practical situations. In this paper,
we focus on these non-destructive one-shot devices and we study some inference
methods for analyzing their lifetime characteristics. The non-destructiveness
condition allows surviving units to continue in the experiment, providing
extra information about their lifetime distribution.

In this context, step-stress ALTs, which increase the stress level
progressively during the experiment at certain pre-specified times (known as
times of stress change), make the best use of the non-destructive devices
under tests. Here, we assume that the lifetime distribution of the one-shot
device at one stress level is related to the distribution at preceding stress
levels by assuming the residual life of the device depends only on the
cumulative exposure it had experienced, with no memory of how this exposure
was accumulated. We consider a multiple step-stress ALT with $k$ ordered
stress levels, $x_{1}<x_{2}<\dots<x_{k}$ and their corresponding times of
stress change $\tau_{1}<\tau_{2}\dots<\tau_{k}.$ We assume the lifetimes of
one-shot devices follows an exponential distribution, which is widely used as
a simple lifetime model in engineering and physical sciences. The cumulative
exposure model describes the lifetime distribution of a device as

$G_{T}(t)=\begin{cases}G_{1}(t)=1-e^{-\lambda_{1}t},&0<t<\tau_{1}\\\
G_{2}\left(t+a_{1}-\tau_{1}\right)=1-e^{-\lambda_{2}(t+a_{1}-\tau_{1})},&\tau_{1}\leq
t<\tau_{2}\\\ \vdots&\vdots\\\
G_{k}\left(t+a_{k-1}-\tau_{k-1}\right)=1-e^{-\lambda_{k}(t+a_{k-1}-\tau_{k-1})},&\tau_{k-1}\leq
t<\infty,\\\ \end{cases}$ (1)

with

$a_{i-1}=\frac{\sum_{l=1}^{i-1}\left(\tau_{l}-\tau_{l-1}\right)\lambda_{l}}{\lambda_{i}},\hskip
8.5359pti=1,...,k-1.$ (2)

and

$\lambda_{i}(\boldsymbol{\theta})=\theta_{0}\exp(\theta_{1}x_{i}),\hskip
8.5359pti=1,..,k,$ (3)

where
$\boldsymbol{\theta}=(\theta_{0},\theta_{1})\in\mathbb{R}^{+}\times\mathbb{R}=\Theta$
is an unknown parameter vector of the model. The log-linear relation in (3) is
frequently assumed in accelerated life test models, as it can be shown to be
equivalent to the well-known inverse power law model or the Arrhenius reaction
rate model.

Now, let consider a grid of inspection times, $t_{1}<t_{2}<\dots<t_{L}$,
containing all times of stress change. The probability of failure within the
interval $(t_{j-1},t_{j}]$ is given by

$\pi_{j}(\boldsymbol{\theta})=G_{T}(t_{j})-G_{T}(t_{j-1}),\hskip
8.5359ptj=1,..,L,$ (4)

and the probability of survival at the end of the experiment is
$\pi_{L+1}(\boldsymbol{\theta})=1-G_{T}(t_{L}).$

Classical inferential methods for one-shot are based on the maximum likelihood
estimator (MLE), which is very efficient by it lacks of robustness. To
overcome the robustness drawback, Balakrishnan et al. (2022) proposed robust
estimators for one-shot devices based on the popular density power divergence
(DPD) (Basu et al. 1998) under exponential lifetimes. They developed minimum
DPD estimators (MDPPE) as well as Wald-type test based on them, and studied
theoretically and empirically their asymptotic and robustness properties.

On the other hand, some inferential procedures, as Rao-type tests are based on
restricted estimators. Basu et al. (2018) developed robust restricted
estimators based on the DPD for general statistical models, and derived their
asymptotic distribution and robustness properties. Jaenada et al. (2022)
extended the theory using the Rényi pseudistance and developed some testing
procedures based on the restricted estimators.

In this paper, we develop restricted MDPDE under linearly constrained
subspaces for non-destructive one-shot devices tested under step-stress ALT.
In Section 2 we define the restricted MDPDE, and we state its asymptotic
distribution. Section 3 theoretically analyzes the robustness of the
restricted MDPDEs through its Influence Function (IF). Finally, in Section 4 a
simulation study is carried out to evaluate the performance of the proposed
estimators under different scenarios of contamination.

##  2 Minimum density power divergence estimator and Restricted minimum
density power divergence estimator

The density DPD family represents a rich class of density based divergences.
It is indexed by a tuning parameter $\beta\geq 0$ controlling the trade-off
between robustness and efficiency. Let consider $(n_{1},...,n_{L+1})$ a sample
of one-shot data. The empirical probability vector of a multinomial model can
be defined as

$\widehat{\boldsymbol{p}}=(\frac{n_{1}}{N},...,\frac{n_{L+1}}{N}).$ (5)

For step-stress ALT for one-shot devices under exponential lifetime
distributions, the DPD between the the empirical and theoretical probability
vectors, defined in (4) and (5), respectively, is given by

$d_{\beta}\left(\widehat{\boldsymbol{p}},\boldsymbol{\pi}\left(\boldsymbol{\theta}\right)\right)=\sum_{j=1}^{L+1}\left(\pi_{j}(\boldsymbol{\theta})^{1+\beta}-\left(1+\frac{1}{\beta}\right)\widehat{p}_{j}\pi_{j}(\boldsymbol{\theta})^{\beta}+\frac{1}{\beta}\widehat{p}_{j}^{\beta+1}\right),$
(6)

From the above, we define the MDPDE for the step-stress ALT model with one-
shot devices as

$\boldsymbol{\widehat{\theta}}^{\beta}=\left(\widehat{\theta}_{0}^{\beta},\widehat{\theta}_{1}^{\beta}\right)=\operatorname{arg}\operatorname{min}_{\boldsymbol{\theta}\in\Theta}d_{\beta}\left(\widehat{\boldsymbol{p}},\boldsymbol{\pi}\left(\boldsymbol{\theta}\right)\right).$
(7)

Note that, at $\beta=0$, the DPD coincides with the Kullback-Leibler
divergence and so the MDPDE for $\beta=0$ coincides with the MLE.

In many practical situations it may be of interest to reduce the parameter
space to values of $\boldsymbol{\theta}$ satisfying a linear constraint of the
form

$g(\boldsymbol{\theta})=\boldsymbol{m}^{T}\boldsymbol{\theta}-d=0,$ (8)

with $\boldsymbol{m}=(m_{0},m_{1})^{T}\in\mathbb{R}^{2}$ and $d\in\mathbb{R}.$
Accordingly, the restricted MDPDE, $\widetilde{\boldsymbol{\theta}}^{\beta}$
is defined by

$\widetilde{\boldsymbol{\theta}}^{\beta}=\operatorname{arg}\operatorname{min}_{\boldsymbol{\theta\in\Theta_{0}}}d_{\beta}(\widehat{\boldsymbol{p}},\boldsymbol{\pi}(\boldsymbol{\theta})).$
(9)

Since the restricted MDPPE is a constrained minimum, its estimating equations
can be written in terms of Lagrange multipliers. That is, the MDPDE restricted
to the the linear constraint (8), $\widetilde{\boldsymbol{\theta}}^{\beta},$
must satisfy the restricted equations

$\boldsymbol{W}^{T}\boldsymbol{D}_{\boldsymbol{\pi}(\widetilde{\boldsymbol{\theta}}^{\beta})}^{\beta-1}\left(\widehat{\boldsymbol{p}}-\boldsymbol{\pi}(\widetilde{\boldsymbol{\theta}}^{\beta})\right)+\boldsymbol{m}\widetilde{\boldsymbol{\lambda}}=\boldsymbol{0}_{2},$
(10)

for some vector $\widetilde{\boldsymbol{\lambda}}$ of Lagrangian multipliers,
where $\boldsymbol{0}_{2}$ is the 2-dimensional null vector,
$\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta})}$ denotes a
$(L+1)\times(L+1)$ diagonal matrix with diagonal entries
$\pi_{j}(\boldsymbol{\theta}),$ $j=1,...,L+1,$ and $\boldsymbol{W}$ is a
$(L+1)\times 2$ matrix with rows
$\boldsymbol{w}_{j}=\boldsymbol{z}_{j}-\boldsymbol{z}_{j-1},$ where

$\displaystyle\boldsymbol{z}_{j}$
$\displaystyle=g_{T}(t_{j})\begin{pmatrix}\frac{t_{j}+a_{i-1}-\tau_{i-1}}{\theta_{0}}\\\
(t_{j}+a_{i-1}-\tau_{i-1})x_{i}+a_{i-1}^{\ast}\end{pmatrix},\hskip
8.5359ptj=1,...,L,$ (11) $\displaystyle a_{i-1}^{\ast}$
$\displaystyle=\frac{1}{\lambda_{i}}\sum_{l=1}^{i-1}\lambda_{l}\left(\tau_{l}-\tau_{l-1}\right)(-x_{i}+x_{l}),\hskip
8.5359pti=2,..,k,$ (12)

$\boldsymbol{z}_{-1}=\boldsymbol{z}_{L+1}=\boldsymbol{0}$ and $i$ is the
stress level at which the units are tested after the $j-$th inspection time.

The next theorem states the asymptotic distribution of the restricted MDPDE
for non-destructive one-shot devices under the step-stress ALT model.

###### Theorem 1

Let $\boldsymbol{\theta}_{0}$ be the true value of the parameter
$\boldsymbol{\theta}$ and assume that $g(\boldsymbol{\theta}_{0})=0$ with
$g(\cdot)$ defined in (8). The asymptotic distribution of the restricted MDPDE
for the step-stress ALT model under exponential lifetimes,
$\tilde{\boldsymbol{\theta}}^{\beta},$ obtained under the constraint
$g(\boldsymbol{\theta})=0,$ is given by

$\sqrt{N}\left(\widetilde{\boldsymbol{\theta}}^{\beta}-\boldsymbol{\theta}_{0}\right)\xrightarrow[N\rightarrow\infty]{L}\mathcal{N}\left(\boldsymbol{0},\Sigma_{\beta}(\boldsymbol{\theta}_{0})\right)$

where

$\displaystyle\Sigma_{\beta}(\boldsymbol{\theta}_{0})$
$\displaystyle=\boldsymbol{P}_{\beta}(\boldsymbol{\theta}_{0})\boldsymbol{K}_{\beta}(\boldsymbol{\theta}_{0})\boldsymbol{P}_{\beta}(\boldsymbol{\theta}_{0}),$
(13) $\displaystyle\boldsymbol{P}_{\beta}(\boldsymbol{\theta}_{0})$
$\displaystyle=\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})^{-1}-\boldsymbol{Q}_{\beta}(\boldsymbol{\theta}_{0})\boldsymbol{m}^{T}\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})^{-1},$
$\displaystyle\boldsymbol{Q}_{\beta}(\boldsymbol{\theta}_{0})$
$\displaystyle=\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})^{-1}\boldsymbol{m}(\boldsymbol{m}^{T}\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})^{-1}\boldsymbol{m})^{-1},$

with

$\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})=\boldsymbol{W}^{T}D_{\boldsymbol{\pi}(\boldsymbol{\theta_{0}})}^{\beta-1}\boldsymbol{W},\hskip
8.5359pt\hskip
8.5359pt\boldsymbol{K}_{\beta}(\boldsymbol{\theta}_{0})=\boldsymbol{W}^{T}\left(D_{\boldsymbol{\pi}(\boldsymbol{\theta_{0}})}^{2\beta-1}-\boldsymbol{\pi}(\boldsymbol{\theta}_{0})^{\beta}\boldsymbol{\pi}(\boldsymbol{\theta}_{0})^{\beta
T}\right)\boldsymbol{W},$ (14)

$D_{\boldsymbol{\pi}(\boldsymbol{\theta_{0}})}$ denotes the diagonal matrix
with entries $\pi_{j}(\boldsymbol{\theta_{0}}),$ $j=1,...,L+1,$ and
$\boldsymbol{\pi}(\boldsymbol{\theta}_{0})^{\beta}$ denotes the vector with
components $\pi_{j}(\boldsymbol{\theta}_{0})^{\beta}.$

The proof follows from Theorem 2 of Basu et al. (2018) and Result 3 of
Balakrishnan et al. (2022).

##  3 Influence function of the restricted minimum density power divergence
estimator

The robustness of an estimator is widely analyzed using the concept of
Influence Function (IF), first introduced in Hampel et al. (1986).
Intuitively, the IF describes the effect of an infinitesimal contamination of
the model on the estimate. Therefore, IFs associated to locally robust
estimators should be bounded. The IF of the MDPDE for the step-stress ALT
model with non-destructive one-shot devices was established in Balakrishnan et
al. (2022), and the boundedness of the function was discussing there,
concluding that the IF of the MDPDE is always bounded for positive values of
the tuning parameter. Here we derive the IF of the restricted MDPDE,
$\widetilde{\boldsymbol{\theta}}^{\beta},$ defined in Section 2. Observe that
in this case the functional associated to the restricted estimator must also
satisfied the subspace constraint. The statistical functional and influence
function of the estimators under parametric restrictions have been rigorously
studied in Ghosh (2015). Here, we study the IF of the restricted MDPDE when
the subspace constraint has a linear form. We consider $F_{\theta}$ and the
$G$ the assumed and real distribution functions with associated mass functions
$\boldsymbol{\pi}(\boldsymbol{\theta})$ and $\boldsymbol{g},$ respectively. We
define $\widetilde{\boldsymbol{T}}_{\beta}$ the functional associated to the
restricted MDPDE, $\widetilde{\boldsymbol{\theta}}^{\beta},$ computed as the
minimizer of the DPD given in (6) between the mass functions
$\boldsymbol{\pi}(\boldsymbol{\theta})$ and $\boldsymbol{g}$ subject to the
linear constraint $\boldsymbol{m}^{T}\boldsymbol{\theta}-d=0.$

For influence function analysis, one could derive the IF expression from the
estimating equations of the restricted MDPDE in terms of Lagrange multipliers
given in (10). However, Ghosh (2015) proposed an alternative approach where
the functional $\widetilde{\boldsymbol{T}}_{\beta}$ associated to the
restricted MDPDE is calculated as a solution of the estimating equations of
the (unconstrained) MDPDE over the subspace $\Theta_{0}.$ The existence of
such solution is guaranteed by the Implicit Function Theorem. Hence, the IF of
the restricted MDPDE at the contamination point $\boldsymbol{n}$ and the model
distribution with true parameter value $\boldsymbol{\theta}_{0}$,
$F_{\boldsymbol{\theta}_{0}},$ must simultaneously verify the expression of
the IF of the MDPDE stated in Balakrishnan et al. (2022),

$\text{IF}\left(\boldsymbol{n},\widetilde{\boldsymbol{T}}_{\beta},F_{\boldsymbol{\theta}_{0}}\right)=\boldsymbol{J}_{\beta}^{-1}(\boldsymbol{\theta}_{0})\boldsymbol{W}^{T}\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta}_{0})}^{\beta-1}\left(-\boldsymbol{\pi}(\boldsymbol{\theta}_{0})+\Delta_{\boldsymbol{n}}\right)$

and the subspace constraint
$\boldsymbol{m}^{T}\widetilde{\boldsymbol{T}}_{\beta}-d=0.$ Differentiating on
the previous subspace constraint, we have that

$\boldsymbol{m}^{T}\text{IF}\left(\boldsymbol{n},\widetilde{\boldsymbol{T}}_{\beta},F_{\boldsymbol{\theta}_{0}}\right)=0$

and therefore, combining both equations, we get

$\begin{pmatrix}\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})\\\
\boldsymbol{m}^{T}\end{pmatrix}\text{IF}\left(\boldsymbol{n},\widetilde{\boldsymbol{T}}_{\beta},F_{\boldsymbol{\theta}_{0}}\right)=\begin{pmatrix}\boldsymbol{W}^{T}\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta}_{0})}^{\beta-1}\left(-\boldsymbol{\pi}(\boldsymbol{\theta}_{0})+\Delta_{\boldsymbol{n}}\right)\\\
0\end{pmatrix}.$

Now, multiplying both terms by
$\left(\boldsymbol{J}_{\beta}(\boldsymbol{\theta})^{T},\boldsymbol{m}\right)$
and inverting in both sizes of the equation, the expression of the IF of the
restricted MDPDE is given by

$\text{IF}\left(\boldsymbol{n},\widetilde{\boldsymbol{T}}_{\beta},F_{\boldsymbol{\theta}_{0}}\right)=\left(\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})^{T}\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})+\boldsymbol{m}\boldsymbol{m}^{T}\right)^{-1}\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})^{T}\boldsymbol{W}^{T}\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta}_{0})}^{\beta-1}\left(-\boldsymbol{\pi}(\boldsymbol{\theta}_{0})+\Delta_{\boldsymbol{n}}\right).$
(15)

Since the matrix
$\left(\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})^{T}\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})+\boldsymbol{m}\boldsymbol{m}^{T}\right)^{-1}\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})^{T}$
is typically assumed to be bounded, the robustness of the restricted MDPDE
depends only on the boundedness of the IF of the (unrestricted) MDPDE.
Therefore, restricted MDPDE are robust for all type of outliers when using
positives values of $\beta$, whereas the restricted MLE (corresponding to
$\beta=0$) lacks of robustness against stress level or inspection times
contamination, i.e., bad leverage points.

##  4 Applications of the restricted MDPDE

An interesting application of the restricted MDPDE are robust testing
procedures based on the DPD for testing linear null hypothesis of the form

$\operatorname{H}_{0}:\boldsymbol{m}^{T}\boldsymbol{\theta}=d.$ (16)

with $\boldsymbol{m}\in\mathbb{R}^{2}$ and $d\in\mathbb{R}.$ In this section,
we develop two families of test statistics based on the DPD for testing (16)
for one-shot devices under the step-stress ALT model, namely Rao-type test
statistics and DPD-based tests statistics. These two families were studied for
general statistical models in Basu et al. (2018). Let consider
$\widetilde{\boldsymbol{\theta}}^{\beta}$ the restricted MDPDE with restricted
parameter space defined by the null hypothesis in (16),

$\Theta_{0}=\\{\boldsymbol{\theta}|\hskip
5.69046pt\boldsymbol{m}^{T}\boldsymbol{\theta}=d\\}$

and recall $\widehat{\boldsymbol{\theta}}^{\beta}$ denotes the MDPDE for
$\boldsymbol{\theta}$ computed in all parameter space.

###  4.1 Rao-type tests statistics

Let us consider the score of the DPD loss function for the step-stress ALT
model

$\boldsymbol{U}_{\beta,N}(\boldsymbol{\theta})=\boldsymbol{W}^{T}\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta})}^{\beta-1}(\widehat{\boldsymbol{p}}-\boldsymbol{\pi}(\boldsymbol{\theta}))$
(17)

where matrices $\boldsymbol{W}$ and
$\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta})}$ are defined in
Section 2. That is, the MDPDE verifies the estimating equations given by

$\boldsymbol{U}_{\beta,N}(\widehat{\boldsymbol{\theta}}^{\beta})=\boldsymbol{0}$

We define Rao-type test statistics for testing linear null hypothesis (16) as

###### Definition 2

The Rao-type statistics, based on the restricted to the linear null hypothesis
(16) MDPDE, $\widetilde{\boldsymbol{\theta}}^{\beta},$ for testing (16) is
given by

$\boldsymbol{R}_{\beta,N}(\widetilde{\boldsymbol{\theta}}^{\beta})=N\boldsymbol{U}_{\beta,N}(\widetilde{\boldsymbol{\theta}}^{\beta})^{T}\boldsymbol{Q}_{\beta}(\widetilde{\boldsymbol{\theta}}^{\beta})\left[\boldsymbol{Q}_{\beta}(\widetilde{\boldsymbol{\theta}}^{\beta})^{T}\boldsymbol{K}_{\beta}(\widetilde{\boldsymbol{\theta}}^{\beta})\boldsymbol{Q}_{\beta}(\widetilde{\boldsymbol{\theta}}^{\beta})\right]^{-1}\boldsymbol{Q}_{\beta}(\widetilde{\boldsymbol{\theta}}^{\beta})^{T}\boldsymbol{U}_{\beta,N}(\widetilde{\boldsymbol{\theta}}^{\beta}),$
(18)

where matrices $\boldsymbol{K}_{\beta}(\boldsymbol{\theta})$,
$\boldsymbol{Q}_{\beta}(\boldsymbol{\theta})$ and
$\boldsymbol{U}_{\beta,N}(\boldsymbol{\theta}),$ is defined in (14), (13) and
(17), respectively.

Here, the matrix $\boldsymbol{Q}_{\beta}(\boldsymbol{\theta})$ depends on the
null hypothesis trough $\boldsymbol{m},$ and the term $d$ is only used to
obtain the restricted MDPDE. Moreover, if $\boldsymbol{m}=(0,1)$ (simple null
hypothesis) then the restricted estimate of $\theta_{1}$ must be necessarily
$d.$

Before presenting the asymptotic distribution of the Rao-type test statistics,
$\boldsymbol{R}_{\beta,N}(\widetilde{\boldsymbol{\theta}}^{\beta}),$ we shall
establish the asymptotic distribution of the score
$\boldsymbol{U}_{\beta,N}(\widetilde{\boldsymbol{\theta}}^{\beta}).$

###### Theorem 3

The asymptotic distribution of the score
$\boldsymbol{U}_{\beta,N}(\widetilde{\boldsymbol{\theta}}^{\beta})$ for the
step-stress ALT model under exponential lifetimes, is given by

$\sqrt{N}\boldsymbol{U}_{\beta,N}(\widetilde{\boldsymbol{\theta}}^{\beta})\xrightarrow[N\rightarrow\infty]{L}\mathcal{N}\left(\boldsymbol{0},\boldsymbol{K}_{\beta}(\boldsymbol{\theta})\right)$

where the variance-covariance matrix
$\boldsymbol{K}_{\beta}(\boldsymbol{\theta})$ is defined in (14).

Proof. It is well known that

$\sqrt{N}(\widehat{\boldsymbol{p}}-\boldsymbol{\pi}(\boldsymbol{\theta}))\xrightarrow[N\rightarrow\infty]{L}\mathcal{N}\left(\boldsymbol{0},\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta})}-\boldsymbol{\pi}(\boldsymbol{\theta})\boldsymbol{\pi}(\boldsymbol{\theta})^{T}\right).$

since $\widehat{\boldsymbol{p}}$ is the MLE of the multinomial model, and
$\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta})}-\boldsymbol{\pi}(\boldsymbol{\theta})\boldsymbol{\pi}(\boldsymbol{\theta})^{T}$
is the inverse of the Fisher information matrix of that model. Therefore, the
score $\sqrt{N}\boldsymbol{U}_{\beta,N}(\boldsymbol{\theta})$ is
asymptotically normal with mean vector

$\mathbb{E}\left[\boldsymbol{U}_{\beta,N}(\boldsymbol{\theta})\right]=\mathbb{E}\left[\boldsymbol{W}^{T}\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta})}^{\beta-1}(\widehat{\boldsymbol{p}}-\boldsymbol{\pi}(\boldsymbol{\theta}))\right]=\boldsymbol{0}$

and variance-covariance matrix

$\displaystyle\text{Cov}\left[\boldsymbol{U}_{\beta,N}(\boldsymbol{\theta})\right]$
$\displaystyle=\boldsymbol{W}^{T}\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta})}^{\beta-1}\text{Cov}\left[\widehat{\boldsymbol{p}}-\boldsymbol{\pi}(\boldsymbol{\theta})\right]\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta})}^{\beta-1}\boldsymbol{W}$
$\displaystyle=\boldsymbol{W}^{T}\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta})}^{\beta-1}\left[\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta})}-\boldsymbol{\pi}(\boldsymbol{\theta})\boldsymbol{\pi}(\boldsymbol{\theta})^{T}\right]\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta})}^{\beta-1}\boldsymbol{W}$
$\displaystyle=\boldsymbol{W}^{T}\left[\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta})}^{2\beta-1}-\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta})}^{\beta-1}\boldsymbol{\pi}(\boldsymbol{\theta})\boldsymbol{\pi}(\boldsymbol{\theta})^{T}\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta})}^{2\beta-1}\right]\boldsymbol{W}$
$\displaystyle=\boldsymbol{W}^{T}\left[\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta})}^{2\beta-1}-\boldsymbol{\pi}(\boldsymbol{\theta})^{\beta}\boldsymbol{\pi}(\boldsymbol{\theta})^{\beta
T}\right]\boldsymbol{W}$
$\displaystyle=\boldsymbol{K}_{\beta}(\boldsymbol{\theta}).$

Now, the following results states the asymptotic distribution of the Rao-type
test statistics

###### Theorem 4

The asymptotic distribution of the Rao-type test statistics defined in (18)
under the linear null hypothesis (16) is a chi-square with 1 degree of
freedom.

Proof. The MDPDE restricted to the null hypothesis (16),
$\widetilde{\boldsymbol{\theta}}^{\beta},$ is defined as the minimum of the
DPD loss restricted to the condition
$\boldsymbol{m}^{T}\boldsymbol{\theta}=d$. Then, it satisfies the restricted
equations

$\widetilde{\boldsymbol{U}}_{\beta,N}(\widetilde{\boldsymbol{\theta}}^{\beta})+\boldsymbol{m}\widetilde{\boldsymbol{\lambda}}=\boldsymbol{0},$
(19)

for some vector $\widetilde{\boldsymbol{\lambda}}$ of Lagrangian multipliers.
Then, we can write
$\boldsymbol{U}_{\beta,N}(\widetilde{\boldsymbol{\theta}}^{\beta})=-\boldsymbol{m}\widetilde{\boldsymbol{\lambda}}$
and consequently

$\boldsymbol{U}_{\beta,N}(\widetilde{\boldsymbol{\theta}}^{\beta})^{T}\boldsymbol{Q}_{\beta}(\widetilde{\boldsymbol{\theta}}^{\beta})=-\widetilde{\boldsymbol{\lambda}}\boldsymbol{m}^{T}\boldsymbol{Q}_{\beta}(\widetilde{\boldsymbol{\theta}}^{\beta})=-\boldsymbol{\widetilde{\lambda}}$

where $\boldsymbol{Q}_{\beta}(\widetilde{\boldsymbol{\theta}}^{\beta})$ is
defined in (13). Further, the Rao-type test statistics defined in (18) can be
computed in terms of the vector of Lagrange multipliers as follows,

$\boldsymbol{R}_{\beta,N}(\boldsymbol{\theta})=N\boldsymbol{\widetilde{\lambda}}^{T}\left[\boldsymbol{Q}_{\beta}(\widetilde{\boldsymbol{\theta}}^{\beta})^{T}\boldsymbol{K}_{\beta}(\widetilde{\boldsymbol{\theta}}^{\beta})\boldsymbol{Q}_{\beta}(\widetilde{\boldsymbol{\theta}}^{\beta})\right]^{-1}\boldsymbol{\widetilde{\lambda}}.$

In order to obtain the asymptotic distribution of the Rao-type test
statistics, we first derive the asymptotic distribution of the vector of
Lagrangian multipliers $\widetilde{\boldsymbol{\lambda}}.$ We consider the
second order Taylor expansion series of the score function
$\boldsymbol{U}_{\beta,N}(\boldsymbol{\theta})$ around the true parameter
value $\boldsymbol{\theta}_{0},$

$\boldsymbol{U}_{\beta,N}(\widetilde{\boldsymbol{\theta}}^{\beta})=\boldsymbol{U}_{\beta,N}(\boldsymbol{\theta}_{0})+\frac{\partial\boldsymbol{U}_{\beta,N}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\bigg{|}_{\boldsymbol{\theta}=\boldsymbol{\theta}_{0}}\left(\widetilde{\boldsymbol{\theta}}^{\beta}-\boldsymbol{\theta}_{0}\right)+o\left(||\widetilde{\boldsymbol{\theta}}^{\beta}-\boldsymbol{\theta}_{0}||^{2}\boldsymbol{1}_{2}\right).$

On the other hand, since
$\widehat{p}\xrightarrow[N\rightarrow\infty]{P}\boldsymbol{\pi}(\boldsymbol{\theta}_{0})$,
it is not difficult to show that

$\frac{\partial\boldsymbol{U}_{\beta,N}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\bigg{|}_{\boldsymbol{\theta}=\boldsymbol{\theta}_{0}}\xrightarrow[N\rightarrow\infty]{P}-\boldsymbol{J}_{\beta}\left(\boldsymbol{\theta}_{0}\right),$

where the matrix $\boldsymbol{J}_{\beta}\left(\boldsymbol{\theta}\right)$ is
defined in (14). Therefore, we can approximate the score function at the
restricted MDPDE by

$\boldsymbol{U}_{\beta,N}(\widetilde{\boldsymbol{\theta}}^{\beta})=\boldsymbol{U}_{\beta,N}(\boldsymbol{\theta}_{0})-\boldsymbol{J}_{\beta}\left(\boldsymbol{\theta}_{0}\right)\left(\widetilde{\boldsymbol{\theta}}^{\beta}-\boldsymbol{\theta}_{0}\right)+o\left(||\widetilde{\boldsymbol{\theta}}^{\beta}-\boldsymbol{\theta}_{0}||^{2}\boldsymbol{1}_{2}\right)+o\left(\boldsymbol{1}_{2}\right).$

Based on (19), we have that

$\boldsymbol{U}_{\beta,N}(\boldsymbol{\theta}_{0})-\boldsymbol{J}_{\beta}\left(\boldsymbol{\theta}_{0}\right)\left(\widetilde{\boldsymbol{\theta}}^{\beta}-\boldsymbol{\theta}_{0}\right)+\boldsymbol{m}\widetilde{\boldsymbol{\lambda}}=o\left(\boldsymbol{1}_{2}\right)$

so under the null hypothesis we can write,

$\boldsymbol{m}^{T}\widetilde{\boldsymbol{\theta}}^{\beta}-d=\boldsymbol{m}^{T}\left(\widetilde{\boldsymbol{\theta}}^{\beta}-\boldsymbol{\theta}_{0}\right)=0.$

Joining both equations, we get

$\begin{pmatrix}-\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})&\boldsymbol{m}\\\
\boldsymbol{m}^{T}&\boldsymbol{0}\end{pmatrix}\begin{pmatrix}\widetilde{\boldsymbol{\theta}}^{\beta}-\boldsymbol{\theta}_{0}\\\
\widetilde{\boldsymbol{\lambda}}\end{pmatrix}=\begin{pmatrix}-\boldsymbol{U}_{\beta,N}(\boldsymbol{\theta}_{0})\\\
0\end{pmatrix}+\begin{pmatrix}o(\boldsymbol{1}_{2})\\\ 0\end{pmatrix}$

and solving the previous equation, we have that

$\begin{pmatrix}\widetilde{\boldsymbol{\theta}}^{\beta}-\boldsymbol{\theta}_{0}\\\
\widetilde{\boldsymbol{\lambda}}\end{pmatrix}=\begin{pmatrix}-\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})&\boldsymbol{m}\\\
\boldsymbol{m}^{T}&\boldsymbol{0}\end{pmatrix}^{-1}\begin{pmatrix}-\boldsymbol{U}_{\beta,N}(\boldsymbol{\theta}_{0})\\\
\boldsymbol{0}\end{pmatrix}+\begin{pmatrix}o(\boldsymbol{1}_{2})\\\
0\end{pmatrix}.$

Now, computing the inverse matrix

$\begin{pmatrix}-\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})&\boldsymbol{m}\\\
\boldsymbol{m}^{T}&\boldsymbol{0}\end{pmatrix}^{-1}=\begin{pmatrix}\boldsymbol{P}_{\beta}(\boldsymbol{\theta}_{0})&\boldsymbol{Q}_{\beta}(\boldsymbol{\theta}_{0})\\\
\boldsymbol{Q}_{\beta}(\boldsymbol{\theta}_{0})^{T}&\left(\boldsymbol{m}^{T}\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})^{-1}\boldsymbol{m}\right)^{-1}\end{pmatrix}$

with $\boldsymbol{P}_{\beta}(\boldsymbol{\theta}_{0})$ and
$\boldsymbol{Q}_{\beta}(\boldsymbol{\theta}_{0})$ defined in (13). But from
Theorem 3,

$\begin{pmatrix}\sqrt{N}\boldsymbol{U}_{\beta,N}(\boldsymbol{\theta}_{0})\\\
\boldsymbol{0}\end{pmatrix}\xrightarrow[N\rightarrow\infty]{L}\mathcal{N}\left(\boldsymbol{0}_{3},\begin{pmatrix}\boldsymbol{K}_{\beta}(\boldsymbol{\theta}_{0})&\boldsymbol{0}\\\
\boldsymbol{0}^{T}&0\end{pmatrix}\right)$

and hence,

$\begin{pmatrix}\sqrt{N}(\widetilde{\boldsymbol{\theta}}^{\beta}-\boldsymbol{\theta}_{0})\\\
\sqrt{N}\widetilde{\boldsymbol{\lambda}}\end{pmatrix}\xrightarrow[N\rightarrow\infty]{L}\mathcal{N}\left(\boldsymbol{0}_{3},\boldsymbol{V}_{\beta}(\boldsymbol{\theta}_{0})\right)$

with

$\boldsymbol{V}_{\beta}(\boldsymbol{\theta}_{0})=\begin{pmatrix}\boldsymbol{P}_{\beta}(\boldsymbol{\theta}_{0})&\boldsymbol{Q}_{\beta}(\boldsymbol{\theta}_{0})\\\
\boldsymbol{Q}_{\beta}(\boldsymbol{\theta}_{0})^{T}&\left(\boldsymbol{m}^{T}\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})^{-1}\boldsymbol{m}\right)^{-1}\end{pmatrix}\begin{pmatrix}\boldsymbol{K}_{\beta}(\boldsymbol{\theta}_{0})&\boldsymbol{0}\\\
\boldsymbol{0}^{T}&0\end{pmatrix}\begin{pmatrix}\boldsymbol{P}_{\beta}(\boldsymbol{\theta}_{0})&\boldsymbol{Q}_{\beta}(\boldsymbol{\theta}_{0})\\\
\boldsymbol{Q}_{\beta}(\boldsymbol{\theta}_{0})^{T}&\left(\boldsymbol{m}^{T}\boldsymbol{J}_{\beta}(\boldsymbol{\theta}_{0})^{-1}\boldsymbol{m}\right)^{-1}\end{pmatrix}.$

Thus, the asymptotic distribution of the vector of Lagrangian multipliers is
given by

$\sqrt{N}\widetilde{\boldsymbol{\lambda}}\xrightarrow[N\rightarrow\infty]{L}\mathcal{N}\left(0,\boldsymbol{Q}_{\beta}(\boldsymbol{\theta}_{0})^{T}\boldsymbol{K}_{\beta}(\boldsymbol{\theta}_{0})\boldsymbol{Q}_{\beta}(\boldsymbol{\theta}_{0})\right).$

Using the previous convergence and the consistency of the restricted MDPDE, it
follows that the asymptotic distribution of the Rao-type test statistics,
$\boldsymbol{R}_{\beta,N}(\widetilde{\boldsymbol{\theta}}^{\beta}),$ is a chi-
square distribution with 1 degree of freedom.

Based on Theorem 4, for any $\beta\geq 0$ and
$\boldsymbol{m}\in\mathbb{R}^{2},$ the critical region with significance level
$\alpha$ for the hypothesis test with null hypothesis (16) is given by

$\mathcal{R}_{\alpha}=\\{(n_{1},...,n_{L+1})\text{ s.t.
}\boldsymbol{R}_{\beta,N}(\boldsymbol{\theta})>\chi^{2}_{1,\alpha}\\}$ (20)

where $\chi^{2}_{1,\alpha}$ denotes the lower $\alpha$-quantile of a chi-
square with 1 degree of freedom.

###### Remark 5

We could extent the scope to more general linear null hypothesis,

$\operatorname{H}_{0}:\boldsymbol{M}^{T}\boldsymbol{\theta}=\boldsymbol{d},$

with $\boldsymbol{M}$ a $2\times 2$ matrix of range $2$ and
$\boldsymbol{d}\in\mathbb{R}^{2}.$ However, in this case the restricted MDPDE
is explicitly determinate by the constraint
$\boldsymbol{M}^{T}\boldsymbol{\theta}=\boldsymbol{d}.$ Consequently, the Rao-
type test statistics is completely defined by the null hypothesis and it is
independent of the value of $\beta$. In particular, if we define
$\boldsymbol{M}$ to be the identity matrix, we get the simple null hypothesis

$\operatorname{H}_{0}:(\theta_{0},\theta_{1})=(d_{0},d_{1}),$

and the associated Rao-type test is given by

$\boldsymbol{R}_{\beta,N}^{\ast}(\boldsymbol{d})=N\boldsymbol{U}_{\beta,N}(\boldsymbol{d})^{T}\boldsymbol{K}_{\beta}(\boldsymbol{d})^{-1}\boldsymbol{U}_{\beta,N}(\boldsymbol{d}),$

since $\boldsymbol{Q}_{\beta}(\boldsymbol{d})$ is the identity matrix.
Following similar steps than in Theorem 4, it is not difficult to establish
that

$\boldsymbol{R}_{\beta,N}^{\ast}(\boldsymbol{d})\xrightarrow[N\rightarrow\infty]{L}\chi^{2}_{2}.$

Thus, for any $\beta\geq 0,$ the critical region with significance level
$\alpha$ for the hypothesis test with simple null hypothesis is given by

$\mathcal{R}_{\alpha}=\\{(n_{1},...,n_{L+1})\text{ s.t.
}\boldsymbol{R}_{\beta,N}^{\ast}(\boldsymbol{d})>\chi_{2,\alpha}\\},$

where $\chi_{2,\alpha}$ denotes the lower $\alpha$-quantile of a chi-square
with 2 degree of freedom.

###  4.2 DPD-based statistics

We develop a class of test statistics based on the DPD between the model
evaluated under the null hypothesis and under the whole parameter space,
respectively. Let $\boldsymbol{\pi}(\widehat{\boldsymbol{\theta}}^{\beta})$
and $\boldsymbol{\pi}(\widetilde{\boldsymbol{\theta}}^{\beta})$ denote the
probability vector of the multinomial model estimated in the whole parameter
space and the restricted parameter space, respectively. The DPD for $\tau>0$
between these two mass function is given by

$d_{\tau}(\widehat{\boldsymbol{\theta}}^{\beta},\widetilde{\boldsymbol{\theta}}^{\beta})=d_{\tau}(\boldsymbol{\pi}(\widehat{\boldsymbol{\theta}}^{\beta}),\boldsymbol{\pi}(\widetilde{\boldsymbol{\theta}}^{\beta}))=\sum_{i=1}^{L+1}\left(\pi_{j}(\widehat{\boldsymbol{\theta}}^{\beta})^{1+\tau}+\left(1+\frac{1}{\tau}\right)\pi_{j}(\widehat{\boldsymbol{\theta}}^{\beta})\pi_{j}(\widetilde{\boldsymbol{\theta}}^{\beta})^{\tau}+\frac{1}{\tau}\pi_{j}(\widetilde{\boldsymbol{\theta}}^{\beta})^{\tau+1}\right).$
(21)

Note that here we should deal with two different tuning parameters controlling
the trade-off between robustness and efficiency in the estimation ($\beta$)
and in the DPD between the estimated mass functions ($\tau$). Many authors
choose $\tau=\beta$ for the seek of simplicity. If the true parameter
$\boldsymbol{\theta}_{0}$ verifies the null hypothesis, then the distance in
(21) must near zero. Then, the reject region for the test with null hypothesis
(16) is given by

$RC=\\{(n_{1},...,n_{L+1})s.t.\hskip
5.69046ptd_{\tau}(\widehat{\boldsymbol{\theta}}^{\beta},\widetilde{\boldsymbol{\theta}}^{\beta})>k\\}$

with $k$ a constant computed such that the test has size $\alpha.$ The exact
distribution of the test statistic
$d_{\tau}(\widehat{\boldsymbol{\theta}}^{\beta},\widetilde{\boldsymbol{\theta}}^{\beta})$
is not easy to get, but using similar arguments as in Jaenada et al. (2022) is
it not difficult to establish that

$T_{\tau}(\widehat{\boldsymbol{\theta}}^{\beta},\widetilde{\boldsymbol{\theta}}^{\beta})=2nd_{\tau}(\widehat{\boldsymbol{\theta}}^{\beta},\widetilde{\boldsymbol{\theta}}^{\beta})$

is asymptotically distributed, under the null hypothesis, as a linear
combination of chi-squared random variables with a degree of freedom.
Furthermore, asymptotic results under some particular kind of alternative
hypothesis, such as contiguous alternative hypothesis, may be also of
interest.

##  5 Simulation study

We empirically analyze the performance of the restricted MDPDEs for one-shot
devices data under the step-stress model with exponential lifetime
distributions, $\widetilde{\boldsymbol{\theta}}^{\beta}.$ Further, we evaluate
their robustness properties under different scenarios of contamination.

For the multinomial model, we should consider “outlying cells” rather than
“outlying devices”. Then, we introduce contamination by increasing (or
decreasing) the probability of failure in (4) for (at least) one interval
(i.e., one cell). The probability of failure is switched in such contaminated
cells as

$\tilde{\pi}_{j}(\boldsymbol{\theta})=G_{\boldsymbol{\theta}}(\textit{IT}_{j})-G_{\boldsymbol{\tilde{\theta}}}(\textit{IT}_{j-1})$
(22)

for some $j=2,...,L$, where
$\boldsymbol{\tilde{\theta}}=(\widetilde{\theta}_{0},\widetilde{\theta}_{1})$
is a contaminated parameter with $\tilde{\theta}_{0}\leq\theta_{0}$ and
$\tilde{\theta}_{1}\leq\theta_{1}.$ The resulting probability vector must
normalized after introducing contamination.

Let us consider a 2-step stress ALT experiment with $L=11$ inspection times
and a total of $N=180$ one-shot devices. The devices are tested at two stress
levels, $x_{1}=35$ and $x_{2}=45$. The first stress level is maintained from
the beginning of the experiment until $\tau_{1}=25,$ and the experiment ends
at $\tau_{2}=70.$ During the experiment, devices inspection is performed at a
grid of inspection times, containing the times of stress change,
$\text{IT}=(10,15,20,25,30,35,40,45,50,60,70).$ At each inspection time, all
surviving units under test are examined. We set the true value of the
parameter $\boldsymbol{\theta}_{0}=(0.003,0.03),$ and then generate data from
the corresponding multinomial model described in Section 2. Additionally, we
contaminate the described model by increasing the probability of failure in
the third interval following (22), with
$\widetilde{\boldsymbol{\theta}}=(\varepsilon\theta_{0},\theta_{1})$ for the
first scenario of contamination and
$\widetilde{\boldsymbol{\theta}}=(\theta_{0},\varepsilon\theta_{1})$ for the
second scenario of contamination. Note that in both scenarios, the mean
lifetime is decreased for the outlying cell. Further, we consider the linearly
restricted parameter space of the form

$\Theta_{0}=\\{\theta\in\Theta|\hskip 5.69046pt\theta_{1}=d\\}$

with $d\in\mathbb{R}.$ We evaluate the performance of the restricted
estimators when the true parameter verifies the subspace constraints as well
as when $\boldsymbol{\theta}_{0}$ does not belong to the restricted subspace
$\Theta_{0}.$ For the first scenario ($\boldsymbol{\theta}_{0}\in\Theta_{0}$)
we set $d=0.03$ and for the second one
($\boldsymbol{\theta}_{0}\notin\Theta_{0}$) we set $d=0.027.$ Note that, in
the second case, the true parameter is close to the restricted space.

Figure 1 shows the mean squared error (MSE) of the restricted MDPDE when the
true parameter value satisfies the subspace restrictions over $R=10.000$
repetitions, for the two scenarios of contamination and different values of
$\beta.$ Larger values of the tuning parameter produces more robust estimates,
although less efficient. From this empirical results, moderately large values
of the tuning parameter $\beta$ over 0.4 could produce the best trade-off
between efficiency and robustness. Similarly, Figure 2 shows the MSE of the
restricted MDPDE for similar scenarios, when the true parameter value does not
belong to the restricted parameter space. Similar conclusions are drawn, but
the MSE on the estimation increases considerably in this context.

(a) $\theta_{0}$-contaminated cell

(b) $\theta_{1}$-contaminated cell

Figure 1:  MSE of the restricted MDPDE when the true parameter belongs to the
restricted space over $R=10.000$ repetitions

(a) $\theta_{0}$-contaminated cell

(b) $\theta_{1}$-contaminated cell

Figure 2:  MSE of the restricted MDPDE when the true parameter does not belong
to the restricted space over $R=10.000$ repetitions

Finally, we examine the performance of the Rao-type test statistics against
the sample size $N$ in three contamination scenarios (Figure 3), in the
absence of contamination (top), $\theta_{0}$-contaminated third cell with a
$40\%$ reduction of the parameter (middle) and $\theta_{1}$-contaminated third
cell with a $40\%$ reduction of the parameter (bottom). Again, we found a
similar performance of the Rao-type test statistics based on the restricted
MDPDE with different values of the tuning parameter $\beta,$ whereas positives
values of $\beta$ produce more robust statistics when introducing
contamination in either of the model parameters, $\theta_{0}$ and
$\theta_{1}$. Moreover, the lack of robustness of the Rao-type test statistics
based on the restricted MLE is highlighted with large sample sizes, in both
empirical level and power.

(a) Absence of contamination

(b) Absence of contamination

(c) $\theta_{0}$-contaminated cell

(d) $\theta_{0}$-contaminated cell

(e) $\theta_{1}$-contaminated cell

(f) $\theta_{1}$-contaminated cell

Figure 3: Empirical level and power against sample size with different
contaminated scenarios in $R=1000$ replications

##  6 Conclusions

In this paper with have developed the restricted minimum MDPDE for one-shot
devices data tested under step-stress ALTs. We have derived its asymptotic
distribution and analyzed its robustness properties through its IF. Further,
we have defined two families of robust testing procedures based on the
restricted DPD estimators, Rao-type test statistics and DPD-based statistics,
and we have established the asymptotic distribution for the first one.
Finally, all properties stated theoretically have been illustrated empirically
through simulation, showing the certain advantage in terms of robustness of
DPD-based inference methods.

The robustness of the the two proposed families of tests statistics would be
interesting to study theoretically through their IF analysis, and furthermore,
empirical performance comparison of these two families, together with well-
known Wald-type test statistics based on the DPD for one-shot devices tested
under step-stress ALT model, will be interesting to be examined in future
researches.

## References

  * [1] Balakrishnan, N., Castilla, E., Jaenada M. and Pardo, L. (2022). Robust inference for non-destructive one-shot devicetesting under step-stress model with exponential lifetimes
  * [2] Basu, A., Harris, I. R., Hjort, N. L., and Jones, M. C. (1998). Robust and efficient estimation by minimising a density power divergence. Biometrika, 85(3), 549-559.
  * [3] Basu, A. , Mandal, A., Martin, N. and Pardo, L. (2018). Testing Composite Hypothesis Based on the Density Power Divergence _Sankhya B: The Indian Journal of Statistics_ , 80(2), 222-262.
  * [4] Ghosh, A. (2015). Influence function analysis of the restricted minimum divergence estimators: A general form. _Electronic Journal of Statistics,_ 9, 1017-1040.
  * [5] Hampel, F.R., Ronchetti, E., Rousseauw, P.J., and Stahel, W. (1986). Robust Statistics: The Approach Based on Influence Functions John Wiley & Sons.
  * [6] Jaenada, M., Miranda, P. and Pardo, L. (2022). Robust test statistics based on Restricted minimum Rényi’s pseudodistance estimators. Entropy, 24(5), 616.