# A Bernstein-von Mises Theorem for Generalized Fiducial Distributions

J.E. Borgert  University of North Carolina at Chapel Hill<EMAIL_ADDRESS>and  Jan Hannig  University of North Carolina at Chapel Hill
<EMAIL_ADDRESS>
###### Abstract.

We prove a Bernstein-von Mises result for generalized fiducial distributions
following the approach based on quadratic mean differentiability in Le Cam,
(1986); van der Vaart, (1998). Building on their approach, we introduce only
two additional conditions for the generalized fiducial setting. While
asymptotic normality of generalized fiducial distributions has been studied
before, particularly in Hannig, (2009) and Sonderegger and Hannig, (2014),
this work significantly extends the usefulness of such a result by the much
more general condition of quadratic mean differentiability. We demonstrate the
applicability of our result with two examples that necessitate these more
general assumptions: the triangular distributions and free-knot spline models.

###### Key words and phrases:

Generalized Fiducial Inference, Bernstein-von Mises Theorem

###### 2020 Mathematics Subject Classification:

62A01, 62E20, 62F15

##  1\. Introduction

The Bernstein-von Mises (BvM) theorem asserts that under mild smoothness
conditions, Bayesian posterior distributions asymptotically coincide with
frequentist estimators. The classical result is proved in the context of a
correctly specified, independent, identically distributed (i.i.d.), and
differentiable parametric model in which the dimension of the parameter is
fixed and finite. In finer detail, the theorem establishes that posterior
distributions in that setting are asymptotically Gaussian, centered at locally
sufficient statistics, and with variance inversely proportional to the Fisher
Information. Moreover, the limiting distribution does not depend on the
Bayesian prior. The result allows for frequentist interpretation of Bayesian
credible sets as confidence regions with the same asymptotic coverage,
unifying classical asymptotic theory with Bayesian models and providing
frequentist justification for Bayesian inference.

To formalize the above mathematically, let $\boldsymbol{\theta}$ denote a
statistical parameter belonging to a measurable subset of a Euclidean space,
namely $\boldsymbol{\theta}\in\boldsymbol{\Theta}\subset\mathbb{R}^{d}$.
Suppose $\pi_{n}(\boldsymbol{\theta}|y_{1},\ldots,y_{n})$ is the posterior
distribution of $\boldsymbol{\theta}$ based on observations
$\mathbf{y}=(y_{1},\ldots,y_{n})$ randomly sampled from a distribution
$P_{\boldsymbol{\theta}}$ that admits a density $p_{\boldsymbol{\theta}}$ with
respect to a measure $\mu$ on a measurable space $(\mathcal{Y},\mathcal{A})$
and a prior measure $\pi$ on the parameter space $\boldsymbol{\Theta}$. Assume
also that the map $(\boldsymbol{\theta},\textbf{y})\mapsto
p_{\boldsymbol{\theta}}(\textbf{y})$ is measurable. The Bernstein-von Mises
theorem states that if $\mathbf{y}=(y_{1},\ldots,y_{n})$ is a random sample
from the density $p_{\boldsymbol{\theta}_{0}}$, the model
$(\boldsymbol{\theta},\textbf{y})\mapsto
p_{\boldsymbol{\theta}_{0}}(\textbf{y})$ is sufficiently smooth and
identifiable, and the prior puts positive mass around the true parameter
$\boldsymbol{\theta}_{0}$, then

$\left\lVert\pi_{n}(\boldsymbol{\theta}|y_{1},\ldots,y_{n})-N(\hat{\boldsymbol{\theta}}_{n},I_{\boldsymbol{\theta}_{0}}^{-1})\right\rVert_{TV}\overset{P_{\boldsymbol{\theta}_{0}}^{n}}{\rightarrow}0,$

where $N(\hat{\boldsymbol{\theta}}_{n},I_{\boldsymbol{\theta}_{0}}^{-1})$, is
the Gaussian distribution centered on $\hat{\boldsymbol{\theta}}_{n}$, which
can be any efficient estimator sequence of the parameter, and with the inverse
Fisher Information as variance.

The BvM phenomenon is named for Sergei Bernstein (Bernstein,, 1917) and
Richard von Mises (Von Mises,, 1931) and even dates back to Laplace (Laplace,,
1810), though Doob (Doob,, 1949) provided the first formal proof.
Generalizations relying on second, third, or higher order derivatives of the
log-likelihood are formulated in Le Cam, (1953), Walker, (1969) and Dawid,
(1970) , while a similar limiting theorem for posterior means is proved in
Bickel and Yahav, (1969). It was later shown in the seminal paper Le Cam,
(1970) that the classical differentiability conditions for the BvM result
could be replaced by a more general condition, differentiability in quadratic
mean (DQM), which affords a quadratic expression but requires only a
(particular) first-order derivative. The authors of Cam and Schwartz, (1960)
and Schwartz, (1965) developed the existence of uniformly consistent tests as
an assumption that guarantees that the true parameter is distinguishable. The
proof strategy under DQM and existence of uniformly consistent tests in the
canonical setting is streamlined in the book van der Vaart, (1998).

The literature explores asymptotic results for posterior distributions in
various settings. Although BvM does not fully generalize to non- and semi-
parametric estimation problems, several studies have presented rigorous
results in these scenarios (Castillo and Nickl,, 2013, 2014; Leahu,, 2011;
Castillo,, 2012; Jonge and Zanten,, 2013). A very general BvM result for
finite sample semiparametric models is presented in Panov and Spokoiny,
(2015), even when the full parameter dimension grows with the sample size.
High- and infinite-dimensional parameters have been thoroughly investigated,
with works examining conditions for and counterexamples of BvM behavior
(Freedman,, 1999; Ghosal,, 2000; Johnstone,, 2010). Extensions to dependent
settings are given in Heyde and Johnstone, (1979) and Sweeting and Adekola,
(1987), while a more recent manuscript offers a general BvM for the weakly
dependent case (Connault,, 2014). Furthermore, a BvM theorem specifically
addressing the i.i.d. case under misspecification is provided in Kleijn and
van der Vaart, (2012).

Asymptotic normality of generalized fiducial distributions has also been
studied previously in the literature. In Hannig, (2009), sufficient conditions
for the univariate fiducial distribution to converge to the Bayesian posterior
are given, which is in turn shown to be close to the distribution of the
maximum likelihood estimator (MLE) using the classical conditions found in
Cramér, (1999); Lehmann and Casella, (2006). The approach in Hannig, (2009) is
extended to the multivariate setting in Sonderegger and Hannig, (2014).

The objective of this work is to formulate conditions for asymptotic normality
of generalized fiducial distributions comparable to those in Le Cam, (1986);
van der Vaart, (1998). We build on the approach of those authors and require
only two additional conditions in the generalized fiducial setting. Before
discussing conditions for our theorem, we review generalized fiducial
inference in Section 2. The main BvM result and discussion of the necessary
conditions are then presented in Section 3. We apply our theorem to the
triangular distribution in Section 4 and provide a simulation study to compare
our solution with a Bayesian approach. In Section 5, we establish asymptotic
normality for free-knot spline models of degree $p\geq 1$.

##  2\. Generalized fiducial inference

Fiducial inference, introduced by Fisher (Fisher,, 1935), provides a
posterior-like distribution for a parameter without requiring a prior. Despite
its initial lack of endorsement due to non-exactness issues in multivariate
cases, the approach has regained favor through the development of generalized
fiducial inference (GFI) Hannig et al., (2016); Murph et al., (2023), which
was motivated by connections between the fiducial perspective and the
generalized confidence intervals of Weerahandi, (1995); Hannig et al., (2006).
GFI defines a data-dependent measure on the parameter space by inverting a
deterministic data generating algorithm (DGA) without using Bayes’ theorem.
This perspective involves associating the data Y with the parameters
$\boldsymbol{\theta}$ and a random component U with a known distribution
through the DGA. By inverting the DGA, we obtain the generalized fiducial
distribution (GFD). Since the DGA is a function of a random variable with a
known distribution, it immediately determines the likelihood
$f(\textbf{y}|\boldsymbol{\theta})$. Solving the DGA for $\boldsymbol{\theta}$
results in a distribution estimator (instead of a point or interval estimator)
of the unknown parameter of interest.

Explained in plain language, a smoothly varying DGA
$G(\textbf{U},\boldsymbol{\theta})$ can be locally approximated as a linear
function near the observed data value y. Thus, given an independent copy of U,
denoted $\textbf{U}^{*}$, there exists a value $\boldsymbol{\theta}^{*}$ such
that $G(\textbf{U}^{*},\boldsymbol{\theta}^{*})$ is closest to y. The GFD can
be computed as the distribution of $\boldsymbol{\theta}^{*}$ using the
implicit function theorem, based on the distribution of $\textbf{U}^{*}$
conditional on the event
$\\{G(\textbf{U}^{*},\boldsymbol{\theta}^{*})\approx\textbf{y}\\}$.

More precisely, a GFD is a probability measure on the parameter space
$\boldsymbol{\Theta}$ if it can be obtained as a weak limit in the following
way (see also Murph et al.,, 2023). We first define a pseudo-inverse of the
DGA by the optimization problem

$Q_{y}(u)=\arg\min_{\boldsymbol{\theta}^{*}}\left\lVert\textbf{y}-G(\textbf{u},\boldsymbol{\theta}^{*})\right\rVert,$
(2.1)

where $\left\lVert\cdot\right\rVert$ is typically either the $\ell_{2}$ or
$\ell_{\infty}$ norm. This is used to find the closest point
$\boldsymbol{\theta}^{*}$ described above. Next, for each small
$\varepsilon>0$, define the random variable
$\boldsymbol{\theta}^{*}_{\varepsilon}=Q_{\textbf{y}}(\textbf{U}^{*}_{\varepsilon})$,
where $\textbf{U}^{*}$ follows the distribution of U truncated to the set

$\mathcal{U}_{\textbf{y},\varepsilon}=\\{\textbf{U}^{*}:\left\lVert\textbf{y}-G(\textbf{U}^{*},Q_{\textbf{y}}(\textbf{U}^{*}))\right\rVert\leq\varepsilon\\},$

i.e., having the density

$\dfrac{f_{\textbf{U}}(\textbf{u}^{*})\mathbb{I}_{\mathcal{U}_{\textbf{y},\varepsilon}(\textbf{u}^{*})}}{\int_{\mathcal{U}_{\textbf{y},\varepsilon}}f_{\textbf{U}}(\textbf{u})d\textbf{u}},$

where $f_{\textbf{U}}$ is the original density of U. Then, if the random
variable $\boldsymbol{\theta}^{*}$ converges in distribution as
$\varepsilon\to 0$, the GFD is defined as the limiting distribution of
$\lim_{\varepsilon\to 0}\boldsymbol{\theta}^{*}$. Under mild conditions
(Assumptions B.1-B.4 in Hannig et al., (2016)), that limiting distribution has
density

$r(\boldsymbol{\theta}|\textbf{y})=\dfrac{p(\textbf{y},\boldsymbol{\theta})J(\textbf{y},\boldsymbol{\theta})}{\int_{\boldsymbol{\Theta}}p(\textbf{y},\boldsymbol{\theta}^{\prime})J(y,\boldsymbol{\theta}^{\prime})d\boldsymbol{\theta}^{\prime}}$

where
$J(\textbf{y},\boldsymbol{\theta})=D\,\nabla_{\boldsymbol{\theta}}G(\textbf{u},\boldsymbol{\theta})\Big{|}_{\textbf{u}=G^{-1}(\textbf{y},\boldsymbol{\theta})}$.
The gradient matrix
$\nabla_{\boldsymbol{\theta}}G(\textbf{u},\boldsymbol{\theta})$ is computed
with respect to $\boldsymbol{\theta}$, and $D$ is a determinant-like operator
that depends on the norm used in (2.1). When the $\ell_{2}$ norm is used,
$D\,M=(\det(n^{-1}M^{\top}M))^{1/2}$.

######  Example 2.1.

As an example, we derive the generalized fiducial solution for the triangular
distribution defined on the interval $(0,1)$ further studied in Section 4. The
probability density function for this triangular distribution is given by

$f(y,\theta)=\dfrac{2y}{\theta}I_{(0,\theta]}(y)+\dfrac{2-2y}{1-\theta}I_{(\theta,1)}(y),\quad\theta\in(0,1).$
(2.2)

To generate random data points, denoted as Y, following the triangular
distribution, one can utilize a random variate U drawn from a uniform
distribution on the interval $(0,1)$. The DGA for Y is defined as

$\textbf{Y}=G(\textbf{U},\theta)=\sqrt{\textbf{U}\theta}\hskip
1.42271ptI_{(0,\theta]}(\textbf{U})+(1-\sqrt{(1-\textbf{U})(1-\theta)}\hskip
1.42271ptI_{(\theta,1)}(\textbf{U})).$

To proceed, we compute the Jacobian function $J(\textbf{y},\theta)$. Note that
the gradient of $G(\textbf{u},\theta)$ with respect to $\theta$ at
$\textbf{u}=G^{-1}(\textbf{y},\theta)$ takes the form

$\nabla_{\theta}G(\textbf{u},\theta)\Big{|}_{\textbf{u}=G^{-1}(\textbf{y},\theta)}=\begin{pmatrix}\frac{y_{1}}{2\theta}I_{(0,\theta]}(y_{1})+\frac{1-y_{1}}{2(1-\theta)}I_{(\theta,1)}(y_{1})\\\
\vdots\\\
\frac{y_{n}}{2\theta}I_{(0,\theta]}(y_{n})+\frac{1-y_{n}}{2(1-\theta)}I_{(\theta,1)}(y_{n})\end{pmatrix}.$

Consequently

$J(\textbf{y},\theta)=\left(\frac{1}{n}\sum_{i=1}^{n}\left(\frac{y_{i}}{2\theta}\right)^{2}I_{(0,\theta]}(y_{i})+\frac{1}{n}\sum_{i=1}^{n}\left(\frac{1-y_{i}}{2(1-\theta)}\right)^{2}I_{(\theta,1)}(y_{i})\right)^{1/2}.$
(2.3)

We will continue studying the asymptotic behavior of this fiducial
distribution in Section 4.

##  3\. A general Bernstein-von Mises theorem for generalized fiducial
distributions

###  3.1. Local Asymptotic Normality

To introduce relevant notation and establish the problem setting, we begin
with a review of local asymptotic normality. We consider a sequence of
statistical models denoted by
$\\{P^{n}_{\boldsymbol{\theta}}:\boldsymbol{\theta}\in\boldsymbol{\Theta}\\}$
where $\boldsymbol{\theta}$ is a parameter belonging to $\boldsymbol{\Theta}$,
an open subset of $\mathbb{R}^{d}$. For each value of $n$, assume that there
exists a single dominating measure for all $P^{n}_{\boldsymbol{\theta}}$.
Additionally, suppose there exists a “true” measure
$P^{n}_{\boldsymbol{\theta}_{0}}$, where the parameter
$\boldsymbol{\theta}_{0}$ is a fixed, non-random value in the interior of
$\boldsymbol{\Theta}$. Further, assume each $P^{n}_{\boldsymbol{\theta}}$ and
$P^{n}_{\boldsymbol{\theta}_{0}}$ have corresponding densities jointly
measurable in $(\boldsymbol{\theta},\textbf{y})$. We assume the observations
$(y_{1},\ldots,y_{n})$ are randomly sampled from the density corresponding to
$P^{n}_{\boldsymbol{\theta}_{0}}$.

Classical asymptotic theory hinges primarily on the ability to make local
quadratic approximations to log-likelihood ratios. In this context, the term
“local” refers to alternative parameter values that are in such close
proximity to the true parameter $\boldsymbol{\theta}_{0}$ that their
distributions cannot be easily distinguished. We consider models that satisfy
a property known as local asymptotic normality (LAN) Cam and Schwartz, (1960).
This property enables us to approximate the sequence of models asymptotically
by a normal location model, following a local rescaling of the parameter.

Given an interior point $\boldsymbol{\theta}\in\boldsymbol{\Theta}$, the
sequence
$\\{P_{n,\boldsymbol{\theta}}:\boldsymbol{\theta}\in\boldsymbol{\Theta}\\}$ is
locally asymptotically normal at $\boldsymbol{\theta}$ if there exists
matrices $\textbf{r}_{n}$ and $I_{\boldsymbol{\theta}}$ and a random vector
$\Delta_{n,\boldsymbol{\theta}}\sim N(0,I_{\boldsymbol{\theta}})$ such that,
for every converging sequence $\textbf{h}_{n}\rightarrow\textbf{h}$,

$\ln\frac{dP_{n,\boldsymbol{\theta}+\textbf{r}_{n}^{-1}\textbf{h}_{n}}}{dP_{n,\boldsymbol{\theta}}}=\textbf{h}^{\prime}\Delta_{n,\boldsymbol{\theta}}-\frac{1}{2}\textbf{h}^{\prime}I_{\boldsymbol{\theta}}\textbf{h}+o_{P_{n,\boldsymbol{\theta}}}(1).$

The classical regularity conditions of twice continuous differentiability of
the log-likelihood with respect to $\boldsymbol{\theta}$ along with a
dominated second derivative gives the log-likelihood ratio the LAN property. A
more general smoothness condition that guarantees the LAN property is defined
in Le Cam, (1970), requiring that a Taylor expansion of the square root
densities exist only in mean-square. In particular, a family
$\\{P_{\boldsymbol{\theta}}:\boldsymbol{\theta}\in\boldsymbol{\Theta}\\}$ is
differentiable in quadratic mean (DQM) at
$\boldsymbol{\theta}\in\boldsymbol{\Theta}$ if there exists a score function
$\dot{\ell}_{\boldsymbol{\theta}}:\mathcal{Y}\rightarrow\mathbb{R}^{d}$ such
that

$\int\left(\sqrt{p_{\boldsymbol{\theta}+\textbf{h}}}-\sqrt{p_{\boldsymbol{\theta}}}-\frac{1}{2}\textbf{h}^{\top}\dot{\ell}\sqrt{p_{\boldsymbol{\theta}}}\right)^{2}d\mu=o(\left\lVert\textbf{h}\right\rVert^{2})$

as $\textbf{h}\rightarrow\textbf{0}$. Lemma 7.6 in van der Vaart, (1998) gives
simple conditions for DQM. These conditions involve demonstrating
differentiability of the map
$\boldsymbol{\theta}\mapsto\sqrt{p_{\boldsymbol{\theta}}(\textbf{y})}$ for
almost every y, plus uniform integrability. The covariance matrix
$I_{\boldsymbol{\theta}}=P_{\boldsymbol{\theta}}\dot{\ell}_{\boldsymbol{\theta}}\dot{\ell}_{\boldsymbol{\theta}}^{T}$
is called the Fisher Information matrix. When the Fisher Information matrix is
invertible, the DQM property implies an asymptotic expansion of the log-
likelihood ratio statistic which indicates local asymptotic normality of the
sequence of statistical experiments.

A key concept in establishing asymptotic results under local alternatives is
contiguity, introduced in Le Cam, (1960). Contiguity is (roughly) an
asymptotic counterpart to absolute continuity of measures that permits an
analogous asymptotic statement. When coupled with the LAN property, contiguity
enables the evaluation of the distribution of any (regular) estimator under a
local alternative. A measure $Q_{n}$ is contiguous with respect to a measure
$P_{n}$, written $Q_{n}\lhd P_{n}$, if for any sequence of sets $A_{n}$ such
that $P_{n}(A_{n})\rightarrow 0$ implies $Q_{n}(A_{n})\rightarrow 0$. We
denote mutually contiguous measures $P_{n}$ and $Q_{n}$ by $P_{n}\lhd\rhd
Q_{n}$.

###  3.2. Asymptotic Normality of GFDs

In this paper we consider the case where the generalized fiducial distribution
is formed relative to the model $\boldsymbol{\theta}_{0}\mapsto
p_{\boldsymbol{\theta}_{0}}$. We examine a sequence of local alternatives
based on parameters
$\textbf{h}=\sqrt{n}(\boldsymbol{\theta}-\boldsymbol{\theta}_{0})$,
representing each model as
$P_{n,\textbf{h}}=P^{n}_{\boldsymbol{\theta}_{0}+\textbf{h}/\sqrt{n}}$ with
corresponding density denoted as
$p_{n,\textbf{h}}(\textbf{y})=p(y,\boldsymbol{\theta}_{0}+\textbf{h}/\sqrt{n}).$
Note the distinction in notation to reflect the local parameter space. We use
the notation $P_{n,\textbf{0}}$ in the specific case of
$\textbf{h}=\textbf{0}$. For convenience, let $\textbf{H}_{n}$ denote the
space of local alternatives. Following this convention, we write the Jacobian
in the fiducial solution with parameter h as
$J_{n}(\textbf{h})=J(\textbf{y},\boldsymbol{\theta}_{0}+\textbf{h}/\sqrt{n})$
and the density of its limiting measure (in $n$), when it exists, as
$\pi_{n}(\textbf{h})=\pi(\boldsymbol{\theta}_{0}+\textbf{h}/\sqrt{n})$. For an
arbitrary compact set $C\subset\mathbb{R}^{d}$, we also consider a conditional
version $\pi_{n}^{C}(\textbf{h})$ of $\pi_{n}(\textbf{h})$,

$\pi_{n}^{C}(\textbf{h})=\dfrac{\mathbbm{1}_{C}\pi_{n}(\textbf{h})}{\pi_{n}(C)}.$

The marginal distribution of the data under the limit measure conditional on
$C$ can then be defined as $P_{n,C}=\int
p_{n,\textbf{h}}d\pi_{n}^{C}(\textbf{h})$. Similarly, the conditional fiducial
distribution is

$r^{C}_{\textbf{H}_{n}|\textbf{Y}_{n}}=\dfrac{\mathbbm{1}_{C}(\textbf{h})p_{n,\textbf{h}}J_{n}(\textbf{h})}{\int_{C}p_{n,\textbf{g}}J_{n}(\textbf{g})d\textbf{g}}.$

To study the LAN property, we define sufficient statistics for the local
parameter,

$\Delta_{n,\boldsymbol{\theta}_{0}}=\dfrac{1}{\sqrt{n}}\sum\limits_{i=1}^{n}I_{\boldsymbol{\theta}_{0}}^{-1}\dot{\ell}_{\boldsymbol{\theta}_{0}}(y_{i}),$

where $I_{\boldsymbol{\theta}_{0}}$ is the Fisher Information matrix. Lastly,
let $\phi_{n}$ denote the density of the Gaussian distribution
$N(\Delta_{n,\boldsymbol{\theta}_{0}},I_{\boldsymbol{\theta}_{0}}^{-1})$,
centered on the locally sufficient statistics and with the inverse Fisher
Information as variance. We will also use
$N^{C}(\Delta_{n,\boldsymbol{\theta}_{0}},I_{\boldsymbol{\theta}_{0}}^{-1})$
to denote this normal distribution restricted and renormalized to $C$, and
$\phi^{C}_{n}$ to denote its corresponding density function.

Theorem 3.1 below states that the sequence of GFDs converges to a normal
sequence in total variation norm. It is important to note that this
convergence occurs in $P^{n}_{\boldsymbol{\theta}_{0}}$-probability,
emphasizing the sample-dependent nature of this convergence. The proof,
following the approach in van der Vaart, (1998), involves a decomposition of
the parameter space into balls centered at $\boldsymbol{\theta}_{0}$ and their
respective complements. Local asymptotic normality guarantees the convergence
of the terms truncated to the balls around $\boldsymbol{\theta}_{0}$, while a
separation hypothesis achieves bounds on their complements.

The classical BvM result in Section 10.2 of van der Vaart, (1998) is proved
under three conditions. The models must exhibit differentiability in quadratic
mean to ensure the existence of the LAN property. Furthermore, tests that are
exponentially consistent against local alternatives are required to ensure the
rate of convergence of the posterior distribution aligns with the one dictated
by the LAN property. This sequence of tests effectively separates the point of
convergence from the complements of shrinking balls with the same rate
specified in the LAN property. The final necessary assumption is that the
prior measure on $\boldsymbol{\Theta}$ assigns adequate mass to sufficiently
small balls around the true parameter, which allows for optimal rates of
convergence of the posterior.

For Theorem 3.1, we require analogues of the three conditions discussed above
and make only two additional assumptions. For balls centered at
$\boldsymbol{\theta}_{0}$, we assume that the quantity
$J(\textbf{y},\boldsymbol{\theta})$ has a uniform limit in
$\boldsymbol{\theta}$. A similar notion of convergence is demonstrated in
Theorem 2 of Hannig, (2009). On the complements, we stipulate a decomposition
of the likelihood into two terms. The second term plays the role of a minimal
training sample (Berger and Pericchi,, 2004) and is used to control the tails
of the Jacobian $J(\textbf{y},\boldsymbol{\theta})$. This assumption is both
necessary and reasonable, as $J(\textbf{y},\boldsymbol{\theta})$ and its limit
may not always be finite, while the integral of the GFD with a minimal
training sample is guaranteed to be finite.

######  Assumption 1 (Differentiability in quadratic mean).

The experiment
$\\{P_{\boldsymbol{\theta}}:\boldsymbol{\theta}\in\boldsymbol{\Theta}\\}$ is
differentiable in quadratic mean at $\boldsymbol{\theta}_{0}$ with non-
singular Fisher Information matrix $I_{\boldsymbol{\theta}_{0}}$.

######  Assumption 2 (Convergence of Jacobian function).

There exists a $\sigma$-finite measure absolutely continuous with respect to
the Lebesgue measure on
$\left\lVert\boldsymbol{\theta}-\boldsymbol{\theta}_{0}\right\rVert\leq D$ for
some $D>0$, with density $\pi$ such that

$\sup\limits_{\boldsymbol{\theta}:\left\lVert\boldsymbol{\theta}-\boldsymbol{\theta}_{0}\right\rVert\leq
D}\frac{\lvert
J(\textbf{y},\boldsymbol{\theta})-\pi(\boldsymbol{\theta})\rvert}{J(\textbf{y},\boldsymbol{\theta})}\stackrel{{\scriptstyle
P_{\boldsymbol{\theta}_{0}}}}{{\rightarrow}}0.$

######  Assumption 3 (Mass of limiting measure).

The limiting density $\pi(\boldsymbol{\theta})$ is continuous and positive at
$\boldsymbol{\theta}_{0}$.

######  Assumption 4 (Likelihood splitting).

There exist finite measures
$P^{1}_{\boldsymbol{\theta}},P^{2}_{\boldsymbol{\theta}}$ absolutely
continuous with respect to the Lebesgue measure having densities
$p^{1}(\textbf{y},\boldsymbol{\theta}),p^{2}(\textbf{y},\boldsymbol{\theta})$,
respectively, such that

$p(\textbf{y},\boldsymbol{\theta})=p^{1}(\textbf{y},\boldsymbol{\theta})p^{2}(\textbf{y},\boldsymbol{\theta}).$

Moreover, there exists a finite measure on $\theta$, absolutely continuous
with respect to the Lebesgue measure, with density
$\gamma(\boldsymbol{\theta})$ so that for the same $D>0$ as in Assumption 2,

$P_{\boldsymbol{\theta}_{0}}\big{(}p^{2}(\textbf{y},\boldsymbol{\theta})J(\textbf{y},\boldsymbol{\theta})I_{\\{\left\lVert\boldsymbol{\theta}-\boldsymbol{\theta}_{0}\right\rVert>D\\}}{\leq}\gamma(\boldsymbol{\theta})\big{)}\rightarrow
1.$

######  Assumption 5 (Exponentially consistent tests).

For every $\epsilon>0$, there exists a sequence of tests $\delta_{n}$ and a
constant $c>0$ such that for
$\left\lVert\boldsymbol{\theta}-\boldsymbol{\theta}_{0}\right\rVert\geq\epsilon$,

$P_{\boldsymbol{\theta}_{0}}\delta_{n}\rightarrow 0\quad\text{and}\quad
P_{\boldsymbol{\theta}}(1-\delta_{n})\leq
e^{-cn(\left\lVert\boldsymbol{\theta}-\boldsymbol{\theta}_{0}\right\rVert^{2}\wedge
1)}\quad\text{and}\quad P_{\boldsymbol{\theta}}^{1}(1-\delta_{n})\leq
e^{-cn(\left\lVert\boldsymbol{\theta}-\boldsymbol{\theta}_{0}\right\rVert^{2}\wedge
1)}.$

######  Remark 3.1.

We emphasize that our result does not require the assumption of i.i.d data.
However, in the specific i.i.d setting, we can directly construct uniformly
exponentially consistent tests as assumed in Assumption 5 under a milder
condition. The condition states that for every $\epsilon>0$, there exists a
sequence of tests $\delta_{n}$ such that
$P_{\boldsymbol{\theta}_{0}}\delta_{n}\rightarrow 0$ and
$\sup\limits_{\boldsymbol{\theta}:\left\lVert\boldsymbol{\theta}-\boldsymbol{\theta}_{0}\right\rVert\geq\epsilon}P_{\boldsymbol{\theta}}(1-\delta_{n})\rightarrow
0$. Using only these uniformly consistent tests, exponentially consistent
versions can be constructed in the i.i.d setting. This relaxed assumption is
convenient, as uniformly consistent estimators often exist. In fact, Lemma
10.6 in van der Vaart, (1998) states that uniformly consistent estimators
exist when $\boldsymbol{\Theta}$ is $\sigma$-compact,
$P_{\boldsymbol{\theta}}$ is distinguishable at different values of the
parameter, and the maps $\boldsymbol{\theta}\mapsto P_{\boldsymbol{\theta}}$
are continuous for the total variation norm. Uniformly exponentially
consistent tests can analogously be constructed for
$P_{\boldsymbol{\theta}}^{1}$ in the i.i.d setting under the relaxed
assumption.

######  Theorem 3.1.

Under Assumptions 1-5, the fiducial distribution satisfies

$\left\lVert
r_{\textbf{H}_{n}|\textbf{Y}_{n}}-N(\Delta_{n,\boldsymbol{\theta}_{0}},I_{\boldsymbol{\theta}_{0}}^{-1})\right\rVert_{TV}\overset{P_{\boldsymbol{\theta}_{0}}^{n}}{\rightarrow}0.$

Proof. Consider a sequence of balls $C_{n}$ centered at $0$ with diameters
$M_{n}\rightarrow\infty$. The proof relies on the following decomposition:

$\left\lVert
r_{\textbf{H}_{n}|\textbf{Y}_{n}}-N(\Delta_{n,\boldsymbol{\theta}_{0}},I_{\boldsymbol{\theta}_{0}}^{-1})\right\rVert_{TV}\leq\left\lVert
r_{\textbf{H}_{n}|\textbf{Y}_{n}}^{C_{n}}-N^{C_{n}}(\Delta_{n,\boldsymbol{\theta}_{0}},I_{\boldsymbol{\theta}_{0}}^{-1})\right\rVert_{TV}+\left\lVert
r_{\textbf{H}_{n}|\textbf{Y}_{n}}-r_{\textbf{H}_{n}|\textbf{Y}_{n}}^{C_{n}}\right\rVert_{TV}\\\
+\left\lVert
N^{C_{n}}(\Delta_{n,\boldsymbol{\theta}_{0}},I_{\boldsymbol{\theta}_{0}}^{-1})-N(\Delta_{n,\boldsymbol{\theta}_{0}},I_{\boldsymbol{\theta}_{0}}^{-1})\right\rVert_{TV}.$

For a particular $C_{n}$, we can establish convergence of the fully truncated
term,

$\left\lVert
r_{\textbf{H}_{n}|\textbf{Y}_{n}}^{C_{n}}-N^{C_{n}}(\Delta_{n,\boldsymbol{\theta}_{0}},I_{\boldsymbol{\theta}_{0}}^{-1})\right\rVert_{TV},$

via the LAN property. To do so, first fix a ball $C$ centered at $0$ with
radius $M<D\sqrt{n}$ where $D$ is as in Assumption 2. By this assumption,
there exists $\pi$ such that for sufficiently large $n$, $\pi_{n}(\textbf{h})$
is dominated by the Lebesgue measure and we can define a posterior
distribution as

$\pi^{C}_{\textbf{H}_{n}|\textbf{Y}_{n}}=\dfrac{\mathbbm{1}_{C}(\textbf{h})p_{n,\textbf{h}}\pi_{n}(\textbf{h})}{\int_{C}p_{n,\textbf{g}}\pi_{n}(\textbf{g})d\textbf{g}}.$

Then, the truncated term can be further decomposed as follows:

$\left\lVert
r_{\textbf{H}_{n}|\textbf{Y}_{n}}^{C}-N^{C}(\Delta_{n,\boldsymbol{\theta}_{0}},I_{\boldsymbol{\theta}_{0}}^{-1})\right\rVert_{TV}\leq\left\lVert\pi^{C}_{\textbf{H}_{n}|\textbf{Y}_{n}}-N^{C}(\Delta_{n,\boldsymbol{\theta}_{0}},I_{\boldsymbol{\theta}_{0}}^{-1})\right\rVert_{TV}+\left\lVert
r_{\textbf{H}_{n}|\textbf{Y}_{n}}^{C}-\pi^{C}_{\textbf{H}_{n}|\textbf{Y}_{n}}\right\rVert_{TV}.$
(3.1)

Recall, that for any two measures $P$ and $Q$, $\left\lVert
P-Q\right\rVert_{TV}=2\int(1-\frac{p}{q})_{+}dQ$, where $p$ and $q$ are the
corresponding densities with respect to a dominating measure and
$(u)_{+}=\max(u,0).$ Let
$\phi_{n}^{C}(\textbf{h})=\dfrac{\mathbbm{1}_{C}\phi_{n}(\textbf{h})}{\phi_{n}(C)}$
where $\phi_{n}$ denotes the density of the Gaussian distribution
$N(\Delta_{n,\boldsymbol{\theta}_{0}},I_{\boldsymbol{\theta}_{0}}^{-1})$.
Using this and suppressing y’s in the notation for convenience, we can then
write the first term of (3.1) as

$\displaystyle\dfrac{1}{2}\left\lVert\pi^{C}_{\textbf{H}_{n}|\textbf{Y}_{n}}-N^{C}(\Delta_{n,\boldsymbol{\theta}_{0}},I_{\boldsymbol{\theta}_{0}}^{-1})\right\rVert_{TV}$
$\displaystyle=\bigintss\left(1-\dfrac{\phi_{n}^{C}(\textbf{h})}{\dfrac{\mathbbm{1}_{C}p_{n,\textbf{h}}\pi_{n}(\textbf{h})}{\int_{C}p_{n,\textbf{g}}\pi_{n}(\textbf{g})\,d\textbf{g}}}\right)_{+}\,d\pi^{C}_{\textbf{H}_{n}|\textbf{Y}_{n}}(\textbf{h})$
$\displaystyle\leq\iint\left(1-\dfrac{\phi_{n}(\textbf{h})p_{n,\textbf{g}}\pi_{n}(\textbf{g})}{\phi_{n}(\textbf{g})p_{n,\textbf{h}}\pi_{n}(\textbf{h})}\right)_{+}\phi_{n}^{C}(\textbf{g})\,d\textbf{g}\,d\pi^{C}_{\textbf{H}_{n}|\textbf{Y}_{n}}(h).$

The last line is obtained by multiplying by
$\dfrac{\phi_{n}^{C}(\textbf{g})}{\phi_{n}^{C}(\textbf{g})}$ and applying
Jensen’s inequality. From the above, define

$I(\textbf{h},\textbf{g},\textbf{y}):=\iint\left(1-\dfrac{\phi_{n}(\textbf{h})p_{n,\textbf{g}}\pi_{n}(\textbf{g})}{\phi_{n,\textbf{g}}p_{n,\textbf{h}}\pi_{n}(\textbf{h})}\right)_{+}\phi_{n}^{C}(\textbf{g})\,d\textbf{g}\,d\pi^{C}_{\textbf{H}_{n}|\textbf{Y}_{n}}(\textbf{h}).$

We show $I(\textbf{h},\textbf{g},\textbf{y})\stackrel{{\scriptstyle
L_{1}}}{{\rightarrow}}0$ by integrating $I(\textbf{h},\textbf{g},\textbf{y})$
with respect to the truncated marginal $P_{n,C}$. Applying
$\pi^{C}_{\textbf{H}_{n}|\textbf{Y}_{n}}P_{n,C}=p_{n,\textbf{h}}\pi^{C}_{n}(\textbf{h})$:

$\int
I(\textbf{h},\textbf{g},\textbf{y})dp_{n,C}(\textbf{y})=\iiint\left(1-\dfrac{\phi_{n}(\textbf{h})p_{n,\textbf{g}}\pi_{n}(\textbf{g})}{\phi_{n,\textbf{g}}p_{n,\textbf{h}}\pi_{n}(\textbf{h})}\right)_{+}\phi_{n}^{C}(\textbf{g})\,d\textbf{g}\,p_{n,\textbf{h}}\pi^{C}_{n}(\textbf{h})d\textbf{h}d\mu_{n}(\mathbf{y}).$

Using $P_{n,\textbf{h}}\vartriangleleft\vartriangleright P_{n,\textbf{0}}$ and
that by the boundedness of $C$, $\pi^{C}_{n}(\textbf{h})$ and
$\phi^{C}_{n}(\textbf{g})$ are contiguous with the uniform distribution on
$C$, we get

$\phi_{n}^{C}(\textbf{g})\,\,p_{n,\textbf{h}}\pi^{C}_{n}(\textbf{h})\vartriangleleft\vartriangleright\lambda^{C}(\textbf{g})\,\,p_{n,\textbf{0}}\lambda^{C}(\textbf{h}).$

Note that by continuity of $\pi$,

$\lim\limits_{n\rightarrow\infty}\dfrac{\pi_{n}(\textbf{h})}{\pi_{n}(\textbf{g})}=\lim\limits_{n\rightarrow\infty}\dfrac{\pi(\boldsymbol{\theta}_{0}+\textbf{h}/\sqrt{n})}{\pi(\boldsymbol{\theta}_{0}+\textbf{g}/\sqrt{n})}=\dfrac{\pi(\boldsymbol{\theta}_{0})}{\pi(\boldsymbol{\theta}_{0})}=1.$

It follows from Lemma A.2 that
$\left(1-\dfrac{\pi_{n}(\textbf{g})}{\pi_{n}(\textbf{h})}\cdot\dfrac{p_{n,\textbf{g}}\phi_{n}(\textbf{h})}{p_{n,\textbf{h}}\phi_{n}(\textbf{g})}\right)_{+}\stackrel{{\scriptstyle
P_{n,\textbf{0}}}}{{\rightarrow}}0$ for each g and h. Uniform integrability
gives convergence in $P_{n,\textbf{0}}$ mean for each g and h, and the
dominated convergence theorem gives convergence in
$\lambda^{C}(\textbf{g})\,\,p_{n,\textbf{0}}\lambda^{C}(\textbf{h})$ mean (and
hence probability). By contiguity, convergence in probability holds under
$\phi_{n}^{C}(\textbf{g})\,\,p_{n,\textbf{h}}\pi^{C}_{n}(\textbf{h})$. Again
by uniform integrability, this implies convergence in
$\phi_{n}^{C}(\textbf{g})\,\,p_{n,\textbf{h}}\pi^{C}_{n}(\textbf{h})$ mean, or
$P_{n,C}$ mean. Convergence in $P_{n,\textbf{0}}$ probability follows by
contiguity, as shown in Lemma A.1.

For the second term of (3.1):

$\frac{1}{2}\left\lVert
r_{\textbf{H}_{n}|\textbf{Y}_{n}}^{C}-\pi^{C}_{\textbf{H}_{n}|\textbf{Y}_{n}}\right\rVert_{TV}=\bigintss\left(1-\dfrac{\dfrac{\mathbbm{1}_{C}(\textbf{h})p_{n,\textbf{h}}J_{n}(\textbf{h})}{\int_{C}p_{n,\textbf{g}}J_{n}(\textbf{g})d\textbf{g}}}{\dfrac{\mathbbm{1}_{C}(h)p_{n,\textbf{h}}\pi_{n}(\textbf{h})}{\int_{C}p_{n,\textbf{g}}\pi_{n}(\textbf{g})d\textbf{g}}}\right)_{+}\,d\pi^{C}_{\textbf{H}_{n}|\textbf{Y}_{n}}(\textbf{h}).$

Since $C$ is a ball around $0$, it follows from Assumption 2 that
$\dfrac{\mathbbm{1}_{C}J_{n}(\textbf{h})}{\mathbbm{1}_{C}\pi_{n}(\textbf{h})}\stackrel{{\scriptstyle
P_{n,\textbf{0}}}}{{\rightarrow}}1$ and

$\displaystyle\dfrac{p_{n,\textbf{h}}\int_{C}p_{n,\textbf{g}}\pi_{n}(\textbf{g})d\textbf{g}}{p_{n,\textbf{h}}\int_{C}p_{n,\textbf{g}}J_{n}(\textbf{g})d\textbf{g}}$
$\displaystyle=\dfrac{\int_{C}p_{n,\textbf{g}}\pi_{n}(\textbf{g})d\textbf{g}}{\int_{C}p_{n,\textbf{g}}J_{n}(\textbf{g})d\textbf{g}}=\dfrac{\int_{C}p_{n,\textbf{g}}\left(\pi_{n}(\textbf{g})-J_{n}(\textbf{g})+J_{n}(\textbf{g})\right)dg}{\int_{C}p_{n,\textbf{g}}J_{n}(\textbf{g})d\textbf{g}}$
$\displaystyle\leq 1+\dfrac{\sup\limits_{\textbf{g}\in C}\frac{\lvert
J_{n}(\textbf{g})-\pi_{n}(\textbf{g})\rvert}{J_{n}(\textbf{g})}\int_{C}p_{n,\textbf{g}}J_{n}(\textbf{g})d\textbf{g}}{\int_{C}p_{n,\textbf{g}}J_{n}(\textbf{g})d\textbf{g}}\stackrel{{\scriptstyle
P_{n,\textbf{0}}}}{{\rightarrow}}1.$

Hence, we can show $L_{1}$ convergence by a similar argument as for the first
term to get the desired $\left\lVert
r_{\textbf{H}_{n}|\textbf{Y}_{n}}^{C}-\pi^{C}_{\textbf{H}_{n}|\textbf{Y}_{n}}\right\rVert_{TV}\stackrel{{\scriptstyle
P_{n,\textbf{0}}}}{{\rightarrow}}0$.

We have now shown that

$\left\lVert\pi^{C}_{\textbf{H}_{n}|\textbf{Y}_{n}}-N^{C}(\Delta_{n,\boldsymbol{\theta}_{0}},I_{\boldsymbol{\theta}_{0}}^{-1})\right\rVert_{TV}\stackrel{{\scriptstyle
P_{n,\textbf{0}}}}{{\rightarrow}}0$

for every fixed $C$. Finally, note that we can extract a subsequence $C_{n}$
with radius $M_{n}\rightarrow\infty$ at a rate slower than $\sqrt{n}$, such
that

$\left\lVert\pi^{C}_{\textbf{H}_{n}|\textbf{Y}_{n}}-N^{C_{n}}(\Delta_{n,\boldsymbol{\theta}_{0}},I_{\boldsymbol{\theta}_{0}}^{-1})\right\rVert_{TV}\stackrel{{\scriptstyle
P_{n,\textbf{0}}}}{{\rightarrow}}0.$

Next, we show that the GFD is close to the truncated GFD. A similar argument
can also be constructed to show that the normal distribution is close to the
truncated normal, so we do not repeat it. The proof for these terms relies on
uniformly exponentially consistent tests for testing
$H_{0}:\boldsymbol{\theta}=\boldsymbol{\theta}_{0}$ against
$H_{1}:\left\lVert\boldsymbol{\theta}-\boldsymbol{\theta}_{0}\right\rVert\geq\varepsilon$,
for every $\varepsilon>0$, as in Assumption 5.

We begin by noting that for any set $B$ and a particular $C$,

$r_{\textbf{H}_{n}|\textbf{Y}_{n}}(B)-r^{C}_{\textbf{H}_{n}|\textbf{Y}_{n}}(B)=r_{\textbf{H}_{n}|\textbf{Y}_{n}}(B\cap
C^{c})-r^{C}_{\textbf{H}_{n}|\textbf{Y}_{n}}(B)r_{\textbf{H}_{n}|\textbf{Y}_{n}}(C^{c}).$

Taking the supremum over $B$ in the above yields

$\left\lVert
r_{\textbf{H}_{n}|\textbf{Y}_{n}}-r^{C}_{\textbf{H}_{n}|\textbf{Y}_{n}}\right\rVert_{TV}\leq
2\,r_{\textbf{H}_{n}|\textbf{Y}_{n}}(C^{c}).$

Now take $C_{n}$ to be any sequence of balls centered at $0$ with radius
$M_{n}\rightarrow\infty$ and let $C$ be a ball around $0$ of fixed radius $M$.
We will show
$r_{\textbf{H}_{n}|\textbf{Y}_{n}}(C_{n}^{c})\stackrel{{\scriptstyle
P_{n,\textbf{0}}}}{{\rightarrow}}0$ by establishing the stronger $L_{1}$
convergence,
$E_{P_{n,C}}(r_{\textbf{H}_{n}|\textbf{Y}_{n}}(C_{n}^{c}))\rightarrow 0.$ For
this, let $\delta_{n}$ be a sequence of exponentially consistent tests as in
Assumption 5. By contiguity, $E_{P_{n,C}}(\delta_{n})\rightarrow 0$.
Consequently, we obtain the following:

$E_{n,C}(r_{\textbf{H}_{n}|\textbf{Y}_{n}}(C_{n}^{c}))=E_{n,C}(r_{\textbf{H}_{n}|\textbf{Y}_{n}}(C_{n}^{c})(1-\delta_{n}(\textbf{Y}_{n})))+o(1).$

By compactness and absolute continuity of measure, $\pi(\theta)$ is uniformly
bounded by a constant $c_{1}$ on
$\left\lVert\boldsymbol{\theta}-\boldsymbol{\theta}_{0}\right\rVert\leq D$.
Defining $S_{1}:=\\{\textbf{h}:M_{n}\leq\left\lVert\textbf{h}\right\rVert\leq
D\sqrt{n}\\}$ and
$S_{2}:=\\{\textbf{h}:\left\lVert\textbf{h}\right\rVert>D\sqrt{n}\\}$ allows
us to partition the complement of $C_{n}$ into two distinct regions. Utilizing
this partition, we can express the right-hand side of the equality above as

$\displaystyle
E_{n,C}(r_{\textbf{H}_{n}|\textbf{Y}_{n}}(C_{n}^{c}))(1-\delta_{n}(\textbf{Y}_{n}))$
$\displaystyle=\int\int_{C}\int_{C_{n}^{c}}\dfrac{p_{n,\textbf{h}}(\textbf{y})J_{n}(\textbf{h})d\textbf{h}}{\int
p_{n,\textbf{q}}(\textbf{y})J_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}d\mu_{n}(\textbf{y})$
$\displaystyle=\int\int_{C}\int_{S_{1}}\dfrac{p_{n,\textbf{h}}(\textbf{y})J_{n}(\textbf{h})d\textbf{{h}}}{\int
p_{n,\textbf{q}}(\textbf{y})J_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}d\mu_{n}(\textbf{y})$
(3.2) $\displaystyle\hskip
28.45274pt+\int\int_{C}\int_{S_{2}}\dfrac{p_{n,\textbf{h}}(\textbf{y})J_{n}(\textbf{h})d\textbf{h}}{\int
p_{n,\textbf{q}}(\textbf{y})J_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}d\mu_{n}(\textbf{y}).$
(3.3)

By Assumption 2, for all $\epsilon_{1}>0$, there is large enough $n$ such that
the term

$\displaystyle\eqref{eq:S1}$
$\displaystyle=\int\int_{C}\int_{S_{1}}\dfrac{p_{n,\textbf{h}}(\textbf{y})J_{n}(\textbf{h})d\textbf{h}}{\int
p_{n,\textbf{q}}(\textbf{y})J_{n}(\textbf{q})dq}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}\mathbbm{1}_{\left|\dfrac{J_{n}(\textbf{h})}{\pi_{n}(\textbf{h})}\right|\leq
1+\varepsilon_{1}}d\mu_{n}(\textbf{y})$ $\displaystyle\hskip
28.45274pt+\int\int_{C}\int_{S_{1}}\dfrac{p_{n,\textbf{h}}(\textbf{y})J_{n}(\textbf{h})d\textbf{h}}{\int
p_{n,\textbf{q}}(\textbf{y})J_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}\mathbbm{1}_{\left|\dfrac{J_{n}(\textbf{h})}{\pi_{n}(\textbf{h})}\right|>1+\varepsilon_{1}}d\mu_{n}(\textbf{y})$
$\displaystyle\leq(1+\varepsilon_{1})\int\int_{C}\int_{S_{1}}\dfrac{p_{n,\textbf{h}}(\textbf{y})\pi_{n}(\textbf{h})d\textbf{h}}{\int
p_{n,\textbf{q}}(\textbf{y})J_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}\mathbbm{1}_{\left|\dfrac{J_{n}(\textbf{h})}{\pi_{n}(\textbf{h})}\right|\leq
1+\varepsilon_{1}}d\mu_{n}(\textbf{y})$ $\displaystyle\hskip
28.45274pt+E_{n,C}\left(r_{\textbf{H}_{n}|\textbf{Y}_{n}}(S_{1})(1-\delta_{n}(\textbf{Y}_{n}))\mathbbm{1}_{\left|\dfrac{J_{n}(\textbf{h})}{\pi_{n}(\textbf{h})}\right|>1+\varepsilon_{1}}\right).$

For all $\epsilon_{2}>0$, there is large enough $n$ such that

$E_{n,C}\left(r_{\textbf{H}_{n}|\textbf{Y}_{n}}(S_{1})(1-\delta_{n}(\textbf{Y}_{n}))\mathbbm{1}_{\left|\dfrac{J_{n}(\textbf{h})}{\pi_{n}(\textbf{h})}\right|>1+\varepsilon_{1}}\right)\leq
P_{n,C}\left(\dfrac{J_{n}(\textbf{h})}{\pi_{n}(\textbf{h})}>\varepsilon_{1}\right)<\varepsilon_{2}.$

By Lemma A.3, we know that
$\frac{\bigg{|}\int_{\textbf{H}_{n}}p_{n,\textbf{h}}J_{n}(\textbf{h})d\textbf{h}-\int_{\textbf{H}_{n}}p_{n,\textbf{h}}\pi_{n}(\textbf{h})d\textbf{h}\bigg{|}}{\int_{\textbf{H}_{n}}p_{n,\textbf{h}}J_{n}(\textbf{h})d\textbf{h}}\stackrel{{\scriptstyle
P_{n,\textbf{0}}}}{{\rightarrow}}0$. Repeating the same steps as above, we
obtain for all $\epsilon_{3}>0$ and $\epsilon_{4}>0$:

$\displaystyle\int\int_{C}\int_{S_{1}}\dfrac{p_{n,\textbf{h}}(\textbf{y})\pi_{n}(\textbf{h})d\textbf{h}}{\int
p_{n,\textbf{q}}(\textbf{y})J_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}d\mu_{n}(\textbf{y})$
$\displaystyle\leq(1+\varepsilon_{3})\int\int_{C}\int_{S_{1}}\dfrac{p_{n,\textbf{h}}(\textbf{y})\pi_{n}(\textbf{h})d\textbf{h}}{\int
p_{n,\textbf{q}}(\textbf{y})\pi_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}\mathbbm{1}_{\left|\dfrac{\int
p_{n,\textbf{q}}(\textbf{y})\pi_{n}(\textbf{q})d\textbf{q}}{\int
p_{n,\textbf{q}}(\textbf{y})J_{n}(\textbf{q})d\textbf{q}}\right|\leq
1+\varepsilon_{3}}d\mu_{n}(\textbf{y})+\varepsilon_{4}.$

Defining $\eta_{1}=\max\\{\varepsilon_{1},\varepsilon_{3}\\}$,
$\delta_{2}=(1+\eta_{1})\cdot\max\\{\varepsilon_{2},\varepsilon_{4}\\}+\eta_{1}$
yields the following inequality for the whole expression in (3.2):

$\displaystyle\eqref{eq:S1}\leq$
$\displaystyle(1+\eta_{1})^{2}\int\int_{C}\int_{S_{1}}\dfrac{p_{n,\textbf{h}}(\textbf{y})\pi_{n}(\textbf{h})d\textbf{h}}{\int
p_{n,\textbf{q}}(\textbf{y})\pi_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}d\mu_{n}(\textbf{y})+\eta_{2}.$

Next, we address the expression in (3.3). For this, we first apply Fubini’s
lemma to change the order of integration as follows:

$\displaystyle\int\int_{C}$
$\displaystyle\int_{S_{1}}\dfrac{p_{n,\textbf{h}}(\textbf{y})\pi_{n}(\textbf{h})d\textbf{h}}{\int
p_{n,\textbf{q}}(\textbf{y})\pi_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}d\mu_{n}(\textbf{y})$
$\displaystyle=$
$\displaystyle\dfrac{\pi_{n}(S_{1})}{\pi_{n}(C)}\int\int_{S_{1}}\int_{C}\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\int
p_{n,\textbf{q}}(\textbf{y})\pi_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{h}}(\textbf{y})\pi_{n}(\textbf{h})d\textbf{h}}{\pi_{n}(S_{1})}d\mu_{n}(\textbf{y})$
$\displaystyle=$
$\displaystyle\dfrac{\pi_{n}(S_{1})}{\pi_{n}(C)}\int\int_{S_{1}}\pi_{\textbf{H}_{n}|\textbf{Y}_{n}}(C)(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{h}}(\textbf{y})\pi_{n}(\textbf{h})d\textbf{h}}{\pi_{n}(S_{1})}d\mu_{n}(\textbf{y})$
$\displaystyle\leq$
$\displaystyle\dfrac{\pi_{n}(S_{1})}{\pi_{n}(C)}\int\int_{S_{1}}(1-\delta_{n}(\textbf{Y}))\dfrac{p_{n,\textbf{h}}(\textbf{y})\pi_{n}(\textbf{h})d\textbf{h}}{\pi_{n}(S_{1})}$
$\displaystyle=$
$\displaystyle\dfrac{1}{\pi_{n}(C)}\int_{S_{1}}E_{n,\textbf{h}}(1-\delta_{n}(\textbf{Y}))\pi_{n}(\textbf{h})d\textbf{h}.$

Recall that $d$ denotes the dimension of $\Theta$ and note that
$\pi_{n}(C)>c_{2}/\sqrt{n}^{d}$. Using this fact and applying the definition
of $S_{1}$ and the uniform exponential consistency property:

$\displaystyle\dfrac{1}{\pi_{n}(C)}$
$\displaystyle\int_{S_{1}}E_{n,\textbf{h}}(1-\delta_{n}(\textbf{Y}))\pi_{n}(\textbf{h})d\textbf{h}$
$\displaystyle\leq$ $\displaystyle
c_{2}\sqrt{n}^{d}\int_{M_{n}\leq\left\lVert\textbf{h}\right\rVert\leq
D\sqrt{n}}E_{n,\textbf{h}}(1-\delta_{n}(\textbf{Y}))\pi_{n}(\textbf{h})d\textbf{h}$
$\displaystyle\leq
c_{2}\sqrt{n}^{d}\int_{M_{n}\leq\left\lVert\textbf{h}\right\rVert\leq
D\sqrt{n}}c_{3}\dfrac{c_{1}}{\sqrt{n}^{d}}e^{-c_{4}(\left\lVert\textbf{h}^{2}\right\rVert\wedge
n)}d\textbf{h}$ $\displaystyle\leq$ $\displaystyle
c_{5}\int_{M_{n}\leq\left\lVert\textbf{h}\right\rVert\leq
D\sqrt{n}}e^{-c_{4}(\left\lVert\textbf{h}^{2}\right\rVert)}d\textbf{h}.$

We again invoke Lemma A.3 and following similar steps in applying this lemma
as we did for the expression in (3.2), we obtain:

$\eqref{eq:S2}\leq(1+\varepsilon_{3})\int\int_{C}\int_{S_{2}}\dfrac{p_{n,\textbf{h}}(\textbf{y})J_{n}(\textbf{h})d\textbf{h}}{\int
p_{n,\textbf{q}}(\textbf{y})\pi_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}d\mu_{n}(\textbf{y})+\varepsilon_{4}.$

Per Assumption 4, we can split the likelihood and achieve a probabilistic
upper bound on
$p^{2}(\textbf{y},\boldsymbol{\theta})J(\textbf{y},\boldsymbol{\theta})I_{\\{\left\lVert\theta-\theta_{0}\right\rVert>D\\}}.$
We apply this, followed by Fubini’s lemma, to bound the above integral
expression as follows:

$\displaystyle\int\int_{C}\int_{S_{2}}$
$\displaystyle\dfrac{p_{n,\textbf{h}}(\textbf{y})J_{n}(\textbf{h})d\textbf{h}}{\int
p_{n,\textbf{q}}(\textbf{y})\pi_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}d\mu_{n}(\textbf{y})$
$\displaystyle=\int\int_{C}\int_{S_{2}}\dfrac{p_{n,\textbf{h}}^{1}p_{n,\textbf{h}}^{2}(\textbf{y})J_{n}(\textbf{h})d\textbf{h}}{\int
p_{n,\textbf{q}}(\textbf{y})\pi_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}d\mu_{n}(\textbf{y})$
$\displaystyle=\int\int_{C}\int_{S_{2}}\dfrac{p_{n,\textbf{h}}^{1}p_{n,\textbf{h}}^{2}(\textbf{y})J_{n}(\textbf{h})d\textbf{h}}{\int
p_{n,\textbf{q}}(\textbf{y})\pi_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}d\mu_{n}(\textbf{y})\mathbbm{1}_{p^{2}_{n,\textbf{h}}(\textbf{y})J_{n}(\textbf{h})\leq\gamma_{n}(\textbf{h})}$
$\displaystyle\hskip
28.45274pt+\int\int_{C}\int_{S_{2}}\dfrac{p_{n,\textbf{h}}^{1}p_{n,\textbf{h}}^{2}(\textbf{y})J_{n}(\textbf{h})d\textbf{h}}{\int
p_{n,\textbf{q}}(\textbf{y})\pi_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}d\mu_{n}(\textbf{y})\mathbbm{1}_{p^{2}_{n,\textbf{h}}(\textbf{y})J_{n}(\textbf{h})>\gamma_{n}(\textbf{h})}$
$\displaystyle\leq\int\int_{C}\int_{S_{2}}\dfrac{p_{n,\textbf{h}}^{1}\gamma_{n}(\textbf{h})d\textbf{h}}{\int
p_{n,\textbf{q}}(\textbf{y})\pi_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}d\mu_{n}(\textbf{y})+\varepsilon_{5}$
$\displaystyle=\dfrac{\gamma_{n}(S_{2})}{\pi_{n}(C)}\int\int_{S_{2}}\int_{C}\dfrac{p_{n,\textbf{g}}\pi_{n}(\textbf{g})d\textbf{g}}{\int
p_{n,\textbf{q}}(\textbf{y})\pi_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{h}}^{1}(\textbf{y})\gamma_{n}(\textbf{h})d\textbf{h}}{\gamma_{n}(S_{2})}d\mu_{n}(\textbf{y})+\varepsilon_{5}$
$\displaystyle=\dfrac{\gamma_{n}(S_{2})}{\pi_{n}(C)}\int\int_{S_{2}}\pi_{\textbf{H}_{n}|\textbf{Y}_{n}}(C)(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{h}}^{1}(\textbf{y})\gamma_{n}(\textbf{h})d\textbf{h}}{\gamma_{n}(S_{2})}d\mu_{n}(\textbf{y})+\varepsilon_{5}$
$\displaystyle\leq\dfrac{1}{\pi_{n}(C)}\int_{S_{2}}E_{n,\textbf{h}}^{1}(1-\delta_{n}(\textbf{Y}))\gamma_{n}(\textbf{h})d\textbf{h}+\varepsilon_{5}.$

Recalling that $\pi_{n}(C)>c_{2}/\sqrt{n}^{d}$ and applying the uniform
exponential consistency property:

$\displaystyle\leq
c_{2}\sqrt{n}^{d}\int_{\left\lVert\textbf{h}\right\rVert\geq
D\sqrt{n}}e^{-c_{4}(\left\lVert\textbf{h}^{2}\right\rVert\wedge
n)}\gamma_{n}(\textbf{h})d\textbf{h}+\varepsilon_{5}$ $\displaystyle\leq
c_{2}\sqrt{n}^{d}e^{-c_{4}D^{2}n}+\varepsilon_{5}.$

Finally, combining the calculations for (3.2) and (3.3) results in the overall
inequality for our initial expression:

$\displaystyle E_{n,C}(r_{\textbf{H}_{n}|\textbf{Y}_{n}}(C_{n}^{c}))$
$\displaystyle(1-\delta_{n}(\textbf{Y}_{n}))\leq$
$\displaystyle(1+\eta_{1})^{2}\left(c_{5}\int_{M_{n}\leq\left\lVert\textbf{h}\right\rVert\leq
D\sqrt{n}}e^{-c_{4}(\left\lVert\textbf{h}^{2}\right\rVert)}d\textbf{h}+\eta_{2}\right)$
$\displaystyle+(1+\varepsilon_{3})\left(c_{2}\sqrt{n}^{d}e^{-c_{4}D^{2}n}+\varepsilon_{5}\right)+\varepsilon_{4}.$

Thus, as $M_{n}\rightarrow\infty$,

$E_{n,C}(r_{\textbf{H}_{n}|\textbf{Y}_{n}}(C_{n}^{c}))(1-\delta_{n}(\textbf{Y}_{n}))\rightarrow
0,$

and moreover, by the contiguity result shown in Lemma A.1,
$r_{\textbf{H}_{n}|\textbf{Y}_{n}}(C_{n}^{c})\stackrel{{\scriptstyle
P_{n,\textbf{0}}}}{{\rightarrow}}0$.∎

##  4\. Triangular distribution

We again consider the triangular distribution defined on the interval $(0,1)$,
for which the fiducial distribution has been previously derived in Section 2.
Recall the probability density function in (2.2). When $\theta=1/2$, it
corresponds to the symmetric triangular distribution over the interval
$(0,1)$. This distribution represents the probability density of the mean of
two i.i.d uniform random variables over the same interval $(0,1)$. Its origins
can be traced back to the 18th century, specifically in Simpson, (1755) and
Schmidt, (1934).

Following this work, the general non-symmetric triangular distribution was
initially explored in Ayyangar, (1941). Later on, the authors of Johnson and
Kotz, (1999) revisited non-symmetric triangular distributions within the
context of modeling prices. The triangular density has a unique mode at the
value $\theta$ and satisfies the property $P(y\leq\theta)=\theta$. This
property can be leveraged to derive an estimate of $\theta$ based on the
empirical distribution function. It is worth noting that the non-symmetric
triangular distribution lacks a useful reduced sufficient statistic.

The probability density function for this distribution does not meet the
requirement of possessing third partial derivatives, as specified for
asymptotic normality of GFDs in Sonderegger and Hannig, (2014). In what
follows, we demonstrate that it does, however, fulfill the criteria for DQM
and verify that its fiducial solution satisfies the other conditions of
Theorem 3.1 for asymptotic normality.

Assumption 1 (Differentiability in quadratic mean). We want to show that

$\int_{0}^{1}\left(\sqrt{P_{\theta+h}}-\sqrt{P_{\theta}}-\dfrac{1}{2}h\dot{\ell}_{\theta}\sqrt{P_{\theta}}\right)^{2}\,dy=o(h^{2}).$

Substituting the triangluar density function for $P_{\theta}$, $P_{\theta+h}$:

$\displaystyle\int_{0}^{1}(\sqrt{P_{\theta+h}}-$
$\displaystyle\sqrt{P_{\theta}}-\dfrac{1}{2}h\dot{\ell}_{\theta}\sqrt{P_{\theta}})^{2}\,dy=$
$\displaystyle=\int_{0}^{\theta}\left(\sqrt{\dfrac{2y}{\theta+h}}-\sqrt{\dfrac{2y}{\theta}}+\dfrac{1}{2}h\dfrac{1}{\theta}\sqrt{\dfrac{2y}{\theta}}\right)^{2}\,dy$
$\displaystyle+\int_{\theta}^{\theta+h}\left(\sqrt{\dfrac{2-2y}{1-(\theta+h)}}-\sqrt{\dfrac{2-2y}{1-\theta}}-\dfrac{1}{2}h\dfrac{1}{1-\theta}\sqrt{\dfrac{2-2y}{1-\theta}}\right)^{2}\,dy$
$\displaystyle+\int_{\theta+h}^{1}\left(\sqrt{\dfrac{2-2y}{1-(\theta+h)}}-\sqrt{\dfrac{2-2y}{1-\theta}}-\dfrac{1}{2}h\dfrac{1}{1-\theta}\sqrt{\dfrac{2-2y}{1-\theta}}\right)^{2}\,dy.$

By direct expansion of the square and computation of the integrals, the first
and third summand can be shown to have limit 0 as $h\rightarrow 0$.

The middle term can be upper bounded to show it is $o(h^{2})$. To see this,
note that the function inside the square is increasing in $y$, so we can
achieve this bound by evaluating the function at the limits of integration and
then computing the integral of the bound. Define

$f(y)=\sqrt{\dfrac{2-2y}{1-(\theta+h)}}-\sqrt{\dfrac{2-2y}{1-\theta}}-\dfrac{1}{2}h\dfrac{1}{1-\theta}\sqrt{\dfrac{2-2y}{1-\theta}}$

and evaluate

$f(\theta+h)=\sqrt{2}-\sqrt{2-\dfrac{2h}{1-\theta}}-\dfrac{1}{2}h\dfrac{1}{1-\theta}\sqrt{2-\dfrac{2h}{1-\theta}}.$

Taylor expansion at $h=0$ gives

$\sqrt{2-\dfrac{2h}{1-\theta}}=\sqrt{2}+\dfrac{h}{\sqrt{2}(\theta-1)}-\dfrac{h^{2}}{4(\sqrt{2}(\theta-1)^{2})}+\dfrac{h^{3}}{8\sqrt{2}(\theta-1)^{3}}+O(h^{4}),$

and by substituting into the integral, one can verify that
$\int_{\theta}^{\theta+h}(f(\theta+h))^{2}\,dy=o(h^{2})$.

Similarly, evaluate

$f(\theta)=\sqrt{\dfrac{2\theta}{\theta+h}}-\sqrt{2}-\dfrac{1}{2}h\dfrac{1}{1-\theta}\sqrt{2}.$

Taylor expansion of $\sqrt{\dfrac{2\theta}{\theta+h}}$ at $h=0$ gives

$\sqrt{\dfrac{2\theta}{\theta+h}}=\sqrt{2}-\dfrac{h}{\sqrt{2}\theta}+\dfrac{3h^{2}}{4\sqrt{2}\theta}-\dfrac{5h^{3}}{8(\sqrt{2}\theta^{3})}+O(h^{4}),$

and again, by substituting into the integral, one can show
$\int_{\theta}^{\theta+h}(f(\theta))^{2}\,dy=o(h^{2}).$

Thus, the experiment is DQM and has Fisher Information

$I_{\theta}=\int_{0}^{\theta}\left(-\dfrac{1}{\theta}\right)^{2}\dfrac{2y}{\theta}dy+\int_{\theta}^{1}\left(\dfrac{1}{1-\theta}\right)^{2}\dfrac{2-2y}{1-\theta}dy=\dfrac{1}{\theta}+\dfrac{1}{1-\theta}=\left(\theta(1-\theta)\right)^{-1}\neq
0,\hskip 7.11317pt\theta\in(0,1).$

Assumption 2 (Convergence of Jacobian function) and Assumption 3 (Mass of
limiting measure). Recall (2.3) and observe that the Law of Large Numbers and
continuous mapping theorem gives,

$\displaystyle J(\textbf{y},\theta)$
$\displaystyle=\bigg{(}\frac{1}{n}\sum_{i=1}^{n}\Big{(}\frac{y_{i}}{2\theta}\Big{)}^{2}I_{(0,\theta]}(y_{i})+\frac{1}{n}\sum_{i=1}^{n}\Big{(}\frac{1-y_{i}}{2(1-\theta)}\Big{)}^{2}I_{(\theta,1)}(y_{i})\bigg{)}^{1/2}\rightarrow$
$\displaystyle\hskip
28.45274pt\Bigg{(}E\Big{(}\frac{y_{i}^{2}}{4\theta^{2}}I_{(0,\theta]}(y_{i})\Big{)}+E\Big{(}\frac{(1-y_{i})^{2}}{4(1-\theta)^{2}}I_{(\theta,1)}(y_{i})\Big{)}\Bigg{)}^{1/2}=\Big{(}\frac{\theta}{8}+\frac{1-\theta}{8}\Big{)}^{1/2}=\sqrt{\frac{1}{8}}.$

Also, note that

$J(\textbf{y},\theta)=\Big{(}\frac{1}{n}\sum_{i=1}^{n}\big{(}\frac{y_{i}}{2\theta}\big{)}^{2}I_{(0,\theta]}(y_{i})+\frac{1}{n}\sum_{i=1}^{n}\big{(}\frac{1-y_{i}}{2(1-\theta)}\big{)}^{2}I_{(\theta,1)}(y_{i})\Big{)}^{1/2}\leq\frac{1}{4}.$

So, $\pi(\theta)=\sqrt{2}$ satisfies the condition. Also notice that for large
$n$, the GFD should behave similarly to the Bayesian posterior computed using
a flat prior.

Assumption 4 (Likelihood splitting). Let $p^{1}_{\theta}=P_{\theta}$ and
$p^{2}_{\theta}=1$. We previously showed $J(y,\theta)\leq\frac{1}{4}$. Thus,
for $\gamma(\theta)=1$, $p^{2}(y,\theta)\cdot J(y,\theta)\leq\gamma(\theta).$

Assumption 5 (Exponentially consistent tests). Recall from Remark 3.1, we need
only show the existence of uniformly consistent estimators in the i.i.d
setting. To do so, we can verify the conditions from Lemma 10.6 in van der
Vaart, (1998). Indeed, $\Theta=(0,1)$ is $\sigma$-compact and $P_{\theta}\neq
P_{\theta^{\prime}}$ for every pair $\theta\neq\theta^{\prime}$. To check
whether the maps $\theta\rightarrow P_{\theta}$ are continuous for the total
variation norm, observe

$\left\lVert
P_{\theta}-P_{\theta^{\prime}}\right\rVert_{TV}=2\int_{\Theta}\left(1-\dfrac{P_{\theta}}{P_{\theta^{\prime}}}\right)_{+}d\mathbb{P}_{\theta}=2\int_{0}^{1}\left(1-\dfrac{P_{\theta}}{P_{\theta^{\prime}}}\right)_{+}P_{\theta^{\prime}}\,d\lambda.$
(4.1)

Now,
$\left(1-\dfrac{P_{\theta}}{P_{\theta^{\prime}}}\right)_{+}P_{\theta^{\prime}}\rightarrow
0$ and
$\left(1-\dfrac{P_{\theta}}{P_{\theta^{\prime}}}\right)_{+}P_{\theta^{\prime}}\leq
2$, so by dominated convergence theorem we get that the right-hand side of
(4.1) converges to $0$, hence, $\left\lVert
P_{\theta}-P_{\theta^{\prime}}\right\rVert_{TV}\rightarrow 0$.

###  4.1. Simulation study

In this simulation study, we compare the frequentist behavior of credible
intervals derived using the generalized fiducial approach with those obtained
by Bayesian methods. Specifically, we consider two different non-informative
priors in Bayesian analysis: the Jeffreys prior and a uniform prior.

The widely used non-informative prior in Bayesian analysis is the Jeffreys
prior (Jeffreys,, 1998), which is proportional to the positive square root of
the determinant of the Fisher Information matrix. For the triangular
distribution, the authors of Berger et al., (2009) conjectured that the
Jeffreys prior corresponds to a $\text{Beta}(1/2,1/2)$ distribution based on
numerical evaluation, but could not derive it analytically due to
differentiability issues. As previously demonstrated, $f(\textbf{y},\theta)$
is DQM, implying that the Fisher Information satisfies
$I_{\boldsymbol{\theta}}=P_{\boldsymbol{\theta}}\dot{\ell}_{\boldsymbol{\theta}}\dot{\ell}_{\boldsymbol{\theta}}^{T}$
as in the definition of DQM in Section 3.1. This computation, given above,
shows that the Jeffreys prior is proportional to
$I^{-1/2}_{\theta}=(\theta(1-\theta))^{-1/2}$, as conjectured by the authors
of Berger et al., (2009). Note that the definition of Fisher Information under
DQM does not impose a formal derivative but is instead defined through the
score function, which often coincides with the derivative of the log
likelihood when it exists. In this example, the more general score function
allows us to circumvent differentiability issues due to the indicator
functions.

A careful examination reveals an issue in the generalized fiducial solution
derived in Example 2.1. In the case $n=1$, the DGA (2.1) is not invertible for
some values of $u$ and $y$. Indeed, its inverse image is

$Q_{y}(u)=\begin{cases}\emptyset&u<y^{2}\\\ \frac{y^{2}}{u}&y^{2}\leq u<y\\\
1-\frac{(1-y)^{2}}{1-u}&y\leq u\leq 1-(1-y)^{2}\\\
\emptyset&1-(1-y)^{2}<u.\end{cases}$

As a result, the GFD has poor frequentist properties for small sample sizes
when the true $\theta$ is close to $0$ or $1$. This can be remedied by the
following modification of the inverse image:

$\tilde{Q}_{y}(u)=\begin{cases}1&u<y^{2}\\\ \frac{y^{2}}{u}&y^{2}\leq u<y\\\
1-\frac{(1-y)^{2}}{1-u}&y\leq u\leq 1-(1-y)^{2}\\\
0&1-(1-y)^{2}<u.\end{cases}$

The above leads to the following modification of the GFD that is continuous on
$(0,1)$ and has positive probability on $\\{0,1\\}$:

$r^{\text{mod}}(\boldsymbol{\theta}|\mathbf{y})=\left(\min_{i}(1-y_{i})^{2}\right)\delta_{\\{0\\}}+\left(\min_{i}y_{i}^{2}\right)\delta_{\\{1\\}}+\left(1-\min_{i}y_{i}^{2}-\min_{i}(1-y_{i})^{2}\right)r(\boldsymbol{\theta}|\mathbf{y}),$

where $r(\boldsymbol{\theta}|\mathbf{y})$ is the non-modified GFD. We found
this modification to perform better in simulation and still possess the
Bernstein von-Mises property since we can show

$\left\lVert
r^{\text{mod}}(\boldsymbol{\theta}|\mathbf{y})-r(\boldsymbol{\theta}|\mathbf{y})\right\rVert_{TV}\leq\min_{i}y_{i}^{2}+\min_{i}(1-y_{i})^{2}\to
0\quad\mbox{a.s.}$

We used the same software to evaluate the generalized fiducial and Bayesian
approaches, with the only difference in the implementation being whether the
likelihood was multiplied by the Bayesian prior distribution or by the
Jacobian function in the GFD and, in the case of the modified GFD, scaled and
summed with the appropriate constants. We compared the methods across
different values of the parameter and 10 sample sizes ranging from $n=1$ to
$n=1000$. The results, presented in Table 1, summarize the frequentist
coverage and length of the $95\%$ credible intervals.

The Jeffreys Bayesian method is conservative for small sample sizes and
achieves competitive coverage for medium sample sizes. We found the modified
generalized fiducial method obtains the shortest intervals while maintaining
the desired coverage. In contrast, the GFD and the Bayesian method with a
uniform behave similarly, but the GFD obtains much shorter intervals at small
sample sizes. As anticipated from large sample theory, all four methods
perform similarly for larger sample sizes.

##  5\. Free-knot spline

A common approach to curve estimation is a spline model. A spline model of
degree $p$ is characterized by piece-wise degree $p$ polynomials with the
requirement that the resulting function be, in some sense, smooth at the
connection points, or knot points. A typical smoothness requirement is for the
$p-1$ derivative to exist. An important consideration of the spline model is
placement of the knot points. In the free-knot spline setting, the problem of
interest is to estimate the location of a small number of carefully placed
knot points (Toms and Lesperance,, 2003; Sonderegger et al.,, 2009). A
Bayesian solution for the arbitrary degree $p$ problem with a fixed number of
knot points is provided in DiMatteo et al., (2001) . The authors suggest
utilizing a prior of
$p(\boldsymbol{\alpha},\mathbf{t},\sigma^{2})\propto\sigma^{2}$, where
$\sigma^{2}$ represents the typical variance term, $\boldsymbol{\alpha}$
denotes the polynomial coefficients, and $\mathbf{t}$ represents the vector of
knot points. A generalized fiducial solution to the free-knot spline problem
was previously derived and thoroughly investigated in Sonderegger and Hannig,
(2014). There, splines of degree $p\geq 4$ are shown to satisfy sufficient
conditions for asymptotic normality of a multivariate generalized fiducial
estimator. Additional simulation experiments in Chapters 4 and 5 of
Sonderegger, (2010) indicate that the fiducial solution for the free-knot
spline problem is asymptotically accurate and demonstrates strong performance
in simulation studies, including for lower degree splines. In particular, the
simulation study suggests asymptotic normality should hold for splines of
degree $p\leq 3$. It is therefore important to extend the theoretical findings
in Sonderegger and Hannig, (2014); Sonderegger, (2010) to more realistic
splines. Here, we show that Theorem 3.1 applies for splines of degree $p\geq
1$, solving the conjecture in Sonderegger and Hannig, (2014); Sonderegger,
(2010).

Consider data points $\\{x_{i},y_{i}\\}$ generated from

$y_{i}=g(x_{i}|\boldsymbol{\alpha},\mathbf{t})+\sigma\varepsilon_{i},$

where $\varepsilon_{i}$ are independent $N(0,1)$ and the design points $x_{i}$
are assumed independent with their distribution having positive density,
denoted $P_{x}$, on $[a,b]$ such that $P_{x}>\zeta$ for some $\zeta>0$.
Consequently, the data points $\\{x_{i},y_{i}\\}$ are i.i.d.

In this setting, function $g(x_{i}|\boldsymbol{\alpha},\mathbf{t})$ is a
spline with $\kappa$ ordered knot points
$\mathbf{t}=(t_{1},\ldots,t_{\kappa})^{\top}$ and $p+\kappa+1$ polynomial
coefficients $\boldsymbol{\alpha}$. Suppose $\kappa$ is known but the
locations of the knot points $\mathbf{t}$ are unknown and are to be estimated.
Moreover, assume that $t_{k}\in[a+\delta,b-\delta]$ and
$|t_{k}-t_{k+1}|>\delta$ for all $k=1,\ldots,\kappa$ and some $\delta>0$.
Define the set of vectors with such knot points as
$T_{\delta}=\\{\mathbf{t}=(t_{1},\ldots,t_{\kappa})^{\top}\hskip
2.84544pt\big{|}\hskip 2.84544ptt_{k+1}>t_{k},\hskip
2.84544pt|t_{k}-t_{k+1}|>\delta,\hskip 2.84544ptt_{1}\geq a-\delta,\hskip
2.84544ptt_{\kappa}\leq b-\delta,\hskip 2.84544ptk=1,\ldots,\kappa\\}$. We
require $p\geq 1$ and consider the piecewise truncated polynomial basis:

$g(x_{i}|\boldsymbol{\alpha},\mathbf{t})=\sum_{j=0}^{p}\alpha_{j}x_{i}^{j}+\sum_{k=1}^{\kappa}\alpha_{p+k}(x_{i}-t_{k})_{+}^{p},$

where $(u)_{+}=\max\\{u,0\\}$ is the truncation operator and has higher
precedence than the exponentiation. We require that $\boldsymbol{\alpha}$ be
bounded away from $0$, i.e. $|\alpha_{p+k}|>\xi>0$ for all
$k=1,\ldots,\kappa$, to ensure each knot can be estimated. Note that if
$\alpha_{p+k}=0$ for any $k$, we could not estimate the knot $t_{k}$;
moreover, if $t_{k}$ were already known, it would not be a parameter of
interest. Denote the truncated polynomial spline basis functions as

$\displaystyle N(x,\mathbf{t})=$
$\displaystyle\\{N_{1}(x,\mathbf{t}),\ldots,N_{p+\kappa+1}(x,\mathbf{t})\\}$
$\displaystyle=$
$\displaystyle\\{1,x,\ldots,x^{p},(x-t_{1})^{p}_{+},\ldots,(x-t_{\kappa})^{p}_{+}\\}.$

Then, the density function is

$f(y,\mathbf{\theta})=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\bigg{(}-\frac{1}{2\sigma^{2}}(y-N(x,\mathbf{t})\boldsymbol{\alpha})^{2}\bigg{)},$

where $\mathbf{\theta}=(\mathbf{t},\boldsymbol{\alpha},\sigma^{2})^{\top}$.

To find the generalized fiducial solution to this problem, we invert the data
generating algorithm and solve for $\varepsilon_{i}$. The Jacobian is found by
taking derivatives with respect to the parameters of interest of the DGA and
substituting the inversion result for $\varepsilon_{i}$. Denote the DGA by
$G(\varepsilon_{i},\boldsymbol{\theta})$ and let
$\boldsymbol{\eta}=\\{\boldsymbol{\alpha},\mathbf{t}\\}$. Note that

$\varepsilon_{i}=G^{-1}(y_{i},\boldsymbol{\theta})=\frac{1}{\sigma}(y_{i}-g(x_{i}|\boldsymbol{\eta})),$

and

$\frac{\partial
G(\varepsilon_{i},\boldsymbol{\theta})}{\partial\boldsymbol{\alpha}}=\big{(}1,x_{i},\ldots,x_{i}^{p},(x_{i}-t_{1})^{p}_{+},\ldots,(x_{i}-t_{\kappa})^{p}_{+}\big{)},$

$\frac{\partial
G(\varepsilon_{i},\boldsymbol{\theta})}{\partial\mathbf{t}}=p\big{(}\alpha_{p+1}(x_{i}-t_{1})^{p-1}_{+},\ldots,\alpha_{p+\kappa}(x_{i}-t_{\kappa})^{p-1}_{+}\big{)},$

$\frac{\partial
G(\varepsilon_{i},\boldsymbol{\theta})}{\partial\sigma}=\frac{1}{\sigma}\big{(}y_{i}-g(x_{i}|\boldsymbol{\eta})\big{)}.$

Thus, the Jacobian is

$J(\mathbf{y},\boldsymbol{\theta})=\frac{p^{\kappa}}{\sigma}\prod_{k=1}^{\kappa}|\alpha_{p+k}|\det\bigg{[}\frac{1}{n}\mathbf{A}^{\top}\mathbf{A}\bigg{]}^{1/2},$

where
$\mathbf{A}=[\mathbf{B}_{\boldsymbol{\alpha}},\mathbf{B}_{\mathbf{t}},\mathbf{B}_{\sigma}]$
and

$\mathbf{B}_{\boldsymbol{\alpha}}=\begin{pmatrix}1&x_{1}&\ldots&x_{1}^{p}&(x_{1}-t_{1})_{+}^{p}&\ldots&(x_{1}-t_{\kappa})_{+}^{p}\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
1&x_{n}&\ldots&x_{n}^{p}&(x_{n}-t_{1})_{+}^{p}&\ldots&(x_{n}-t_{\kappa})_{+}^{p}\end{pmatrix},$

$\mathbf{B}_{\mathbf{t}}=\begin{pmatrix}(x_{1}-t_{1})_{+}^{p-1}&\ldots&(x_{1}-t_{\kappa})_{+}^{p-1}\\\
\vdots&\ddots&\vdots\\\
(x_{n}-t_{1})_{+}^{p-1}&\ldots&(x_{n}-t_{\kappa})_{+}^{p-1}\end{pmatrix},$

and

$\mathbf{B}_{\boldsymbol{\sigma}}=\begin{pmatrix}y_{1}-g(x_{1}|\boldsymbol{\theta})\\\
\vdots\\\ y_{n}-g(x_{n}|\boldsymbol{\theta})\end{pmatrix}.$

Since $\mathbf{B}_{\boldsymbol{\sigma}}$ contains a subtraction of a linear
combination of columns from $\mathbf{B}_{\boldsymbol{\alpha}}$, the
subtraction does not change the determinant and we can therefore replace
$\mathbf{B}_{\boldsymbol{\sigma}}$ with
$\tilde{\mathbf{B}}_{\boldsymbol{\sigma}}$ in the calculation of the
determinant, where

$\tilde{\mathbf{B}}_{\boldsymbol{\sigma}}=\begin{pmatrix}y_{1}\\\ \vdots\\\
y_{n}\end{pmatrix}.$

Therefore,

$J(\mathbf{y},\boldsymbol{\theta})=\frac{p^{\kappa}}{\sigma}\prod_{k=1}^{\kappa}|\alpha_{p+k}|\det\bigg{[}\frac{1}{n}\mathbf{A}^{\top}\mathbf{A}\bigg{]}^{1/2}=\frac{p^{\kappa}}{\sigma}\prod_{k=1}^{\kappa}|\alpha_{p+k}|\det\bigg{[}\frac{1}{n}\tilde{\mathbf{A}}^{\top}\tilde{\mathbf{A}}\bigg{]}^{1/2},$

where
$\tilde{\mathbf{A}}=[\mathbf{B}_{\boldsymbol{\alpha}},\mathbf{B}_{\mathbf{t}},\tilde{\mathbf{B}}_{\boldsymbol{\sigma}}]$.

We now verify that the free-knot spline model of degree $p\geq 1$ and its
generalized fiducial solution derived above satisfy the assumptions for
Theorem 3.1.

Assumption 1(Differentiability in quadratic mean). It is straightforward to
show that exponential families are QMD. Below we compute the Fisher
Information matrix and verify that it is non-singular. First, observe

$\displaystyle
E_{\boldsymbol{\theta}}\bigg{(}\frac{\partial}{\partial\sigma^{2}}L(\boldsymbol{\theta},y)\frac{\partial}{\partial\sigma^{2}}L(\boldsymbol{\theta},y)\bigg{)}$
$\displaystyle=E_{\boldsymbol{\theta}}\bigg{(}\bigg{[}\frac{-1}{2\sigma^{2}}+\frac{1}{2(\sigma^{2})^{2}}\big{(}y-N(x,\mathbf{t})^{\top}\boldsymbol{\alpha}\big{)}^{2}\bigg{]}\bigg{[}\frac{-1}{2\sigma^{2}}+\frac{1}{2(\sigma^{2})^{2}}\big{(}y-N(x,\mathbf{t})^{\top}\boldsymbol{\alpha}\big{)}^{2}\bigg{]}\bigg{)}$
$\displaystyle=E_{\boldsymbol{\theta}}\bigg{(}\frac{1}{4\sigma^{4}}-\frac{2}{4\sigma^{6}}\big{(}y-N(x,\mathbf{t})^{\top}\boldsymbol{\alpha}\big{)}^{2}+\frac{1}{4\sigma^{8}}\big{(}y-N(x,\mathbf{t})^{\top}\boldsymbol{\alpha}\big{)}^{4}\bigg{)}$
$\displaystyle=\frac{1}{4\sigma_{0}^{4}}-\frac{2}{4\sigma_{0}^{6}}\sigma_{0}^{2}+\frac{1}{4\sigma_{0}^{8}}3\sigma_{0}^{4}$
$\displaystyle=\frac{1}{2\sigma_{0}^{4}}>0.$

Next, note that

$\displaystyle
E_{\boldsymbol{\theta}}\bigg{(}\frac{\partial}{\partial\eta_{j}}L(\boldsymbol{\theta},y)\frac{\partial}{\partial\sigma^{2}}L(\boldsymbol{\theta},y)\bigg{)}$
$\displaystyle=E_{\boldsymbol{\theta}}\bigg{(}\frac{1}{\sigma^{2}}\big{(}y-N(x,\mathbf{t})^{\top}\boldsymbol{\alpha}\big{)}\frac{\partial}{\partial\eta_{j}}N(x,\mathbf{t})^{\top}\boldsymbol{\alpha}\bigg{[}\frac{-1}{2\sigma^{2}}+\frac{1}{2\sigma^{4}}\big{(}y-N(x,\mathbf{t})^{\top}\boldsymbol{\alpha}\big{)}^{2}\bigg{]}\bigg{)}$
$\displaystyle=E_{\boldsymbol{\theta}}\bigg{(}\frac{-1}{2\sigma^{4}}\big{(}y-N(x,\mathbf{t})^{\top}\boldsymbol{\alpha}\big{)}\frac{\partial}{\partial\eta_{j}}N(x,\mathbf{t})^{\top}\boldsymbol{\alpha}$
$\displaystyle\hskip
113.81102pt+\frac{1}{2\sigma^{6}}\big{(}y-N(x,\mathbf{t})^{\top}\boldsymbol{\alpha}\big{)}^{3}\frac{\partial}{\partial\eta_{j}}N(x,\mathbf{t})^{\top}\boldsymbol{\alpha}\bigg{)}$
$\displaystyle=0,$

so, we can focus on the following sub-matrix:

$\displaystyle I_{j,k}(\boldsymbol{\eta})$
$\displaystyle=\sum_{i=1}^{n}E_{\boldsymbol{\theta}}\bigg{[}\frac{\partial}{\partial\eta_{j}}L(\boldsymbol{\theta},y_{i})\frac{\partial}{\partial\eta_{k}}L(\boldsymbol{\theta},y_{i})\bigg{]}$
$\displaystyle=\frac{1}{\sigma^{2}}\sum_{i=1}^{n}\frac{\partial}{\partial\eta_{j}}\big{(}N(x_{i},\mathbf{t})^{\top}\boldsymbol{\alpha}\big{)}\frac{\partial}{\partial\eta_{k}}\big{(}N(x_{i},\mathbf{t})^{\top}\boldsymbol{\alpha}\big{)}.$

Note that,

$\frac{\partial}{\partial
t_{j}}N(x_{i},\mathbf{t})^{\top}\boldsymbol{\alpha}=-p(x_{i}-t_{j})^{p-1}_{+}\alpha_{j+p+1},$
$\frac{\partial}{\partial\alpha_{j}}N(x_{i},\mathbf{t})^{\top}\boldsymbol{\alpha}=N_{j}(x_{i},\mathbf{t}).$

Define a matrix $\mathbf{X}$ as

$\mathbf{X}=\begin{pmatrix}N_{1}(x_{1},\mathbf{t})&\ldots&N_{p+\kappa+1}(x_{1},\mathbf{t})&\frac{\partial}{\partial
t_{1}}N(x_{1},\mathbf{t})^{\top}\boldsymbol{\alpha}&\ldots&\frac{\partial}{\partial
t_{\kappa}}N(x_{1},\mathbf{t})^{\top}\boldsymbol{\alpha}\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
N_{1}(x_{n},\mathbf{t})&\ldots&N_{p+\kappa+1}(x_{n},\mathbf{t})&\frac{\partial}{\partial
t_{1}}N(x_{n},\mathbf{t})^{\top}\boldsymbol{\alpha}&\ldots&\frac{\partial}{\partial
t_{\kappa}}N(x_{n},\mathbf{t})^{\top}\boldsymbol{\alpha}\end{pmatrix}.$

Then the Fisher Information for $\boldsymbol{\eta}$ is
$I(\boldsymbol{\eta})=\mathbf{X}^{\top}\mathbf{X}$. This matrix is non-
singular if the columns of $\mathbf{X}$ are linearly independent, which holds
when $t_{j}\neq t_{k}$ and $\alpha_{j+p+1}\neq 0$.

Assumption 2 (Convergence of Jacobian function) and Assumption 3 (Mass of
limiting measure). Note that
$\frac{1}{n}\mathbf{\tilde{A}}^{\top}\mathbf{\tilde{A}}$ is given by

$n^{-1}\setcounter{MaxMatrixCols}{11}\begin{pmatrix}n&\sum\limits_{i=1}^{n}x_{i}&\ldots&\sum\limits_{i=1}^{n}x_{i}^{p}&\sum\limits_{i=1}^{n}(x_{i}-t_{1})_{+}^{p}&\ldots&\sum\limits_{i=1}^{n}(x_{i}-t_{\kappa})_{+}^{p}&\sum\limits_{i=1}^{n}(x_{i}-t_{1})_{+}^{p-1}&\ldots&\sum\limits_{i=1}^{n}(x_{i}-t_{\kappa})_{+}^{p-1}&\sum\limits_{i=1}^{n}y_{i}\\\
\sum\limits_{i=1}^{n}x_{i}&\sum\limits_{i=1}^{n}x_{i}x_{i}&\ldots&\sum\limits_{i=1}^{n}x_{i}x_{i}^{p}&\sum\limits_{i=1}^{n}x_{i}(x_{i}-t_{1})_{+}^{p}&\ldots&\sum\limits_{i=1}^{n}x_{i}(x_{i}-t_{\kappa})_{+}^{p}&\sum\limits_{i=1}^{n}x_{i}(x_{i}-t_{1})_{+}^{p-1}&\ldots&\sum\limits_{i=1}^{n}x_{i}(x_{i}-t_{\kappa})_{+}^{p-1}&\sum\limits_{i=1}^{n}x_{i}y_{i}\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
\sum\limits_{i=1}^{n}x_{i}^{p}&\sum\limits_{i=1}^{n}x_{i}x_{i}^{p}&\ldots&\sum\limits_{i=1}^{n}x_{i}^{p}x_{i}^{p}&\sum\limits_{i=1}^{n}x_{i}^{p}(x_{i}-t_{1})_{+}^{p}&\ldots&\sum\limits_{i=1}^{n}x_{i}^{p}(x_{i}-t_{\kappa})_{+}^{p}&\sum\limits_{i=1}^{n}x_{i}^{p}(x_{i}-t_{1})_{+}^{p-1}&\ldots&\sum\limits_{i=1}^{n}x_{i}^{p}(x_{i}-t_{\kappa})_{+}^{p-1}&\sum\limits_{i=1}^{n}x_{i}^{p}y_{i}\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
\sum\limits_{i=1}^{n}(x_{i}-t_{1})_{+}^{p}&\sum\limits_{i=1}^{n}x_{i}(x_{i}-t_{1})_{+}^{p}&\ldots&\sum\limits_{i=1}^{n}x_{i}^{p}(x_{i}-t_{1})_{+}^{p}&\sum\limits_{i=1}^{n}(x_{i}-t_{1})_{+}^{p}(x_{i}-t_{1})_{+}^{p}&\ldots&\sum\limits_{i=1}^{n}(x_{i}-t_{1})_{+}^{p}(x_{i}-t_{\kappa})_{+}^{p}&\sum\limits_{i=1}^{n}(x_{i}-t_{1})^{p}(x_{i}-t_{1})_{+}^{p-1}&\ldots&\sum\limits_{i=1}^{n}(x_{i}-t_{1})^{p}(x_{i}-t_{\kappa})_{+}^{p-1}&\sum\limits_{i=1}^{n}(x_{i}-t_{1})^{p}y_{i}\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
\sum\limits_{i=1}^{n}(x_{i}-t_{\kappa})_{+}^{p}&\sum\limits_{i=1}^{n}x_{i}(x_{i}-t_{\kappa})_{+}^{p}&\ldots&\sum\limits_{i=1}^{n}x_{i}^{p}(x_{i}-t_{\kappa})_{+}^{p}&\sum\limits_{i=1}^{n}(x_{i}-t_{1})_{+}^{p}(x_{i}-t_{\kappa})_{+}^{p}&\ldots&\sum\limits_{i=1}^{n}(x_{i}-t_{\kappa})_{+}^{p}(x_{i}-t_{\kappa})_{+}^{p}&\sum\limits_{i=1}^{n}(x_{i}-t_{1})^{p}(x_{i}-t_{\kappa})_{+}^{p-1}&\ldots&\sum\limits_{i=1}^{n}(x_{i}-t_{\kappa})^{p}(x_{i}-t_{\kappa})_{+}^{p-1}&\sum\limits_{i=1}^{n}(x_{i}-t_{\kappa})^{p}y_{i}\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
\sum\limits_{i=1}^{n}y_{i}&\sum\limits_{i=1}^{n}x_{i}y_{i}^{p}&\ldots&\sum\limits_{i=1}^{n}x_{i}^{p}y_{i}&\sum\limits_{i=1}^{n}(x_{i}-t_{1})_{+}^{p}y_{i}&\ldots&\sum\limits_{i=1}^{n}(x_{i}-t_{\kappa})_{+}^{p}y_{i}&\sum\limits_{i=1}^{n}(x_{i}-t_{1})_{+}^{p-1}y_{i}&\ldots&\sum\limits_{i=1}^{n}(x_{i}-t_{\kappa})_{+}^{p-1}y_{i}&\sum\limits_{i=1}^{n}y_{i}y_{i}\end{pmatrix}$.

Note also that $(x_{i}-t_{k})_{+}^{j}\leq(b-a)$ for $i=1,\ldots,n$,
$k=1,\ldots,\kappa$, and $j=1,\ldots,p$. Since the $(x_{i},y_{i})$ are i.i.d.,
the Uniform Law of Large Numbers implies that

$\pi(\theta)=\prod_{k=1}^{\kappa}|\alpha_{p+k}|\frac{p^{\kappa}}{\sigma}\det(E(\frac{1}{n}\mathbf{\tilde{A}}^{\top}\mathbf{\tilde{A}}))$

where $E(\frac{1}{n}\mathbf{\tilde{A}}^{\top}\mathbf{\tilde{A}})$ denotes the
matrix of (element-wise) expected values.

Assumption 4 (Likelihood splitting). We consider artificial data chosen
according to the following criteria. First, we obtain artificial
$x^{\sharp}_{i}$ by selecting at least two points between consecutive true
knot points such that $|x^{\sharp}_{i}-x^{\sharp}_{j}|>\delta/2$ for $i\neq
j.$ The exact number of artificial data points needed, denoted $m$, depends on
the number of parameters to be estimated, i.e., $m=p+\kappa+2$. This ensures
that we have sufficient artificial data to estimate the $m=p+\kappa+2$
parameters. Then, for each $i=1,\ldots,m$, we set $y^{\sharp}_{i}$ to be any
value in the bounded interval $[-q,q]$.

Let $\tilde{\mathbf{A}}^{\sharp}$ denote the analogue to the matrix
$\tilde{\mathbf{A}}$ for the artificial data and denote the corresponding
Jacobian calculated from this matrix by
$J(\mathbf{y}^{\sharp},\boldsymbol{\theta}).$ By setting $x^{\sharp}_{i}$ on
either side of each knot point, the columns of
$\tilde{\mathbf{A}}^{\sharp^{\top}}\tilde{\mathbf{A}}^{\sharp}$ will be
linearly independent and since $|\alpha_{p+k}|>\xi>0$ for all
$k=1,\ldots,\kappa$, we know $J(\mathbf{y}^{\sharp},\boldsymbol{\theta})$ is
bounded away from $0$.

Let

$p^{2}(\mathbf{y},\boldsymbol{\theta})=p(\mathbf{y}^{\sharp},\boldsymbol{\theta})=\prod_{i=1}^{m}\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\bigg{(}-\frac{1}{2\sigma^{2}}(y^{\sharp}_{i}-N(x^{\sharp}_{i},\mathbf{t})\boldsymbol{\alpha})^{2}\bigg{)}$

and

$p^{1}(\mathbf{y},\boldsymbol{\theta})=\frac{p(\mathbf{y},\boldsymbol{\theta})}{p(\mathbf{y}^{\sharp},\boldsymbol{\theta})}=\frac{\prod_{i=1}^{n}\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\bigg{(}-\frac{1}{2\sigma^{2}}(y_{i}-N(x_{i},\mathbf{t})^{\top}\boldsymbol{\alpha})^{2}\bigg{)}}{\prod_{i=1}^{m}\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\bigg{(}-\frac{1}{2\sigma^{2}}(y^{\sharp}_{i}-N(x^{\sharp}_{i},\mathbf{t})^{\top}\boldsymbol{\alpha})^{2}\bigg{)}}.$
(5.1)

Next, observe that

$\displaystyle
p^{2}(\mathbf{y},\boldsymbol{\theta})J(\mathbf{y},\boldsymbol{\theta})$
$\displaystyle=p(\mathbf{y}^{\sharp},\boldsymbol{\theta})J(\mathbf{y}^{\sharp},\boldsymbol{\theta})\frac{J(\mathbf{y},\boldsymbol{\theta})}{J(\mathbf{y}^{\sharp},\boldsymbol{\theta})}=p(\mathbf{y}^{\sharp},\boldsymbol{\theta})J(\mathbf{y}^{\sharp},\boldsymbol{\theta})\frac{\det\bigg{[}\frac{1}{n}\tilde{\mathbf{A}}^{\top}\tilde{\mathbf{A}}\bigg{]}}{\det\bigg{[}\frac{1}{m}\tilde{\mathbf{A}}^{\sharp^{\top}}\tilde{\mathbf{A}}^{\sharp}\bigg{]}}.$

Note that $\underset{t_{k}\in
T_{\delta}}{\inf}\det\bigg{[}\frac{1}{m}\tilde{\mathbf{A}}^{\sharp^{\top}}\tilde{\mathbf{A}}^{\sharp}\bigg{]}>0$
and $\underset{t_{k}\in
T_{\delta}}{\sup}\det\bigg{[}\frac{1}{s}\tilde{\mathbf{A}}^{\sharp^{\top}}\tilde{\mathbf{A}}^{\sharp}\bigg{]}<C_{1}$
by our selection of the data points. Recall that
$\frac{1}{n}\tilde{\mathbf{A}}^{\top}\tilde{\mathbf{A}}$ has a uniform limit,
which we denoted $E(\frac{1}{n}\mathbf{\tilde{A}}^{\top}\mathbf{\tilde{A}})$,
and that both depend on $\mathbf{t}$. Then, given $\nu>0$,

$P\left(\left\lVert\underset{t_{k}\in
T_{\delta}}{\sup}\det\bigg{[}\frac{1}{n}\tilde{\mathbf{A}}^{\top}\tilde{\mathbf{A}}-E(\frac{1}{n}\mathbf{\tilde{A}}^{\top}\mathbf{\tilde{A}})\right\rVert_{F}<\nu\right)\rightarrow
1,$

where $\left\lVert\cdot\right\rVert_{F}$ denotes the Frobenius norm. Thus, we
can find a constant $C_{2}$ such that,

$P_{\theta_{0}}\Big{(}p^{2}(\mathbf{y},\boldsymbol{\theta})J(\mathbf{y},\boldsymbol{\theta})<C_{2}\cdot
p(\mathbf{y}^{\sharp},\boldsymbol{\theta})J(\mathbf{y}^{\sharp},\boldsymbol{\theta})\Big{)}\rightarrow
1.$

Since $\int
p(\mathbf{y}^{\sharp},\boldsymbol{\theta})J(\mathbf{y}^{\sharp},\boldsymbol{\theta})\,d\boldsymbol{\theta}=1$,
we can define $\gamma(\theta)=C_{2}\cdot
p(\mathbf{y}^{\sharp},\boldsymbol{\theta})J(\mathbf{y}^{\sharp},\boldsymbol{\theta})$.

Assumption 5 (Exponentially consistent tests). The exponential bound is
straightforward to verify for exponential families by Lemma 10.6 in van der
Vaart, (1998), so Assumption 5 is satisfied for $P_{\theta}$.

We now show the bound also holds for $P^{1}_{\theta}$. We can rewrite (5.1) by
separating the first $M$ data points in the numerator as follows:

$p^{1}(\mathbf{y},\boldsymbol{\theta})=\left(\frac{1}{\sqrt{2\pi\sigma^{2}}}\right)^{M-m}\exp\left(\frac{1}{2\sigma^{2}}\left(\sum_{i=1}^{m}(y^{\sharp}_{i}-N(x^{\sharp}_{i},\mathbf{t})^{\top}\boldsymbol{\alpha})^{2}-\sum_{i=1}^{M}(y_{i}-N(x_{i},\mathbf{t})^{\top}\boldsymbol{\alpha})^{2}\right)\right)\\\
\times\prod_{i=M+1}^{n}\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\bigg{(}-\frac{1}{2\sigma^{2}}(y_{i}-N(x_{i},\mathbf{t})^{\top}\boldsymbol{\alpha})^{2}\bigg{)}.$
(5.2)

Note that
$\sum_{i=1}^{m}(y_{i}^{\sharp}-N(x_{i}^{\sharp},\mathbf{t})^{\top}\boldsymbol{\alpha}))^{2}$
and $\sum_{i=1}^{M}(y_{i}-N(x_{i},\mathbf{t})^{\top}\boldsymbol{\alpha})^{2}$
are positive definite quadratic functions of $\boldsymbol{\alpha}$. With
probability $1$, there exists a finite $M\geq m$ that the eigenvalues of
$\sum_{i=1}^{M}(y_{i}-N(x_{i},\mathbf{t})^{\top}\boldsymbol{\alpha})^{2}$ are
larger than the eigenvalues of
$\sum_{i=1}^{m}(y_{i}^{\sharp}-N(x_{i}^{\sharp},\mathbf{t})^{\top}\boldsymbol{\alpha})^{2},$
and thus,
$\sum_{i=1}^{m}(y_{i}^{\sharp}-N(x_{i}^{\sharp},\mathbf{t})^{\top}\boldsymbol{\alpha})^{2}-\sum_{i=1}^{M}(y_{i}-N(x_{i},\mathbf{t})^{\top}\boldsymbol{\alpha})^{2}$
is a negative definite quadratic function of $\boldsymbol{\alpha}$. Therefore,
since the knot points $\boldsymbol{t}$ are in a compact, for almost all y, the
first term (5.2) is bounded by a constant. The second term in (5.2)
corresponds to the original $P_{\theta}$ with exponential bound
$e^{-b(n-M)\left\lVert\boldsymbol{\theta}-\boldsymbol{\theta}_{0}\right\rVert}\leq
e^{-cn\left\lVert\boldsymbol{\theta}-\boldsymbol{\theta}_{0}\right\rVert}$ for
$c\geq\frac{b(n-M)}{n}.$

##  Appendix A Proofs of Lemmas

######  Lemma A.1.

Under Assumptions 1-3, given an arbitrary compact set
$C\subset\mathbb{R}^{d}$,

$P_{n,0}\lhd\rhd P_{n,C}.$

###### Proof.

Suppose $P_{n,\textbf{0}}(A_{n})\rightarrow 0$ for some set $A_{n}$. Theorem
7.2 in van der Vaart, (1998) establishes that for every h,
$P_{n,\textbf{0}}\lhd\rhd P_{n,\textbf{h}}$. Thus, by contiguity,
$P_{n,\textbf{h}}(A_{n})\rightarrow 0$ for any $\textbf{h}\in C$. Note that

$P_{n,C}(A_{n})=\int_{C}P_{n,\textbf{h}}(A_{n})\pi_{n}(\textbf{h})d\textbf{h}.$

Since $|P_{n,\textbf{h}}(A_{n})|\leq 1$ for all h, we can apply DCT to get
$P_{n,C}(A_{n})\rightarrow 0.$

Now suppose $P_{n,C}(A_{n})\rightarrow 0$ for some $A_{n}$. Let
$\textbf{h}_{n}\in C$ such that $P_{n,\textbf{h}_{n}}(A_{n})\leq
P_{n,C}(A_{n})$. Thus, $P_{n,\textbf{h}_{n}}(A_{n})\rightarrow 0$ and it
follows by contiguity $P_{n,0}(A_{n})\rightarrow 0$.

∎

######  Lemma A.2.

Suppose Assumptions 1-3 hold. Let $\textbf{h},\textbf{g}\in\textbf{H}_{n}$ be
distinct local alternatives as defined in Section 3.1. Then,
$\dfrac{p_{n,\textbf{h}}(\textbf{y})\phi_{n}(\textbf{g})}{p_{n,\textbf{g}}(\textbf{y})\phi_{n}(\textbf{h})}\rightarrow
1$ in $P_{n,\textbf{h}}$, where $P_{n,\textbf{h}}$, $p_{n,\textbf{h}}$, and
$\phi_{n}(\textbf{h})$ are as defined in Section 3.1 (analogously for g).

###### Proof.

By the LAN property that follows from Assumption 1,

$\displaystyle\dfrac{p_{n,\textbf{h}}(\textbf{y})\phi_{n}(\textbf{g})}{p_{n,\textbf{g}}(\textbf{y})\phi_{n}(\textbf{h})}$
$\displaystyle=\dfrac{\dfrac{p_{n,\textbf{h}}(\textbf{y})}{p_{n,\textbf{0}}(\textbf{y})}}{\dfrac{p_{n,\textbf{g}}(\textbf{y})}{p_{n,\textbf{0}}(\textbf{y})}}\cdot\dfrac{\phi_{n}(\textbf{g})}{\phi_{n}(\textbf{h})}$
$\displaystyle=\dfrac{\exp\left(\dfrac{1}{\sqrt{n}}\sum\limits_{i=1}^{n}\textbf{h}^{\top}\dot{\ell}_{\boldsymbol{\theta}_{0}}(y_{i})-\dfrac{1}{2}\textbf{h}^{\top}I_{\boldsymbol{\theta}_{0}}\textbf{h}+o_{p_{n,\textbf{0}}}(1)\right)}{\exp\left(\dfrac{1}{\sqrt{n}}\sum\limits_{i=1}^{n}\textbf{g}^{\top}\dot{\ell}_{\boldsymbol{\theta}_{0}}(y_{i})-\dfrac{1}{2}\textbf{g}^{\top}I_{\boldsymbol{\theta}_{0}}\textbf{g}+o_{p_{n,\textbf{0}}}(1)\right)}\cdot\dfrac{\exp\left(-\dfrac{1}{2}\left(\textbf{g}-\Delta_{n,\boldsymbol{\theta}_{0}}\right)^{\top}I_{\boldsymbol{\theta}_{0}}(\textbf{g}-\Delta_{n,\boldsymbol{\theta}_{0}})\right)}{\exp\left(-\dfrac{1}{2}\left(\textbf{h}-\Delta_{n,\boldsymbol{\theta}_{0}}\right)^{\top}I_{\boldsymbol{\theta}_{0}}(\textbf{h}-\Delta_{n,\boldsymbol{\theta}_{0}})\right)}$
$\displaystyle=\dfrac{\exp\left(\dfrac{1}{\sqrt{n}}\sum\limits_{i=1}^{n}\textbf{h}^{\top}\dot{\ell}_{\boldsymbol{\theta}_{0}}(y_{i})-\dfrac{1}{2}\textbf{h}^{\top}I_{\boldsymbol{\theta}_{0}}\textbf{h}+o_{p_{n,\textbf{0}}}(1)\right)}{\exp\left(\dfrac{1}{\sqrt{n}}\sum\limits_{i=1}^{n}\textbf{g}^{\top}\dot{\ell}_{\boldsymbol{\theta}_{0}}(y_{i})-\dfrac{1}{2}g^{\top}I_{\boldsymbol{\theta}_{0}}\textbf{g}+o_{p_{n,\textbf{0}}}(1)\right)}\cdot\dfrac{\exp\left(-\dfrac{1}{2}\textbf{g}^{\top}I_{\boldsymbol{\theta}_{0}}\textbf{g}\right)}{\exp\left(-\dfrac{1}{2}\textbf{h}^{\top}I_{\boldsymbol{\theta}_{0}}\textbf{h}\right)}\cdot\dfrac{\exp\left(\dfrac{1}{\sqrt{n}}\textbf{g}^{\top}\dot{\ell}_{\boldsymbol{\theta}_{0}}(y_{i})\right)}{\exp\left(\dfrac{1}{\sqrt{n}}\textbf{h}^{\top}\dot{\ell}_{\boldsymbol{\theta}_{0}}(y_{i})\right)}$
$\displaystyle=\dfrac{e^{o_{p_{n,\textbf{0}}}(1)}}{e^{o_{p_{n,\textbf{0}}}(1)}}\stackrel{{\scriptstyle
P_{n,\textbf{0}}}}{{\rightarrow}}1.$

Since $P_{n,\textbf{0}}\lhd\rhd P_{n,\textbf{h}}$, it follows that
$\dfrac{p_{n,\textbf{h}}(\textbf{y})\phi_{n}(\textbf{g})}{p_{n,\textbf{g}}(\textbf{y})\phi_{n}(\textbf{h})}\stackrel{{\scriptstyle
P_{n,\textbf{h}}}}{{\rightarrow}}1$. ∎

######  Lemma A.3.

Suppose Assumptions 1-5 hold and let $\textbf{h}\in\textbf{H}_{n}$ denote the
local parameter defined by
$\textbf{h}=\sqrt{n}(\boldsymbol{\theta}-\boldsymbol{\theta}_{0})$. Then,

$\frac{\bigg{|}\int_{\textbf{H}_{n}}p_{n,\textbf{h}}J_{n}(\textbf{h})d\textbf{h}-\int_{\textbf{H}_{n}}p_{n,\textbf{h}}\pi_{n}(\textbf{h})d\textbf{h}\bigg{|}}{\int_{\textbf{H}_{n}}p_{n,\textbf{h}}J_{n}(\textbf{h})d\textbf{h}}\stackrel{{\scriptstyle
P_{n,\textbf{0}}}}{{\rightarrow}}0,$

where $P_{n,\textbf{0}}$, $p_{n,\textbf{h}}$, $J_{n}(\textbf{h})$, and
$\pi_{n}(h)$ are as defined in Section 3.1.

###### Proof.

Let $C$ be a ball around $0$ of fixed radius $M$ and $C_{n}$ a sequence of
balls with radius $M_{n}\rightarrow\infty$. Let $D>0$ and define
$C_{n}^{c}=\\{\textbf{h}:\left\lVert\textbf{h}\right\rVert>D\sqrt{n}\\}$.
Observe that,

$\displaystyle\begin{split}\bigg{|}\int_{\textbf{H}_{n}}p_{n,\textbf{h}}J_{n}(\textbf{h})d\textbf{h}-\int_{\textbf{H}_{n}}p_{n,\textbf{h}}\pi_{n}(\textbf{h})d\textbf{h}\bigg{|}=\Bigg{|}\bigg{(}&\int_{\left\lVert\textbf{h}\right\rVert\leq
D\sqrt{n}}{}p_{n,\textbf{h}}J_{n}(\textbf{h})d\textbf{h}+\int_{\left\lVert\textbf{h}\right\rVert>D\sqrt{n}}p_{n,\textbf{h}}J_{n}(\textbf{h})d\textbf{h}\bigg{)}\\\
&-\bigg{(}\int_{\left\lVert\textbf{h}\right\rVert\leq
D\sqrt{n}}p_{n,\textbf{h}}\pi_{n}(\textbf{h})d\textbf{h}+\int_{\left\lVert\textbf{h}\right\rVert>D\sqrt{n}}p_{n,\textbf{h}}\pi_{n}(\textbf{h})d\textbf{h}\bigg{)}\Bigg{|}\end{split}$
$\displaystyle\begin{split}=\Bigg{|}\bigg{(}\int_{\left\lVert\textbf{h}\right\rVert\leq
D\sqrt{n}}{}&p_{n,\textbf{h}}J_{n}(\textbf{h})d\textbf{h}-\int_{\left\lVert\textbf{h}\right\rVert\leq
D\sqrt{n}}p_{n,\textbf{h}}\pi_{n}(\textbf{h})d\textbf{h}\bigg{)}\\\
&+\bigg{(}\int_{\left\lVert\textbf{h}\right\rVert>D\sqrt{n}}p_{n,\textbf{h}}J_{n}(\textbf{h})d\textbf{h}-\int_{\left\lVert\textbf{h}\right\rVert>D\sqrt{n}}p_{n,\textbf{h}}\pi_{n}(\textbf{h})d\textbf{h}\bigg{)}\Bigg{|}\end{split}$
$\displaystyle\begin{split}\leq\bigg{|}\int_{\left\lVert\textbf{h}\right\rVert\leq
D\sqrt{n}}{}&p_{n,\textbf{h}}(J_{n}(\textbf{h})-\pi_{n}(\textbf{h}))d\textbf{h}\bigg{|}\\\
&+\bigg{|}\int_{\left\lVert\textbf{h}\right\rVert>D\sqrt{n}}p_{n,\textbf{h}}J_{n}(\textbf{h})d\textbf{h}-\int_{\left\lVert\textbf{h}\right\rVert>D\sqrt{n}}p_{n,\textbf{h}}\pi_{n}(\textbf{h})d\textbf{h}\bigg{|}\end{split}$
$\displaystyle\begin{split}\leq\sup\limits_{\left\lVert\textbf{h}\right\rVert\leq
D\sqrt{n}}{}&\frac{\lvert
J_{n}(\textbf{h})-\pi_{n}(\textbf{h})\rvert}{J_{n}(\textbf{h})}\int_{\left\lVert\textbf{h}\right\rVert\leq
D\sqrt{n}}p_{n,\textbf{h}}J_{n}(\textbf{h})d\textbf{h}\\\
&+\int_{\left\lVert\textbf{h}\right\rVert>D\sqrt{n}}p_{n,\textbf{h}}\big{|}J_{n}(\textbf{h})-\pi_{n}(\textbf{h})\big{|}d\textbf{h}.\end{split}$

Since $\frac{\int_{\left\lVert\textbf{h}\right\rVert\leq
D\sqrt{n}}p_{n,\textbf{h}}J_{n}(\textbf{h})d\textbf{h}}{\int_{\textbf{H}_{n}}p_{n,\textbf{h}}J_{n}(\textbf{h})d\textbf{h}}\leq
1$ and $\sup\limits_{\left\lVert\textbf{h}\right\rVert\leq
D\sqrt{n}}\frac{\lvert
J_{n}(\textbf{h})-\pi_{n}(\textbf{h})\rvert}{J_{n}(\textbf{h})}\stackrel{{\scriptstyle
P_{n,\textbf{0}}}}{{\rightarrow}}0$ by Assumption 2, the first term in the
last line above converges to 0 in $P_{n,\textbf{0}}$ probability. Recall from
the previous lemma that convergence in $P_{n,\textbf{0}}$ is implied by
convergence in $P_{n,C}$ probability by contiguity.

To show that the second term in the last line above also converges to 0 in
$P_{n,\textbf{0}}$ probability, we show convergence in $P_{n,C}$ probability
by the stronger $L^{1}$ convergence,

$E_{n,C}\left(\frac{\int_{\left\lVert\textbf{h}\right\rVert>D\sqrt{n}}p_{n,\textbf{h}}\big{|}J_{n}(\textbf{h})-\pi_{n}(\textbf{h})\big{|}d\textbf{h}}{\int_{\textbf{H}_{n}}p_{n,\textbf{h}}J_{n}(\textbf{h})d\textbf{h}}\right)\rightarrow
0.$

We follow similar steps as in the proof of the main result. Let $\delta_{n}$
be a sequence of exponentially consistent tests as in Assumption 5. By
contiguity, $E_{P_{n,C}}(\delta_{n})\rightarrow 0$. Consequently,

$E_{n,C}\left(\frac{\int_{\left\lVert\textbf{h}\right\rVert>D\sqrt{n}}p_{n,\textbf{h}}\big{|}J_{n}(\textbf{h})-\pi_{n}(\textbf{h})\big{|}d\textbf{h}}{\int_{\textbf{H}_{n}}p_{n,\textbf{h}}J_{n}(\textbf{h})d\textbf{h}}(1-\delta_{n}(\textbf{Y}_{n}))\right)+o(1).$

Then, defining $\pi_{n}(\textbf{h}):=0$ for $\textbf{h}\in C_{n}^{c}$,

$\displaystyle E_{n,C}$
$\displaystyle\left(\frac{\int_{\left\lVert\textbf{h}\right\rVert>D\sqrt{n}}p_{n,\textbf{h}}\big{|}J_{n}(\textbf{h})-\pi_{n}(\textbf{h})\big{|}d\textbf{h}}{\int_{\textbf{H}_{n}}p_{n,\textbf{h}}J_{n}(\textbf{h})d\textbf{h}}(1-\delta_{n}(\textbf{Y}_{n}))\right)$
$\displaystyle=\int\int_{C}\int_{C_{n}^{c}}\dfrac{p_{n,\textbf{h}}(\textbf{y})J_{n}(\textbf{h})dh}{\int
p_{n,\textbf{q}}(\textbf{y})J_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}d\mu_{n}(\textbf{y})$
$\displaystyle=\int\int_{C}\int_{C_{n}^{c}}\dfrac{p_{n,\textbf{h}}(\textbf{y})J_{n}(\textbf{h})d\textbf{h}}{\int
p_{n,\textbf{q}}(\textbf{y})J_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{g}}(\textbf{y})\pi_{n}(\textbf{g})d\textbf{g}}{\pi_{n}(C)}\frac{J_{n}(\textbf{g})}{J_{n}(\textbf{g})}d\mu_{n}(\textbf{y})$
$\displaystyle=\int\int_{C_{n}^{c}}\int_{C}\dfrac{p_{n,\textbf{g}}(\textbf{y})J_{n}(\textbf{g})d\textbf{g}}{\int
p_{n,\textbf{q}}(\textbf{y})J_{n}(\textbf{q})d\textbf{q}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{h}}(\textbf{y})J_{n}(\textbf{h})d\textbf{h}}{\pi_{n}(C)}d\mu_{n}(\textbf{y})$
$\displaystyle\leq\int\int_{C_{n}^{c}}(1-\delta_{n}(\textbf{y}))\dfrac{p_{n,\textbf{h}}(\textbf{y})J_{n}(\textbf{h})d\textbf{h}}{\pi_{n}(C)}d\mu_{n}(\textbf{y})$
$\displaystyle\leq\frac{\gamma_{n}(C_{n}^{c})}{\pi_{n}(C)}\int\int_{C_{n}^{c}}(1-\delta_{n}(\textbf{y}))\dfrac{p^{1}_{n,\textbf{h}}(\textbf{y})\gamma_{n}(\textbf{h})d\textbf{h}}{\gamma_{n}(C_{n}^{c})}d\mu_{n}(\textbf{y})$
$\displaystyle=\dfrac{1}{\pi_{n}(C))}\int_{C_{n}^{c}}E^{1}_{n,\textbf{h}}(1-\delta_{n}(\textbf{y}))\gamma_{n}(\textbf{h})d\textbf{h}$
$\displaystyle\leq c_{2}\sqrt{n}^{d}e^{-c_{4}D^{2}n}.$

The third equality follows from
$\frac{\pi_{n}(\textbf{g})}{J_{n}(\textbf{g})}\rightarrow 1$ for
$\textbf{g}\in C$. The first inequality holds since
$r_{\textbf{H}_{n}|\textbf{Y}_{n}}(C)\leq 1$. The next inequality follows from
Assumption 4, and the final inequality from Assumption 5. ∎

##  Appendix B Tables

| $\theta$
---|---
$n$ | 0.01 | 0.03 | 0.05 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50
Generalized Fiducial 95% CI |
1 | 0.298 (0.781) | 0.603 (0.793) | 0.753 (0.802) | 0.895 ( 0.82) | 0.966 (0.841) | 0.983 (0.852) | 0.990 (0.858) | 0.992 ( 0.86)
2 | 0.389 ( 0.75) | 0.710 (0.758) | 0.826 (0.766) | 0.925 (0.783) | 0.973 ( 0.81) | 0.989 (0.828) | 0.992 (0.839) | 0.992 (0.843)
5 | 0.557 (0.589) | 0.802 (0.605) | 0.875 (0.619) | 0.936 ( 0.65) | 0.970 (0.698) | 0.980 (0.731) | 0.984 ( 0.75) | 0.986 (0.757)
10 | 0.689 (0.394) | 0.864 (0.417) | 0.908 (0.438) | 0.946 (0.483) | 0.963 (0.553) | 0.969 ( 0.6) | 0.969 (0.628) | 0.969 (0.636)
20 | 0.808 (0.229) | 0.911 (0.256) | 0.935 ( 0.28) | 0.955 ( 0.33) | 0.956 (0.404) | 0.957 (0.453) | 0.957 (0.482) | 0.957 (0.491)
50 | 0.902 (0.107) | 0.946 (0.133) | 0.953 (0.155) | 0.954 (0.198) | 0.953 (0.254) | 0.954 (0.287) | 0.954 (0.304) | 0.955 ( 0.31)
100 | 0.928 (0.0611) | 0.951 (0.0843) | 0.955 (0.102) | 0.954 (0.136) | 0.953 (0.175) | 0.951 (0.197) | 0.949 (0.209) | 0.951 (0.214)
200 | 0.949 (0.0373) | 0.952 (0.0563) | 0.950 (0.0699) | 0.952 (0.0932) | 0.950 (0.121) | 0.950 (0.137) | 0.953 (0.145) | 0.949 (0.148)
500 | 0.955 (0.0213) | 0.954 (0.0341) | 0.951 (0.0424) | 0.950 (0.0565) | 0.953 (0.0742) | 0.950 (0.0841) | 0.952 (0.0897) | 0.953 (0.0916)
1000 | 0.950 (0.0144) | 0.951 (0.0233) | 0.951 (0.0291) | 0.950 (0.0392) | 0.952 (0.0516) | 0.949 (0.0588) | 0.950 (0.0626) | 0.949 (0.0639)
Modified Generalized Fiducial 95% CI
1 | 0.950 (0.798) | 0.950 (0.814) | 0.950 ( 0.83) | 0.950 ( 0.87) | 0.950 (0.922) | 0.950 (0.941) | 0.950 (0.948) | 0.950 (0.951)
2 | 0.968 (0.816) | 0.977 ( 0.83) | 0.980 (0.844) | 0.983 ( 0.88) | 0.986 (0.935) | 0.986 (0.958) | 0.986 (0.968) | 0.986 (0.971)
5 | 0.955 (0.656) | 0.973 (0.682) | 0.980 (0.706) | 0.986 (0.764) | 0.989 (0.862) | 0.989 ( 0.91) | 0.989 (0.933) | 0.989 (0.939)
10 | 0.946 (0.426) | 0.973 (0.457) | 0.980 (0.487) | 0.986 (0.562) | 0.986 (0.709) | 0.984 (0.804) | 0.982 ( 0.85) | 0.981 (0.865)
20 | 0.944 (0.237) | 0.974 (0.268) | 0.980 (0.296) | 0.983 (0.364) | 0.976 (0.499) | 0.974 (0.612) | 0.972 (0.679) | 0.972 (0.702)
50 | 0.953 (0.109) | 0.973 (0.138) | 0.974 (0.163) | 0.972 (0.215) | 0.966 ( 0.29) | 0.966 (0.342) | 0.963 (0.376) | 0.964 (0.388)
100 | 0.951 (0.062) | 0.965 (0.0866) | 0.966 (0.106) | 0.963 (0.143) | 0.961 (0.188) | 0.959 (0.212) | 0.957 (0.225) | 0.957 (0.229)
200 | 0.959 (0.0377) | 0.959 (0.0574) | 0.956 (0.0715) | 0.957 (0.0956) | 0.955 (0.124) | 0.954 ( 0.14) | 0.957 (0.149) | 0.954 (0.151)
500 | 0.957 (0.0214) | 0.956 (0.0344) | 0.954 (0.0427) | 0.952 (0.057) | 0.955 (0.0748) | 0.952 (0.0849) | 0.953 (0.0905) | 0.954 (0.0924)
1000 | 0.952 (0.0144) | 0.952 (0.0233) | 0.952 (0.0292) | 0.951 (0.0393) | 0.953 (0.0518) | 0.950 (0.059) | 0.951 (0.0629) | 0.949 (0.0641)
Flat Bayesian 95% CI
1 | 0.208 (0.921) | 0.672 (0.922) | 0.909 (0.923) | 0.999 (0.924) | 1.000 (0.926) | 1.000 (0.927) | 1.000 (0.928) | 1.000 (0.928)
2 | 0.347 (0.851) | 0.736 (0.858) | 0.879 (0.863) | 0.975 (0.872) | 0.998 (0.885) | 1.000 (0.892) | 1.000 (0.896) | 1.000 (0.897)
5 | 0.552 (0.627) | 0.817 (0.644) | 0.894 (0.659) | 0.952 (0.691) | 0.981 (0.738) | 0.989 ( 0.77) | 0.991 (0.788) | 0.992 (0.794)
10 | 0.694 (0.408) | 0.873 (0.432) | 0.915 (0.454) | 0.954 ( 0.5) | 0.971 (0.572) | 0.975 (0.621) | 0.976 ( 0.65) | 0.976 (0.659)
20 | 0.813 (0.233) | 0.915 ( 0.26) | 0.940 (0.284) | 0.959 (0.336) | 0.961 (0.412) | 0.960 (0.463) | 0.962 (0.492) | 0.961 (0.502)
50 | 0.904 (0.108) | 0.947 (0.134) | 0.954 (0.156) | 0.956 ( 0.2) | 0.955 (0.257) | 0.956 (0.289) | 0.956 (0.307) | 0.956 (0.313)
100 | 0.929 (0.0613) | 0.952 (0.0846) | 0.955 (0.103) | 0.955 (0.136) | 0.954 (0.176) | 0.952 (0.198) | 0.950 ( 0.21) | 0.951 (0.215)
200 | 0.950 (0.0374) | 0.953 (0.0564) | 0.951 (0.0701) | 0.952 (0.0934) | 0.951 (0.121) | 0.951 (0.137) | 0.953 (0.145) | 0.950 (0.148)
500 | 0.955 (0.0213) | 0.954 (0.0342) | 0.951 (0.0424) | 0.950 (0.0566) | 0.953 (0.0742) | 0.951 (0.0842) | 0.952 (0.0898) | 0.953 (0.0917)
1000 | 0.950 (0.0144) | 0.951 (0.0233) | 0.951 (0.0291) | 0.950 (0.0392) | 0.953 (0.0516) | 0.949 (0.0588) | 0.950 (0.0626) | 0.949 (0.0639)
Jeffrey’s Bayesian 95% CI
1 | 0.961 ( 0.98) | 0.994 (0.983) | 0.998 (0.985) | 0.999 (0.988) | 1.000 ( 0.99) | 1.000 (0.991) | 1.000 (0.991) | 1.000 (0.991)
2 | 0.939 ( 0.89) | 0.981 (0.904) | 0.989 (0.915) | 0.995 (0.935) | 0.998 (0.958) | 0.998 (0.969) | 0.999 (0.974) | 0.998 (0.976)
5 | 0.942 (0.606) | 0.971 (0.634) | 0.979 (0.658) | 0.984 ( 0.71) | 0.985 (0.788) | 0.984 (0.843) | 0.983 (0.877) | 0.983 (0.888)
10 | 0.960 (0.365) | 0.977 (0.398) | 0.981 (0.427) | 0.980 (0.492) | 0.972 (0.598) | 0.966 (0.675) | 0.963 (0.721) | 0.961 (0.737)
20 | 0.971 (0.199) | 0.979 (0.232) | 0.978 (0.262) | 0.970 (0.326) | 0.959 (0.427) | 0.954 (0.491) | 0.952 (0.528) | 0.951 ( 0.54)
50 | 0.979 (0.0924) | 0.973 (0.123) | 0.965 ( 0.15) | 0.955 (0.203) | 0.952 (0.264) | 0.953 (0.297) | 0.952 (0.314) | 0.953 (0.319)
100 | 0.979 (0.0544) | 0.964 (0.0818) | 0.956 (0.103) | 0.952 (0.139) | 0.952 (0.178) | 0.950 ( 0.2) | 0.949 (0.212) | 0.950 (0.216)
200 | 0.971 (0.0348) | 0.951 (0.0569) | 0.946 (0.0713) | 0.951 (0.0943) | 0.950 (0.121) | 0.950 (0.138) | 0.953 (0.146) | 0.949 (0.148)
500 | 0.956 (0.0212) | 0.951 (0.0346) | 0.950 (0.0427) | 0.951 (0.0567) | 0.954 (0.0743) | 0.950 (0.0843) | 0.952 (0.0899) | 0.953 (0.0918)
1000 | 0.948 (0.0146) | 0.950 (0.0233) | 0.951 (0.0291) | 0.950 (0.0392) | 0.951 (0.0517) | 0.950 (0.0588) | 0.950 (0.0627) | 0.949 (0.0639)
Table 1. Coverage and mean length of credible intervals for $\theta$.

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