# Survival functions versus conditional aggregation-based survival functions
on discrete space

Basarik Stanislav11footnotemark: 1<EMAIL_ADDRESS>Borzová
Jana22footnotemark: 2<EMAIL_ADDRESS>Halčinová Lenka33footnotemark: 3
<EMAIL_ADDRESS>Institute of Mathematics, P. J. Šafárik University in
Košice, Jesenná 5, 040 01 Košice, Slovakia

###### Abstract

In this paper we deal with conditional aggregation-based survival functions
recently introduced by Boczek et al. (2020). The concept is worth to study
because of its possible implementation in real-life situations and
mathematical theory as well. The aim of this paper is the comparison of this
new notion with the standard survival function. We state sufficient and
necessary conditions under which the generalized and the standard survival
function equal. The main result is the characterization of the family of
conditional aggregation operators (on discrete space) for which these
functions coincide.

###### keywords:

aggregation, survival function, nonadditive measure, visualization, size

###### MSC:

[2010] 28A12

††journal: Information Sciencesfn2fn2footnotetext: Supported by the grants
APVV-16-0337, VEGA 1/0657/22, bilateral call Slovak-Poland grant scheme No.
SK-PL-18-0032 and grant scheme VVGS-PF-2021-1782.

##  1 Introduction

We continue to study the novel survival functions introduced in [1] as a
generalization of size-based level measure developed for the use in
nonadditive analysis in [3, 12, 13]. The concept appeared initially in time-
frequency analysis [8]. As the main result, in Theorem 4.7 we show that the
generalized survival function is equal to the original notion (for any
monotone measure and any input vector) just in very particular case. The
concept of the novel survival function is useful in many real-life situations
and pure theory as well. In fact, the standard survival function (also known
in the literature as the standard level measure [13], strict level measure [5]
or decumulative distribution function [10]) is the crucial ingredient of many
definitions in mathematical analysis. Many well-known integrals are based on
the survival function, e.g. the Choquet integral, the Sugeno integral, the
Shilkret integral, the seminormed integral [5], universal integrals [14], etc.
Also, the convergence of a sequence of functions in measure is based on the
same concept. Hence a reasonable generalization of the survival function leads
to the generalizations of all mentioned concepts. For more on applications of
the generalized survival function, see [1, 8].

Due to the number of factors needed in the definition of the generalized
survival function, it is quite difficult to understand this concept. In order
to understand it more deeply, in the following we shall focus on the graphical
visualization of inputs, see [4]. Moreover, the graphical representation will
help us to formulate basic results of this paper. In the whole paper, we
restrict ourselves to discrete settings. We consider finite basic set

$\displaystyle[n]:=\\{1,2,\dots,n\\},\,\,n\geq 1$

and a monotone measure $\displaystyle\mu$ on $\displaystyle 2^{[n]}$. If
$\displaystyle\mathbf{x}=(x_{1},\dots,x_{n})$ is a nonnegative real-valued
function on $\displaystyle[n]$, i.e., a vector, then the survival function (or
standard survival function) of the vector $\displaystyle\mathbf{x}$ with
respect to $\displaystyle\mu$, see [1, 9], is defined by

$\displaystyle\mu(\\{\mathbf{x}>\alpha\\}):=\mu\left(\\{i\in[n]:x_{i}>\alpha\\}\right),\quad\alpha\in[0,\infty).$

For the thorough exposition see Preliminaries. To avoid too abstract setting
in the following visual representations, let us consider the input vector
$\displaystyle\mathbf{x}=(2,3,4)$ and the monotone measure $\displaystyle\mu$
on $\displaystyle 2^{[3]}$ defined in Table 1.

$\displaystyle E$ |  $\displaystyle\\{1,2,3\\}$ |  $\displaystyle\\{2,3\\}$ |  $\displaystyle\\{1,3\\}$ |  $\displaystyle\\{1,2\\}$ |  $\displaystyle\\{3\\}$ |  $\displaystyle\\{2\\}$ |  $\displaystyle\\{1\\}$ |  $\displaystyle\emptyset$
---|---|---|---|---|---|---|---|---
$\displaystyle E^{c}$ |  $\displaystyle\emptyset$ |  $\displaystyle\\{1\\}$ |  $\displaystyle\\{2\\}$ |  $\displaystyle\\{3\\}$ |  $\displaystyle\\{1,2\\}$ |  $\displaystyle\\{1,3\\}$ |  $\displaystyle\\{2,3\\}$ |  $\displaystyle\\{1,2,3\\}$
$\displaystyle\mu(E^{c})$ |  $\displaystyle 0$ |  $\displaystyle 0.25$ |  $\displaystyle 0.25$ |  $\displaystyle 0.4$ |  $\displaystyle 0.75$ |  $\displaystyle 0.75$ |  $\displaystyle 0.75$ |  $\displaystyle 1$
$\displaystyle\max_{i\in E}x_{i}$ |  $\displaystyle 4$ |  $\displaystyle 4$ |  $\displaystyle 4$ |  $\displaystyle 3$ |  $\displaystyle 4$ |  $\displaystyle 3$ |  $\displaystyle 2$ |  $\displaystyle 0$
$\displaystyle\sum_{i\in E}x_{i}$ |  $\displaystyle 9$ |  $\displaystyle 7$ |  $\displaystyle 6$ |  $\displaystyle 5$ |  $\displaystyle 4$ |  $\displaystyle 3$ |  $\displaystyle 2$ |  $\displaystyle 0$
Table 1: Sample measure $\displaystyle\mu$ and two conditional aggregation
operators for vector $\displaystyle\mathbf{x}=(2,3,4)$

#### The survival functions visual representation

We begin with a nonstandard representation of standard survival function, as a
stepping stone to its generalization. Before, let us introduce the following
equivalent representation of survival function:

$\displaystyle\displaystyle\begin{split}\mu(\\{\mathbf{x}>\alpha\\})=\mu([n]\setminus\\{i\in[n]:x_{i}\leq\alpha\\})&=\min\big{\\{}\mu(E^{c}):(\forall
i\in E)\,\,x_{i}\leq\alpha,\,E\in 2^{[n]}\big{\\}}\\\
&=\min\big{\\{}\mu(E^{c}):\max_{i\in E}x_{i}\leq\alpha,\,E\in
2^{[n]}\big{\\}},\end{split}$ (1)

where $\displaystyle E^{c}=[n]\setminus E$, see motivation problem 1 in [1].
Let us start the visualization with inputs from Table 1.

|
---|---
Figure 1: The survival function visualization for
$\displaystyle\mathbf{x}=(2,3,4)$ and $\displaystyle\mu$ given in Table 1.

Let us depict all maximal values of $\displaystyle\mathbf{x}$ on
$\displaystyle E$, for each set $\displaystyle E\in 2^{[3]}$ on the lower
axis, see left image of Figure 1, in decreasing order and the corresponding
values of monotone measure of complement, i.e. $\displaystyle\mu(E^{c})$, on
the upper axis. In this picture of Figure 1, the number on lower axis is
linked with the number on the upper one via a straight line once they
correspond to the same set, i.e., $\displaystyle a$ is linked with
$\displaystyle b$ if there is $\displaystyle E\in 2^{[3]}$ such that

$\displaystyle a=\max\limits_{i\in E}x_{i}\hskip 14.22636pt\text{ and }\hskip
14.22636ptb=\mu(E^{c}).$

Finally, the value $\displaystyle\mu(\\{\mathbf{x}>\alpha\\})$ at some
$\displaystyle\alpha\in[0,\infty)$ can be read from the left image of Figure 1
considering the minimal value on the upper axis which is linked to a value
smaller than $\displaystyle\alpha$ (i.e., right-hand side value) on the lower
one. Thus considering an illustrative example in the left image of Figure 1,
the value of survival function at $\displaystyle 2{.}5$ is $\displaystyle
0{.}75$. Indeed, there are just 2 values on the right hand side of
$\displaystyle 2{.}5$, namely numbers 2 and 0. These are linked to
$\displaystyle 0{.}75$ and 1, respectively. Hence, $\displaystyle 0{.}75$ is a
smaller one. The graph of survival function is in the right image of Figure 1.

#### The generalized survival functions visual representation

In the modification of the survival function, the previously described
computational procedure stays. However, we allow to use any conditional
aggregation operator, not just maximum operator. The standard example of
conditional aggregation is the sum of components of $\displaystyle\mathbf{x}$,
see the last line in Table 1 and the corresponding visualisation in Figure 2.
Applying the described computational procedure we obtain the sum-based
survival function of vector $\displaystyle\mathbf{x}$, i.e., the generalized
survival function of vector $\displaystyle\mathbf{x}$ studied in [1, 3, 12,
13]. The formula linked to this procedure is the following:

$\displaystyle\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=\min\left\\{\mu(E^{c}):\mathsf{A}^{\mathrm{sum}}(\mathbf{x}|E)\leq\alpha,\,E\in\mathscr{E}\right\\}$

with $\displaystyle\mathsf{A}^{\mathrm{sum}}(\mathbf{x}|E)=\sum\limits_{i\in
E}x_{i}$ and $\displaystyle\\{\emptyset\\}\subseteq\,\mathscr{E}\subseteq
2^{[n]}$ (in the illustrative example $\displaystyle\mathscr{E}=2^{[3]}$).

The corresponding graph is: Considering discrete space, the computation of the
generalized survival function studied in [1, 3, 12, 13] may be always
represented via the corresponding diagrams similar to those in Figures 1 and
2.

|
---|---
Figure 2: Generalized survival function visualization for
$\displaystyle\mathbf{x}=(2,3,4)$, $\displaystyle\mu$ given in Table 1 and
$\displaystyle\mathsf{A}=\mathsf{A}^{\mathrm{sum}}$

Except for a better understanding of survival functions, the visual
representation may help us to answer the problem of their
indistinguishability. With the introduction of novel survival function a
natural question arises: When does the generalized survival function coincide
with the survival function? The motivation for answering these questions is
not only to know the relationship between mentioned concepts for given inputs,
but it will help us to compare the corresponding integrals based on them, see
[1, Definition 5.1, Definition 5.4]. In the literature, there are known some
families of conditional aggregation operators together with the collection
$\displaystyle\mathscr{E}$ when the generalized survival function equals to
the survival function. In the following we list them:

  * 1.

(cf. [13, Corollary 4.15])
$\displaystyle{\mathscr{A}}={\mathscr{A}}^{\rm{size}}$ with size
$\displaystyle\mathsf{s}$ being the weighted sum444
$\displaystyle{\mathsf{s}}_{{\\#},p}(\mathbf{x})(E)=\left(\frac{1}{{\\#}(E)}\cdot\sum\limits_{x_{i}\in
E}x_{i}^{p}\right)^{\frac{1}{p}}$ for $\displaystyle E\neq\emptyset$,
$\displaystyle{\mathsf{s}}_{{\\#},p}(\mathbf{x})(\emptyset)=0$ and
$\displaystyle p>0$., $\displaystyle\mathscr{D}$ contains all singletons of
$\displaystyle[n]$ and $\displaystyle\mathscr{E}=2^{[n]}$;

  * 2.

(cf. [1, Example 4.2] or [13, Section 5])
$\displaystyle{\mathscr{A}}={\mathscr{A}}^{\rm{max}}$ with
$\displaystyle\mathscr{E}=2^{[n]}$;

  * 3.

(cf. [1, Proposition 4.6])
$\displaystyle{\mathscr{A}}={\mathscr{A}}^{\mu-\mathrm{ess}}$ with
$\displaystyle\mathscr{E}=2^{[n]}$.

Although the first two items appear to be different, in fact, under the above
conditions, they are equal
$\displaystyle{\mathscr{A}}^{\rm{size}}={\mathscr{A}}^{\rm{max}}$. Settings of
above mentioned examples lead to the survival function regardless of the
choice of monotone measure $\displaystyle\mu$. However, the identity between
generalized survival function and survival function may happen also for other
families of conditional aggregation operators (a FCA for short), but with
specific monotone measures, e.g. $\displaystyle{\mathscr{A}}^{\mathrm{sum}}$
with the weakest monotone measure555$\displaystyle\mu_{*}\colon
2^{[n]}\to[0,\infty)$ given by
$\displaystyle\mu_{*}(E)=\begin{cases}\mu([n]),&E=[n],\\\
0,&\textrm{otherwise}.\end{cases}$ shrinks to survival function for any input
vector $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$ and
$\displaystyle\mathscr{E}=2^{[n]}$. In this paper we shall treat the following
problems:

Problem 1: Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$,
$\displaystyle\mu$ be a monotone measure on $\displaystyle 2^{[n]}$, and
$\displaystyle{\mathscr{A}}$ be FCA. What are sufficient and necessary
conditions on $\displaystyle\mathbf{x}$, $\displaystyle\mu$ and
$\displaystyle{\mathscr{A}}$ to hold
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$?

Problem 2: Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$, and
$\displaystyle{\mathscr{A}}$ be FCA. What are sufficient and necessary
conditions on $\displaystyle\mathbf{x}$ and $\displaystyle{\mathscr{A}}$ to
hold
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any monotone measure $\displaystyle\mu$?

Problem 3: Let $\displaystyle{\mathscr{A}}$ be FCA. What are sufficient and
necessary conditions on $\displaystyle{\mathscr{A}}$ to hold
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any monotone measure $\displaystyle\mu$ and
$\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$?

The paper is organized as follows. We continue with preliminary section
containing needed definitions and notations. In Section 3 we solve Problem 1,
see e.g. Corollary 3.7, Corollary 3.11, Remark 3.12, Proposition 3.15 and
Theorem 3.17. In Section 4 we provide quite surprising result, see Theorem 4.7
that characterizes the family of conditional aggregation operators (in
discrete setting) for which the generalized survival function coincides with
the standard survival function. Thus we answer Problem 3. In Section 4 we also
treat Problem 2, see Theorem 4.2 and Theorem 4.6. Many our results are
supported by appropriate examples.

##  2 Background and preliminaries

In order to be self-contained as far as possible, we recall in this section
necessary definitions and all basic notations. In the whole paper, we restrict
ourselves to discrete settings. As we have already mentioned, we shall
consider a finite set

$\displaystyle X=[n]:=\\{1,2,\dots,n\\},\,\,n\geq 1.$

We shall denote by $\displaystyle 2^{[n]}$ the power set of
$\displaystyle[n]$. A monotone or nonadditive measure on $\displaystyle
2^{[n]}$ is a nondecreasing set function $\displaystyle\mu\colon
2^{[n]}\to{{[0,\infty)}},$ i.e., $\displaystyle\mu(E)\leq\mu(F)$ whenever
$\displaystyle E\subseteq F$, with $\displaystyle\mu(\emptyset)=0.$ Moreover,
we shall suppose $\displaystyle\mu([n])>0$. The set of monotone measures on
$\displaystyle 2^{[n]}$ we shall denote by $\displaystyle\mathbf{M}$. The
monotone measure satisfying the equality $\displaystyle\mu([n])=1$ will be
called the normalized monotone measure (also known as a capacity in [15]). In
this paper we shall always work with monotone measures being defined on
$\displaystyle 2^{[n]}$, although, on several places the domain of
$\displaystyle\mu$ can be smaller. Also, we shall need special properties of
$\displaystyle\mu$ on a system $\displaystyle\mathscr{S}\subseteq 2^{[n]}$.
The monotone measure $\displaystyle\mu\in\mathbf{M}$ with the property
$\displaystyle\mu(E)\neq\mu(F)$ for any $\displaystyle
E,F\in\mathscr{S}\subseteq 2^{[n]}$, $\displaystyle E\neq F$ will be called
strictly monotone measure on $\displaystyle\mathscr{S}$. The counting measure
will be denoted by $\displaystyle{\\#}$. Further, we put
$\displaystyle\max\emptyset=0$ and $\displaystyle\sum_{i\in\emptyset}x_{i}=0$.

We shall work with nonnegative real-valued vectors, we shall use the notation
$\displaystyle\mathbf{x}=(x_{1},\dots,x_{n})$, $\displaystyle
x_{i}\in[0,\infty)$, $\displaystyle i=1,2,\dots,n$. The set
$\displaystyle[0,\infty)^{[n]}$ is the family of all nonnegative real-valued
functions on $\displaystyle[n]$, i.e. vectors. For any
$\displaystyle\mathbf{x}=(x_{1},\dots,x_{n})\in[0,\infty)^{[n]}$ we denote by
$\displaystyle(\cdot)$ a permutation $\displaystyle(\cdot)\colon[n]\to[n]$
such that $\displaystyle x_{(1)}\leq x_{(2)}\leq\dots\leq x_{(n)}$ and
$\displaystyle x_{(0)}=0$, $\displaystyle x_{(n+1)}=\infty$ by convention. Let
us remark that the permutation $\displaystyle(\cdot)$ need not be unique (this
happens if there are some ties in the sample $\displaystyle(x_{1},...,x_{n})$,
see [7]). For a fixed input vector $\displaystyle\mathbf{x}$ and a fixed
permutation $\displaystyle(\cdot)$ we shall denote by $\displaystyle E_{(i)}$
the set of the form $\displaystyle E_{(i)}=\\{(i),\dots,(n)\\}$ for any
$\displaystyle i\in[n]$ with the convention $\displaystyle
E_{(n+1)}=\emptyset$. By $\displaystyle\mathbf{1}_{E}$ we shall denote the
indicator function of a set $\displaystyle E\subseteq Y$, $\displaystyle
Y\subseteq[0,\infty)$, i.e., $\displaystyle\mathbf{1}_{E}(x)=1$ if
$\displaystyle x\in E$ and $\displaystyle\mathbf{1}_{E}(x)=0$ if
$\displaystyle x\notin E$. Especially,
$\displaystyle\mathbf{1}_{\emptyset}(x)=0$ for each $\displaystyle x\in Y$. We
shall work with indicator function with respect to two different sets. We
shall work with $\displaystyle Y=[n]$ when dealing with vectors (i.e.
$\displaystyle\mathbf{1}_{E}$ is a characteristic vector of $\displaystyle
E\subseteq[n]$ in $\displaystyle\\{0,1\\}^{{[n]}}$) and $\displaystyle
Y=[0,\infty)$ when dealing with survival functions.

In the following we list several definitions (adopted to discrete settings).
Firstly, the concept of the conditional aggregation operator is presented. Its
crucial feature is that the validity of properties is required only on
conditional set, not on the whole set. The inspiration for its introduction
came from the conditional expectation, which is the fundamental notion of
probability theory. Let us also remark that this operator generalizes the
aggregation operator introduced earlier by Calvo et al. in [6, Definition 1]
and it is the crucial ingredient in the definition of the generalized survival
function.

###### Definition 2.1

(cf. [1, Definition 3.1]) A map
$\displaystyle\mathsf{A}(\cdot|B)\colon[0,\infty)^{[n]}\to[0,\infty)$ is said
to be a conditional aggregation operator with respect to a set $\displaystyle
B\in 2^{[n]}\setminus\\{\emptyset$} if it satisfies the following conditions:

  * i)

$\displaystyle\mathsf{A}(\mathbf{x}|B)\leq\mathsf{A}(\mathbf{y}|B)$ for any
$\displaystyle\mathbf{x},\mathbf{y}\in[0,\infty)^{[n]}$ such that
$\displaystyle x_{i}\leq y_{i}$ for any $\displaystyle i\in B$;

  * ii)

$\displaystyle\mathsf{A}(\mathbf{1}_{B^{c}}|B)=0.$

Let us compare the settings of the previous definition with the settings of
the original definition, see [1, Definition 3.1]). We consider the greatest
$\displaystyle\sigma$-algebra as the domain of $\displaystyle\mu$ in
comparison with the original arbitrary $\displaystyle\sigma$-algebra
$\displaystyle\Sigma$. Then all vectors are measurable and this assumption may
be omitted from the definition. The measurability of each vector is desired
property mainly from the application point of view. Because of the property
$\displaystyle\mathsf{A}(\mathbf{x}|B)=\mathsf{A}(\mathbf{x}\mathbf{1}_{B}|B)$
for any $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$ with fixed $\displaystyle
B\in 2^{[n]}\setminus\\{\emptyset\\}$ the value
$\displaystyle\mathsf{A}(\mathbf{x}|B)$ can be interpreted as an aggregated
value of $\displaystyle\mathbf{x}$ on $\displaystyle B$, see [1]. In the
following we list several examples of conditional aggregation operators we
shall use in this paper. For further examples and some properties of
conditional aggregation operators we recommend [1, Section 3].

###### Example 2.2

Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$, $\displaystyle B\in
2^{[n]}\setminus\\{\emptyset\\}$ and $\displaystyle m\in\mathbf{M}$.

  1. i)

$\displaystyle\mathsf{A}^{m-\mathrm{ess}}(\mathbf{x}|B)=\mathrm{ess}\sup_{m}(\mathbf{x}\mathbf{1}_{B})$,
where $\displaystyle\mathrm{ess}\sup_{m}(\mathbf{x})=\min\\{\alpha\geq
0:\,\\{\mathbf{x}>\alpha\\}\in\mathscr{N}_{m}\\}$.666A set $\displaystyle N\in
2^{[n]}$ is said to be a null set with respect to a monotone measure
$\displaystyle m$ if $\displaystyle m(E\cup N)=m(E)$ for all $\displaystyle
E\in 2^{[n]}.$ By $\displaystyle{{\mathscr{N}}_{m}}$ we denote the family of
null sets with respect to $\displaystyle m.$

  2. ii)

$\displaystyle\mathsf{A}(\mathbf{x}|B)=\mathrm{J}(\mathbf{x}\mathbf{1}_{B},m)$,
(the multiplication of vectors is meant by components) where
$\displaystyle\mathrm{J}$ is an integral defined in [2, Definition 2.2].
Namely,

     * a)

$\displaystyle\mathsf{A}^{\mathrm{Ch}_{m}}(\mathbf{x}|B)=\sum\limits_{i=1}^{n}x_{(i)}\left(m(E_{(i)}{\cap
B})-m(E_{(i+1)}{\cap B})\right)$;

     * b)

$\displaystyle\mathsf{A}^{\mathrm{Sh}_{m}}(\mathbf{x}|B)=\max\limits_{i\in[n]}\left\\{x_{(i)}\cdot
m(E_{(i)}{\cap B})\right\\}$;

     * c)

$\displaystyle\mathsf{A}^{\mathrm{Su}_{m}}(\mathbf{x}|B)=\max\limits_{i\in[n]}\left\\{\min\\{x_{(i)},m(E_{(i)}{\cap
B})\\}\right\\}$.

  3. iii)

$\displaystyle\mathsf{A}(\mathbf{x}|B)=\frac{\max_{i\in B}(x_{i}\cdot
w_{i})}{\max_{i\in B}z_{i}},$ where $\displaystyle\mathbf{w}\in[0,1]^{[n]}$ is
a fixed weight vector, $\displaystyle\mathbf{z}\in(0,1]^{[n]}$ is fixed vector
such that $\displaystyle\max_{i\in[n]}z_{i}=1$. We note, that for
$\displaystyle\mathbf{w}=\mathbf{z}=\mathbf{1}_{[n]}$ we get
$\displaystyle\mathsf{A}^{\mathrm{max}}(\mathbf{x}|B)=\max_{i\in B}x_{i}$.

  4. iv)

$\displaystyle\mathsf{A}^{p-\mathrm{mean}}(\mathbf{x}|B)=\left(\frac{1}{{\\#}(B)}\cdot\sum\limits_{i\in
B}(x_{i})^{p}\right)^{\frac{1}{p}}$ with $\displaystyle p\in(0,\infty)$. For
$\displaystyle p=1$ we get the arithmetic mean.

  5. v)

$\displaystyle\mathsf{A}^{\mathrm{size}}(\mathbf{x}|B)=\max\limits_{D\in\mathscr{D}}\mathsf{s}(\mathbf{x}\mathbf{1}_{B})(D)$
with $\displaystyle\mathsf{s}$ being a size, see [3, 12, 13], is the outer
essential supremum of $\displaystyle\mathbf{x}$ over $\displaystyle B$ with
respect to a size $\displaystyle\mathsf{s}$ and a collection
$\displaystyle\mathscr{D}\subseteq 2^{[n]}$. In particular, for the sum as a
size, i.e.,
$\displaystyle\mathsf{s}_{\mathrm{sum}}(\mathbf{x})(G)=\sum\limits_{i\in
G}x_{i}$ for any $\displaystyle G\in 2^{[n]}$ and for
$\displaystyle\mathscr{D}$ such that there is a set $\displaystyle C\supseteq
B,C\in\mathscr{D}$ we get
$\displaystyle\mathsf{A}^{\mathrm{sum}}(\mathbf{x}|B)=\sum\limits_{i\in
B}x_{i}$.

Observe that the empty set is not included in the Definition 2.1. The reason
for that is the fact that the empty set does not provide any additional
information for aggregation. However, in order to have the concept of the
generalized survival function correctly introduced, it is necessary to add the
assumption $\displaystyle\mathsf{A}(\cdot|\emptyset)=0$, see [1, Section 4].
From now on, we shall consider only these conditional aggregation operators.
Let us remark, that all mappings from Example 2.2 with the convention
?$\displaystyle 0/0=0$? satisfy this property. In the following we shall
provide the definition of the generalized survival function, see [1,
Definition 4.1.]. Let us consider a collection $\displaystyle\mathscr{E}$,
$\displaystyle{{\\{}}\emptyset{{\\}}}\subseteq\mathscr{E}\subseteq 2^{[n]}$
and conditional aggregation operators on sets from $\displaystyle\mathscr{E}$
with $\displaystyle\mathsf{A}(\cdot|\emptyset)=0$. The set of such aggregation
operators we shall denote by

$\displaystyle{\mathscr{A}}=\\{\mathsf{A}(\cdot|E):E\in\mathscr{E}\\}$

and we shall call it a family of conditional aggregation operators (FCA for
short). For example,
$\displaystyle{\mathscr{A}}^{\mathrm{sum}}=\\{\mathsf{A}^{\mathrm{sum}}(\cdot|E):E\in
2^{[n]}\\}$,
$\displaystyle{\mathscr{A}}^{\mathrm{max}}=\\{\mathsf{A}^{\mathrm{max}}(\cdot|E):E\in\\{\emptyset,\\{1\\},\\{2\\},\dots,\\{n\\}\\}\\}$,
$\displaystyle\widehat{{\mathscr{A}}}^{\mathrm{max}}=\\{\mathsf{A}^{\mathrm{max}}(\cdot|E):E\in\\{\emptyset\\}\\}$
or $\displaystyle{\mathscr{A}}=\\{\mathsf{A}(\cdot|E):E\in 2^{[n]}\\}$,
$\displaystyle n\geq 2$, where

$\displaystyle\mathsf{A}(\mathbf{x}|E)=\begin{cases}\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E),&E\in\\{\\{1\\},\\{2\\},\dots,\\{n\\}\\},\\\
0,&E=\emptyset,\\\
\mathsf{A}^{\mathrm{sum}}(\mathbf{x}|E),&\text{otherwise}\end{cases}$

for any $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$.

###### Definition 2.3

(cf. [1, Definition 4.1.]) Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$,
$\displaystyle\mu\in\mathbf{M}$. The generalized survival function with
respect to $\displaystyle{\mathscr{A}}$ is defined as

$\displaystyle\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\min\left\\{\mu(E^{c}):\mathsf{A}(\mathbf{x}|E)\leq\alpha,\,E\in\mathscr{E}\right\\}$

for any $\displaystyle\alpha\in[0,\infty)$.

The presented definition is correct. Really, for any $\displaystyle
E\in\mathscr{E}$ it holds that $\displaystyle E^{c}\in 2^{[n]}$ is a
measurable set. Moreover, the set
$\displaystyle\\{E\in\mathscr{E}:\mathsf{A}(\mathbf{x}|E)\leq\alpha\\}$ is
nonempty for all $\displaystyle\alpha\in[0,\infty)$, because
$\displaystyle\mathsf{A}(\cdot|\emptyset)=0$ by convention and
$\displaystyle\emptyset\in\mathscr{E}$. Immediately it is seen, that for
$\displaystyle\mathscr{E}=2^{[n]}$ and
$\displaystyle{\mathscr{A}}^{\mathrm{max}}$ we get the standard survival
function, compare with (1). When it will be necessary we shall emphasize the
collection $\displaystyle\mathscr{E}$ in the notation of generalized survival
function, i.e. we shall use $\displaystyle{\mathscr{A}}^{\mathscr{E}}$.

On several places in this paper we shall work with the FCA that is
nondecreasing w.r.t sets, i.e. the map $\displaystyle
E\mapsto\mathsf{A}(\cdot|E)$ will be nondecreasing. Many FCA satisfy this
property, e.g.
$\displaystyle{\mathscr{A}}^{m-\mathrm{ess}}=\\{\mathsf{A}^{m-\mathrm{ess}}(\cdot|E):E\in\mathscr{E}\\}$,
$\displaystyle{\mathscr{A}}^{\mathrm{Ch}_{m}}=\\{\mathsf{A}^{\mathrm{Ch}_{m}}(\cdot|E):E\in\mathscr{E}\\}$,
$\displaystyle{\mathscr{A}}^{\mathrm{Su}_{m}}=\\{\mathsf{A}^{\mathrm{Su}_{m}}(\cdot|E):E\in\mathscr{E}\\}$,
$\displaystyle{\mathscr{A}}^{\mathrm{Sh}_{m}}=\\{\mathsf{A}^{\mathrm{Sh}_{m}}(\cdot|E):E\in\mathscr{E}\\}$,
$\displaystyle{\mathscr{A}}^{\mathrm{max}}=\\{\mathsf{A}^{\mathrm{max}}(\cdot|E):E\in\mathscr{E}\\}$,
see Example 2.2 i), ii), iii).

##  3 Equality and inequalities of the generalized and standard survival
function

In this section we shall treat Problem 1. We provide sufficient and necessary
conditions on $\displaystyle\mathbf{x}$, $\displaystyle\mu$ and
$\displaystyle{\mathscr{A}}$ under which the generalized survival function and
survival function coincide. The important knowledge we use is the standard
survival function formula. In what follows we shall work with the expression
of the survival function on a finite set in the form

$\mu(\\{\mathbf{x}>\alpha\\})=\sum_{i=0}^{n-1}\mu\left(E_{(i+1)}\right)\cdot\mathbf{1}_{[x_{(i)},x_{(i+1)})}(\alpha)\text{}$
(2)

with the permutation $\displaystyle(\cdot)$ such that $\displaystyle
0=x_{(0)}\leq x_{(1)}\leq x_{(2)}\leq\dots\leq x_{(n)}$ and $\displaystyle
E_{(i)}=\\{(i),\dots,(n)\\}$ for $\displaystyle i\in[n]$. However, one can
easily see that some summands in the formula (2) can be redundant. For
example, for vectors with the property $\displaystyle x_{(i)}=x_{(i+1)}$ for
some $\displaystyle i\in[n-1]\cup\\{0\\}$ we have
$\displaystyle\mu\left(E_{(i+1)}\right)\cdot\mathbf{1}_{[x_{(i)},x_{(i+1)})}(\alpha)=0$
for any $\displaystyle\alpha\in[0,\infty)$, i.e., this summand does not change
the values of survival function and can be omitted.

Let us consider an arbitrary (fixed) input vector $\displaystyle\mathbf{x}$
together with a permutation $\displaystyle(\cdot)$ such that $\displaystyle
0=x_{(0)}\leq x_{(1)}\leq x_{(2)}\leq\dots\leq x_{(n)}$. Let us denote

$\displaystyle\displaystyle\Psi_{\mathbf{x}}:=\\{i\in[n-1]\cup\\{0\\}:x_{(i)}<x_{(i+1)}\\}\cup\\{n\\}.$
(3)

For example, for the input vector $\displaystyle\mathbf{x}=(3,2,3,1)$ and the
permutation $\displaystyle(\cdot)$ such that $\displaystyle x_{(0)}=0$,
$\displaystyle x_{(1)}=1$, $\displaystyle x_{(2)}=2$, $\displaystyle
x_{(3)}=3$, $\displaystyle x_{(4)}=3$, we get
$\displaystyle\Psi_{\mathbf{x}}=\\{0,1,2,4\\}$. The following proposition
includes the very basic properties of system $\displaystyle\Psi_{\mathbf{x}}$
needed for further results.

###### Proposition 3.1

Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$.

  1. i)

$\displaystyle\Psi_{\mathbf{x}}$ is independent on permutation
$\displaystyle(\cdot)$ of $\displaystyle\mathbf{x}$, i.e.,
$\displaystyle\Psi_{\mathbf{x}}$ contains the same values for any permutation
$\displaystyle(\cdot)$ of $\displaystyle\mathbf{x}$ such that $\displaystyle
0=x_{(0)}\leq x_{(1)}\leq x_{(2)}\leq\dots\leq x_{(n)}$.

  2. ii)

For any $\displaystyle i\in[n]$ there exists $\displaystyle
k_{i}\in\Psi_{\mathbf{x}}\setminus\\{0\\}$ such that $\displaystyle
x_{i}=x_{(k_{i})}$, i.e.
$\displaystyle\\{x_{(k_{i})}:k_{i}\in\Psi_{\mathbf{x}}\setminus\\{0\\}\\}$
contains all different values of $\displaystyle\mathbf{x}$.

  3. iii)

$\displaystyle x_{(\min\Psi_{\mathbf{x}})}=0$.

  4. iv)

$\displaystyle\left\\{[x_{(k)},x_{(k+1)}):k\in\Psi_{\mathbf{x}}\right\\}$ is a
decomposition of interval $\displaystyle[0,\infty)$ into nonempty pairwise
disjoint sets.

Proof.

  1. i)

Let us consider two different permutations of $\displaystyle\mathbf{x}$ (if
they exist) $\displaystyle(\cdot)_{1}$ and $\displaystyle(\cdot)_{2}$ with the
required property. Let us denote

$\displaystyle\displaystyle\Psi_{\mathbf{x}}$
$\displaystyle\displaystyle:=\\{i\in[n-1]\cup\\{0\\}:x_{(i)_{1}}<x_{(i+1)_{1}}\\}\cup\\{n\\},$
$\displaystyle\displaystyle\Phi_{\mathbf{x}}$
$\displaystyle\displaystyle:=\\{i\in[n-1]\cup\\{0\\}:x_{(i)_{2}}<x_{(i+1)_{2}}\\}\cup\\{n\\}.$

We show that $\displaystyle\Psi_{\mathbf{x}}=\Phi_{\mathbf{x}}$. Indeed,
$\displaystyle n\in\Psi_{\mathbf{x}},n\in\Phi_{\mathbf{x}}$. If $\displaystyle
i\in\Psi_{\mathbf{x}}\setminus\\{n\\}$, then $\displaystyle
x_{(i)_{1}}<x_{(i+1)_{1}}$. Because of nondecreasing rearangement of
$\displaystyle\mathbf{x}$ with respect to $\displaystyle(\cdot)_{1}$,
$\displaystyle(\cdot)_{2}$ we get $\displaystyle x_{(i)_{2}}<x_{(i+1)_{2}}$,
therefore $\displaystyle i\in\Phi_{\mathbf{x}}$ and
$\displaystyle\Psi_{\mathbf{x}}\subseteq\Phi_{\mathbf{x}}$. By analogy it
holds $\displaystyle\Phi_{\mathbf{x}}\subseteq\Psi_{\mathbf{x}}$.

  2. ii)

Since any $\displaystyle i\in[n]$ can be represented via permutation as
$\displaystyle i={(j_{i})}$, $\displaystyle j_{i}\in[n]$, let us set

$\displaystyle k_{i}=\max\\{j_{i}\in[n]:\,x_{i}=x_{(j_{i})}\\}.$

As for any $\displaystyle k_{i}<n$ it holds that $\displaystyle
x_{(k_{i})}<x_{(k_{i}+1)}$, then $\displaystyle
k_{i}\in\Psi_{\mathbf{x}}\setminus\\{0\\}$. Moreover, $\displaystyle
k_{i}=n\in\Psi_{\mathbf{x}}$ because of the definition of
$\displaystyle\Psi_{\mathbf{x}}$, see (3).

  3. iii)

It follows immediately from the fact that
$\displaystyle\min\Psi_{\mathbf{x}}=\max\\{i\in[n]\cup\\{0\\}:x_{(i)}=x_{(0)}=0\\}$.

  4. iv)

It follows from part ii), iii) and from definition of system
$\displaystyle\Psi_{\mathbf{x}}$, since $\displaystyle x_{(k)}<x_{(k+1)}$ for
any $\displaystyle k\in\Psi_{\mathbf{x}}$ and $\displaystyle x_{(k_{1})}\neq
x_{(k_{2})}$ for any $\displaystyle k_{1},k_{2}\in\Psi_{\mathbf{x}}$.

Since we have shown that the system $\displaystyle\Psi_{\mathbf{x}}$ is
independent of the chosen permutation, henceforward we shall not explicitly
mention the permutation in assumptions of presented results. The following
proposition states that the formula (2) can be rewritten by the system
$\displaystyle\Psi_{\mathbf{x}}$ in the simpler form, see part
$\displaystyle{\rm{i)}}$. Moreover, in the second part of the proposition we
show that for a fixed $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$ it is
$\displaystyle\mu_{\mathscr{A}^{\mathrm{max},{{\mathscr{E}}}}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
with smaller collection $\displaystyle\mathscr{E}$ than the whole powerset
$\displaystyle 2^{[n]}$ (compare with the known result [1, Example 4.2] or see
(1)). The collection $\displaystyle\mathscr{E}$ depends on
$\displaystyle\mathbf{x}$ (equivalently on $\displaystyle\Psi_{\mathbf{x}}$).

###### Proposition 3.2

Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$,
$\displaystyle\mu\in\mathbf{M}$.

  1. i)

Then

$\mu(\\{\mathbf{x}>\alpha\\})=\sum_{k\in\Psi_{\mathbf{x}}}\mu\left(E_{(k+1)}\right)\cdot\mathbf{1}_{[x_{(k)},x_{(k+1)})}(\alpha)$
(4)

for any $\displaystyle\alpha\in[0,\infty)$ with the convention $\displaystyle
x_{(n+1)}=\infty$.

  2. ii)

If
$\displaystyle\mathscr{E}\supseteq\\{E_{(k+1)}^{c}:k\in\Psi_{\mathbf{x}}\\}$,
then

$\displaystyle\mu_{\mathscr{A}^{\mathrm{max}}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\}).$

Proof.

  1. i)

For $\displaystyle k\in[n-1]\cup\\{0\\},\,k\notin\Psi_{\mathbf{x}}$, we have
$\displaystyle x_{(k)}=x_{(k+1)}$. This leads to the fact that
$\displaystyle\mu\left(E_{(k+1)}\right)\cdot\mathbf{1}_{[x_{(k)},x_{(k+1)})}(\alpha)=0$
for any $\displaystyle\alpha\in[0,\infty)$. Using the Proposition 3.1 iv) we
have the required assertion.

  2. ii)

According to Proposition 3.1 (iv) let us divide interval
$\displaystyle[0,\infty)$ into disjoint sets

$\displaystyle[0,\infty)=\bigcup_{k\in\Psi_{\mathbf{x}}}[x_{(k)},x_{(k+1)}).$

Let us consider an arbitrary (fixed) $\displaystyle k\in\Psi_{\mathbf{x}}$.
Then from the fact, that $\displaystyle E_{(k+1)}^{c}\in\mathscr{E}$ and
$\displaystyle\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E_{(k+1)}^{c})=x_{(k)}$ we
have $\displaystyle
E_{(k+1)}^{c}\in\\{E:\mathsf{A}(\mathbf{x}|E)\leq\alpha\\}$ for any
$\displaystyle\alpha\in[x_{(k)},x_{(k+1)})$. Therefore we get

$\displaystyle\mu_{\mathscr{A}^{\mathrm{max},{{\mathscr{E}}}}}(\mathbf{x},\alpha)=\min\\{\mu(E^{c}):\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)\leq\alpha,E\in\mathscr{E}\\}\leq\mu(E_{(k+1)})=\mu(\\{\mathbf{x}>\alpha\\})$

for any $\displaystyle\alpha\in[x_{(k)},x_{(k+1)})$, where the last equality
follows from part i). On the other hand, as $\displaystyle\mathscr{E}\subseteq
2^{[n]}$ from properties of minimum we have

$\displaystyle\mu_{\mathscr{A}^{\mathrm{max},{{\mathscr{E}}}}}(\mathbf{x},\alpha)\geq\mu_{\mathscr{A}^{\mathrm{max},{{2^{[n]}}}}}(\mathbf{x},\alpha)=\mu(E_{(k+1)})=\mu(\\{\mathbf{x}>\alpha\\})$

for any $\displaystyle\alpha\in[x_{(k)},x_{(k+1)})$. To sum it up,
$\displaystyle\mu_{\mathscr{A}^{\mathrm{max},{{\mathscr{E}}}}}(\mathbf{x},\alpha)=\mu(E_{(k+1)})=\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[x_{(k)},x_{(k+1)})$.

###### Remark 3.3

Let us remark that in the whole paper we suppose $\displaystyle\mu$ is defined
on $\displaystyle 2^{[n]}$. However, in fact, it is enough to have a smaller
$\displaystyle\mathtt{Dom}{(\mu)}$. For example, in part ii) of the previous
proposition it is enough to consider the domain of $\displaystyle\mu$ being
$\displaystyle\\{E^{c}:E\in\mathscr{E}\\}$.

Let us note that in (4) the last summand is always equal to $\displaystyle 0$
because $\displaystyle\mu\left(E_{(n+1)}\right)=\mu(\emptyset)=0$. However, it
is useful to consider the form of survival function in (4) with sum over the
whole set $\displaystyle\Psi_{\mathbf{x}}$ not
$\displaystyle\Psi_{\mathbf{x}}\setminus\\{n\\}$ because of some technical
details in presented proofs in this paper.

###### Example 3.4

Let us take
$\displaystyle{\mathscr{A}}^{\mathrm{max}}=\\{\mathsf{A}^{\mathrm{max}}(\cdot|E):E\in\mathscr{E}\\}$
and normalized monotone measure $\displaystyle\mu$ on $\displaystyle 2^{[3]}$
given in the following table:

$\displaystyle E$ | $\displaystyle\emptyset$ | $\displaystyle\\{1\\}$ | $\displaystyle\\{2\\}$ | $\displaystyle\\{3\\}$ | $\displaystyle\\{1,2\\}$ | $\displaystyle\\{1,3\\}$ | $\displaystyle\\{2,3\\}$ | $\displaystyle\\{1,2,3\\}$
---|---|---|---|---|---|---|---|---
$\displaystyle\mu(E)$ | $\displaystyle 0$ | $\displaystyle 0$ | $\displaystyle 0.5$ | $\displaystyle 0$ | $\displaystyle 0.5$ | $\displaystyle 0.5$ | $\displaystyle 0.5$ | $\displaystyle 1$

Further, let us take the input vector $\displaystyle\mathbf{x}=(1,2,1)$ with
the permutation $\displaystyle(1)\\!=\\!1$, $\displaystyle(2)\\!=\\!3$,
$\displaystyle(3)\\!=\\!2$.

Then $\displaystyle\Psi_{\mathbf{x}}=\\{0,2,3\\}$ and the collection
guarantying the equality between survival function and generalized survival
function (of input $\displaystyle\mathbf{x}$) is according to Proposition 3.2
ii) e.g.

$\displaystyle\displaystyle\mathscr{E}$
$\displaystyle\displaystyle=\\{E_{(k+1)}^{c}:\,k\in\Psi_{\mathbf{x}}\\}=\\{E_{(1)}^{c},E_{(3)}^{c},E_{(4)}^{c}\\}=\\{\emptyset,\\{(1),(2)\\},\\{(1),(2),(3)\\}\\}$
$\displaystyle\displaystyle=\\{\emptyset,\\{1,3\\},\\{1,2,3\\}\\}.$

Indeed,

$\displaystyle\mu_{\mathscr{A}^{\mathrm{max}}}(\mathbf{x},\alpha)=1\cdot\mathbf{1}_{[0,1)}(\alpha)+0.5\cdot\mathbf{1}_{[1,2)}(\alpha)=\mu(\\{\mathbf{x}>\alpha\\}).$

Figure 3: The survival function visualization
$\displaystyle\mu_{{\mathscr{A}}^{\mathrm{max}}}$ with
$\displaystyle{\mathscr{A}}=\\{\mathsf{A}^{\mathrm{max}}(\cdot|E):E\in\\{E_{(k+1)}^{c}:k\in\Psi_{\mathbf{x}}\\}\\}$

From the previous result it follows that the standard survival function can be
represented by the formula

$\displaystyle\displaystyle\mu(\\{\mathbf{x}>\alpha\\})$
$\displaystyle\displaystyle=\min\big{\\{}\mu(E^{c}):\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)\leq\alpha,\,E\in\\{E_{(k+1)}^{c}:k\in\Psi_{\mathbf{x}}\\}\big{\\}}$
(5)

with the system $\displaystyle\Psi_{\mathbf{x}}$ given by the input vector
$\displaystyle\mathbf{x}$. This formula can be visualized by Figure 3. Let us
remark that since
$\displaystyle\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E_{(k+1)}^{c})=x_{(k)}$ on
the upper line we measure sets $\displaystyle(E_{(k+1)}^{c})^{c}$. The
calculation of (generalized) survival function is processed as we have
described in the Introduction. Let us remark that the essence of the following
results is the pointwise comparison of the generalized survival function with
the standard survival function having in mind the representation (5) together
with its visualization, see Figure 3.

It is obvious that the equality of survival functions (standard and
generalized) means that they have to achieve the same values, i.e.,
$\displaystyle\mu\left(E_{(k+1)}\right)$, $\displaystyle
k\in\Psi_{\mathbf{x}}$, on the same corresponding intervals
$\displaystyle[x_{(k)},x_{(k+1)})$, $\displaystyle k\in\Psi_{\mathbf{x}}$.
Having in mind the formula (4), the survival function representation given by
(5) and the visualization, see Figure 3, we can formulate the following
sufficient conditions. While (C1) ensures that the generalized survival
function will be able to achieve the same values as the survival function,
(C2) guarantees it. Let $\displaystyle{\mathscr{A}}$ be FCA.

  1. (C1)

For any $\displaystyle k\in\Psi_{\mathbf{x}}$ there exists $\displaystyle
G_{k}\in\mathscr{E}$ such that

$\displaystyle\mathsf{A}(\mathbf{x}|G_{k})=x_{(k)}\quad\text{and}\quad\mu(G_{k}^{c})=\mu(E_{(k+1)}).$

  2. (C2)

For any $\displaystyle k\in\Psi_{\mathbf{x}}$ and for any $\displaystyle
E\in\mathscr{E}$ it holds:

$\displaystyle\mathsf{A}(\mathbf{x}|E)<x_{(k+1)}\Rightarrow\mu(E^{c})\geq\mu(E_{(k+1)}).$

Figure 4: The visualization of conditions $\displaystyle\mathrm{(C1)}$ and
$\displaystyle\mathrm{(C2)}$

The visualization of conditions $\displaystyle\mathrm{(C1)}$,
$\displaystyle\mathrm{(C2)}$ via two parallel lines is drawn in Figure 4. Let
us remark that for $\displaystyle k=n$ (C2) holds trivially. Also, for
$\displaystyle k=\min\Psi_{\mathbf{x}}$ (C1) holds trivially with
$\displaystyle G_{\min\Psi_{\mathbf{x}}}=\emptyset$.

###### Remark 3.5

In accordance with the above written, it can be easily seen that for
$\displaystyle{\mathscr{A}}^{\mathrm{max}}$ with
$\displaystyle\mathscr{E}\supseteq\\{E_{(k+1)}^{c}:k\in\Psi_{\mathbf{x}}\\}$
it holds $\displaystyle G_{k}=E_{(k+1)}^{c}$ for any $\displaystyle
k\in\Psi_{\mathbf{x}}$ regardless of the choice of $\displaystyle\mu$ in
$\displaystyle\mathrm{(C1)}$. Of course, for specific classes of monotone
measures $\displaystyle\mu$ also other sets $\displaystyle G_{k}$ can satisfy
$\displaystyle\rm{(C1)}$. Similarly, the validity of $\displaystyle\rm{(C2)}$
is clear. Indeed, if
$\displaystyle\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)<x_{(k+1)}$, then we have
$\displaystyle E\subseteq E_{(k+1)}^{c}$, i.e., $\displaystyle E^{c}\supseteq
E_{(k+1)}$. From the monotonicity of $\displaystyle\mu$ we have
$\displaystyle\mu(E^{c})\geq\mu(E_{(k+1)})$ for any $\displaystyle
E\in\mathscr{E}$.

Conditions $\displaystyle\mathrm{(C1)}$ and $\displaystyle\mathrm{(C2)}$
guarantee inequalities between survival functions. Thus the equality of
survival function is a consequence.

###### Proposition 3.6

Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$,
$\displaystyle\mu\in{\mathbf{M}}$, and let $\displaystyle{\mathscr{A}}$ be
FCA.

  1. i)

If $\displaystyle\mathrm{(C1)}$ holds, then
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)\leq\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

  2. ii)

$\displaystyle\mathrm{(C2)}$ holds if and only if
$\displaystyle\mu(\\{\mathbf{x}>\alpha\\})\leq\mu_{\mathscr{A}}(\mathbf{x},\alpha)$
for any $\displaystyle\alpha\in[0,\infty)$.

Proof. According to Proposition 3.1 (iv) let us divide interval
$\displaystyle[0,\infty)$ into disjoint sets

$\displaystyle[0,\infty)=\bigcup_{k\in\Psi_{\mathbf{x}}}[x_{(k)},x_{(k+1)}).$

Let us consider an arbitrary (fixed) $\displaystyle k\in\Psi_{\mathbf{x}}$.

Let us prove part i). According to $\displaystyle\mathrm{(C1)}$ there exists
the set $\displaystyle G_{k}\in\mathscr{E}$ such that
$\displaystyle\mathsf{A}(\mathbf{x}|G_{k})=x_{(k)}$ and
$\displaystyle\mu(G_{k}^{c})=\mu(E_{(k+1)})$. From the fact that
$\displaystyle\mu(E_{(k+1)})=\mu(G_{k}^{c})\in\left\\{\mu(E^{c}):\,\mathsf{A}(\mathbf{x}|E)\leq
x_{(k)}\right\\}$ and since
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)$ is nonincreasing (see [1,
Proposition 4.3 (a)]) we have

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)\leq\mu_{\mathscr{A}}(\mathbf{x},x_{(k)})\leq\mu(E_{(k+1)})=\mu(\\{\mathbf{x}>\alpha\\})$

for any $\displaystyle\alpha\in[x_{(k)},x_{(k+1)})$, where the last equality
follows from (4).

Let us prove part ii). From $\displaystyle\mathrm{(C2)}$ it follows that for
any $\displaystyle E\in\mathscr{E}$ if
$\displaystyle\mathsf{A}(\mathbf{x}|E)<x_{(k+1)}$, then
$\displaystyle\mu(E^{c})\geq\mu(E_{(k+1)})$. Therefore,

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)\geq\mu(E_{(k+1)})=\mu(\\{\mathbf{x}>\alpha\\})$

for any $\displaystyle\alpha\in[x_{(k)},x_{(k+1)})$, where the last equality
follows from (4). It is enough to prove the implication
$\displaystyle\Leftarrow$. Since

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\min\left\\{\mu(E^{c}):\mathsf{A}(\mathbf{x}|E)\leq\alpha<x_{(k+1)},E\in\mathscr{E}\right\\}\geq\mu(E_{(k+1)})=\mu(\\{\mathbf{x}>\alpha\\})$

for any $\displaystyle\alpha\in[x_{(k)},x_{(k+1)})$, then for any
$\displaystyle E\in\mathscr{E}$ it has to hold: if
$\displaystyle\mathsf{A}(\mathbf{x}|E)<x_{(k+1)}$, then
$\displaystyle\mu(E^{c})\geq\mu(E_{(k+1)})$.

###### Corollary 3.7

Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$,
$\displaystyle\mu\in{\mathbf{M}}$, and let $\displaystyle{\mathscr{A}}$ be
FCA. If $\displaystyle\mathrm{(C1)}$ and $\displaystyle\mathrm{(C2)}$ are
satisfied, then
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

The application of the previous result is illustrated in the following
example. The second example proves that $\displaystyle\mathrm{(C1)}$ and
$\displaystyle\mathrm{(C2)}$ are only sufficient and not necessary.

|
---|---
Figure 5: Generalized survival function and visualization from Example 3.8

###### Example 3.8

Let us consider
$\displaystyle{\mathscr{A}}^{\mathrm{sum}}=\\{\mathsf{A}^{\mathrm{sum}}(\cdot|E):E\in
2^{[3]}\\}$, and normalized monotone measure $\displaystyle\mu$ on
$\displaystyle 2^{[3]}$ with the following values:

$\displaystyle E$ | $\displaystyle\emptyset$ | $\displaystyle\\{1\\}$ | $\displaystyle\\{2\\}$ | $\displaystyle\\{3\\}$ | $\displaystyle\\{1,2\\}$ | $\displaystyle\\{1,3\\}$ | $\displaystyle\\{2,3\\}$ | $\displaystyle\\{1,2,3\\}$
---|---|---|---|---|---|---|---|---
$\displaystyle\mu(E)$ | $\displaystyle 0$ | $\displaystyle 0$ | $\displaystyle 0.5$ | $\displaystyle 0$ | $\displaystyle 0.5$ | $\displaystyle 0$ | $\displaystyle 0.7$ | $\displaystyle 1$
$\displaystyle\mathsf{A}^{\mathrm{sum}}(\mathbf{x}|E)$ | $\displaystyle 0$ | $\displaystyle 1$ | $\displaystyle 3$ | $\displaystyle 1$ | $\displaystyle 4$ | $\displaystyle 2$ | $\displaystyle 4$ | $\displaystyle 5$

Further, let us take the input vector $\displaystyle\mathbf{x}=(1,3,1)$ with
the permutation $\displaystyle(1)=1$, $\displaystyle(2)=3$,
$\displaystyle(3)=2$. Then $\displaystyle x_{(0)}=0$, $\displaystyle
x_{(1)}=1$, $\displaystyle x_{(2)}=1$, $\displaystyle x_{(3)}=3$, therefore
$\displaystyle\Psi_{\mathbf{x}}=\\{0,2,3\\}$ and

$\displaystyle E_{(1)}=\\{(1),(2),(3)\\}=\\{1,2,3\\},\quad
E_{(3)}=\\{(3)\\}=\\{2\\},\quad E_{(4)}=\emptyset.$

We can see, that the assertion $\displaystyle\mathrm{(C1)}$ of Corollary 3.7
is satisfied with

$\displaystyle G_{0}=\emptyset$, $\displaystyle G_{2}=\\{3\\}$, $\displaystyle
G_{3}=\\{2\\}$.

Indeed, $\displaystyle\mathsf{A}^{\mathrm{sum}}(\mathbf{x}|G_{0})=0=x_{(0)}$
and $\displaystyle\mu(G_{0}^{c})=\mu(E_{(1)})$. Further,
$\displaystyle\mathsf{A}^{\mathrm{sum}}(\mathbf{x}|G_{2})=1=x_{(2)}$ and
$\displaystyle\mu(G_{2}^{c})=\mu(\\{1,2\\})=\mu(E_{(3)})$. Finally,
$\displaystyle\mathsf{A}^{\mathrm{sum}}(\mathbf{x}|G_{3})=3=x_{(3)}$ and
$\displaystyle\mu(G_{3}^{c})=\mu(\\{1,3\\})=\mu(E_{(4)})$. The assertion
$\displaystyle\mathrm{(C2)}$ is also satisfied, see the visualisation of
generalized survival function via parallel lines in Figure 5. Discussed
survival functions coincide and take the form

$\displaystyle\mu(\\{\mathbf{x}>\alpha\\})=\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=\mathbf{1}_{[0,1)}(\alpha)+0{.}5\cdot\mathbf{1}_{[1,3)}(\alpha)$

for $\displaystyle\alpha\in[0,\infty)$. The plot of (generalized) survival
function is in Figure 5.

###### Example 3.9

Let us consider
$\displaystyle{\mathscr{A}}^{\mathrm{sum}}=\\{\mathsf{A}^{\mathrm{sum}}(\cdot|E):E\in
2^{[3]}\\}$, and normalized monotone measure $\displaystyle\mu$ on
$\displaystyle 2^{[3]}$ with the following values:

$\displaystyle E$ | $\displaystyle\emptyset$ | $\displaystyle\\{1\\}$ | $\displaystyle\\{2\\}$ | $\displaystyle\\{3\\}$ | $\displaystyle\\{1,2\\}$ | $\displaystyle\\{1,3\\}$ | $\displaystyle\\{2,3\\}$ | $\displaystyle\\{1,2,3\\}$
---|---|---|---|---|---|---|---|---
$\displaystyle\mu(E)$ | $\displaystyle 0$ | $\displaystyle 0$ | $\displaystyle 0$ | $\displaystyle 0.7$ | $\displaystyle 0$ | $\displaystyle 0.8$ | $\displaystyle 0.7$ | $\displaystyle 1$
$\displaystyle\mathsf{A}^{\mathrm{sum}}(\mathbf{x}|E)$ | $\displaystyle 0$ | $\displaystyle 2$ | $\displaystyle 3$ | $\displaystyle 4$ | $\displaystyle 5$ | $\displaystyle 6$ | $\displaystyle 7$ | $\displaystyle 9$

Further, let us take the input vector $\displaystyle\mathbf{x}=(2,3,4)$ with
the permutation being the identity. Then survival functions coincide

$\displaystyle\mu(\\{\mathbf{x}>\alpha\\})=\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=\mathbf{1}_{[0,2)}(\alpha)+0{.}7\cdot\mathbf{1}_{[2,4)}(\alpha).$

Here, $\displaystyle G_{0}=\emptyset$, $\displaystyle G_{1}=\\{1\\}$,
$\displaystyle G_{2}=\\{2\\}$, $\displaystyle G_{3}=\\{3\\}$ are the only sets
that satisfy the equality
$\displaystyle\mathsf{A}^{\mathrm{sum}}(\mathbf{x}|G_{k})=x_{(k)}$ for
$\displaystyle k\in\Psi_{\mathbf{x}}=\\{0,1,2,3\\}$. However,

$\displaystyle 0{.}8=\mu(G_{2}^{c})\neq\mu(E_{(3)})=0{.}7.$

Thus, a sufficient condition in Corollary 3.7 is not a necessary condition.

Let us return to Proposition 3.6. While $\displaystyle\mathrm{(C2)}$ is the
necessary and sufficient condition under which the generalized survival
function is greater or equal to the survival function,
$\displaystyle\mathrm{(C1)}$ is only sufficient for the reverse inequality.
Since this condition seems too strict, let us define conditions
$\displaystyle\mathrm{(C3)}$ and $\displaystyle\mathrm{(C4)}$ as follows:

  1. (C3)

For any $\displaystyle k\in\Psi_{\mathbf{x}}$ there exists $\displaystyle
F_{k}\in\mathscr{E}$ such that $\displaystyle\mathsf{A}(\mathbf{x}|F_{k})\leq
x_{(k)}$ and $\displaystyle\mu(F_{k}^{c})\leq\mu(E_{(k+1)})$.

  2. (C4)

For any $\displaystyle k\in\Psi_{\mathbf{x}}$ there exists $\displaystyle
F_{k}\in\mathscr{E}$ such that $\displaystyle\mathsf{A}(\mathbf{x}|F_{k})\leq
x_{(k)}$ and $\displaystyle\mu(F_{k}^{c})=\mu(E_{(k+1)})$.

The visualization of condition $\displaystyle\mathrm{(C3)}$ is drawn in Figure
6. In the following we show that exactly $\displaystyle\mathrm{(C3)}$ improves
Proposition 3.6 ii). As a consequence we also get improvement of Corollary
3.7. Replacing $\displaystyle\mathrm{(C1)}$ with $\displaystyle\mathrm{(C3)}$,
we obtain sufficient and necessary condition for equality between survival
functions. However, it will turn out that under the
$\displaystyle\mathrm{(C2)}$ assumption $\displaystyle\mathrm{(C3)}$ will be
reduced to $\displaystyle\mathrm{(C4)}$.

Figure 6: The visualization of condition $\displaystyle\mathrm{(C3)}$ from
Proposition 3.10

###### Proposition 3.10

Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$,
$\displaystyle\mu\in{\mathbf{M}}$, and let $\displaystyle{\mathscr{A}}$ be
FCA. $\displaystyle\mathrm{(C3)}$ holds if and only if
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)\leq\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

Proof. Let us prove the implication $\displaystyle\Rightarrow$. According to
Proposition 3.1 (iv) let us divide interval $\displaystyle[0,\infty)$ into
disjoint sets

$\displaystyle[0,\infty)=\bigcup_{k\in\Psi_{\mathbf{x}}}[x_{(k)},x_{(k+1)}).$

Let us consider an arbitrary (fixed) $\displaystyle k\in\Psi_{\mathbf{x}}$.
Then by assumptions, there is $\displaystyle F_{k}\in\mathscr{E}$ such that
$\displaystyle\mathsf{A}(\mathbf{x}|F_{k})\leq x_{(k)}$ and
$\displaystyle\mu(F_{k}^{c})\leq\mu(E_{(k+1)})$. Thus
$\displaystyle\mu(F_{k}^{c})\in\\{\mu(E^{c}):\,\mathsf{A}(\mathbf{x}|E)\leq\alpha,\,E\in\mathscr{E}\\}$
for any $\displaystyle\alpha\in[x_{(k)},x_{(k+1)})$. Hence,

$\displaystyle\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\min\left\\{\mu(E^{c}):\mathsf{A}(\mathbf{x}|E)\leq\alpha,E\in\mathscr{E}\right\\}\leq\mu(F_{k}^{c})\leq\mu(E_{(k+1)})=\mu(\\{\mathbf{x}>\alpha\\})\text{}$

for any $\displaystyle\alpha\in[x_{(k)},x_{(k+1)})$.

Let us prove the reverse implication $\displaystyle\Leftarrow$. Let
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)\leq\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$. Then from this fact and from (4)
it follows:

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},x_{(k)})\leq\mu(\\{\mathbf{x}>x_{(k)}\\})=\mu(E_{(k+1)})$

for any $\displaystyle k\in\Psi_{\mathbf{x}}$. As
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},x_{(k)})=\min\left\\{\mu(E^{c}):\mathsf{A}(\mathbf{x}|E)\leq
x_{(k)},E\in\mathscr{E}\right\\}$, there exists $\displaystyle
F_{k}\in\mathscr{E}$ such that $\displaystyle\mathsf{A}(\mathbf{x}|F_{k})\leq
x_{(k)}$ and $\displaystyle\mu(F_{k}^{c})\leq\mu(E_{(k+1)})$.

###### Corollary 3.11

Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$,
$\displaystyle\mu\in{\mathbf{M}}$, and let $\displaystyle{\mathscr{A}}$ be
FCA.

  1. i)

If $\displaystyle\mathrm{(C2)}$ holds, then $\displaystyle\mathrm{(C3)}$ is
equivalent to $\displaystyle\mathrm{(C4)}$.

  2. ii)

$\displaystyle\mathrm{(C2)}$ and $\displaystyle\mathrm{(C3)}$ hold if and only
if
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

  3. iii)

$\displaystyle\mathrm{(C2)}$ and $\displaystyle\mathrm{(C4)}$ hold if and only
if
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

Proof. It is enough to prove part i), more precisely, the implication
$\displaystyle\mathrm{(C3)}\Rightarrow\mathrm{(C4)}$. Let
$\displaystyle\mathrm{(C3)}$ is satisfied, we show that
$\displaystyle\mu(F_{k}^{c})=\mu(E_{(k+1)})$ holds for any $\displaystyle
k\in\Psi_{\mathbf{x}}$. Since for any $\displaystyle F_{k}\in\mathscr{E}$,
$\displaystyle k\in\Psi_{\mathbf{x}}$ we have
$\displaystyle\mathsf{A}(F_{k}|x)\leq x_{(k)}<x_{(k+1)}$, then from
$\displaystyle\mathrm{(C2)}$ we have
$\displaystyle\mu(F_{k}^{c})\geq\mu(E_{(k+1)})$. On the other hand, from
$\displaystyle\mathrm{(C3)}$ we have
$\displaystyle\mu(F_{k}^{c})\leq\mu(E_{(k+1)})$.

###### Remark 3.12

At the end of this main part let us remark that some above mentioned results
are true also without constructing $\displaystyle\Psi_{\mathbf{x}}$ system.
Let us denote:

  1. $\displaystyle(\widetilde{\mathrm{C}}1)$

For any $\displaystyle i\in[n]\cup\\{0\\}$ there exists $\displaystyle
G_{i}\in\mathscr{E}$ such that
$\displaystyle\mathsf{A}(\mathbf{x}|G_{i})=x_{(i)}$ and
$\displaystyle\mu(F_{i}^{c})=\leavevmode\nobreak\ \mu(E_{(i+1)})$.

  2. $\displaystyle(\widetilde{\mathrm{C}}2)$

For any $\displaystyle i\in[n]\cup\\{0\\}$ and for any $\displaystyle
E\in\mathscr{E}$ it holds:
$\displaystyle\mathsf{A}(\mathbf{x}|E)<x_{(i+1)}\Rightarrow\mu(E^{c})\geq\mu(E_{(i+1)}).$

  3. $\displaystyle(\widetilde{\mathrm{C}}3)$

For any $\displaystyle i\in[n]\cup\\{0\\}$ there exists $\displaystyle
F_{i}\in\mathscr{E}$ such that $\displaystyle\mathsf{A}(\mathbf{x}|F_{i})\leq
x_{(i)}$ and $\displaystyle\mu(F_{i}^{c})\leq\mu(E_{(i+1)})$.

Then Proposition 3.6 and Corollary 3.11 (ii) remain to be true, although,
requirements in $\displaystyle(\widetilde{\mathrm{C}}1)$,
$\displaystyle(\widetilde{\mathrm{C}}2)$,
$\displaystyle(\widetilde{\mathrm{C}}3)$ will be for some $\displaystyle
i\in[n]\cup\\{0\\}$ redundant 888They will be redundant for $\displaystyle
i\in[n]\cup\\{0\\}$ such that $\displaystyle x_{(i)}=x_{(i+1)}$, compare with
the motivation of $\displaystyle\Psi_{\mathbf{x}}$ system introduction.. On
the other hand, Corollary 3.11 (i), (iii) need not be satisfied in general.

Inequalities: Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$,
$\displaystyle\mu\in{\mathbf{M}}$, and let $\displaystyle{\mathscr{A}}$ be
FCA.

  1. i)

If $\displaystyle(\widetilde{\mathrm{C}}1)$ holds, then
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)\leq\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

  2. ii)

$\displaystyle(\widetilde{\mathrm{C}}2)$ holds if and only if
$\displaystyle\mu(\\{\mathbf{x}>\alpha\\})\leq\mu_{\mathscr{A}}(\mathbf{x},\alpha)$
for any $\displaystyle\alpha\in[0,\infty)$.

Sufficient and necessary condition: Let
$\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$,
$\displaystyle\mu\in{\mathbf{M}}$, and let $\displaystyle{\mathscr{A}}$ be
FCA. $\displaystyle(\widetilde{\mathrm{C}}2)$ and
$\displaystyle(\widetilde{\mathrm{C}}3)$ hold if and only if
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

###  3.1 Equality of generalized survival function and standard survival
function, further results

In this subsection we provide further results on indistinguishability of
survival functions. Considering the formula of standard survival function (4)
one can observe that the same value of monotone measure may be achieved on
several intervals. These intervals can be joined together. Thus we obtain
again a shorter formula of survival function, see Proposition 3.14 i), which
allows us to formulate further results. Let us define system
$\displaystyle\Psi_{\mathbf{x}}^{*}\subseteq\Psi_{\mathbf{x}}$ as follows:

$\displaystyle\displaystyle\Psi_{\mathbf{x}}^{*}:=\\{k\in\Psi_{\mathbf{x}}\setminus\\{\min\Psi_{\mathbf{x}}\\}:\,\mu(E_{(j+1)})>\mu(E_{(k+1)}),j<k,j\in\Psi_{\mathbf{x}}\\}\cup\\{\min\Psi_{\mathbf{x}}\\}$
(6)

(compare with the definition of system $\displaystyle\Psi_{\mathbf{x}}$ which
is analogous, however the main condition is concentrated on components of
$\displaystyle\mathbf{x}$ instead of values of $\displaystyle\mu$). Let us
give an example of the $\displaystyle\Psi_{\mathbf{x}}^{*}$ system calculation
considering inputs from Example 3.9. For given input
$\displaystyle\Psi_{\mathbf{x}}=\\{0,1,2,3\\}$. Then by definition of
$\displaystyle\Psi_{\mathbf{x}}^{*}$ we have
$\displaystyle\min\Psi_{\mathbf{x}}=0\in\Psi_{\mathbf{x}}^{*}$. For
$\displaystyle k=1,3$ the inequality
$\displaystyle\mu(E_{(j+1)})>\mu(E_{(k+1)})$, $\displaystyle j<k$ holds,
however, for $\displaystyle k=2$ we have
$\displaystyle\mu(E_{(2)})=\mu(E_{(3)})$. Thus $\displaystyle
2\notin\Psi_{\mathbf{x}}^{*}$. In summary,
$\displaystyle\Psi_{\mathbf{x}}^{*}=\\{0,1,3\\}$.

For purpose of this subsection for any $\displaystyle
k\in\Psi_{\mathbf{x}}^{*}$ let us denote

$\displaystyle\displaystyle
l_{k}:=\max\\{j\in\Psi_{\mathbf{x}}:\,\mu(E_{(j+1)})=\mu(E_{(k+1)})\\}.$ (7)

###### Proposition 3.13

Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$,
$\displaystyle\mu\in\mathbf{M}$.

  1. i)

$\displaystyle x_{(\min\Psi_{\mathbf{x}}^{*})}=0$.

  2. ii)

$\displaystyle\left\\{[x_{(k)},x_{(l_{k}+1)}):k\in\Psi_{\mathbf{x}}^{*}\right\\}$
with $\displaystyle l_{k}$ given by (7) and with the convention $\displaystyle
x_{(n+1)}=\infty$ is a decomposition of interval $\displaystyle[0,\infty)$
into nonempty pairwise disjoint sets.

  3. iii)

$\displaystyle(\forall
k\in\Psi_{\mathbf{x}}^{*}\setminus\\{\min\Psi_{\mathbf{x}}^{*}\\})$
$\displaystyle(\exists r\in\Psi_{\mathbf{x}}^{*},r<k)$ $\displaystyle
x_{(k)}=x_{(l_{r}+1)}$. Moreover,
$\displaystyle\mu(E_{(k+1)})<\mu(E_{(l_{r}+1)})$.

  4. iv)

If $\displaystyle\mu$ is such that it is strictly monotone on
$\displaystyle\\{E_{(k+1)}:k\in\Psi_{\mathbf{x}}\\}$, then
$\displaystyle\Psi_{\mathbf{x}}=\Psi_{\mathbf{x}}^{*}$.

Proof. Part i) follows from Proposition 3.1 iii). Since
$\displaystyle\Psi_{\mathbf{x}}^{*}\subseteq\Psi_{\mathbf{x}}$ and
$\displaystyle\min\Psi_{\mathbf{x}}\in\Psi_{\mathbf{x}}^{*}$, then
$\displaystyle\min\Psi_{\mathbf{x}}=\min\Psi_{\mathbf{x}}^{*}$. The proof of
ii) follows from Proposition 3.1 part iv) and from the fact that each partial
interval $\displaystyle[x_{(k)},x_{(l_{k}+1)})$, $\displaystyle
k\in\Psi_{\mathbf{x}}^{*}$ can be rewritten as follows

$\displaystyle[x_{(k)},x_{(l_{k}+1)})=\bigcup_{j=k,j\in\Psi_{\mathbf{x}}}^{l_{k}}[x_{(j)},x_{(j+1)}).$

The equality $\displaystyle x_{(k)}=x_{(l_{r}+1)}$ in part iii) follows from
ii) with $\displaystyle
r=\max\\{j\in\Psi_{\mathbf{x}}^{*}:x_{(j)}<x_{(k)}\\}$. Moreover, it holds
$\displaystyle\mu(E_{(l_{r}+1)})=\mu(E_{(r+1)})>\mu(E_{(k+1)})$ where the
first equality holds because of (7), the second inequality is true due to
$\displaystyle r<k$, $\displaystyle r,k\in\Psi_{\mathbf{x}}^{*}$. Part iv)
follows from (6).

###### Proposition 3.14

Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$,
$\displaystyle\mu\in\mathbf{M}$.

  1. i)

Then

$\mu(\\{\mathbf{x}>\alpha\\})=\sum_{k\in\Psi_{\mathbf{x}}^{*}}\mu\left(E_{(k+1)}\right)\cdot\mathbf{1}_{[x_{(k)},x_{(l_{k}+1)})}(\alpha)$
(8)

for any $\displaystyle\alpha\in[0,\infty)$ with $\displaystyle l_{k}$ given by
(7) and with the convention $\displaystyle x_{(n+1)}=\infty$.

  2. ii)

If
$\displaystyle\mathscr{E}\supseteq\\{E_{(k+1)}^{c}:k\in\Psi_{\mathbf{x}}^{*}\\}$,
then
$\displaystyle\mu_{\mathscr{A}^{\mathrm{max}}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

Proof. Part ii) can be proved analogously as Proposition 3.2 part ii). Part i)
follows from the fact that each partial interval
$\displaystyle[x_{(k)},x_{(l_{k}+1)})$, $\displaystyle
k\in\Psi_{\mathbf{x}}^{*}$ can be rewritten as follows
$\displaystyle[x_{(k)},x_{(l_{k}+1)})=\bigcup_{j=k,j\in\Psi_{\mathbf{x}}}^{l_{k}}[x_{(j)},x_{(j+1)}).$
From the formula (4) and from definition of $\displaystyle l_{k}$ we get

$\displaystyle\mu(\\{\mathbf{x}>\alpha\\})=\mu(E_{(j+1)})=\mu(E_{(k+1)}).$

for any $\displaystyle\alpha\in[x_{(j)},x_{(j+1)})$.

All results from the previous subsection will also be true under a slight
modification of conditions (C1), (C2), (C3) and (C4) as follows:

  1. (C1∗)

For any $\displaystyle k\in\Psi_{\mathbf{x}}^{*}$ there exists $\displaystyle
G_{k}\in\mathscr{E}$ such that
$\displaystyle\mathsf{A}(\mathbf{x}|G_{k})=x_{(k)}$ and
$\displaystyle\mu(G_{k}^{c})=\mu(E_{(k+1)}).$

  2. (C2∗)

For any $\displaystyle k\in\Psi_{\mathbf{x}}^{*}$ and for any $\displaystyle
E\in\mathscr{E}$ it holds:
$\displaystyle\mathsf{A}(\mathbf{x}|E)<x_{(l_{k}+1)}\Rightarrow\mu(E^{c})\geq\mu(E_{(l_{k}+1)}).$

  3. (C3∗)

For any $\displaystyle k\in\Psi_{\mathbf{x}}^{*}$ there exists $\displaystyle
F_{k}\in\mathscr{E}$ such that $\displaystyle\mathsf{A}(\mathbf{x}|F_{k})\leq
x_{(k)}$ and $\displaystyle\mu(F_{k}^{c})\leq\mu(E_{(k+1)})$.

  4. (C4∗)

For any $\displaystyle k\in\Psi_{\mathbf{x}}^{*}$ there exists $\displaystyle
F_{k}\in\mathscr{E}$ such that $\displaystyle\mathsf{A}(\mathbf{x}|F_{k})\leq
x_{(k)}$ and $\displaystyle\mu(F_{k}^{c})=\mu(E_{(k+1)})$.

In the following we summarize all modifications of results from the main part
of this section. Since proofs of parts i) – vii) are based on the same ideas,
we omit them. The comparison of these results with those obtained in the main
part can be found in Remark 3.16.

###### Proposition 3.15

Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$,
$\displaystyle\mu\in{\mathbf{M}}$, and let $\displaystyle{\mathscr{A}}$ be
FCA.

  1. i)

If $\displaystyle\mathrm{(C1^{*})}$ holds, then
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)\leq\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

  2. ii)

$\displaystyle\mathrm{(C2^{*})}$ holds if and only if
$\displaystyle\mu(\\{\mathbf{x}>\alpha\\})\leq\mu_{\mathscr{A}}(\mathbf{x},\alpha)$
for any $\displaystyle\alpha\in[0,\infty)$.

  3. iii)

If $\displaystyle\mathrm{(C1^{*})}$ and $\displaystyle\mathrm{(C2^{*})}$ are
satisfied, then
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

  4. iv)

$\displaystyle\mathrm{(C3^{*})}$ holds if and only if
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)\leq\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

  5. v)

$\displaystyle\mathrm{(C2^{*})}$ and $\displaystyle\mathrm{(C3^{*})}$ hold if
and only if
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

  6. vi)

If $\displaystyle\mathrm{(C2^{*})}$ holds, then
$\displaystyle\mathrm{(C3^{*})}$ is equivalent to
$\displaystyle\mathrm{(C4^{*})}$.

  7. vii)

$\displaystyle\mathrm{(C2^{*})}$ and $\displaystyle\mathrm{(C4^{*})}$ hold if
and only if
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

  8. viii)

$\displaystyle\mathrm{(C2)}$ holds if and only if
$\displaystyle\mathrm{(C2^{*})}$ holds.

Proof. The implication (C2) $\displaystyle\Rightarrow$ (C2∗) of part viii) is
clear. We prove the reverse implication. Let us consider any set
$\displaystyle E\in\mathscr{E}$ such that
$\displaystyle\mathsf{A}(\mathbf{x}|E)<x_{(k+1)}$ for some $\displaystyle
k\in\Psi_{\mathbf{x}}$. Let us define

$\displaystyle
j_{k}=\min\\{j\in\Psi_{\mathbf{x}}:\mu(E_{(j+1)})=\mu(E_{(k+1)})\\}.$

It is easy to see that $\displaystyle j_{k}\in\Psi_{\mathbf{x}}^{*}$,
$\displaystyle l_{j_{k}}\geq k\geq j_{k}$. Moreover,
$\displaystyle\mu(E_{(l_{j_{k}}+1)})=\mu(E_{(k+1)})=\mu(E_{(j_{k}+1)})$ and
$\displaystyle x_{(k+1)}\leq x_{(l_{j_{k}}+1)}$. Then from (C2∗) we have
$\displaystyle\mu(E^{c})\geq\mu(E_{(l_{j_{k}}+1)})=\mu(E_{(k+1)})$.

###### Remark 3.16

In comparison with results in the main part of this section, the advantage of
previous statements lies in their efficiency for survival functions equality
or inequality testing. In particular, Proposition 3.15 vii) requires to hold
the same properties as Corollary 3.11 iii), however for a smaller number of
sets, $\displaystyle k\in\Psi_{\mathbf{x}}^{*}\subseteq\Psi_{\mathbf{x}}$. On
the other hand, the equality (inequality) of survival functions implies more
information than those included in the Proposition 3.15, the results are true
for any $\displaystyle k\in\Psi_{\mathbf{x}}$ not only for $\displaystyle
k\in\Psi_{\mathbf{x}}^{*}$. Moreover, system $\displaystyle\Psi_{\mathbf{x}}$
is also easier in definition.

We have seen in the main part of this section that
$\displaystyle\mathrm{(C1)}$, $\displaystyle\mathrm{(C2)}$ are not necessary
for equality between survival functions in general, see Corollary 3.7, Example
3.9. This result we have improved by replacing $\displaystyle\mathrm{(C1)}$
with $\displaystyle\mathrm{(C4)}$. Also, Corollary 3.7 can be improved as it
follows.

###### Theorem 3.17

Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$,
$\displaystyle\mu\in{\mathbf{M}}$, and let $\displaystyle{\mathscr{A}}$ be
FCA. Then the following assertions are equivalent:

  1. i)

$\displaystyle\mathrm{(C1^{*})}$, $\displaystyle\mathrm{(C2^{*})}$ are
satisfied.

  2. ii)

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

Proof. The implication $\displaystyle\text{i)}\Rightarrow\text{ii)}$ follows
from Proposition 3.15 iii). In order to prove the reverse implication, assume
that survival functions are equal. Then (C2∗) follows from Proposition 3.15
vii). It is enough to prove (C1∗). From Proposition 3.15 vii) we have:

For any $\displaystyle k\in\Psi_{\mathbf{x}}^{*}$ there exists $\displaystyle
F_{k}\in\mathscr{E}$ such that $\displaystyle\mathsf{A}(\mathbf{x}|F_{k})\leq
x_{(k)}$ and $\displaystyle\mu(F_{k}^{c})=\mu(E_{(k+1)})$.

We show that $\displaystyle\mathsf{A}(\mathbf{x}|F_{k})=x_{(k)}$. Indeed, for
$\displaystyle k=\min\Psi_{\mathbf{x}}^{*}$ is the result immediate since
$\displaystyle
0\leq\mathsf{A}(\mathbf{x}|F_{\min\Psi_{\mathbf{x}}^{*}})\leq\leavevmode\nobreak\
x_{(\min\Psi_{\mathbf{x}}^{*})}=0$, where the last inequality follows from
Proposition 3.13 i). Let $\displaystyle k>\min\Psi_{\mathbf{x}}^{*}$,
$\displaystyle k\in\Psi_{\mathbf{x}}^{*}$. From Proposition 3.13 iii) there
exists $\displaystyle r\in\Psi_{\mathbf{x}}^{*},r<k$ such that $\displaystyle
x_{(l_{r}+1)}=x_{(k)}$ and
$\displaystyle\mu(F_{k}^{c})=\mu(E_{(k+1)})<\mu(E_{(l_{r}+1)})$. From
contraposition to (C2∗) we have $\displaystyle\mathsf{A}(\mathbf{x}|F_{k})\geq
x_{(l_{r}+1)}=x_{(k)}$.

###### Corollary 3.18

Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$,
$\displaystyle\mu\in{\mathbf{M}}$ such that it is strictly monotone on
$\displaystyle\\{E_{(k+1)}:k\in\Psi_{\mathbf{x}}\\}$, and let
$\displaystyle{\mathscr{A}}$ be FCA. Then the following assertions are
equivalent:

  1. i)

$\displaystyle\mathrm{(C1)}$, $\displaystyle\mathrm{(C2)}$ are satisfied.

  2. ii)

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

Proof. It follows from Proposition 3.13 iv) and Theorem 3.17.

A summary of relationships among some conditions as well as the summary of
sufficient and necessary conditions under which survival functions coincide or
under which they are pointwise comparable with respect to $\displaystyle\leq$,
$\displaystyle\geq$ can be found in the Appendix, see Table 2.

##  4 Equality characterization

Results of the previous section stated conditions depended on FCA
$\displaystyle{\mathscr{A}}$, input vector $\displaystyle\mathbf{x}$ and
monotone measure $\displaystyle\mu$ to hold equality between survival
functions. Of course, when one changes the monotone measure and the other
inputs stay the same, the equality can violate as the following example shows.

###### Example 4.1

Let us consider
$\displaystyle{\mathscr{A}}^{\mathrm{sum}}=\\{\mathsf{A}^{\mathrm{sum}}(\cdot|E):E\in
2^{[3]}\\}$, and normalized monotone measure $\displaystyle\mu$ on
$\displaystyle 2^{[3]}$ with the following values:

$\displaystyle E$ | $\displaystyle\\{1,2,3\\}$ | $\displaystyle\\{2,3\\}$ | $\displaystyle\\{1,3\\}$ | $\displaystyle\\{1,2\\}$ | $\displaystyle\\{3\\}$ | $\displaystyle\\{2\\}$ | $\displaystyle\\{1\\}$ | $\displaystyle\emptyset$
---|---|---|---|---|---|---|---|---
$\displaystyle E^{c}$ | $\displaystyle\emptyset$ | $\displaystyle\\{1\\}$ | $\displaystyle\\{2\\}$ | $\displaystyle\\{3\\}$ | $\displaystyle\\{1,2\\}$ | $\displaystyle\\{1,3\\}$ | $\displaystyle\\{2,3\\}$ | $\displaystyle\\{1,2,3\\}$
$\displaystyle\mu(E^{c})$ | $\displaystyle 0$ | $\displaystyle 0$ | $\displaystyle 0$ | $\displaystyle 0$ | $\displaystyle 0$ | $\displaystyle 0.5$ | $\displaystyle 0.5$ | $\displaystyle 1$
$\displaystyle\nu(E^{c})$ | $\displaystyle 0$ | $\displaystyle 0$ | $\displaystyle 0$ | $\displaystyle 0$ | $\displaystyle 0.5$ | $\displaystyle 0.5$ | $\displaystyle 0.5$ | $\displaystyle 1$
$\displaystyle\mathsf{A}^{\mathrm{sum}}(\mathbf{x}|E)$ | $\displaystyle 4$ | $\displaystyle 3$ | $\displaystyle 2$ | $\displaystyle 3$ | $\displaystyle 1$ | $\displaystyle 2$ | $\displaystyle 1$ | $\displaystyle 0$
$\displaystyle\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)$ | $\displaystyle 2$ | $\displaystyle 2$ | $\displaystyle 1$ | $\displaystyle 2$ | $\displaystyle 1$ | $\displaystyle 2$ | $\displaystyle 1$ | $\displaystyle 0$

Further, let us take the input vector $\displaystyle\mathbf{x}=(1,2,1)$.Then
we can see

$\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=\mathbf{1}_{[0,1)}(\alpha)=\mu(\\{\mathbf{x}>\alpha\\}),\quad\alpha\in[0,\infty),$

but

$\displaystyle\nu_{{\mathscr{A}}^{\mathrm{sum}}}(\mathbf{x},\alpha)=\mathbf{1}_{[0,1)}(\alpha)+0{.}5\,\mathbf{1}_{[1,2)}(\alpha)\not=\mathbf{1}_{[0,1)}(\alpha)=\nu(\\{\mathbf{x}>\alpha\\}),\quad\alpha\in[0,\infty).$

In the following we shall find sufficient and necessary conditions on
$\displaystyle{\mathscr{A}}$ and $\displaystyle\mathbf{x}$ under which
survival functions equal for any monotone measure. So, we answer Problem 2,
see Theorem 4.2, Theorem 4.6. In the second step we characterize FCA for which
survival functions equal (for any monotone measure and any input vector). We
answer Problem 3.

###### Theorem 4.2

Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$, and
$\displaystyle{\mathscr{A}}$ be FCA. Then the following assertions are
equivalent:

  1. i)

$\displaystyle\mathscr{E}\supseteq\\{E_{(k+1)}^{c}:k\in\Psi_{\mathbf{x}}\\}$
and
$\displaystyle\mathsf{A}(\mathbf{x}|E)=\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)$
for any $\displaystyle E=E_{(k+1)}^{c}$ with $\displaystyle
k\in\Psi_{\mathbf{x}}$,
$\displaystyle\mathsf{A}(\mathbf{x}|E)\geq\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)$
otherwise.

  2. ii)

For each $\displaystyle\mu\in\mathbf{M}$ such that it is strictly monotone on
$\displaystyle\\{E_{(k+1)}:k\in\Psi_{\mathbf{x}}\\}$ it holds
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

  3. iii)

For each $\displaystyle\mu\in\mathbf{M}$ it holds
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

Proof. The implication $\displaystyle\text{i)}\Rightarrow\text{iii)}$ we
easily prove by Corollary 3.7. Indeed, for any $\displaystyle
k\in\Psi_{\mathbf{x}}$ (C1) is satisfied with $\displaystyle
G_{k}=E_{(k+1)}^{c}$. If $\displaystyle\mathsf{A}(\mathbf{x}|E)<x_{(k+1)}$,
$\displaystyle k\in\Psi_{\mathbf{x}}$ and $\displaystyle E\in\mathscr{E}$,
then from assumptions we have

$\displaystyle\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)\leq\mathsf{A}(\mathbf{x}|E)<x_{(k+1)}.$

Then we get $\displaystyle E\subseteq E_{(k+1)}^{c}$, i.e., $\displaystyle
E^{c}\supseteq E_{(k+1)}$ and for each monotone measure $\displaystyle\mu$ we
have $\displaystyle\mu(E^{c})\geq\mu(E_{(k+1)})$. Thus (C2) is also satisfied.

Let us prove the implication $\displaystyle\text{ii)}\Rightarrow\text{i)}$.
Since the assumption holds for any $\displaystyle\mu\colon
2^{[n]}\to[0,\infty)$ such that it is strictly monotone measure on
$\displaystyle\\{E_{(k+1)}:k\in\Psi_{\mathbf{x}}\\}$, it holds for
$\displaystyle\mu$ such that it is strictly monotone measure on $\displaystyle
2^{[n]}$ (not only on $\displaystyle\\{E_{(k+1)}:k\in\Psi_{\mathbf{x}}\\}$).
From Corollary 3.18 (C1) holds. Moreover, since sets $\displaystyle E_{(k+1)}$
are the only sets with value equal to $\displaystyle\mu(E_{(k+1)})$, we get
$\displaystyle G_{k}=E_{(k+1)}^{c}$ and
$\displaystyle\mathscr{E}\supseteq\\{E_{(k+1)}^{c}:k\in\Psi_{\mathbf{x}}\\}$.
So, from (C1) we have

$\displaystyle\mathsf{A}(\mathbf{x}|E_{(k+1)}^{c})=x_{(k)}=\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E_{(k+1)}^{c})$

for any $\displaystyle k\in\Psi_{\mathbf{x}}$. Let us prove the second part of
i). Again, if the equality between survival functions holds for any strictly
monotone measure $\displaystyle\mu$ on
$\displaystyle\\{E_{(k+1)}^{c}:k\in\Psi_{\mathbf{x}}\\}$, then it holds for
$\displaystyle\mu\colon 2^{[n]}\to[0,\infty)$ being strictly monotone on the
above mentioned collection with values:

$\displaystyle\mu(E)=\mu(E_{(k+1)})\,\,\text{for any set}\,\,E\,\,\text{such
that}\,\,\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E^{c})=x_{(k)},\text{
}k\in\Psi_{\mathbf{x}}.$

Let $\displaystyle E\in\mathscr{E}$. Then according to Proposition 3.1 ii)
there exists $\displaystyle k\in\Psi_{\mathbf{x}}\setminus\\{0\\}$ such that

$\displaystyle\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)=x_{(k)}.$

Since $\displaystyle\mu$ is strictly monotone on
$\displaystyle\\{E_{(k+1)}:k\in\Psi_{\mathbf{x}}\\}$, then from Proposition
3.13 iv) we have $\displaystyle\Psi_{\mathbf{x}}=\Psi_{\mathbf{x}}^{*}$.
Further, from Proposition 3.13 i), if
$\displaystyle\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)=x_{(\min\Psi_{\mathbf{x}}^{*})}=0$
the result is trivial. Let $\displaystyle k>\min\Psi_{\mathbf{x}}^{*}$. Then
from Proposition 3.13 iii) there exists $\displaystyle
r\in\Psi_{\mathbf{x}}^{*},r<k$ such that $\displaystyle x_{(l_{r}+1)}=x_{(k)}$
and $\displaystyle\mu(E_{(k+1)})<\mu(E_{(l_{r}+1)})$. Therefore
$\displaystyle\mu(E^{c})=\mu(E_{(k+1)})<\mu(E_{(l_{r}+1)})$ and from
contraposition to (C2∗) we have $\displaystyle\mathsf{A}(\mathbf{x}|E)\geq
x_{(l_{r}+1)}=x_{(k)}=\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)$.

###### Remark 4.3

From the previous theorem one can see the other sufficient condition under
which the standard and generalized survival functions coincide, i.e., the
condition i). Let us remark that this sufficient condition is more strict than
$\displaystyle\mathrm{(C1)}$ and $\displaystyle\mathrm{(C2)}$, i.e., if i) is
satisfied then $\displaystyle\mathrm{(C1)}$, $\displaystyle\mathrm{(C2)}$ are
true, however, the reverse implication need not be true in general, see
Example 3.8.

According to previous result there are vectors for which the equality between
survival functions (for any $\displaystyle\mu$) do not lead to
$\displaystyle\mathsf{A}^{\text{max}}$.

###### Example 4.4

Let us consider $\displaystyle{\mathscr{A}}=\\{\mathsf{A}(\cdot|E):E\in
2^{[3]}\\}$ with conditional aggregation operator from Example 2.2 iii) with
$\displaystyle\mathbf{w}=(0.5,0.5,1)$, $\displaystyle\mathbf{z}=(0.5,0.25,1)$.
Let us take the input vector $\displaystyle\mathbf{x}=(2,3,4)$. The values of
$\displaystyle\mathsf{A}(\mathbf{x}|E)$, $\displaystyle E\in 2^{[3]}$ are
summarized in following table:

$\displaystyle E$ | $\displaystyle\\{1,2,3\\}$ | $\displaystyle\\{2,3\\}$ | $\displaystyle\\{1,3\\}$ | $\displaystyle\\{1,2\\}$ | $\displaystyle\\{3\\}$ | $\displaystyle\\{2\\}$ | $\displaystyle\\{1\\}$ | $\displaystyle\emptyset$
---|---|---|---|---|---|---|---|---
$\displaystyle E^{c}$ | $\displaystyle\emptyset$ | $\displaystyle\\{1\\}$ | $\displaystyle\\{2\\}$ | $\displaystyle\\{3\\}$ | $\displaystyle\\{1,2\\}$ | $\displaystyle\\{1,3\\}$ | $\displaystyle\\{2,3\\}$ | $\displaystyle\\{1,2,3\\}$
$\displaystyle\mathsf{A}(\mathbf{x}|E)$ | $\displaystyle 4$ | $\displaystyle 4$ | $\displaystyle 4$ | $\displaystyle 3$ | $\displaystyle 4$ | $\displaystyle 6$ | $\displaystyle 2$ | $\displaystyle 0$

Then $\displaystyle\Psi_{\mathbf{x}}=\\{0,1,2,3\\}$ and it holds

$\displaystyle\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
$\displaystyle\displaystyle=\mu(\\{1,2,3\\})\cdot\mathbf{1}_{[0,2)}(\alpha)+\mu(\\{2,3\\})\cdot\mathbf{1}_{[2,3)}(\alpha)+\mu(\\{3\\})\cdot\mathbf{1}_{[3,4)}(\alpha)$
$\displaystyle\displaystyle=\mu(E_{(1)})\cdot\mathbf{1}_{[0,2)}(\alpha)+\mu(E_{(2)})\cdot\mathbf{1}_{[2,3)}(\alpha)+\mu(E_{(3)})\cdot\mathbf{1}_{[3,4)}(\alpha)$

for any $\displaystyle\alpha\in[0,\infty)$ and monotone measure
$\displaystyle\mu$. So, we have shown that there is vector
$\displaystyle\mathbf{x}$ and
$\displaystyle{\mathscr{A}}\neq{\mathscr{A}}^{\mathrm{max}}$ such that
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any monotone measure $\displaystyle\mu$. Indeed, $\displaystyle
6=\mathsf{A}(\mathbf{x}|\\{2\\})>\mathsf{A}^{\mathrm{max}}(\mathbf{x}|\\{2\\})=3$
($\displaystyle\mathsf{A}(\mathbf{x}|E)=\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)$
for any $\displaystyle E\in 2^{[3]}\setminus\\{\\{2\\}\\}$).

Aggregation functions with the property being bounded from below by the
maximum are in the literature called disjunctive, see [11]. For FCA
$\displaystyle{\mathscr{A}}$ nondecreasing w.r.t. sets we get an interesting
consequence.

###### Lemma 4.5

Let $\displaystyle{\mathscr{A}}=\\{\mathsf{A}(\cdot|E):E\in 2^{[n]}\\}$ be FCA
nondecreasing w.r.t. sets. If for $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$
it holds that
$\displaystyle\mathsf{A}(\mathbf{x}|E)=\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)$
for any $\displaystyle E=E_{(k+1)}^{c}$ with $\displaystyle
k\in\Psi_{\mathbf{x}}$ and
$\displaystyle\mathsf{A}(\mathbf{x}|E)\geq\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)$
otherwise, then
$\displaystyle\mathsf{A}(\mathbf{x}|E)=\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)$
for any $\displaystyle E\in 2^{[n]}$.

Proof. Let us consider an arbitrary set $\displaystyle E\in\mathscr{E}$ and
let us denote $\displaystyle\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E):=x_{s}$.
Then according to Proposition 3.1 there exists $\displaystyle
k_{s}\in\Psi_{\mathbf{x}}$ such that $\displaystyle x_{s}=x_{(k_{s})}$. Then
$\displaystyle E\subseteq E_{(k_{s}+1)}^{c}$. From above mentioned and from
Theorem 4.2 we have

$\displaystyle
x_{(k_{s})}=\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E_{(k_{s}+1)}^{c})=\mathsf{A}(\mathbf{x}|E_{(k_{s}+1)}^{c})\geq\mathsf{A}(\mathbf{x}|E)\geq\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)=x_{(k_{s})}.$

###### Theorem 4.6

Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$, and
$\displaystyle{\mathscr{A}}$ be FCA nondecreasing w.r.t. sets. Then the
following assertions are equivalent:

  1. i)

$\displaystyle\mathscr{E}\supseteq\\{E_{(k+1)}^{c}:k\in\Psi_{\mathbf{x}}\\}$
and
$\displaystyle\mathsf{A}(\mathbf{x}|E)=\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)$
for any set $\displaystyle E\in\mathscr{E}$.

  2. ii)

For each $\displaystyle\mu\in\mathbf{M}$ it holds
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

Proof. The implication $\displaystyle\text{i)}\Rightarrow\text{ii)}$ follows
from Proposition 3.2 ii). The reverse implication follows from Theorem 4.2 and
Lemma 4.5.

Let us return to Example 4.4. We have shown that for the input vector
$\displaystyle\mathbf{x}=(2,3,4)$ with $\displaystyle{\mathscr{A}}$ given in
example
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\mu$. However, for another vector, let us take
$\displaystyle\mathbf{y}=(2,5,4)$ the equality can violate:

$\displaystyle\displaystyle\mu_{\mathscr{A}}(\mathbf{y},\alpha)$
$\displaystyle\displaystyle=\mu(\\{1,2,3\\})\cdot\mathbf{1}_{[0,2)}(\alpha)+\mu(\\{2,3\\})\cdot\mathbf{1}_{[2,4)}(\alpha),$
$\displaystyle\displaystyle\mu(\\{\mathbf{y}>\alpha\\})$
$\displaystyle\displaystyle=\mu(\\{1,2,3\\})\cdot\mathbf{1}_{[0,2)}(\alpha)+\mu(\\{2,3\\})\cdot\mathbf{1}_{[2,4)}(\alpha)+\mu(\\{2\\})\cdot\mathbf{1}_{[4,5)}(\alpha),$

i.e.
$\displaystyle\mu_{\mathscr{A}}(\mathbf{y},\alpha)=\mu(\\{\mathbf{y}>\alpha\\})$
does not hold for any $\displaystyle\mu$. In the following we shall ask for
FCA $\displaystyle{\mathscr{A}}$ for which the equality holds for any
$\displaystyle\mu$ and for any $\displaystyle\mathbf{x}$. As a last thus we
solve Problem 3.

###### Theorem 4.7

Let $\displaystyle{\mathscr{A}}$ be FCA. The following assertions are
equivalent:

  1. i)

$\displaystyle\mathscr{A}=\\{\mathsf{A}^{\max}(\cdot|E):\,\,E\in 2^{[n]}\\}$.

  2. ii)

For each $\displaystyle\mu\in\mathbf{M}$, and for each
$\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$ it holds
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$
for any $\displaystyle\alpha\in[0,\infty)$.

Proof. The implication $\displaystyle\text{i)}\Rightarrow\text{ii)}$ is
immediate. We prove $\displaystyle\text{ii)}\Rightarrow\text{i)}$. Since the
equality holds for any $\displaystyle\mathbf{x}$, according to Theorem 4.2 we
get

$\displaystyle\mathscr{E}=\bigcup_{\mathbf{x}\in[0,\infty)^{[n]}}\mathscr{E}^{\Psi_{\mathbf{x}}-\text{chain}}=2^{[n]}$

with
$\displaystyle\mathscr{E}^{\Psi_{\mathbf{x}}-\text{chain}}:=\\{E_{(k+1)}^{c}:k\in\Psi_{\mathbf{x}}\\}.$

Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$ be an arbitrary fixed vector.
From Theorem 4.2 we have
$\displaystyle\mathsf{A}(\mathbf{x}|E)=\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)$
for any $\displaystyle E\in\mathscr{E}^{\Psi_{\mathbf{x}}-\text{chain}}$ and
$\displaystyle\mathsf{A}(\mathbf{x}|E)\geq\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)$
for any $\displaystyle E\in
2^{[n]}\setminus\mathscr{E}^{\Psi_{\mathbf{x}}-\text{chain}}$. However, we
show that
$\displaystyle\mathsf{A}(\mathbf{x}|E)=\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)$
for any $\displaystyle E\in 2^{[n]}$. Let us consider an arbitrary fixed
$\displaystyle E\in
2^{[n]}\setminus\mathscr{E}^{\Psi_{\mathbf{x}}-\text{chain}}$ and vector

$\displaystyle\widehat{\mathbf{x}}=\mathbf{x}\mathbf{1}_{E}+a\mathbf{1}_{E^{c}},\,\,a>\max_{i\in
E}x_{i}.$

The set $\displaystyle[n]\in\mathscr{E}^{\Psi_{\mathbf{x}}-\text{chain}}$ by
definition of $\displaystyle\Psi_{\mathbf{x}}$, therefore
$\displaystyle\widehat{\mathbf{x}}\neq\mathbf{x}$. Moreover, there exists
permutation $\displaystyle(\cdot)$ such that $\displaystyle
0=\widehat{x}_{(0)}\leq\widehat{x}_{(1)}\leq\widehat{x}_{(2)}\leq\dots=\widehat{x}_{(\widehat{k})}<\widehat{x}_{(\widehat{k}+1)}=\dots=\widehat{x}_{(n)}=a$
with $\displaystyle\widehat{k}=|E|$. Therefore
$\displaystyle\widehat{k}\in\Psi_{\widehat{\mathbf{x}}}$, and $\displaystyle
E=\\{(1),\dots,(\widehat{k})\\}=E^{c}_{(\widehat{k}+1)}\in\mathscr{E}^{\Psi_{\widehat{\mathbf{x}}}-\text{chain}}$.
Finally, from Theorem 4.2, and because of the property
$\displaystyle\mathsf{A}(\mathbf{y}|E)=\mathsf{A}(\mathbf{y}\mathbf{1}_{E}|E)$
for any $\displaystyle\mathbf{y}\in[0,\infty)^{[n]}$, see [1], we have:

$\displaystyle\displaystyle\mathsf{A}(\mathbf{x}|E)$
$\displaystyle\displaystyle=\mathsf{A}(\mathbf{x}\mathbf{1}_{E}|E)=\mathsf{A}(\widehat{\mathbf{x}}\mathbf{1}_{E}|E)=\mathsf{A}(\widehat{\mathbf{x}}|E)=\mathsf{A}^{\mathrm{max}}(\widehat{\mathbf{x}}|E)=\mathsf{A}^{\mathrm{max}}(\widehat{\mathbf{x}}\mathbf{1}_{E}|E)$
$\displaystyle\displaystyle=\mathsf{A}^{\mathrm{max}}(\mathbf{x}\mathbf{1}_{E}|E)=\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E).$

##  5 Conclusion

In this paper we have solved three problems dealing with the question of
equality between the survival function and the generalized survival function
based on conditional aggregation operators introduced originally in [1] (the
generalization of concepts of papers [8], [13]). We have restricted ourselves
to discrete settings. The most interesting results are Corollary 3.7,
Corollary 3.11, Proposition 3.15 and Theorem 3.17 (solutions of Problem 1),
Theorem 4.2 and Theorem 4.6 (solution of Problem 2). Results were derived from
the well-known formula of the standard survival function with a permutation
$\displaystyle(\cdot)$ playing a crucial role. As the main result, we have
determined the family of conditional aggregation operators with respect to
which the novel survival function is identical to the standard survival
function regardless of the monotone measure and input vector, see Theorem 4.7.

We expect the future extension of our results into the area of integrals
introduced with respect to novel survival functions, see [1, Definition 5.1].
The relationship of studied survival functions (in the sense of equalities or
inequalities) determines also the relationship of corresponding integrals
(based on standard and generalized survival function). The interesting
question for the future work is: Is
$\displaystyle{\mathscr{A}}^{\mathrm{sup}}$ family of conditional operators
also the only one that generates the standard survival function in case of
arbitrary basic set $\displaystyle X$ instead of $\displaystyle[n]$, i.e., is
it true that

$\displaystyle\mu_{\mathscr{A}}(f,\alpha)=\mu(\\{f>\alpha\\})$,
$\displaystyle\alpha\in[0,\infty)$ for any $\displaystyle\mu$ and for any
$\displaystyle f$ if and only if
$\displaystyle{\mathscr{A}}={\mathscr{A}}^{\mathrm{sup}}$?

Up to now there are not known any other families except of
$\displaystyle{\mathscr{A}}^{\mathrm{sup}}$ generating generalized survival
function indistinguishable from survival function (for any $\displaystyle\mu$,
$\displaystyle f$). We believe that new results will be beneficial in some
applications, e.g. in the theory of decision making. The equality between
survival functions of a given alternative $\displaystyle\mathbf{x}$ means that
the overall score of it with respect to the Choquet integral and the
$\displaystyle{\mathscr{A}}$-Choquet integral is the same. Also, immediately
with decision making application the question of
$\displaystyle(\mu,{\mathscr{A}})$-indistinguishability arises, i.e. under
which condition on $\displaystyle\mu$, $\displaystyle{\mathscr{A}}$ it holds
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu_{\mathscr{A}}(\mathbf{y},\alpha)$
for $\displaystyle\mathbf{x},\mathbf{y}\in[0,\infty)^{[n]}$. Then alternatives
$\displaystyle\mathbf{x},\mathbf{y}$ will be
$\displaystyle{\mathscr{A}}$-Choquet integral indistinguishable, i.e. they
achieve the same overall score.

## Appendix

In this subsection we summarize all sufficient and necessary conditions for
equality or inequality between survival functions, see Table 2.

$\displaystyle(\widetilde{\mathrm{C}}1)$ and $\displaystyle(\widetilde{\mathrm{C}}2)$ | $\displaystyle\Rightarrow$ | $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$ |  |  Rem. 3.12
---|---|---|---|---
$\displaystyle(\widetilde{\mathrm{C}}2)$ and $\displaystyle(\widetilde{\mathrm{C}}3)$ | $\displaystyle\Leftrightarrow$ | $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$ |  |  Rem. 3.12
$\displaystyle\mathrm{(C1)}$ and $\displaystyle\mathrm{(C2)}$ | $\displaystyle\Rightarrow$ | $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$ |  |  Cor. 3.7
$\displaystyle\mathrm{(C2)}$ and $\displaystyle\mathrm{(C3)}$ | $\displaystyle\Leftrightarrow$ | $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$ |  |  Cor. 3.11
$\displaystyle\mathrm{(C2)}$ and $\displaystyle\mathrm{(C4)}$ | $\displaystyle\Leftrightarrow$ | $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$ |  |  Cor. 3.11
$\displaystyle\mathrm{(C1)}$ and $\displaystyle\mathrm{(C2)}$ |  $\displaystyle\Rightarrow$ |  $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$ |  |  Cor. 3.18
$\displaystyle\mu\colon 2^{[n]}\to[0,\infty)$ is strictly monotone on
$\displaystyle\\{E_{(k+1)}:k\in\Psi_{\mathbf{x}}\\}$
$\displaystyle\mathrm{(C2^{*})}$ and $\displaystyle\mathrm{(C3^{*})}$ | $\displaystyle\Leftrightarrow$ | $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$ |  |  Prop. 3.15
$\displaystyle\mathrm{(C2^{*})}$ and $\displaystyle\mathrm{(C4^{*})}$ | $\displaystyle\Leftrightarrow$ | $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$ |  |  Prop. 3.15
$\displaystyle\mathrm{(C1^{*})}$ and $\displaystyle\mathrm{(C2^{*})}$ | $\displaystyle\Leftrightarrow$ | $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu(\\{\mathbf{x}>\alpha\\})$ |  |  Th. 3.17
$\displaystyle\mathrm{(C1)}$ | $\displaystyle\Rightarrow$ | $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)\leq\mu(\\{\mathbf{x}>\alpha\\})$ |  |  Prop. 3.6
$\displaystyle\mathrm{(C1^{*})}$ | $\displaystyle\Rightarrow$ | $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)\leq\mu(\\{\mathbf{x}>\alpha\\})$ |  |  Prop. 3.15
$\displaystyle\mathrm{(C3)}$ | $\displaystyle\Leftrightarrow$ | $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)\leq\mu(\\{\mathbf{x}>\alpha\\})$ |  |  Prop. 3.10
$\displaystyle\mathrm{(C3^{*})}$ | $\displaystyle\Leftrightarrow$ | $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)\leq\mu(\\{\mathbf{x}>\alpha\\})$ |  |  Prop. 3.15
$\displaystyle\mathrm{(C2)}$ | $\displaystyle\Leftrightarrow$ | $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)\geq\mu(\\{\mathbf{x}>\alpha\\})$ |  |  Prop. 3.6
$\displaystyle\mathrm{(C2^{*})}$ | $\displaystyle\Leftrightarrow$ | $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)\geq\mu(\\{\mathbf{x}>\alpha\\})$ |  |  Prop. 3.15
Table 2: Sufficient and necessary conditions for pointwise comparison of
survival functions

From the Table 2 the following relationships between conditions (C1), (C2),
(C3), (C4) and its ∗ versions hold.

###### Corollary 5.1

Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$,
$\displaystyle\mu\in{\mathbf{M}}$, and let $\displaystyle{\mathscr{A}}$ be
FCA. Then it holds:

$\displaystyle\displaystyle\big{(}\mathrm{(C1)}\wedge\mathrm{(C2)}\big{)}$
$\displaystyle\displaystyle\Rightarrow\big{(}\mathrm{(C1^{*})}\wedge\mathrm{(C2^{*})}\big{)}\Leftrightarrow\big{(}\mathrm{(C2)}\wedge\mathrm{(C3)}\big{)}\Leftrightarrow\big{(}\mathrm{(C2)}\wedge\mathrm{(C4)}\big{)}\Leftrightarrow\big{(}(\widetilde{\mathrm{C}}2)\wedge(\widetilde{\mathrm{C}}3)\big{)}$
$\displaystyle\displaystyle\Leftrightarrow\big{(}\mathrm{(C2^{*})}\wedge\mathrm{(C3^{*})}\big{)}\Leftrightarrow\big{(}\mathrm{(C2^{*})}\wedge\mathrm{(C4^{*})}\big{)}\Leftrightarrow\big{(}\mathrm{(C1^{*})}\wedge\mathrm{(C2)}\big{)}.$

###### Corollary 5.2

Let $\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$,
$\displaystyle\mu\in{\mathbf{M}}$, and let $\displaystyle{\mathscr{A}}$ be
FCA. If $\displaystyle\mathrm{(C2^{*})}$ holds, then

$\displaystyle\mathrm{(C1^{*})}\Leftrightarrow\mathrm{(C3^{*})}\Leftrightarrow\mathrm{(C4^{*})}.$

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