# A Characterization of Most(More) Powerful Test Statistics with Simple
Nonparametric Applications

Albert Vexler
Department of Biostatistics, The State University of New York at Buffalo,
Buffalo, NY
and
Alan D. Hutson
Department of Biostatistics and Bioinformatics,
Roswell Park Comprehensive Cancer Center, Buffalo, NY

###### Abstract

Data-driven most powerful tests are statistical hypothesis decision-making
tools that deliver the greatest power against a fixed null hypothesis among
all corresponding data-based tests of a given size. When the underlying data
distributions are known, the likelihood ratio principle can be applied to
conduct most powerful tests. Reversing this notion, we consider the following
questions. (a) Assuming a test statistic, say $T$, is given, how can we
transform $T$ to improve the power of the test? (b) Can $T$ be used to
generate the most powerful test? (c) How does one compare test statistics with
respect to an attribute of the desired most powerful decision-making
procedure? To examine these questions, we propose one-to-one mapping of the
term ”most powerful” to the distribution properties of a given test statistic
via matching characterization. This form of characterization has practical
applicability and aligns well with the general principle of sufficiency.
Findings indicate that to improve a given test, we can employ relevant
ancillary statistics that do not have changes in their distributions with
respect to tested hypotheses. As an example, the present method is illustrated
by modifying the usual t-test under nonparametric settings. Numerical studies
based on generated data and a real-data set confirm that the proposed approach
can be useful in practice.

Keywords: Ancillary statistic; Likelihood ratio; Most powerful test;
Nonparametric test; Sufficiency; t-test; Test for median.

##  1 Introduction

Methods for developing and examining data-based decision-making mechanisms
have been widely established in both theoretical and experimental statistical
frameworks. The common approach for evaluating modern data-based testing
algorithms follows the standards and foundations formulated nearly a century
ago. In this context, for an extensive review and associated examples we refer
the reader to Lehmann and Romano (2005). The criteria for which statistical
tests are commonly competed against each other uses the following
prescription: 1) type I error (TIE) rates of considered tests are fixed at the
same level, say $\alpha$; and 2) power levels of the tests are compared. This
classical principle was largely created and advocated by J. Neyman and E. S.
Pearson in a series of substantive papers published during 1928$-$1938 (e.g.,
Lehmann, 1993). In this framework, the likelihood methodology is associated
with the likelihood ratio concept, which allows for the development of
powerful statistical inference tools in decision-making tasks.

In view of this, the likelihood ratio principle can be employed across a wide
range of decision-making problems, although likelihood ratio tests are not
completely specified in many practical applications. Cases exist in which
estimated parametric likelihood ratio statistics can have different
formulations depending upon the underlying schemes of estimations. Other
times, the relevant likelihood functions may be quite complicated when, for
example, the observations belong to correlated longitudinal data subject to
some type of missing data mechanism. There are other nonparametric scenarios
that limit our ability to write corresponding likelihood ratio statistics. A
systematic study of the inherent properties of likelihood ratios is necessary
for proposing policies to advise on procedures for constructing, modifying,
and/or selecting test statistics.

Without loss of generality, and in order to simplify the explanations of the
main aim of this paper, we state the following formal notations. Assume we
observe the underlying data $D$ with the goal of testing a simple null
hypothesis $H_{0}$ against its simple alternative $H_{1}$. To this end, we let
$T=T(D)$ denote a real valued one-dimensional statistic based on $D$ that
supports rejection of $H_{0}$, if $T(D)>C$, where $C$ is a fixed test-
threshold. We write $\Pr_{k}$ and $\text{E}_{k}$ to denote the probability and
expectation under $H_{k}$, $k\in\\{0,1\\}$, respectively. Throughout most of
this paper, we suppose that the probability distributions $\Pr_{0}$, $\Pr_{1}$
of $D$ are absolutely continuous with respect to a given sigma finite measure
$\tau$ defined over $\Upsilon$, where $\Upsilon$ is an additive class of sets
in a space, say $\mathbb{X}$, over which $D$ is distributed. Then, we have
non-negative probability density functions $f_{0}$ and $f_{1}$ with respect to
$\tau$ that satisfy

$\displaystyle{\Pr}_{k}\left(D\in\upsilon\right)$ $\displaystyle=$
$\displaystyle\int_{\upsilon}f_{k}(x)d\tau(x),\quad\upsilon\in\Upsilon,\quad
k\in\\{0,1\\}.$

Note that $f_{0}$ and $f_{1}$ need not belong to the same parametric family of
distributions. Define the probability density functions $f^{T}_{0}$ and
$f^{T}_{1}$ of $T$ such that

$\displaystyle{\Pr}_{k}\left\\{T(D)\leq t\right\\}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{X}}I\left(T(x)\leq
t\right)f_{k}(x)d\tau(x)\,=\,\int_{{{\mathbb{R}}^{1}}}I\left(u\leq
t\right)f^{T}_{k}(u)du,$

where $I(.)$ means the indicator function, $H_{k}$ is assumed to be true, and
$k\in\\{0,1\\}$. In this framework, the likelihood ratio
$\Lambda=\Lambda(D)=f_{1}(D)/f_{0}(D)$ is the most powerful (MP) test
statistic, if $f_{0}(x)>0$ and $f_{1}(x)>0$, for all $x\in\mathbb{X}$. In
scenarios when the random observable data $D$ is multidimensional, $f_{0}$ and
$f_{1}$ are generalized joint probability density with respect to $\tau$
(e.g., Lehmann, 1950).

For the sake of simplicity, it will be assumed that in a case when a
researcher plans to employ a test statistic $S(D)=L\left(\Lambda(D)\right)$,
where $L(u)$ is a monotonically increasing function with the inverse function
$W(L(u))=u$, we suppose the test statistic $T(D)=W(S(D))$ is in use and $T(D)$
is MP.

The starting point of our study is associated with the following property of
$\Lambda$ that can be found in Vexler and Hutson (2018) and is included for
the sake of completeness.

######  Proposition 1.1.

The likelihood ratio statistic $\Lambda$ satisfies
$f_{1}^{\Lambda}(u)=u{\rm\;}f_{0}^{\Lambda}(u),$ for all $u>0$.

The proof is deferred to the online supplementary materials.

An interesting observation is that the likelihood ratio
$f_{1}^{\Lambda}/f_{0}^{\Lambda}$, based on the likelihood ratio $\Lambda$ is
itself, forms the likelihood ratio $\Lambda$, i.e.,
$f_{1}^{\Lambda}(\Lambda)/f_{0}^{\Lambda}(\Lambda)=\Lambda$. Consider the
situation when a value of a statistic or a single data point, say $X$, is
observed. The best transformation of $X$ for making a decision with respect to
$H_{0}$ against $H_{1}$ is the ratio $f_{1}^{X}(X)/f_{0}^{X}(X)$. In the case
$X=\Lambda$, the observed statistic cannot be improved, and the transformation
$f_{1}^{X}(X)/f_{0}^{X}(X)$ is invertible at $X=\Lambda$, since the likelihood
ratio $\Lambda$ is a root of the equation $f_{1}^{X}(X)/f_{0}^{X}(X)=X$.

Let us for a moment assume that we could improve Proposition 1.1 by including
the idiom ”if and only if” in its statement, thereby asserting: a statistic
$T$ has a likelihood ratio form if and only if
$f_{1}^{T}(u)=u{\rm\;}f_{0}^{T}(u)$. Then, since $\Lambda$ is the MP test
statistic, having a given test statistic $T$, we will try to modify the
structure of $T$, minimizing the distance between $f_{1}^{T}(u)$ and
$u{\rm\;}f_{0}^{T}(u)$, for at least some values of $u$. In this framework,
comparing two test statistics, say $T$ and $B$, we can select $T$, if, for
example, $\max_{u}\left|f_{1}^{T}(u)-u{\rm\;}f_{0}^{T}(u)\right|$
$\leq\max_{u}\left|f_{1}^{B}(u)-u{\rm\;}f_{0}^{B}(u)\right|$. Note that in
nonparametric settings we can approximate and/or estimate distribution
functions of test statistics in many scenarios. Unfortunately, the simple
statement of Proposition 1.1 cannot be used to characterize MP test
statistics. For example, when $D=\left(X_{1},X_{2}\right)$, where $X_{1}$ is
from a normal distribution with $\text{E}_{0}\left(X_{1}\right)=0$,
$\text{E}_{1}\left(X_{1}\right)=\mu$, and
$\text{E}_{0}\left(X^{2}_{1}\right)=\text{E}_{1}\left(X_{1}-\mu\right)^{2}$,
the statistic $T=f_{1}\left(X_{1}\right)/f_{0}\left(X_{1}\right)$ satisfies
$f_{1}^{T}(u)=u{\rm\;}f_{0}^{T}(u),$ whereas the MP test statistic is
$f_{1}\left(X_{1},X_{2}\right)/f_{0}\left(X_{1},X_{2}\right)$. That is, to
characterize MP tests, Proposition 1.1 needs to be modified.

######  Remark 1.1.

In this article, we avoid using the term ”Uniformly Most Powerful” in order to
be able to study cases when parameters do not play an essential role in
testing procedures as well as to consider situations where uniformly most
powerful tests do not exist, e.g., nonparametric testing statements. For
example, let $H_{0}$ infer that observations are from a standard normal
distribution vs. that the observations follow a standard logistic
distribution, under $H_{1}$. In this case, we have the likelihood ratio MP
test, whereas invoking the term ”uniformly most powerful” can be misleading.
In Section 3, the MP concept has a hypothetical context to which we aim to
approach when we develop test procedures. (See also item (iii) in Remark 2.2
and note (d) presented in Section 6, in this aspect.)

The goal of the present article is two-fold: (1) through an extension of
Proposition 1.1, we describe a way to characterize MP tests, and (2) we
exemplify the usefulness of the theoretical characterization of MP tests via
corrections of test statistics that can be easily applied in practice. Under
this framework, Section 2 considers one-to-one mapping of distribution
properties of test statistics to their ability to be most powerful. The proven
characterization is shown to be consistent with the principle of sufficiency
in certain decision problems extensively evaluated by Bahadur (1955). In
Section 3, to exemplify potential uses of the proposed MP characterization, we
apply the theoretical concepts shown in Section 2 toward demonstrating an
efficient principle of improving commonly used test procedures via employing
relevant ancillary statistics. Ancillary statistics have distributions that do
not depend on the competing hypotheses. However, we show that ancillary
statistics can make significant contributions to inference about the
hypotheses of interest. For example, although it seems to be very difficult to
compete against the well-known one sample t-test for the mean, we assert that
a simple modification of the t-test statistic can increase its power. This can
be accomplished by accounting for the effect of population skewness on the
distribution of the sample mean. Section 3 demonstrates modifications of
testing procedures that can be implemented under nonparametric assumptions
when there are no MP decision-making mechanisms. Then, in Section 4, we show
experimental evaluations that confirm high efficiency of the presented schemes
in various situations. Furthermore, as described in Section 5, when used to
analyze data from a biomarker study associated with myocardial infarction
disease, the method proposed in Section 3 for one-sample testing about the
median is more sensitive as compared with known methods to detect asymmetry in
the data distributions. Finally, this paper is concluded with a discussion in
Section 6.

##  2 Characterization and Sufficiency

In order to gain some insight into the purpose of this section, the following
illustrative example is offered.

###  2.1 One Sample Test of the Mean

Let $X_{1},X_{2}$ be a random sample from a normal population with mean $\mu$
and variance $\sigma^{2}=1$. We consider testing $H_{0}:\,\mu=0$ versus
$H_{1}:\,\mu=\delta$, where $\delta$ is a fixed value. We present this simple
toy example to illustrate our results shown in Section 2.2, not to offer a
contender to the usual t-test. In this example, the statistic
$\bar{X}=(X_{1}+X_{2})/2$ can be used for MP testing. However, one may feel
that, for example, the statistic $A=wX_{1}+(1-w)X_{2}$ could be reasonable for
assessing the competing hypotheses $H_{0}$ and $H_{1}$, for some $w\in[0,1]$,
$w\neq 0.5$. The vector $[\bar{X},A]^{\top}$ has a bivariate normal density
function, with $\mathrm{E}\left(\bar{X}\right)=\mathrm{E}\left(A\right)=\mu$,
$\mathrm{var}\left(\bar{X}\right)=\mathrm{cov}\left(\bar{X},A\right)=1/2$, and
$\mathrm{var}(A)=w^{2}+(1-w)^{2}$. Then, by defining a joint density function
of $\left(X_{1},X_{2}\right)$-based statistics $A_{1},A_{2}$ in the form
$f_{\mu}^{A_{1},A_{2}}(u,v)$, it is easy to observe that the ratio
$f_{\mu=\delta}^{\bar{X},A}(u,v)/f_{\mu=0}^{\bar{X},A}(u,v)$ does not depend
on $v$, where $v$ is an argument of the joint density $f_{\mu}^{\bar{X},A}$
relating to $A$’s component. In particular, this means that after surveying
the two data points $\bar{X}$ and $A$, we can improve the
$\left(\bar{X},A\right)$-based decision-making mechanism by creating the MP
statistic for testing $H_{0}$ vs. $H_{1}$ in the likelihood ratio form
$f_{\mu=\delta}^{\bar{X},A}(\bar{X},A)/f_{\mu=0}^{\bar{X},A}(\bar{X},A)$,
which only requires the computation of $\bar{X}$. Thus, one might pose the
question: can the observation above be generalized to extend Proposition 1.1?
In this case, it seems to be reasonable that to provide an essential property
of the MP concept, relationships with other $D$-based statistics should be
taken into account.

The second aspect of our approach is to characterize a scenario where, say,
statistic $A_{1}$ is more preferable in the construction of a test than
statistic $A_{2}$. In this context, as will be seen later, $A_{1}$ is superior
to $A_{2}$, if the ratio
$f_{\mu=\delta}^{A_{1},A_{2}}(u,v)/f_{\mu=0}^{A_{1},A_{2}}(u,v)$ does not
depend on $v$. To exemplify the benefits of this rule, let us pretend that it
is unknown that $\bar{X}$ is the best statistic in this subsection such that
we can consider the following task. The problem then is to indicate a value of
$a\in[0,1]$ in the statistic $T=aX_{1}+(1-a)X_{2}$ such that $T$ outperforms
$A=wX_{1}+(1-w)X_{2}$, for all $w\in[0,1]$. Since the density
$f_{\mu}^{T,A}(u,v)$ is bivariate normal, simple algebra shows that
$f_{\mu=\delta}^{A_{1},A_{2}}(u,v)/f_{\mu=0}^{A_{1},A_{2}}(u,v)$ is not a
function of $v$, if $a^{2}+(1-a)^{2}=aw+(1-a)(1-w)$. Then, the solution is
$a=0.5$. In reality, we do know that $T$, with $a=0.5$, is the best statistic
in this framework. This example illustrates our point.

###  2.2 Theoretical Results

In this section, the main results are provided in Propositions 2.1$-$2.5 below
that establish the characterization of MP tests. The proofs of Propositions
2.1$-$2.5 are included in the online supplementary materials for completeness
and contain comments that augment the description of the obtained results.
Proposition 2.5 revisits the characterization of MP tests in the light of
sufficiency.

To extend Proposition 1.1, we define a joint density function of statistics
$A=A(D)$ and $B=B(D)$ in the form $f^{A,B}_{k}(u,v)$, provided that $H_{k}$ is
true, $k\in\\{0,1\\}$. Then, the likelihood ratio test statistic, $\Lambda$,
has the following property.

######  Proposition 2.1.

For any statistic $A=A(D)$, we have
$f_{1}^{\Lambda,A}(u,v)=u\,f_{0}^{\Lambda,A}(u,v)$, for all $u\geq 0$ and
$v\in{{\mathbb{R}}^{1}}$.

Proposition 2.1 emerges as a generalization of Proposition 1.1, since
$f_{1}^{\Lambda,A}(u,v)=u{\rm\;}f_{0}^{\Lambda,A}(u,v)$ yields

$\displaystyle f_{1}^{\Lambda}(u)=\int f_{1}^{\Lambda,A}(u,v)dv=u\,\int
f_{0}^{\Lambda,A}(u,v)dv=u\,f_{0}^{\Lambda}(u).$

In the following claim, it is shown that Proposition 2.1 can be augmented to
imply a necessary and sufficient condition on test statistics distributions to
present MP decision-making techniques. Define $C$ to be a test threshold.

######  Proposition 2.2.

Assume a statistic for testing, $T=T(D)$, satisfies ${\Pr}_{k}(T\geq 0)=1$,
$k\in\\{0,1\\}$, one rejects $H_{0}$ when $T>C$. The test statistic $T$ is MP
if and only if (iff) $f_{1}^{T,A}(u,v)=u\,f_{0}^{T,A}(u,v)$, for any statistic
$A=A(D)$ and all $u\geq 0$, $v\in{{\mathbb{R}}^{1}}$.

Note that the condition ”$T$ is strictly non-negative” is employed in Fisher
and Robbins (2019), where a monotonic logarithmic transformation of $T$, a
test statistic, may improve the power of the $T$-based test when the
corresponding TIE rate is asymptotically controlled at $\alpha$. However, the
requirement $T\geq 0$ is not critical, because if we evaluate a test
statistic, say $G=G(D)$, that can be negative, then a monotonic transformation
$T=g(G)\geq 0$ can assist in this case.

We can remark that, in scenarios where density functions of test statistics do
not exist, the arguments employed in the proof of Proposition 2.2 can be
applied to obtain the next statement.

######  Proposition 2.3.

The test statistic $T(D)>0$ is MP iff
$\mathrm{E}_{1}\left\\{g(D)\right\\}=\mathrm{E}_{0}\left\\{g(D)T(D)\right\\}$,
for every function $g\in[0,1]$ of $D$.

######  Remark 2.1.

Since
$\mathrm{E}_{1}\left\\{g(D)\right\\}=\mathrm{E}_{0}\left\\{g(D)\Lambda(D)\right\\}$,
the condition
$\mathrm{E}_{1}\left\\{g(D)\right\\}=\mathrm{E}_{0}\left\\{g(D)T(D)\right\\}$
implies $\mathrm{E}_{0}\left[\left\\{T(D)-\Lambda(D)\right\\}g(D)\right]=0$,
for every $g\in[0,1]$, which means, with probability one under $f_{0}$, we
have $T=\Lambda$. Note also that, in Proposition 2.3, we can use $g(D)$,
satisfying
$\mathrm{E}\left\\{g(D)\right\\}^{m}=\mathrm{E}\left\\{g(D)\right\\}$, for all
$m>0$.

The scheme used in the proof of Step (2) of Proposition 2.2 yields the
following result.

######  Proposition 2.4.

A statistic $T_{1}\geq 0$ is more powerful than a statistic $T_{2}$, if the
ratio $f_{1}^{T_{1},T_{2}}(u,v)/f_{0}^{T_{1},T_{2}}(u,v)$ $=u$, for all $u\geq
0$, $v\in{{\mathbb{R}}^{1}}$.

It is interesting to note that, by virtue of Propositions 1.1 and 2.2, for any
$D$-based statistic $A=A(D)$, we have $f_{1}^{T,A}(u,v)={f_{0}^{T,A}(u,v)}\,u$
and then

$f_{1}^{T,A}(u,v)$ $={f_{0}^{T,A}(u,v)}f_{1}^{T}(u)/f_{0}^{T}(u),$ if $T$ is
MP. That is to say, $f_{1}^{A|T}(u,v)={\rm\;}f_{0}^{A|T}(u,v)$, where the
notation $f_{k}^{A|T}$ means a conditional density function of $A$ given $T$
under $H_{k}$, $k\in\\{0,1\\}$. In this case, when $A$ is independent of $T$,
we obtain $f_{1}^{A}(v)=f_{0}^{A}(v)$, and then $A=A(D)$ cannot discriminate
the hypotheses. We can write that $A=A(D)$ is ancillary, meaning
$f_{1}^{A}=f_{0}^{A}$. This motivates us to associate the results above with
the principle of sufficiency.

According to Bahadur (1955), in the considered framework, we can call $T=T(D)$
to be a sufficient test statistic, if
$f_{1}^{A|T}(u,v)={\rm\;}f_{0}^{A|T}(u,v)$, for each $A=A(D)$ and all $u\geq
0$, $v\in{{\mathbb{R}}^{1}}$. In this context, the statements mentioned above
assert the next result.

######  Proposition 2.5.

The following claims are equivalent:

  1. (i)i

$T=T(D)$ is sufficient and $f_{1}^{T}(u)/f_{0}^{T}(u)=u$;

  2. (ii)i

$T$ is a MP statistic for testing the competing hypotheses $H_{0}$ and
$H_{1}$.

Proposition 2.5 presents an argument to the reasonableness of making a
statistical inference based solely on the corresponding sufficient statistics.

######  Remark 2.2.

We can note the following facts:

  1. (i)i

Kagan and Shepp (2005) have exemplified a sufficiency paradox, when an
insufficient statistic preserves the Fisher information.

  2. (ii)i

In order to extend Proposition 2.5, statements related to a wide spectrum of
Basu’s theorem-type results (e.g., Ghosh, 2002) can be employed, in certain
situations. To the best of our knowledge, there are no direct applications of
Basu’s theorem to the questions considered in the present article.

  3. (iii)i

In Bayesian styles of testing (e.g., Johnson, 2013), Proposition 1.1 can be
extended to treat Bayes Factors, see, e.g., Proposition 5 of Vexler (2021), in
this context. Then, Propositions 2.1 and 2.2 can be easily modified to
establish integrated MP tests with respect to incorporated prior information
(Vexler et al., 2010).

##  3 Applications

Section 2 carries out the relatively general underlying theoretical framework
for the MP characterization concept. In this section, we outline three
applications of the proposed MP characterization principle, by modifying well-
accepted statistical tests in an easy to implement manner. It is hoped that
the proposed MP characterization can provide different benefits for
developing, improving, and comparing decision-making algorithms in statistical
practice.

A common problem arising in statistical inference is the need for methods to
modify a given test statistic in order to improve the performance of
controlling the TIE rate and power of the corresponding decision-making
scheme. For example, the accuracy of asymptotic approximations for the null
distribution of a test statistic may be increased by incorporating Bartlett
correction type mechanisms or/and location adjustment techniques. In this
context, we refer the reader to the following examples: Hall and La Scala
(1990), for modifying nonparametric empirical likelihood ratios; Chen (1995),
for different transformations of the t-test statistics assessing the mean of
asymmetrical distributions. Recently, Fisher and Robbins (2019) proposed to
use a logarithmic transformation to obtain a potential increase in power of
the transformed statistic-based test.

This section demonstrates use of the considered MP principle, following the
simple idea outlined below. Suppose that we have a reasonable test statistic
$T_{o}$ and we wish to improve $T_{o}$ to be in a form, say $T_{N}$,
approximately satisfying the claim
$f_{1}^{T_{N},A}(u,v)=u\,f_{0}^{T_{N},A}(u,v)$, for any statistic $A=A(D)$ and
all $u\geq 0$, $v\in{{\mathbb{R}}^{1}}$. Given that in general nonparametric
settings there are no MP tests, it would be attractive to reach the MP
property $f_{1}^{T_{N},A}(u,v)=u\,f_{0}^{T_{N},A}(u,v)$ at least for some
statistic $A$, especially for some ancillary statistic. Informally speaking,
by having $A$ with $f_{1}^{A}=f_{0}^{A}$, we can remove the influence of $A$
from $T_{o}$ to create $T_{N}$ such that the ratio
$f_{1}^{T_{N},A}(u,v)/f_{0}^{T_{N},A}(u,v)$ is a function of $u$ only. In this
case, Proposition 2.4 could insure that $T_{N}$ outperforms $A$. This can be
achieved via an independence between $T_{N}$ and $A$ that is exemplified in
Sections 3.3, 3.4, and 3.5 in detail.

Through the following examples, we aim to show our approach in an intuitive
manner.

###  3.1 Examples of the Use of Ancillary Statistics

We begin with displaying the toy examples below that illustrate our key idea.

Let independent data points $X_{1}$ and $X_{2}$ be observed; when it is
assumed that $X_{i}\sim N\left(\mu,\sigma^{2}_{i}\right)$, $i\in[1,2]$ and
$\sigma^{2}_{1}\neq\sigma^{2}_{2}$ are known. One can use the simple statistic
$T=0.5(X_{1}+X_{2})$ to test $H_{0}:$ $\mu=0$ against $H_{1}:$ $\mu>0$.
Easily, one can confirm that $X_{1}-X_{2}$ is an ancillary statistic. We now
consider a mechanism for transforming $T$ and making a modified test statistic
that is independent of $X_{1}-X_{2}$. Define $T_{N}=T+\gamma(X_{1}-X_{2})$, a
transformed version of $T$, where $\gamma$ is a root of the equation
$\mathrm{cov}\left(T_{N},X_{1}-X_{2}\right)=0$. Then, we obtain
$\gamma=0.5\left(\sigma^{2}_{2}-\sigma^{2}_{1}\right)/\left(\sigma^{2}_{2}+\sigma^{2}_{1}\right)$.
Thus, the derived statistic

$\displaystyle T_{N}$ $\displaystyle=$ $\displaystyle
T+\frac{\sigma^{2}_{2}-\sigma^{2}_{1}}{2\left(\sigma^{2}_{2}+\sigma^{2}_{1}\right)}\left(X_{1}-X_{2}\right)=\frac{\sigma^{2}_{2}X_{1}+\sigma^{2}_{1}X_{2}}{\sigma^{2}_{2}+\sigma^{2}_{1}}\,\,=\,\,\frac{X_{1}/\sigma^{2}_{1}+X_{2}/\sigma^{2}_{2}}{1/\sigma^{2}_{1}+1/\sigma^{2}_{2}}$

is certainly a successful transformation of the initial statistic $T$, which
presents the MP test statistic. For instance, we denote the power

$\displaystyle
P(a)={\Pr}_{\mu=5}\left\\{T+a(X_{1}-X_{2})>C(a)\right\\},\,C(a):\,{\Pr}_{\mu=0}\left\\{T+a(X_{1}-X_{2})>C(a)\right\\}=0.05,$

when $\sigma_{1}=1$ and $\sigma_{2}=4$. Figure 1(a) depicts the function
$P(a)-P(0)$, the difference between the power levels of the
$\left(T+a(X_{1}-X_{2})\right)$-based test and those of the $T$-based test at
$\alpha=0.05$, plotted against the function
$Cov(a)=\mathrm{cov}\left(T+a(X_{1}-X_{2}),X_{1}-X_{2}\right)$, for
$a\in[-0.01,0.9]$. As expected, the function $P(a)-P(0)$ reaches its maximum
when $Cov(a)=0$. Moreover, it turns out that we do not need much accuracy in
approximating the equation $Cov(a)=0$ to outperform the $T$-based test when we
use the modified test statistic $T+a(X_{1}-X_{2})$. Then, intuitively, we can
suppose that a transformed test statistic could include estimated elements
while still providing good power characteristics for its decision-making
algorithm.

In various situations, we shall not exclude the possibility that there exists
more than one ancillary statistic for a given testing statement. Let us
exemplify such case, assuming we observe $X_{1}$, $X_{2}$, and $X_{3}$ from
the normal distributions $N\left(\mu,\sigma^{2}_{1}\right)$,
$N\left(\mu,\sigma^{2}_{2}\right)$, and $N\left(\mu,\sigma^{2}_{3}\right)$,
respectively, where $\sigma^{2}_{i}$, $i\in[1,2,3]$, are known. Suppose we are
interested in testing $H_{0}:$ $\mu=0$ vs. $H_{1}:$ $\mu>0$. The statistic to
be modified is $T=(X_{1}+X_{2}+X_{3})/3$. The observation $X_{1}-X_{2}$ is an
ancillary statistic with respect to $\mu$. Define
$T_{N}=T+\gamma(X_{1}-X_{2})$ with
$\gamma=\left(\sigma^{2}_{2}-\sigma^{2}_{1}\right)\left(\sigma^{2}_{2}+\sigma^{2}_{1}\right)^{-1}/3$,
thereby obtaining that
$\mathrm{cov}\left(T+\gamma(X_{1}-X_{2}),X_{1}-X_{2}\right)=0$. Then, it is
clear that $T_{N}$ is somewhat better than $T$, but
$T_{O}=\sum_{i=1}^{3}\left(X_{i}/\sigma^{2}_{i}\right)/\sum_{i=1}^{2}\left(1/\sigma^{2}_{i}\right)$
is superior to $T_{N}$ in the terms of this example. Define the powers
$P_{T}(\mu)={\Pr}_{\mu}\left\\{T>C^{T}_{0.05}\right\\}$,
$P_{T_{N}}(\mu)={\Pr}_{\mu}\left\\{T_{N}>C^{T_{N}}_{0.05}\right\\}$, and
$P_{T_{O}}(\mu)={\Pr}_{\mu}\left\\{T_{O}>C^{T_{O}}_{0.05}\right\\}$, where the
test thresholds $C^{T}_{0.05}$, $C^{T_{N}}_{0.05}$, and $C^{T_{O}}_{0.05}$
satisfy
${\Pr}_{\mu=0}\left\\{T>C^{T}_{0.05}\right\\}={\Pr}_{\mu=0}\left\\{T_{N}>C^{T_{N}}_{0.05}\right\\}={\Pr}_{\mu=0}\left\\{T_{O}>C^{T_{O}}_{0.05}\right\\}=0.05.$
Figure 1(b) exemplifies the behavior of the functions $P_{T}(\mu)$,
$P_{T_{N}}(\mu)$, and $P_{T_{O}}(\mu)$, when $\sigma_{1}=1$, $\sigma_{2}=4$,
and $\sigma_{3}=3$.

Figure 1: Graphical evaluations related to the examples shown in Section 3.1.
Panel (a) plots $P(a)-P(0)$, the power of the
$\left(T+a(X_{1}-X_{2})\right)$-based test minus the power of the $T$-based
test at the $\alpha=0.05$ level, against the covariance
$Cov(a)=\mathrm{cov}\left(T+a(X_{1}-X_{2}),X_{1}-X_{2}\right)$, for
$a\in[-0.01,0.9]$, where $T=0.5(X_{1}+X_{2})$, $X_{1}\sim N(\mu,1)$,
$X_{2}\sim N(\mu,4^{2})$, $\mathrm{E}_{0}(X_{i})=0$,
$\mathrm{E}_{1}(X_{i})=5$, $i\in[1,2]$. Panel (b) plots the powers
$P_{T_{O}}(\mu)={\Pr}_{\mu}\left\\{T_{O}>C^{T_{O}}_{\alpha}\right\\}$ (solid
line), $P_{T_{N}}(\mu)={\Pr}_{\mu}\left\\{T_{N}>C^{T_{N}}_{\alpha}\right\\}$
(longdashed line), and
$P_{T}(\mu)={\Pr}_{\mu}\left\\{T>C^{T}_{\alpha}\right\\}$ (dotted line) at the
$\alpha=0.05$ level, where $T=(X_{1}+X_{2}+X_{3})/3$,
$T_{N}=T+\gamma(X_{1}-X_{2})$,
$T_{O}=\left(X_{1}/\sigma^{2}_{1}+X_{2}/\sigma^{2}_{2}+X_{3}/\sigma^{2}_{3}\right)/\left(1/\sigma^{2}_{1}+1/\sigma^{2}_{2}+1/\sigma^{2}_{3}\right)$,
$\gamma=\left(\sigma^{2}_{2}-\sigma^{2}_{1}\right)\left(\sigma^{2}_{2}+\sigma^{2}_{1}\right)^{-1}/3$,
$X_{1}\sim N(\mu,1)$, $X_{2}\sim N(\mu,4^{2})$, and $X_{3}\sim N(\mu,3^{2})$,
for $\mu\in[0,5]$.

Thus, to improve a given test statistic, say $T$, we can suggest that one pays
attention to a relevant ancillary statistic, say $A$, modifying $T$ to be
independent (or approximately independent) of $A$.

Note that, although the concept of ancillarity asserts that ancillary
statistics do not provide information about the parameters of interest,
different roles of ancillary statistics in parametric estimation have been
dealt with extensively in the literature. In this context, for an extensive
review, we refer the reader to Ghosh et al. (2010). For example, assume we
observe the vectors $[X_{i},Y_{i}]^{\top}$, $i\in[1,\ldots,n]$, from a
bivariate normal distribution with $\mathrm{E}(X_{1})=\mathrm{E}(Y_{1})=0$,
$\mathrm{var}(X_{1})=\mathrm{var}(Y_{1})=1$, and
$\mathrm{corr}(X_{1},Y_{1})=\rho$, where $\rho\in(-1,1)$ is unknown. The
statistics $U_{1}=\sum_{i=1}^{n}X^{2}_{i}$ and $U_{2}=\sum_{i=1}^{n}Y^{2}_{i}$
are ancillary. According to Ghosh et al. (2010), to define unbiased estimators
of $\rho$, it can be recommended to use the statistics
$\sum_{i=1}^{n}X_{i}Y_{i}/U_{j}$, $j\in[1,1]$. As another example, when
ancillary statistics are applied, we outline a case of so-called Monte Carlo
swindles, simulation based methods that allow small numbers of generated
samples to produce statistical accuracy at the level one would expect from
much larger numbers of generated samples. Boos and Hughes-Oliver (1998)
discussed the following procedure. To estimate the variance of the sample
median $M$ of a normally distributed sample $X_{1},\ldots,X_{n}$, the Monte
Carlo swindle approach estimates $\mathrm{var}\left(M-\bar{X}\right)$ (instead
of $\mathrm{var}\left(M\right)$) by using the $N$ Monte Carlo samples of
$X_{1},\ldots,X_{n}$ and then
$\mathrm{var}\left(\bar{X}\right)=\mathrm{var}\left(X_{1}\right)/n$ is added
to obtain an efficient estimate of $\mathrm{var}\left(M\right)$. In order to
justify this framework, we employ that the statistic
$V=(X_{1}-\bar{X},\ldots,X_{n}-\bar{X})$ is ancillary. Now, since $\bar{X}$ is
complete sufficient, $\bar{X}$ and $V$ are independent by Basu’s theorem.
Then, $\bar{X}$ is independent of the sample median of $V$. Therefore,

$\displaystyle\mathrm{var}\left(M\right)\,=\,\mathrm{var}\left(M-\bar{X}+\bar{X}\right)\,=\,\mathrm{var}\left(M-\bar{X}\right)+\mathrm{var}\left(\bar{X}\right).$

It is clear that, when $X_{1}$ has a normal distribution, the contribution
from $\mathrm{var}\left(X_{1}\right)/n$ to $\mathrm{var}\left(M\right)$ is
much larger than the contribution from $\mathrm{var}\left(M-\bar{X}\right)$,
where the component $\mathrm{var}\left(M-\bar{X}\right)$ is proposed to be
estimated by simulation. This limits the error in estimation by simulation to
a small part of $\mathrm{var}\left(M\right)$ (for details, see Boos and
Hughes-Oliver, 1998).

###  3.2 Theoretical Support

The point of view mentioned above can be supported by the following results.
Assume we have a test statistic $Y=Y(D)$, and the ratio
$L^{Y}(u)=f^{Y}_{1}(u)/f^{Y}_{0}(u)$ is a monotonically increasing function
that has an inverse function, say $W(u)$. In this scenario, $Y$ can be
transformed into the form $Y_{N}=L^{Y}(Y)$, thereby implying that

$\displaystyle f^{Y_{N}}_{k}(u)\,=\,\frac{d}{du}{\Pr}_{k}\left\\{L^{Y}(Y)\leq
u\right\\}\,=\,\frac{d}{du}{\Pr}_{k}\left\\{Y\leq
W(u)\right\\}\,=\,f_{k}^{Y}\left(W(u)\right)\frac{d}{du}W(u),\,k\in\\{0,1\\}.$

This means that the likelihood ratio

$\displaystyle
f^{Y_{N}}_{1}\left(Y_{N}\right)/f^{Y_{N}}_{0}\left(Y_{N}\right)\,=\,f_{1}^{Y}\left(W(Y_{N})\right)/f_{0}^{Y}\left(W(Y_{N})\right)\,=\,f_{1}^{Y}\left(Y\right)/f_{0}^{Y}\left(Y\right)\,=\,Y_{N}.$

(See Proposition 1.1, in this context.) Then, we state the next proposition.
Let a statistic $A=A(D)$ satisfy $f_{1}^{A}=f_{0}^{A}$. Suppose we have the
decision-making procedure based on a statistic $T=T(D)$, and we can modify $T$
to $T_{N}$ to achieve $T_{N}$ and $A$ as independent terms under $H_{0}$ and
$H_{1}$, when $T=\psi\left(T_{N},A\right)$, for a bivariate function $\psi$.
We conclude that:

######  Proposition 3.1.

The $T_{N}$-based test considered above is superior to that based on $T$, if
the ratio $L^{T_{N}}(u)=f^{T_{N}}_{1}(u)/f^{T_{N}}_{0}(u)$ is a monotone
function.

The proof is deferred to the online supplementary materials.

Note that Proposition 3.1 gives some insight into the connection between the
power of statistical tests and ancillarity, concepts that seem to be
unrelated, since ancillary statistics cannot solely discriminate the competing
hypotheses $H_{0}$ and $H_{1}$.

Proposition 3.1 depicts the rationale for modifying the following well-known
test statistics.

###  3.3 One Sample t-Test for the Mean

Assume we observe independent and identically distributed (i.i.d.) data points
$X_{1},X_{2},\ldots,X_{n}$ that provide
$D=\left\\{X_{1},\ldots,X_{n}\right\\}$. For testing the hypothesis $H_{0}:$
$\mu=0$ vs. $H_{1}:$ $\mu>0$, where $\mu=\text{E}X_{1}$, the well-accepted
statistic is $T_{o}=n^{0.5}\bar{X}/\sigma$, where
$\bar{X}=\sum_{i=1}^{n}X_{i}/n$, and $\sigma^{2}=\text{var}(X_{1})$.

In this testing statement, it seems that the statistic
$S^{2}=\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}/(n-1)$, the sample
variance, is approximately ancillary with respect to $\mu$. Note also that
$T_{o}$ and $S^{2}$ are independent, when $X_{1}\sim
N\left(\mu,\sigma^{2}\right)$ ($T_{o}$ is MP, in this case). Then, we denote
the statistic $T_{N}(\gamma)=T_{o}+\gamma S^{2}$ and derive a value of
$\gamma$, say $\gamma_{0}$, that insures
$\text{cov}\left(T_{N}(\gamma_{0}),S^{2}\right)=0$. The statistic
$T_{N}(\gamma_{0})$ is a basic ingredient of the modified test statistic we
will propose. To this end, we define
$\mu_{k}=\text{E}\left(X_{1}-\mu\right)^{k},\,k=3,4$, and employ the results
from O’Neill (2014) in order to obtain

$\displaystyle\gamma_{0}$ $\displaystyle=$
$\displaystyle\sigma^{-1}\mu_{3}n^{0.5}/\left(\sigma^{4}\frac{n-3}{n-1}-\mu_{4}\right)\,\approx\,-\sigma^{-1}\mu_{3}n^{0.5}/\text{var}\left\\{\left(X_{1}-\mu\right)^{2}\right\\}.$

The additional argument for using $T_{N}(\gamma_{0})$ in a test for $H_{0}$
vs. $H_{1}$ can be explained in the following simple fashion. It is clear that
the stated testing problem can be treated in the context of a confidence
interval estimation of $\mu$. Thus, there is a relationship between the
quality of testing $H_{0}:$ $\mu=0$ and the variance of an estimator of $\mu$
involved in corresponding decision-making schemes. (For example, the t-test,
$T_{o}$, uses $\bar{X}$ to estimate $\mu$.) For the sake of simplicity,
consider $\tilde{T}_{N}(\gamma)=\bar{X}+\gamma\left(S^{2}-\sigma^{2}\right)$
that satisfies $\mathrm{E}\tilde{T}_{N}(\gamma)=\mu$ and
$\mathrm{var}\left\\{\tilde{T}_{N}(0)\right\\}$
$=\mathrm{var}\left(\bar{X}\right)$. To find a value of $\gamma$ that
minimizes $\mathrm{var}\left\\{\tilde{T}_{N}(\gamma)\right\\}$, we can solve
the equation
$d\left[\mathrm{var}\left\\{\tilde{T}_{N}(\gamma)\right\\}\right]/d\gamma=0$,
where it is assumed we can write

$\displaystyle\frac{d}{d\gamma}\mathrm{var}\left\\{\tilde{T}_{N}(\gamma)\right\\}$
$\displaystyle=$
$\displaystyle\frac{d}{d\gamma}\mathrm{E}\left\\{\bar{X}+\gamma\left(S^{2}-\sigma^{2}\right)-\mu\right\\}^{2}=\mathrm{E}\frac{d}{d\gamma}\left\\{\bar{X}+\gamma\left(S^{2}-\sigma^{2}\right)-\mu\right\\}^{2}$
$\displaystyle=$ $\displaystyle
2\mathrm{E}\left\\{\bar{X}+\gamma\left(S^{2}-\sigma^{2}\right)-\mu\right\\}\left(S^{2}-\sigma^{2}\right).$

Then, the root

$\gamma=-\mathrm{cov}\left(\bar{X},S^{2}\right)/\mathrm{var}\left(S^{2}\right)=\mu_{3}/\left\\{\sigma^{4}\left(n-3\right)/\left(n-1\right)-\mu_{4}\right\\}\approx-\mu_{3}/\mathrm{var}\left\\{\left(X_{1}-\mu\right)^{2}\right\\}$

minimizes $\mathrm{var}\left\\{\tilde{T}_{N}(\gamma)\right\\}$ and is
$\gamma_{0}\sigma/n^{0.5}$, where $\gamma_{0}$ implies
$\mathrm{cov}\left(T_{N}(\gamma_{0}),S^{2}\right)=0$. That is,
$\mathrm{var}\left\\{\tilde{T}_{N}\left(\gamma_{0}\sigma/n^{0.5}\right)\right\\}$
$\leq$ $\mathrm{var}\left(\bar{X}\right)$. The statistic $T_{N}(\gamma_{0})$
includes $\bar{X}$ multiplied by $n^{0.5}/\sigma$. This confirms that the
statistic $T_{N}(\gamma_{0})$ can be somewhat more powerful than $T_{o}$, in
the terms of testing $H_{0}$ vs. $H_{1}$, for various scenarios of $X_{1}$’s
distributions.

Finally, we standardize the test statistic $T_{N}(\gamma_{0})$ to be able to
control its TIE rate, denoting the $\alpha$ level decision-making rule: the
null hypothesis $H_{0}$ is rejected if

$T_{N}=\hat{\Delta}^{-0.5}\left[T_{o}-\frac{\hat{\mu}_{3}n^{0.5}}{\sigma\,\hat{\text{var}}\left\\{\left(X_{1}-\mu\right)^{2}\right\\}}\left(S^{2}-\sigma^{2}\right)\right]>z_{\alpha},$

where $\hat{\mu}_{3}=n^{-1}\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{3}$ and
$\hat{\text{var}}\left\\{\left(X_{1}-\mu\right)^{2}\right\\}$
$=n^{-1}\sum_{i=1}^{n}\left\\{\left(X_{i}-\bar{X}\right)^{2}-\sigma^{4}\right\\}^{2}$
estimate $\mu_{3}$ and
$\text{var}\left\\{\left(X_{1}-\mu\right)^{2}\right\\}$, respectively;

$\hat{\Delta}=1-S^{-2}\hat{\mu}_{3}^{2}\left[n^{-1}\sum_{i=1}^{n}\left\\{\left(X_{i}-\bar{X}\right)^{2}-S^{4}\right\\}^{2}\right]^{-1}$

is the sample estimator of $\Delta=\text{var}\left(T_{N}(\gamma_{0})\right)$;
and the threshold $z_{\alpha}$ satisfies $\Pr(Z>z_{\alpha})=\alpha$ with
$Z\sim N(0,1)$. It can be interesting to rewrite $T_{N}$ in the form
$T_{N}=n^{0.5}\bar{Y}/\sigma$, where
$Y_{i}=\hat{\Delta}^{-0.5}\left[X_{i}-\hat{\mu}_{3}\left\\{\left(X_{i}-\bar{X}\right)^{2}n/(n-1)-\sigma^{2}\right\\}/\hat{\text{var}}\left\\{\left(X_{1}-\mu\right)^{2}\right\\}\right]$.
The statistic $T_{N}$ is asymptotically $N(0,1)$-distributed, under $H_{0}$.
Certainly, in the case of $X_{1}\sim N(\mu,\sigma^{2})$, meaning $T_{o}$ is
MP, we have $\mu_{3}=0$. In the form $T_{N}$, an adjustment for the skewness
of the underlying data is in effect.

Note that, in the transformation of the test statistic shown above, we achieve
uncorrelatedness between $T_{N}$ and $S^{2}$, thus simplifying the development
of the nonparametric procedure. It is clear that the equality
$\text{cov}\left(T_{N},S^{2}\right)=0$ is essential to the asymptotic
independence between $T_{N}$ and $S^{2}$ (e.g., Ghosh et al., 2021, pp.
181–206).

Section 4 uses extensive Monte Carlo evaluations to demonstrate an efficiency
of the statistic $T_{N}$ for testing $H_{0}$ against various alternatives.

The testing procedures based on $T_{o}$ and $T_{N}$ require $\sigma$ to be
known. This restriction can be overcome, for example by using bootstrap type
strategies, Bayesian techniques and/or p-value-based methods introduced by
Bayarri and Berger (2000). In this article, we only note that there are
practical applications in which it is reasonable to assume $\sigma$ is known,
e.g., Maity and Sherman (2006), Boos and Hughes-Oliver (1998, Section 3.2), as
well as Johnson (2013, p. 1729). In many biostatistical studies, biomarkers
values are scaled in such a way that their variance $\sigma^{2}=1$. We also
remark that developments of simple test statistics improving the t-test,
$T_{o}$, can be of a theoretical interest.

###  3.4 One Sample Test for the Median

A sub-problem related to comparisons between mean and quantile effects can be
considered as follows: Let $X_{1},X_{2}\ldots,X_{n}$ be continuous i.i.d.
observations with $\mathrm{E}X_{1}=0$. We are interested in testing the
hypothesis $H_{0}:$ $\nu=0$ vs. $H_{1}:$ $\nu>0$, where $\nu$ denotes the
median of $X_{1}$. This statement of the problem can be found in various
practical applications related to testing linear regression residuals as being
symmetric, and pre-and post-placebo paired comparison of biomarker
measurements as well as, for example, when researchers investigate data
associated with radioactivity detection in drinking water, where the
population mean is known; see Section 4 in Semkow et al. (2019).

To test for $H_{0}$, it is reasonable to use a statistic in the form

$T_{o}=2n^{0.5}X_{(n/2)}f\left(X_{(n/2)}\right),$

where $X_{(n/2)}$ is the sample estimator of $\nu$ based on the order
statistics $X_{(1)}<X_{(2)}<\cdots<X_{(n)}$, and $1/f\left(X_{(n/2)}\right)$
is a measure of scale, with $f(u)$ being the density function of $X_{1}$. The
statistic $A=n^{0.5}\bar{X}/\sigma$ can be selected as approximately ancillary
with respect to $\nu$, since $\mathrm{E}X_{1}=0$. The known facts we use are:
(a) the statistics $T_{o}$ and $A$ have an asymptotic bivariate normal
distribution with the parameters shown in Ferguson (1998); and (b) if
$X_{1}\sim N(0,1)$ then Basu’s theorem asserts that $\bar{X}$ and
$X_{(n/2)}-\bar{X}$ are independent, since $\bar{X}$ is a complete sufficient
statistic and $X_{(n/2)}-\bar{X}$ is ancillary. That is to say, in a similar
manner to the development shown in Section 3.3, we can improve $T_{o}$ by
focusing on the statistic

$T=\left\\{\sigma
T_{o}/\mathrm{E}\left|X_{1}-\nu\right|-A\right\\}\left\\{\sigma^{2}\left(\mathrm{E}\left|X_{1}-\nu\right|\right)^{-2}-1\right\\}^{-0.5}$

that satisfies $\mathrm{var}\left(T\right)\to 1$ and
$\mathrm{cov}\left(T,A\right)\to 0$, as $n\to\infty$. Thus, the test we
propose is as follows: to reject $H_{0}$, if

$T_{N}=\left\\{2n^{0.5}X_{(n/2)}\hat{f}\left(X_{(n/2)}\right)\,S/\hat{w}-n^{0.5}\bar{X}/S\right\\}\left\\{S^{2}\hat{w}^{-2}-1\right\\}^{-0.5}\,\,>\,\,z_{\alpha},$

where $S^{2}=\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}/(n-1)$,
$\hat{w}=\sum_{i=1}^{n}\left|X_{i}-X_{(n/2)}\right|/n$, and $\hat{f}$ means a
kernel estimator of the density function $f$. In order to estimate
$f\left(X_{(n/2)}\right)$, we suggest employment of the R-command (R
Development Core Team, 2012):

$\mathtt{density(X,from=median(X),to=median(X))[[2]][[1]]}$

It is clear that, for two-sided testing $H_{0}:$ $\nu=0$ vs. $H_{1}:$ $\nu\neq
0$, we can apply the rejection rule: $T_{N}^{2}>\chi^{2}_{1}(\alpha)$, where
the threshold $\chi^{2}_{1}(\alpha)$ satisfies
$\Pr\left\\{Z>\chi^{2}_{1}(\alpha)\right\\}=\alpha$ with a random variable $Z$
having a chi-square distribution with one degree of freedom.

It can be remarked that the testing algorithm shown in this section can be
easily extended to make decisions regarding quantiles of underlying data
distributions; see Section 6, for details.

###  3.5 Test for the Center of Symmetry

In many practical applications, e.g., paired testing for pre-and post-
treatment effects, we may be interested in testing that the center of symmetry
of the paired observations is zero. To this end, we assume that i.i.d.
observations $X_{1},\ldots,X_{n}$ are from an unknown symmetric distribution
$F$ with $\sigma^{2}=\text{var}\left(X_{1}\right)<\infty$. According to Bickel
and Lehmann (2012), the natural location parameter, say $\nu$, for $F$ is its
center of symmetry. We are interested in testing the hypothesis $H_{0}:$
$\nu=0$ vs. $H_{1}:$ $\nu>0$.

The statistics $T_{o}=n^{0.5}\bar{X}/S$ and
$T_{1}=2n^{0.5}X_{(n/2)}\hat{f}\left(X_{(n/2)}\right)$ are reasonable to be
employed for testing $H_{0}$, where
$S^{2}=\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}/(n-1)$ and $\hat{f}(u)$
estimates $f(u)=dF(u)/du$. The components of $T_{o}$ and $T_{1}$ are
specialized in Sections 3.3 and 3.4. It is clear that if $F$ were known to be
a normal distribution function, then $T_{o}$ outperforms $T_{1}$, whereas when
$F$ were known to be a distribution of, e.g., the random variable
$\xi_{1}-\xi_{2}$, where $\xi_{1},\xi_{2}$ are independent and identically
$Exp(1)$-distributed, $T_{1}$ outperforms $T_{o}$.

Consider, for example, the statistic $T_{o}$ as a test statistic to be
modified and the statistic $A=\left(\bar{X}-X_{(n/2)}\right)D^{-0.5}\,$ having
a role of an approximately ancillary statistic, where
$D=\sigma^{2}-\text{E}|X_{1}-\nu|/f(\nu)+1/\left(4f^{2}(\nu)\right)$. Then,
following the concept and the notations defined in Sections 3.3 and 3.4, we
propose to reject $H_{0}$, if

$\displaystyle T_{N}$ $\displaystyle=$ $\displaystyle\left\\{T_{o}+\delta
n^{0.5}\left(\bar{X}-X_{(n/2)}\right)\hat{D}^{-0.5}\right\\}V^{-0.5}>z_{\alpha},$
$\displaystyle\hat{D}$ $\displaystyle=$ $\displaystyle
S^{2}-\hat{w}/\hat{f}\left(X_{(n/2)}\right)+\left(4\hat{f}^{2}\left(X_{(n/2)}\right)\right)^{-1},\,\,\delta\,=\,\left\\{\hat{w}/\left(2S\hat{f}\left(X_{(n/2)}\right)\right)-S\right\\}\hat{D}^{-0.5},$
$\displaystyle V$ $\displaystyle=$ $\displaystyle 1+\delta^{2}+\frac{2\delta
S}{\hat{D}}-\frac{\delta\hat{w}}{\hat{D}^{0.5}S\hat{f}\left(X_{(n/2)}\right)}\,=\,1+\frac{2\hat{w}}{\hat{D}\hat{f}\left(X_{(n/2)}\right)}-\frac{1}{\hat{D}}\left(\frac{\hat{w}}{2S\hat{f}\left(X_{(n/2)}\right)}+S\right)^{2}.$

We will experimentally demonstrate that the $T_{N}$-based test can combine
attractive power properties of the $T_{o}$-and $T_{1}$-based tests.

######  Remark 3.1.

Note that, in this section, the test statistics $T_{N}$ are targeted to
improve the statistics $T_{o}$. In this section’s framework, there are no MP
decision-making mechanisms. Thus, in general, it can be assumed we can find
decision-making procedures that outperform the $T_{N}$-based tests in certain
situations.

Section 4 numerically examines properties of the decision-making schemes
derived in this section.

##  4 Numerical Simulations

We conducted a Monte Carlo study to explore the performance of the proposed
transformations of the tests about the mean, median, and center of symmetry as
described in Section 3. In terms related to evaluations of nonparametric
decision-making procedures, it can be noted that there are no MP tests, in the
frameworks of Sections 3.3, 3.4, and 3.5. We therefore compare the tests based
on the given statistics, $T_{o}$, with those based on the corresponding
statistics $T_{N}$, the modifications of $T_{o}$, under various designs of
$H_{0}$/$H_{1}$-underlying data distributions. The aim of the numerical study
is to confirm that the proposed method can provide improvements in the context
of statistical power. In Sections 4.2 and 4.3, for additional comparisons, we
demonstrate the Monte Carlo power of the one-sample Wilcoxon-Mann-Whitney test
that is frequently used in applications, where researchers are interested in
assessing the hypothesis $H_{0}:$ $\nu=0$ when $\nu$ is the median of
observations. Note that, in practice, it is very difficult to find a
nonparametric alternative to the one sample t-test for the mean. Then, in
Section 4.1, where the t-test and its transformation defined in Section 3.3
are evaluated, we include a bootstrapped (nonparametric resampling) version of
the original t-statistic $T_{o}$ to be compared with the corresponding
statistic $T_{N}$, expecting that the bootstrapped t-test may outperform the
original t-test in several nonparametric scenarios (Efron, 1992).

To evaluate the tests, we generated $55,000$ independent samples of size
$n\in\\{n_{1},\ldots,n_{J}\\}$ from different distributions corresponding to,
say, designs $D_{km}$, $k\in\\{0,1\\}$, $m\in\\{1,\ldots,M\\}$. In this
scheme, designs $D_{km}$, $m\in\\{1,\ldots,M\\}$, fit hypotheses $H_{k}$,
$k\in\\{0,1\\}$, respectively. Each of the presented bootstrap simulation
results are based on $55,000$ replications with $1,000$ bootstrap samples.

Let the notation $T\left(D_{km}\right)$ represent a test statistic $T$
conducted with respect to design $D_{km}$, $k\in\\{0,1\\}$,
$m\in\\{1,\ldots,M\\}$. To judge the experimental characteristics of the
proposed tests, we obtained Monte Carlo estimators, say $\mathrm{PowA}$ and
$\mathrm{Pow}$ of the following quantities:
$\mathrm{Pr}_{1}\left\\{T\left(D_{km}\right)>C_{\alpha}\right\\}$ and
$\mathrm{Pr}_{1}\left\\{T\left(D_{km}\right)>q\left(D_{0m}\right)\right\\}$,
where $C_{\alpha}$ is the $\alpha$-level critical value related to the
asymptotic $H_{0}$-distribution of $T$ and $q\left(D_{0m}\right)$ means a
value of the $100(1-\alpha)\%$-quantile of $T\left(D_{0m}\right)$’s
distribution, respectively. The criterion $\mathrm{PowA}$ calculated under
$D_{0m}$, $m\in\\{1,\ldots,M\\}$, examines our current ability to control the
TIE rate of a $T$-based test using an approximate $H_{0}$-distribution of $T$.
In this framework, $\mathrm{PowA}$ calculated under $D_{1m}$,
$m\in\\{1,\ldots,M\\}$, displays the expected power of $T$. Values of
$\mathrm{Pow}$ can be used to evaluate the actual power levels of $T$,
supposing we can accurately control the TIE rates of the corresponding
$T$-based test. It can be theoretically assumed that we can correct $T$ to
produce a statistic, say $T^{\prime}$, in order to minimize the distance
$\left|\mathrm{Pr}_{0}\left\\{T^{\prime}\left(D_{0m}\right)>C_{\alpha}\right\\}-\alpha\right|$,
by employing a method based on, for example, a Bartlett type correction,
location adjustments, and/or bootstrap techniques. In this framework, an
accurate higher order approximation to the $H_{0}$-distribution of
$T\left(D_{0m}\right)$ might be needed. In several situations, $\mathrm{Pow}$
could indicate potential abilities to improve practical implementations of
studied tests.

###  4.1 One Sample t-Test for the Mean

In order to examine the $T_{N}$-based test generated by modifying the t-test,
$T_{o}$, in Section 3.3, the following designs of underlying data
distributions were applied: $D_{k1}:$ $X_{1}\sim N\left(0.1k,1\right)$;
$D_{k2}:$ $X_{i}=1-\eta_{i}+0.1k$ with $\eta_{i}\sim Exp(1)$; $D_{k3}:$
$X_{i}=\eta_{i}-1+0.1k$; $D_{k4}:$ $X_{i}=\left(\xi_{i}-2\right)/2+0.2k$ with
$\xi_{i}\sim\mathrm{Weibull}(1,2)$; where $k\in\\{0,1\\}$ and
$i\in\\{1,\ldots,n\\}$. The experimental results presented in Table 1 are the
power comparisons of the t-test based on $T_{o}$, its modification based on
$T_{N}$ and the bootstrap test $T_{B}$, the bootstrapped version of the
t-test, when the significance level, $\alpha$, of the tests was supposed to be
fixed at 5$\%$.

Table 1:  Monte Carlo rate of rejections at $\alpha=0.05$ of the following
statistics: the t-test statistic $T_{o}$ and its modification $T_{N}$, defined
in Section 3.3; the t-test statistic’s bootstrapped version $T_{B}$.

$n=150$ $n=200$ $n=250$ $n=300$ $n=350$ Design Test PowA Pow PowA Pow PowA Pow
PowA Pow PowA Pow $D_{01}$ $T_{o}$ 0.049 0.050 0.053 0.050 0.049 0.050 0.050
0.050 0.052 0.050 $T_{N}$ 0.050 0.050 0.054 0.050 0.050 0.050 0.051 0.050
0.053 0.050 $T_{B}$ 0.054 0.054 0.052 0.052 0.050 0.050 0.051 0.051 0.052
0.052 $D_{11}$ $T_{o}$ 0.337 0.334 0.409 0.401 0.475 0.478 0.537 0.537 0.588
0.581 $T_{N}$ 0.337 0.333 0.412 0.399 0.477 0.476 0.539 0.533 0.590 0.579
$T_{B}$ 0.332 0.332 0.416 0.416 0.463 0.463 0.539 0.539 0.586 0.586 $D_{02}$
$T_{o}$ 0.044 0.050 0.046 0.050 0.046 0.050 0.046 0.050 0.044 0.050 $T_{N}$
0.044 0.050 0.043 0.050 0.044 0.050 0.043 0.050 0.043 0.050 $T_{B}$ 0.083
0.083 0.074 0.074 0.068 0.068 0.068 0.068 0.069 0.069 $D_{12}$ $T_{o}$ 0.345
0.363 0.417 0.429 0.487 0.499 0.541 0.554 0.597 0.614 $T_{N}$ 0.479 0.503
0.591 0.615 0.679 0.704 0.748 0.774 0.805 0.826 $T_{B}$ 0.386 0.386 0.447
0.447 0.507 0.507 0.559 0.559 0.608 0.608 $D_{03}$ $T_{o}$ 0.055 0.050 0.055
0.050 0.054 0.050 0.053 0.050 0.052 0.050 $T_{N}$ 0.068 0.050 0.068 0.050
0.067 0.050 0.067 0.050 0.065 0.050 $T_{B}$ 0.032 0.032 0.036 0.036 0.037
0.037 0.039 0.039 0.038 0.038 $D_{13}$ $T_{o}$ 0.331 0.316 0.398 0.379 0.464
0.449 0.528 0.517 0.584 0.579 $T_{N}$ 0.644 0.580 0.732 0.663 0.801 0.749
0.854 0.810 0.890 0.856 $T_{B}$ 0.307 0.307 0.383 0.383 0.454 0.454 0.521
0.521 0.586 0.586 $D_{04}$ $T_{o}$ 0.044 0.050 0.046 0.050 0.045 0.050 0.044
0.050 0.046 0.050 $T_{N}$ 0.042 0.050 0.043 0.050 0.044 0.050 0.044 0.050
0.044 0.050 $T_{B}$ 0.077 0.077 0.079 0.079 0.071 0.071 0.069 0.069 0.068
0.068 $D_{14}$ $T_{o}$ 0.790 0.804 0.879 0.885 0.929 0.934 0.961 0.965 0.979
0.980 $T_{N}$ 0.940 0.950 0.981 0.984 0.994 0.995 0.998 0.999 0.999 1.000
$T_{B}$ 0.781 0.781 0.863 0.863 0.920 0.920 0.954 0.954 0.973 0.973

Designs $D_{k1}$, $k\in\\{0,1\\}$, exemplify scenarios, where $T_{o}$ is MP.
In these cases, values of $\mathrm{Pow}$ testify that $T_{o}$ is slightly
superior to $T_{N}$. Designs $D_{k2}$, $k\in\\{0,1\\}$, correspond to
negatively skewed distributions. In these scenarios, $T_{N}$ is clearly
somewhat better than $T_{o}$, having approximately 27$\%$$-$30$\%$ power gains
as compared with $T_{o}$. Designs $D_{k3}$, $k\in\\{0,1\\}$, represent
positively skewed distributions. The proposed test $T_{N}\left(D_{13}\right)$
is about two times more powerful than $T_{o}\left(D_{13}\right)$. However, we
should note that the asymptotic TIE rate control related to
$T_{N}\left(D_{03}\right)$ suffers from the skewness of the
$H_{0}$-distribution. According to the values of $\mathrm{Pow}$ computed under
$D_{13}$, the procedure $T_{N}\left(D_{k3}\right)$, $k\in\\{0,1\\}$, will
clearly dominate the strategy $T_{o}\left(D_{k3}\right)$, $k\in\\{0,1\\}$, if
the TIE rate control related to $T_{N}$ could be improved. To this end, for
example, a Chen (1995)-type approach can be suggested to be applied. The
present paper does not aim to achieve improvements of test-algorithms for
controlling the TIE rate of $T_{N}$. The computed values of the criterion
$\mathrm{Pow}$ shown in Table 1 confirm that the $T_{N}$-based strategy is
reasonable. The results related to $D_{04}$ and $D_{14}$ support the
conclusions above. Although, under $D_{04}$, the corresponding
$\mathrm{PowA}$’s values indicate that the Monte Carlo asymptotic TIE rates of
$T_{N}$ are smaller than those related to $T_{o}$, the proposed test is
superior to $T_{o}$ in both the $\mathrm{PowA}$ and $\mathrm{Pow}$ contexts
under $D_{14}$.

In Table 1, we also report the experimental results related to the Monte Carlo
implementations of the test based on a bootstrapped version of the $T_{o}$
statistic, denoted $T_{B}$, where $X_{1},\ldots,X_{n}$ are resampled with
replacement. In these cases, asymptotic approximations for the corresponding
TIE rates were not applied. Thus, we denote the criterion PowA=Pow. The
applied bootstrap strategy required a substantial computational cost. However,
we cannot confirm that the $T_{B}$-based test is significantly superior to the
t-test based on $T_{o}$, under $D_{01},D_{11},D_{02},\ldots,D_{14}$. Moreover,
under the designs $D_{02}$ and $D_{04}$, the the bootstrap t-test cannot be
suggested to be used.

###  4.2 One Sample Test for the Median

To gain some insight into operating characteristics of the test statistic
$T_{N}$ defined in Section 3.4, we considered various designs of underlying
data distributions corresponding to the hypotheses $H_{0}:$ $\nu=0$ and
$H_{1}:$ $\nu>0$, where $\nu$ denotes the median of $X_{1}$’s distribution. To
exemplify the results of the conducted Monte Carlo study, we employ the
following schemes: $D_{01}:$ $X_{i}=\eta_{i}-\xi_{i}$; $D_{11}:$
$X_{i}=1-\eta_{i}$; $D_{02}:$ $X_{i}\sim N(0,4)$; $D_{12}:$
$X_{i}=\exp(0.5)-\zeta_{i}$; where $\eta_{i}\sim Exp(1)$, $\xi_{i}\sim
Exp(1)$, $\zeta_{i}\sim LN(0,1)$, and $i\in\\{1,\ldots,n\\}$. In this study,
attending to the statements presented in Section 3.4, the one-sample, one-
sided Wilcoxon-Mann-Whitney test, say W, the $T_{o}$-based test and its
modification, the $T_{N}$-based test, were implemented. Note that, for the W
test, the criterion PowA=Pow, since $H_{0}$-distributions of the Wilcoxon-
Mann-Whitney test statistic do not depend on underlying data distributions.
Table 2 represents the typical results observed during the extensive power
evaluations of W, $T_{o}$, and $T_{N}$, when the significance level, $\alpha$,
of the considered tests was supposed to be fixed at 5$\%$.

Table 2:  Monte Carlo rate of rejections at $\alpha=0.05$ of the following
statistics: the one-sample Wilcoxon-Mann-Whitney test statistic (W), $T_{o}$
and its modification, $T_{N}$, defined in Section 3.4.

$n=25$ $n=50$ $n=75$ Design Test PowA Pow PowA Pow PowA Pow $D_{01}$ W 0.048
0.048 0.049 0.049 0.049 0.049 $T_{o}$ 0.029 0.050 0.025 0.050 0.031 0.050
$T_{N}$ 0.030 0.050 0.027 0.050 0.028 0.050 $D_{11}$ W 0.213 0.213 0.319 0.319
0.417 0.417 $T_{o}$ 0.484 0.549 0.674 0.751 0.806 0.862 $T_{N}$ 0.660 0.725
0.899 0.936 0.973 0.985 $D_{02}$ W 0.048 0.048 0.050 0.050 0.050 0.050 $T_{o}$
0.049 0.050 0.044 0.050 0.045 0.050 $T_{N}$ 0.048 0.050 0.042 0.050 0.045
0.050 $D_{12}$ W 0.394 0.394 0.609 0.609 0.755 0.755 $T_{o}$ 0.734 0.735 0.919
0.927 0.977 0.980 $T_{N}$ 0.843 0.849 0.982 0.986 0.998 0.999

For example, in scenario $\left\\{D_{11},\,n=50\right\\}$, $T_{N}$ improves
$T_{0}$ providing about a 25% power gain.

Regarding the two-sided $T_{N}^{2}$-based test derived in Section 3.4, the
following outcomes exemplify the corresponding Monte Carlo power evaluations:
PowA =$0.223$, $0.585$, and $0.838$ provided by the two-sided W-test,
$T_{o}^{2}$-based test and $T_{N}^{2}$-based test, respectively, when $n=50$
and generated data satisfy $D_{11}$. Note that, in the scenario above, we can
employ the method proposed in Fisher and Robbins (2019). According to Fisher
and Robbins (2019), since $T_{o}^{2}>0$, and $T_{N}^{2}>0$ are
$O_{p}\left(n^{k}\right)$, where $k=0$ and $k=1$, under $H_{0}$ and $H_{1}$,
respectively, the test statistics
$T_{o1}^{2}=-n\log\left(1-T_{o}^{2}/n\right)$ and
$T_{N1}^{2}=-n\log\left(1-T_{N}^{2}/n\right)$ are reasonable to be examined.
These monotonic transformations demonstrated slight PowA increases of
approximately $1.4\%$ and $1.1\%$ for the $T_{o}^{2}$\- and $T_{N}^{2}$-based
strategies, respectively.

###  4.3 Test for the Center of Symmetry

In this section, we examine implementations of the proposed
$T_{N}$-modification of the $T_{o}$-based test developed in Section 3.5. The
$T_{o}$\- and $T_{1}$-based tests as well as the one-sample, one-sided
Wilcoxon-Mann-Whitney test (W) were compared with the $T_{N}$-based test with
respect to the setting depicted in Section 3.5. To exemplify the results of
the conducted numerical study, the following designs of data
$D=\left\\{X_{1},\ldots,X_{n}\right\\}$ generations were employed: for
$k\in\\{0,1\\}$ and $i\in\\{1,\ldots,n\\}$, $D_{k1}:$ $X_{i}\sim N(0.1k,1)$,
when $T_{o}$ can be expected to be superior to $T_{1}$, $T_{N}$, and W;
$D_{k2}:$ $X_{i}=\eta_{i}-\xi_{i}+0.1k$, where $\eta_{i}$ and $\xi_{i}$ are
independent $Exp(1)$-distributed random variables, and then $T_{1}$ can be
expected to be superior to $T_{o}$, $T_{N}$, and W; $D_{k3}:$
$X_{i}=\zeta_{i}+0.1k$, where $\zeta_{i}\sim Unif(-1,1)$; $D_{k4}:$
$X_{i}=\epsilon_{i}-0.5+0.1k$, where $\epsilon_{i}\sim Beta(0.5,0.5)$.

Table 3 summarizes the computed Monte Carlo outputs across scenarios $D_{kj}$,
$k\in\\{0,1\\}$, $j\in\\{1,\ldots,4\\}$, when $n=50,150$ and the significance
level, $\alpha$, of the tests is supposed to be fixed at 5$\%$. It is observed
that: under $D_{01}$ and $D_{11}$, $T_{o}$ and $T_{N}$ have very similar
behavior; under $D_{02}$ with $n=50$, $T_{N}$ does improve $T_{o}$ in terms of
the TIE rate control; under $D_{12}$, the values of the measurement Pow
related to $T_{N}$ and $T_{1}$ are close to each other and greater than those
of $T_{o}$; under $D_{kj}$, $k\in\\{0,1\\}$, $j\in\\{3,4\\}$, $T_{N}$ shows
the Monte Carlo power characteristics that outperform those of $T_{o}$, and W.
For example, under $D_{14}$ with $n=50$, $T_{N}$ has approximately $22\%$,
$23\%$, and $65\%$ power gains as compared with W, $T_{o}$, and $T_{1}$,
respectively.

Table 3: Monte Carlo power levels at $\alpha=0.05$ of the one-sample Wilcoxon-
Mann-Whitney test (W) as well as the $T_{o}$, $T_{1}$, and $T_{N}$-based tests
defined in Section 3.5.

$n=50$ $n=150$ $n=50$ $n=150$ Design Test PowA Pow PowA Pow Design Test PowA
Pow PowA Pow $D_{01}$ W 0.049 0.049 0.052 0.052 $D_{03}$ W 0.050 0.050 0.049
0.049 $T_{o}$ 0.052 0.050 0.053 0.050 $T_{o}$ 0.054 0.050 0.050 0.050 $T_{1}$
0.044 0.050 0.047 0.050 $T_{1}$ 0.062 0.050 0.057 0.050 $T_{N}$ 0.054 0.050
0.054 0.050 $T_{N}$ 0.052 0.050 0.048 0.050 $D_{11}$ W 0.169 0.169 0.321 0.321
$D_{13}$ W 0.301 0.301 0.645 0.645 $T_{o}$ 0.183 0.178 0.338 0.329 $T_{o}$
0.328 0.317 0.678 0.676 $T_{1}$ 0.128 0.140 0.231 0.238 $T_{1}$ 0.191 0.160
0.345 0.319 $T_{N}$ 0.189 0.173 0.341 0.328 $T_{N}$ 0.390 0.381 0.774 0.782
$D_{02}$ W 0.051 0.051 0.049 0.049 $D_{04}$ W 0.050 0.050 0.050 0.050 $T_{o}$
0.058 0.050 0.049 0.050 $T_{o}$ 0.054 0.050 0.053 0.050 $T_{1}$ 0.026 0.050
0.024 0.050 $T_{1}$ 0.089 0.050 0.067 0.050 $T_{N}$ 0.044 0.050 0.035 0.050
$T_{N}$ 0.046 0.050 0.048 0.050 $D_{12}$ W 0.144 0.144 0.280 0.280 $D_{14}$ W
0.643 0.643 0.965 0.965 $T_{o}$ 0.132 0.119 0.225 0.225 $T_{o}$ 0.637 0.617
0.965 0.962 $T_{1}$ 0.089 0.149 0.206 0.315 $T_{1}$ 0.292 0.179 0.493 0.431
$T_{N}$ 0.134 0.148 0.243 0.304 $T_{N}$ 0.826 0.836 0.998 0.998

Based on the conducted Monte Carlo study, we conclude that the proposed
testing strategies exhibit high and stable power characteristics under various
designs of alternatives.

##  5 Real Data Example

By blocking the blood flow of the heart, blood clots commonly cause myocardial
infarction (MI) events that lead to heart muscle injury. Heart disease is a
leading cause of death affecting about or higher than 20% of populations
regardless of different ethnicities according to the Centers for Disease
Control and Prevention, e.g., Schisterman et al. (2001).

The application of the proposed approach is illustrated by employing a sample
from a study that evaluates biomarkers associated with MI. The study was
focused on the residents of Erie and Niagara counties, 35$-$79 years of age.
The New York State department of Motor Vehicles drivers’ license rolls was
used as the sampling frame for adults between the age of 35 and 65 years,
while the elderly sample (age 65$-$79) was randomly chosen from the Health
Care Financing Administration database. The biomarkers called ”thiobarbituric
acid-reactive substances” (TBARS) and ”high-density lipoprotein” (HDL)
cholesterol are frequently used as discriminant factors between individuals
with (MI=1) and without (MI=0) myocardial infarction disease, e.g.,
Schisterman et al. (2001).

The sample of $2,910$ biomarkers’ values was used to estimate the parameters
$a$ and $b$ in the linear regression model $Y_{i}=a+bZ_{i}+\epsilon_{i}$
related to {MI$=1$}’s cases, where $Y_{1},\ldots,Y_{2910}$ are log-transformed
HDL-cholesterol measurements, $Z_{1},\ldots,Z_{2910}$ denote log-transformed
TBARS measurements, and $\epsilon_{i},\,i\geq 1,$ represent regression
residuals with $\text{E}\epsilon_{i}=0$. It was concluded that
$\epsilon_{i}\simeq Y_{i}-4.034-0.045Z_{i},\,i\geq 1$ (see Table S1 in the
online supplemental material, for details). Assume we aim to investigate the
distribution of $\epsilon_{i}$ based on $n=100$ biomarkers’ values, when
$\text{MI}=1$. In this case, it was observed that the sample mean and variance
were $\bar{\epsilon}=\sum_{i=1}^{n}\epsilon_{i}/n\simeq-0.002$ and
$\sum_{i=1}^{n}\left(\epsilon_{i}-\bar{\epsilon}\right)^{2}/(n-1)\simeq
0.073$, respectively. Figure 2 depicts the histogram based on corresponding
values of $\epsilon_{1},\ldots,\epsilon_{100}$.

In order to test for $H_{0}:$ $\nu=0$ vs. $H_{1}:$ $\nu\neq 0$, where $\nu$ is
the median of $\epsilon$’s distribution, we implemented the two-sided
$T_{o}^{2}$-based test and its modification, the $T_{N}^{2}$-based test
denoted in Section 3.4, as well as the two-sided Wilcoxon-Mann-Whitney test
(W). Although the histogram shown in Figure 2 displays a relatively asymmetric
distribution about zero, the $T_{o}^{2}$-based test and the W test have
demonstrated a p-value$=0.071$ and p-value$=0.326$, respectively. The proposed
$T_{N}^{2}$-based test has provided p-value$=0.047$. Then, we organized a
Bootstrap/Jackknife type study to examine the power performances of the test
statistics. The conducted strategy was that a sample with size $n_{b}<100$ was
randomly selected with replacement from the data
$\left\\{\epsilon_{1},\ldots,\epsilon_{n}\right\\}$ to be tested for $H_{0}$
at a 5% level of significance. This strategy was repeated $10,000$ times to
calculate the frequencies of the events {$T_{o}^{2}$ rejects $H_{0}$}, {W
rejects $H_{0}$}, and {$T_{N}^{2}$ rejects $H_{0}$}. The obtained experimental
powers of $T_{o}^{2}$, W, and $T_{N}^{2}$ were: $0.238$, $0.106$, $0.535$,
when $n_{b}=90$; $0.198$, $0.104$, $0.463$, when $n_{b}=80$; $0.176$, $0.100$,
$0.415$, when $n_{b}=70$, respectively. The experimental power levels of the
tests increase as the sample size $n_{b}$ increases. This study experimentally
indicates that the $T_{N}^{2}$-based test outperforms the classical procedures
in terms of the power properties when evaluating whether the residuals of the
association $Y_{i}=a+bZ_{i}+\epsilon_{i},\,i\geq 1$, are distributed
asymmetrically about zero. That is, the proposed test can be expected to be
more sensitive as compared with the known methods to rejecting the null
hypothesis $H_{0}:$ $\nu=0$ vs. $H_{1}:$ $\nu\neq 0$, in this study.

Figure 2: Data-based histogram related to regression residuals
$\epsilon_{1},\ldots,\epsilon_{n}$.

##  6 Concluding Remarks

The present article has provided a theoretical framework for evaluating and
constructing powerful data-based tests. The contributions in this article have
touched on the principles of characterizing most powerful statistical
decision-making mechanisms. Proposition 2.2 provides a method for one-to-one
mapping the term ”most powerful” to the properties of test statistics’
distribution functions via analyzing the behavior of corresponding likelihood
ratios. We demonstrated that the derived characterization of MP tests can be
associated with a principle of sufficiency. The concepts shown in Section 2
have been applied to improving test procedures by accounting for the relevant
ancillary statistics. Applications of the presented theoretical framework have
been employed to display efficient modifications of the one-sample t-test, the
test for the median, and the test for the center of symmetry, in nonparametric
settings. The effectiveness of the proposed nonparametric decision-making
procedures in maintaining relatively high power has been confirmed using
simulations and a real data example across various scenarios based on samples
from relatively skewed distributions. We also note the following remarks. (a)
Propositions 2.4 and 3.1 can be applied to different decision-making problems.
(b) Effective corrections of the classical t-test can be of theoretical and
applied interest. The modification of the t-test as per Section 3.3 involves
using the estimation of the third central moment. This moment plays a role in
some corrections of the t-test structure for adjusting its null distribution
when the underlying data are asymmetric, e.g., Chen (1995). Overall, the
proposed modification is somewhat different from those that are used to
improve control of the TIE rates of t-test type procedures. Thus, in general,
basic ingredients of the methods mentioned above can be combined. (c) The
scheme presented in Section 3.4 can be easily revised to develop a test for
quantiles, by using the observation that $\bar{X}$ and the sample $p$th
quantile, say $X_{(pn)}$, are asymptotically bivariate normal with

$\displaystyle\mathrm{cov}\left(\bar{X},X_{(pn)}\right)\,=\,f\left(\nu_{p}\right)^{-1}\mathrm{E}\left(X_{1}-\nu_{p}\right)\left\\{pI\left(X_{1}>\nu_{p}\right)-(1-p)I\left(X_{1}\leq\nu_{p}\right)\right\\},$
$\displaystyle\mathrm{var}\left(X_{(pn)}\right)\,=\,f\left(\nu_{p}\right)^{-2}p(1-p),\,\,\,\mathrm{Pr}\left(X_{1}<\nu_{p}\right)\,=\,p.$

(d) Sections 3$-$5 have exemplified applications of the treated MP principle
in the nonparametric settings. In many parametric problems, the corresponding
likelihood ratios do not have explicit forms or have very complicated shapes,
e.g., when testing statements are based on longitudinal data, dependent
observations, multivariate outcomes, data subject to different sorts of
errors, and/or missing-values mechanisms. In such cases, issues related to
comparing/developing tests via the considered MP principle can be employed.

A plethora of decision-making algorithms touches on most fields of statistical
practice. Thus, it is not practical in one paper to focus on all the relevant
theory and examples. This paper studies only one approach to characterize a
class of MP mechanisms. That is, there are many potential future directions
that seem to be promising targets for research, including, for example: (i)
examinations of the relationships between general MP characterizations, Basu’s
theorem-type results (e.g., Ghosh, 2002), the concepts of sufficiency,
completeness, and ancillarity under different statements of decision-making
policies. In this aspect, for example, a research question can be as follows:
when can we claim that a statistic $T$ is MP iff $T$ and $A$ are independently
distributed, for any ancillary statistic $A$? (ii) Various parametric and
nonparametric applications of MP characterizations in different settings can
be developed. (iii) Proposition 2.4 can be used and extended to compare
different statistical procedures in practice. (iv) In light of the present MP
principle, relevant evaluations of optimal combinations of test statistics
(e.g., Berk and Jones, 1978) can be proposed. (v) Large sample properties of
test statistics modified with respect to MP characterization can be analyzed.
(vi) Perhaps, Proposition 3.1 can be integrated into various testing
developments, where characterizations of underlying data distributions under
corresponding hypotheses can be used to define relevant ancillary statistics.
Leaving these topics to the future, it is hoped that the present paper will
convince the readers of the benefits of studying different aspects and
characterizations related to MP data-based decision-making techniques.

## Supplementary Materials

The online supplementary materials contain: the proofs of the theoretical
results presented in the article; and Table S1 that displays the analysis of
variance related to the linear regression fitted to the observed log-
transformed HDL-cholesterol measurements using the log-transformed TBARS
measurements as a factor, in Section 5.

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Supplemental Materials to ”A Characterization of Most(More) Powerful Test
Statistics with Simple Nonparametric Applications”

## APPENDIX: PROOFS

This Appendix comprises the necessary proofs to establish the propositions
presented in the present paper.

### Proof of Proposition 1.1

Consider

$\displaystyle{\Pr}_{1}\left(u-s\leq\Lambda\leq
u\right)=\text{E}_{1}I\left(u-s\leq\Lambda\leq u\right)\,=\,\int
I\left(u-s\leq\Lambda\leq u\right)f_{1}$ $\displaystyle\quad=\,\int
I\left(u-s\leq\Lambda\leq u\right)\frac{f_{1}}{f_{0}}f_{0}\,\,\,=\,\int
I\left(u-s\leq\Lambda\leq u\right)\,\Lambda\,\,f_{0},$

where $I(.)$ means the indicator function and $u,s$ are not random variables.
This implies the inequalities

$\displaystyle{\Pr}_{1}\left(u-s\leq\Lambda\leq u\right)$ $\displaystyle\leq$
$\displaystyle\int I\left(u-s\leq\Lambda\leq
u\right)uf_{0}\,\,\,\,\,=\,u{\Pr}_{0}\left(u-s\leq\Lambda\leq u\right),$
$\displaystyle{\Pr}_{1}\left(u-s\leq\Lambda\leq u\right)$ $\displaystyle\geq$
$\displaystyle\int I\left(u-s\leq\Lambda\leq u\right)(u-s)f_{0}$
$\displaystyle=\,(u-s){\Pr}_{0}\left(u-s\leq\Lambda\leq u\right).$

Dividing these inequalities by $s$ and employing $s\to 0$, we complete the
proof.

### Proof of Proposition 2.1

We can write

$\displaystyle f_{1}^{\Lambda,A}(u,v)$ $\displaystyle=$
$\displaystyle\frac{d}{dv}\,\,{\mathop{\lim}\limits_{s\to
0}}\,\frac{1}{s}{\Pr}_{1}\left(u-s<\Lambda<u,A<v\right),$

where

$\displaystyle{\Pr}_{1}\left\\{u-s<\Lambda(D)<u,A(D)<v\right\\}=\text{E}_{1}I\left\\{u-s<\Lambda(D)<u,A(D)<v\right\\}$
$\displaystyle\quad=\int I\left\\{u-s<\Lambda(D)<u,A(D)<v\right\\}\,f_{1}(D)$
$\displaystyle\quad=\int
I\left\\{u-s<\Lambda(D)<u,A(D)<v\right\\}\,\Lambda(D)\,\,f_{0}(D).$

Noting that

$\displaystyle\int I\left(u-s<\Lambda<u,A<v\right)\,\Lambda\,\,f_{0}(D)$
$\displaystyle\leq$ $\displaystyle u\int
I\left(u-s<\Lambda<u,A<v\right)\,f_{0}(D),$ $\displaystyle\int
I\left(u-s<\Lambda<u,A<v\right)\,\Lambda\,\,f_{0}(D)$ $\displaystyle\geq$
$\displaystyle\left(u-s\right)\int I\left(u-s<\Lambda<u,A<v\right)\,f_{0}(D),$

we have

$\displaystyle{\Pr}_{1}\left(u-s<\Lambda<u,A<v\right)$ $\displaystyle\leq$
$\displaystyle u\,{\Pr}_{0}\left(u-s<\Lambda<u,A<v\right),$
$\displaystyle{\Pr}_{1}\left(u-s<\Lambda<u,A<v\right)$ $\displaystyle\geq$
$\displaystyle\left(u-s\right)\,{\Pr}_{0}\left(u-s<\Lambda<u,A<v\right).$

This completes the proof in a similar manner to that of Proposition 1.1.

### Proof of Proposition 2.2

The proof of Proposition 2.2 consists of the two steps below: (1) When
$T=\Lambda$, i.e. $T$ is the MP test statistic, Proposition 2.1 says
$f_{1}^{T,A}(u,v)=u{\rm\;}f_{0}^{T,A}(u,v)$. (2) When
$f_{1}^{T,A}(u,v)=u{\rm\;}f_{0}^{T,A}(u,v)$ is satisfied, we consider the
elementary inequality: for all $B$ and $C$,

$(B-C)\left\\{I(B\geq C)-\delta\right\\}\geq 0,$

where $I\left(.\right)$ is the indicator function and
$\delta\in\left[0,1\right]$. Let $\delta=I\left(\Lambda\geq U\right)$ denote
the $H_{0}$-rejection rule of the likelihood ratio test with the fixed
threshold $U$, such that: we reject $H_{0}$, if $\delta=1$. Letting $B=T$, we
have

$\text{E}_{0}\left\\{\left(T-C\right)I\left(T\geq
C\right)\right\\}\geq\text{E}_{0}\left\\{\left(T-C\right)I\left(\Lambda\geq
U\right)\right\\}.$

Thus, fixing the thresholds $C,\,U$ such that the TIE rates
${\Pr}_{0}\left(T\geq C\right)={\Pr}_{0}\left(\Lambda\geq U\right)$, we obtain

$\text{E}_{0}I\left(T\geq C\right)T\,\geq\,\text{E}_{0}I\left(\Lambda\geq
U\right)T,$

where

$\text{E}_{0}I\left(\Lambda\geq U\right)T\,=\,\text{E}_{0}I\left(\Lambda\geq
U\right)f_{1}^{T,\Lambda}(T,\Lambda)/f_{0}^{T,\Lambda}(T,\Lambda),$

since $u=f_{1}^{T,A}(u,v)/f_{0}^{T,A}(u,v)$, for any statistic $A$ and all
$u\geq 0$. That is to say,

$\displaystyle\text{E}_{0}I\left(\Lambda\geq
U\right)T={\int\\!\\!\\!\\!\int}I\left(v\geq
U\right)u\,f_{0}^{T,\Lambda}(u,v)dudv$
$\displaystyle\quad={\int\\!\\!\\!\\!\int}I\left(v\geq
U\right)\frac{f_{1}^{T,\Lambda}(u,v)}{f_{0}^{T,\Lambda}(u,v)}f_{0}^{T,\Lambda}(u,v)dudv={\int\\!\\!\\!\\!\int}I\left(v\geq
U\right)f_{1}^{T,\Lambda}(u,v)dudv$ $\displaystyle\quad=\int I\left(v\geq
U\right)\int f_{1}^{T,\Lambda}(u,v)dudv=\int I\left(v\geq
U\right)f_{1}^{\Lambda}(v)dv\,\,=\,\,{\Pr}_{1}\left(\Lambda\geq U\right).$

Thus,

$\text{E}_{0}I\left(T\geq C\right)T\geq{\Pr}_{1}\left(\Lambda\geq U\right),$

where

$\displaystyle\text{E}_{0}I\left(T\geq C\right)T$ $\displaystyle=$
$\displaystyle\int I\left(u\geq C\right)uf_{0}^{T}(u)du=\int I\left(u\geq
C\right)\frac{f_{1}^{T}(u)}{f_{0}^{T}(u)}f_{0}^{T}(u)du$ $\displaystyle=$
$\displaystyle{\Pr}_{1}\left(T\geq C\right),$

since $f_{1}^{T,A}(u,v)=u{\rm\;}f_{0}^{T,A}(u,v)$ implies that $\int
f_{1}^{T,A}(u,v)dv=u{\rm\;}\int f_{0}^{T,A}(u,v)dv$, i.e.

$f_{1}^{T}(u)/f_{0}^{T}(u)=u$. This provides

${\Pr}_{1}\left(T\geq C\right)\geq{\Pr}_{1}\left(\Lambda\geq U\right),$

meaning that if $f_{1}^{T,A}(u,v)=uf_{0}^{T,A}(u,v)$ then $T$ is MP, since
$\Lambda$ is MP. The proof is complete.

### Proof of Proposition 2.3

It is clear that $T=\Lambda=f_{1}(D)/f_{0}(D)$, i.e. $T$ is MP, implies
$\mathrm{E}_{1}\left\\{g(D)\right\\}=\mathrm{E}_{0}\left\\{g(D)T(D)\right\\}$.

Now, we assume that $T$ satisfies
$\mathrm{E}_{1}\left\\{g(D)\right\\}=\mathrm{E}_{0}\left\\{g(D)T(D)\right\\}$
and the event $T\geq C$ suggests to reject $H_{0}$. Consider the elementary
inequality: for all $B$ and $C$,

$(B-C)\left\\{I(B\geq C)-\delta\right\\}\geq
0,\,\,\,\delta\in\left[0,1\right].$

Suppose $\delta=I\left(\Lambda\geq U\right)$ denotes the $H_{0}$-rejection
rule of the likelihood ratio test with the fixed threshold $U$, such that: we
reject $H_{0}$, if $\delta=1$ and ${\Pr}_{0}\left(T\geq
C\right)={\Pr}_{0}\left(\Lambda\geq U\right)=\alpha$. In the elementary
inequality above, we define $B=T$ and then obtain

$\displaystyle 0\leq\mathrm{E}_{0}(T-C)\left\\{I(T\geq C)-I\left(\Lambda\geq
U\right)\right\\}\,=\,\mathrm{E}_{0}\left\\{I(T\geq
C)T\right\\}-\mathrm{E}_{0}\left\\{I\left(\Lambda\geq U\right)T\right\\}.$

This yields that

$\displaystyle 0\leq\mathrm{E}_{1}I(T\geq C)-\mathrm{E}_{1}I\left(\Lambda\geq
U\right),$

since
$\mathrm{E}_{0}\left\\{g(D)T(D)\right\\}=\mathrm{E}_{1}\left\\{g(D)\right\\}$,
for every $g\in[0,1]$. But, ${\Pr}_{1}\left(\Lambda\geq
U\right)\geq{\Pr}_{1}\left(T\geq C\right)$, since $\Lambda$ is MP. Therefore
${\Pr}_{1}\left(\Lambda\geq U\right)={\Pr}_{1}\left(T\geq C\right)$, meaning
that $T$ is MP. The proof is complete.

### Proof of Proposition 2.4

To prove Proposition 2.4, we note that, as mentioned above (see, e.g., the
proof of Proposition 2.2),
$f_{1}^{T_{1},T_{2}}(u,v)=uf_{0}^{T_{1},T_{2}}(u,v)$ implies
$f_{1}^{T_{1}}(u)=uf_{0}^{T_{1}}(u)$ that leads to
$\text{E}_{0}I\left(T_{1}\geq C\right)T_{1}={\Pr}_{1}\left(T_{1}\geq
C\right)$, for a fixed threshold $C$. Since
$f_{1}^{T_{1},T_{2}}(u,v)=uf_{0}^{T_{1},T_{2}}(u,v)$, we also have that

$\displaystyle\text{E}_{0}I\left(T_{2}\geq
U\right)T_{1}={\int\\!\\!\\!\\!\int}I\left(v\geq
U\right)u\,f_{0}^{T_{1},T_{2}}(u,v)dudv\,\,=\,\,{\Pr}_{1}\left(T_{2}\geq
U\right),$

where a fixed $U$ satisfies ${\Pr}_{0}\left(T_{1}\geq C\right)$
$={\Pr}_{0}\left(T_{2}\geq U\right)=\alpha$, $\alpha$ denotes the TIE rate of
the tests. Then, in a similar manner to the proof of Proposition 2.2, we apply
the inequality $\text{E}_{0}I\left(T_{1}\geq C\right)T_{1}$
$\geq\text{E}_{0}I\left(T_{2}\geq U\right)T_{1}$ to complete the proof.

### Proof of Proposition 2.5

By virtue of the statements presented above Proposition 2.5 in the article,
the proof is straightforward.

### Proof of Proposition 3.1

Consider the bivariate characteristic function

$\displaystyle\text{E}_{1}e^{\mathrm{i}t_{1}T_{N}+\mathrm{i}t_{2}T}\,=\,\text{E}_{1}e^{\mathrm{i}t_{1}T_{N}+\mathrm{i}t_{2}\psi\left(T_{N},A\right)}\,=\,\int\int
e^{\mathrm{i}t_{1}u+\mathrm{i}t_{2}\psi\left(u,x\right)}f_{1}^{T_{N},A}(u,x)dudx$
$\displaystyle\qquad=\,\int\int
e^{\mathrm{i}t_{1}u+\mathrm{i}t_{2}\psi\left(u,x\right)}f_{1}^{T_{N}}(u)f_{1}^{A}(x)dudx\,=\,\int\int
e^{\mathrm{i}t_{1}u+\mathrm{i}t_{2}\psi\left(u,x\right)}f_{1}^{T_{N}}(u)f_{0}^{A}(x)dudx$
$\displaystyle\qquad=\,\int\int
L^{T_{N}}(u)\,e^{\mathrm{i}t_{1}u+\mathrm{i}t_{2}\psi\left(u,x\right)}f_{0}^{T_{N}}(u)f_{0}^{A}(x)dudx\,=\,\text{E}_{0}\left\\{L^{T_{N}}(T_{N})\,e^{\mathrm{i}t_{1}T_{N}+\mathrm{i}t_{2}\psi\left(T_{N},A\right)}\right\\}$
$\displaystyle=\text{E}_{0}\left\\{L^{T_{N}}(T_{N})\,e^{\mathrm{i}t_{1}T_{N}+\mathrm{i}t_{2}T}\right\\},$

where ${{\mathrm{i}}^{2}}=-1$ and $t_{j}\in{{\mathbb{R}}^{1}}$,
$j\in\\{1,2\\}$. In this equation, the Fourier transformations of
$f_{1}^{T_{N},T}(u,v)$ and $L^{T_{N}}(u)\,f_{0}^{T_{N},T}(u,v)$ are equivalent
that means $f_{1}^{T_{N},T}(u,v)$ $=L^{T_{N}}(u)\,f_{0}^{T_{N},T}(u,v)$. As
mentioned above Proposition 3.1 in the article, since the ratio $L^{T_{N}}(u)$
is a monotonically increasing function, we can without loss of generality
assume that $L^{T_{N}}(T_{N})=T_{N}$. Then, we have

$\displaystyle\text{E}_{1}e^{\mathrm{i}t_{1}T_{N}+\mathrm{i}t_{2}T}$
$\displaystyle=$
$\displaystyle\text{E}_{0}\left\\{T_{N}\,e^{\mathrm{i}t_{1}T_{N}+\mathrm{i}t_{2}T}\right\\},$

yielding $f_{1}^{T_{N},T}(u,v)$ $=u\,f_{0}^{T_{N},T}(u,v)$. Now, by virtue of
Proposition 2.4, the proof of Proposition 3.1 is complete, when $L^{Y}(u)$ is
a monotonically increasing function. Similarly, we can consider the case,
where $L^{Y}(u)$ is a monotonically decreasing function. The proof is
complete.

## TABLE S1

Table 4: The analysis of variance related to the linear regression fitted to
the observed log-transformed HDL-cholesterol measurements,
$Y_{i},i=1,\ldots,2910$, using the log-transformed TBARS measurements,
$Z_{i},i=1,\ldots,2910$, as a factor, in Section 5.

Residuals: Min 1Q Median 3Q Max -0.80510 -0.15987 0.03547 0.20073 0.60918
Coefficients: Estimate Std. Error t value $\Pr(>|t|)$ (Intercept) 4.0342
0.03131 207.371 $<$ 2e-16 $Z$ -0.0448 0.00568 -7.786 9.55e-15 Residual
standard error: 0.2693 on 2908 degrees of freedom Multiple R-squared: 0.02042,
Adjusted R-squared: 0.02008 F-statistic: 60.62 on 1 and 2908 DF, p-value:
9.552e-15