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<center>'''Asymptotic Relative Efficiency in Testing'''</center>

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<center>Ya. Yu. Nikitin, yanikit47@gmail.com</center>



<center>''University of Saint-Petersburg''</center>



<center>September, 2010.</center>

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'''Keywords:'''  Pitman efficiency, Bahadur exact slope, Hodges-Lehmann index, Kullback-Leibler information, goodness-of-fit, testing of symmetry, independence test, large deviations.

====Asymptotic relative efficiency of two tests====
Making a substantiated choice of the most efficient statistical
test of several ones being at the disposal of the
statistician is regarded as one of the basic problems of
Statistics. This problem became especially important in the middle
of XX century when appeared computationally simple but
"inefficient" rank tests.

Asymptotic relative efficiency (ARE) is a notion which enables to
implement in large samples the quantitative comparison of two
different tests used for testing of the same statistical
hypothesis. The notion of the asymptotic efficiency of ''tests'' is more complicated than that of asymptotic efficiency of
''estimates.'' Various approaches to this notion were identified
only in late fourties and early fifties, hence, 20--25 years later
than in the estimation theory. We proceed now to their
description.

Let $\,\{T_n\}$ and $\,\{V_n\}$ be two sequences of statistics
based on $\,n$ observations and assigned for testing the
null-hypothesis $\,H$ against the alternative $\,A.$ We assume
that the alternative is characterized by real parameter $\,\theta$
and for $\,\theta=\theta_0$ turns into $\,H.$ Denote by
$\,N_T(\alpha, \beta, \theta)$ the sample size necessary for the
sequence $\,\{T_n\}$ in order to attain the power $\,\beta$
under the level $\,\alpha$ and the alternative value of
parameter $\,\theta.$ The number $\,N_V(\alpha, \beta,\theta)$ is
defined in the same way.

It is natural to prefer the sequence with smaller $N$. Therefore
the relative efficiency of the sequence $\,\{T_n\}$ with respect
to the sequence $\,\{V_n\}$ is specified as the quantity
$$ e_{T, V}(\alpha,\beta, \theta)=N_{V}(\alpha, \beta,
\theta)\big/N_{T}(\alpha,\beta,\theta)\,,$$ so that it is the
reciprocal ratio of sample sizes $\,N_T$ and $\,N_V.$
The merits of the relative efficiency as means for
comparing the tests are universally acknowledged. Unfortunately
it is extremely difficult to explicitly compute
$\,N_T(\alpha,\beta, \theta)$ even for the simplest sequences of
statistics $\,\{T_n\}.$ At present it is recognized that there is
a possibility to avoid this difficulty by calculating the
limiting values $\, e_{T, V}(\alpha,\beta,\theta)$ as
$\,\theta\to\theta_0,$ as $\,\alpha\to 0$ and as $\,\beta\to 1$
keeping two other parameters fixed. These limiting values $\,
e_{T, V}^P,\, e_{T, V}^B$ and $\, e_{T, V}^{HL} $ are called
respectively the Pitman, Bahadur and Hodges--Lehmann asymptotic
relative efficiency (ARE), they were proposed correspondingly in
{{Cite|Pit}}, {{Cite|Bah}} and {{Cite|Hod}}.

Only close
alternatives, high powers and small levels are of the most
interest from the practical point of view. It keeps one assured
that the knowledge of these ARE types will facilitate
comparing concurrent tests, thus producing well-founded
application recommendations.

The calculation of the mentioned three basic types of efficiency
is not easy, see the description of theory and many examples in
{{Cite|Serf1}}, {{Cite|Niki}} and {{Cite|Vaart}}. We only mention here,
that Pitman efficiency is based on the central limit theorem for
test statistics. On the contrary, Bahadur efficiency requires the
large deviation asymptotics of test statistics under the
null-hypothesis, while Hodges-Lehmann efficiency is connected with
large deviation asymptotics under the alternative. Each type of
efficiency has its own merits and drawbacks.

====Pitman efficiency====
Pitman efficiency is the classical notion used most often for the
asymptotic comparison of various tests.Under some regularity
conditions assuming asymptotic normality of test statistics under
$H$ and $A$, it is a number which has been gradually calculated
for numerous pairs of tests.

We quote now as an example one
of the first Pitman's results that stimulated the development
of nonparametric statistics. Consider the two-sample problem when
under the null-hypothesis both samples have the same continuous
distribution and under the alternative differ only in location.
Let $\,e_{W,\,t}^{\, P}$ be the Pitman ARE of the two-sample
Wilcoxon rank sum test with respect to the corresponding Student
test. Pitman proved that for Gaussian samples $ e_{W,\,t}^{\,P} =3
/\pi\approx 0.955\,,$ and it shows that the ARE of the Wilcoxon
test in the comparison with the Student test (being optimal in
this problem) is unexpectedly high. Later Hodges and Lehmann in
{{Cite|Hod}} proved that $$0.864\le e_{W,\,t}^P\le +\,\infty\,,$$ if
one rejects the assumption of normality and, moreover, the lower
bound is attained at the density
$$
f(x)=\begin{cases}
 3(5-x^2)/(20\sqrt 5) &\text{if } |x| \le \sqrt 5, \\
0 &\text{otherwise}.
\end{cases}
$$
Hence the Wilcoxon
rank test can be infinitely better than the parametric test of
Student but their ARE never falls below 0.864. See analogous
results in {{Cite|Serf}} where the calculation of ARE of related
estimators is discussed.

Another example is the comparison of independence tests based on
Spearman and Pearson correlation coefficients in bivariate normal
samples. Then the value of Pitman efficiency is $9/\pi^2 \approx
0.912.$

In numerical comparisons, the Pitman efficiency appears to be more
relevant for moderate sample sizes than other efficiencies
{{Cite|GG}}. On the other hand, Pitman ARE can be insufficient for
the comparison of tests. Suppose, for instance, that we have a
normally distributed sample with the mean $\theta$ and variance 1
and we are testing $H: \theta =0$ against $A:\theta > 0.$ Let us
compare two significance tests based on the sample mean $\bar{X}$
and the Student ratio $t.$ As the $t-$test does not use the
information on the known variance, it should be inferior to the
optimal test using the sample mean. However, from the point of
view of Pitman efficiency, these two tests are equivalent. On the
contrary, Bahadur efficiency $e_{t,\bar{X}}^B(\theta)$ is strictly
less than 1 for any $\theta >0.$

If the condition of asymptotic normality
fails, considerable difficulties arise when calculating the Pitman
ARE as the latter may not at all exist or may depend on
$\,\alpha$ and $\,\beta.$ Usually one considers limiting Pitman
ARE as $\alpha \to 0.$ In {{Cite|Wie}} Wieand has established the
correspondence between this kind of ARE and the limiting
approximate Bahadur efficiency which is easy to calculate.

====Bahadur efficiency====
The Bahadur approach proposed in {{Cite|Bah}}, {{Cite|Bah67}} to
measuring the ARE prescribes one to fix the power of tests and to
compare the exponential rate of decrease of their sizes for the
increasing number of observations and fixed alternative. This
exponential rate for a sequence of statistics $\{T_n\}$ is usually
proportional to some non-random function $c_T(\theta)$ depending
on the alternative parameter $\theta$ which is called the ''exact slope'' of the sequence $\{T_n\}$. The Bahadur ARE $\,
e_{V,T}^{\, B} (\theta)$ of two sequences of statistics
$\,\{V_n\}$ and $\,\{T_n\}$ is defined by means of the formula
$$e_{V,T}^{\,B}(\theta) = c_V(\theta)\,\big/\,c_T(\theta)\,.$$
It is known that for the calculation of exact slopes it is
necessary to determine the large deviation asymptotics of a
sequence $\,\{T_n\}$ under the null-hypothesis. This problem is
always nontrivial, and the calculation of Bahadur efficiency
heavily depends on advancements in large deviation theory, see
{{Cite|DZ}} , {{Cite|DS}}.

It is important to note that there exists an upper bound for exact slopes
$$c_T(\theta) \leq 2K(\theta) $$ in terms of Kullback--Leibler
information number $K(\theta)$ which measures the "statistical
distance" between the alternative and the null-hypothesis. It is
sometimes compared in the literature with the Cramér--Rao
inequality in the estimation theory. Therefore the absolute
(nonrelative) Bahadur efficiency of the sequence $\{T_n\}$ can be
defined as $e_T^B(\theta) = c_T(\theta)/2K(\theta).$

It is proved that under some regularity conditions the likelihood
ratio statistic is asymptotically optimal in Bahadur sense
{{Cite|Bah67}}, {{Cite|Vaart|&sect; 16.6}}, {{Cite|Arc}}.

Often the exact Bahadur ARE is uncomputable for any alternative
$\theta$ but it is possible to calculate the limit of Bahadur ARE as
$\theta$ approaches the null-hypothesis. Then one speaks about the
''local'' Bahadur efficiency.

The indisputable merit of Bahadur efficiency consists in that it
can be calculated for statistics with non-normal asymptotic
distribution such as Kolmogorov-Smirnov, omega-square, Watson and
many other statistics.

Consider, for instance, the sample with the distribution function
(df) $F$ and suppose we are testing the goodness-of-fit hypothesis
$H_0: F=F_0$ for some known continuous df $F_0$ against the
alternative of location. Well-known distribution-free statistics
for this hypothesis are the Kolmogorov statistic $D_n$ and
omega-square statistic $\omega_n^2.$ The following table presents
their local absolute efficiency in case of six standard underlying
distributions: <!-- \vspace{0.3cm} -->

$ \renewcommand{\arraystretch}{1.2} $

<center>

{| border = "1"
|-
| style="text-align:center" |   Statistic  
| colspan="6" align="center" | Distribution 
|-
| style="text-align:center" |    
| style="text-align:center" |  Gauss  
| style="text-align:center" |  Logistic  
| style="text-align:center" |  Laplace  
| style="text-align:center" |  Hyperbolic Cosine  
| style="text-align:center" |  Cauchy  
| style="text-align:center" |  Gumbel  
|-
| style="text-align:center" |   $D_n$  
| style="text-align:center" |  0.637  
| style="text-align:center" |  0.750  
| style="text-align:center" |  1  
| style="text-align:center" |  0.811  
| style="text-align:center" |  0.811  
| style="text-align:center" |  0.541  
|-
| style="text-align:center" |   $\omega^2_n$  
| style="text-align:center" |  0.907  
| style="text-align:center" |  0.987  
| style="text-align:center" |  0.822  
| style="text-align:center" |  1  
| style="text-align:center" |  0.750  
| style="text-align:center" |  0.731  
|}


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Table 1: Some local Bahadur efficiencies.
</center>

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We see from Table 1 that the integral statistic $\omega_n^2$ is in
most cases preferable with respect to the supremum-type statistic
$D_n$. However, in the case of Laplace distribution the Kolmogorov
statistic is locally optimal, the same happens for the
Cramér-von Mises statistic in the case of hyperbolic cosine
distribution. This observation can be explained in the framework of
Bahadur local optimality, see {{Cite|Niki|Ch.6}}.

See also {{Cite|Niki}} for the calculation of local Bahadur
efficiencies in case of many other statistics.

====Hodges-Lehmann efficiency====
This type of the ARE proposed in {{Cite|Hod}}
is in the conformity with the classical Neyman-Pearson approach.
In contrast with Bahadur efficiency, let us fix the level of tests
and let compare the exponential rate of decrease of their
second-kind errors for the increasing number of observations and
fixed alternative. This exponential rate for a sequence of
statistics $\{T_n\}$ is measured by some non-random function
$d_T(\theta)$ which is called the Hodges-Lehmann index of the
sequence $\{T_n\}$. For two such sequences the Hodges--Lehmann ARE
is equal to the ratio of corresponding indices.

The computation of Hodges--Lehmann indices is difficult, as it
requires large deviation asymptotics of test statistics under the
alternative.

There exists an upper bound for
the Hodges--Lehmann indices analogous to the upper bound for
Bahadur exact slopes. As in the Bahadur theory the sequence of
statistics $\,\{T_n\}$ is said to be ''asymptotically optimal in the Hodges–Lehmann sense'' if this upper bound is attained.

The drawback of Hodges-Lehmann efficiency is that most ''two-sided'' tests like Kolmogorov and Cramér-von Mises tests are
asymptotically optimal, and hence this kind of efficiency cannot
discriminate between them. On the other hand, under some
regularity conditions the one-sided tests like linear rank tests
can be compared on the basis of their indices, and their
Hodges-Lehmann efficiency coincides locally with Bahadur
efficiency, see details in {{Cite|Niki}}.

Coupled with three ``basic" approaches to the ARE calculation
described above, intermediate approaches are also possible if the
transition to the limit occurs simultaneously for two parameters
at a controlled way. Thus emerged the Chernoff ARE introduced by
Chernoff {{Cite|chern}}, see also {{Cite|Kal}}; the intermediate, or
the Kallenberg ARE introduced by Kallenberg {{Cite|kall}}, and the
Borovkov-Mogulskii ARE, proposed in {{Cite|BM}}.

Large deviation approach to asymptotic efficiency of tests was
applied in recent years to more general problems. For instance,
the change-point, "signal plus white noise" and regression
problems were treated in {{Cite|PS}}, the tests for spectral
density of a stationary process were discussed in {{Cite|Kak}},
while {{Cite|Tan}} deals with the time series problems, and the
empirical likelihood for testing moment conditions is studied in
{{Cite|Otsu}}.

====References====
{|
|-
|valign="top"|{{Ref|Arc}}||valign="top"|  Arcones, M. (2005). Bahadur efficiency of the likelihood ratio test.  ''Mathematical Methods of Statistics'', '''14''', 163-179. 
|-
|valign="top"|{{Ref|Bah}}||valign="top"|  Bahadur, R.R. (1960). Stochastic comparison of tests. ''Ann. Mathem. Statist.'', '''31''',  276--295. 
|-
|valign="top"|{{Ref|Bah67}}||valign="top"|  Bahadur, R.R. (1967). Rates of convergence of estimates and test statistics.  ''Ann. Mathem. Statist.'', '''38''',  303--324. 
|-
|valign="top"|{{Ref|BM}}||valign="top"|  Borovkov, A. and Mogulskii, A. (1993). Large deviations and testing of statistical hypotheses. Siberian Adv. Math., '''2'''(3, 4); {\bf 3}(1, 2). 
|-
|valign="top"|{{Ref|chern}}||valign="top"|  Chernoff, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on sums of observations,  ''Ann. Mathem. Statist.'', '''23''', 493--507. 
|-
|valign="top"|{{Ref|DZ}}||valign="top"|  Dembo, A. and Zeitouni, O. (1998). Large deviations techniques and applications. 2nd ed. , Springer, New York. 
|-
|valign="top"|{{Ref|DS}}||valign="top"|  Deuschel, J.-D. and Stroock, D.(1989). Large Deviations. Academic Press, Boston. 
|-
|valign="top"|{{Ref|GG}}||valign="top"|  Groeneboom, P. and Oosterhoff, J. (1981). Bahadur efficiency and small sample efficiency.  ''Intern. Statist. Review'', '''49''',  127--141. 
|-
|valign="top"|{{Ref|Hod}}||valign="top"|  Hodges, J. and Lehmann, E.L. (1956). The efficiency of some nonparametric competitors of the $\, t$-test.  ''Ann. Mathem. Statist.'',  '''26''', 324--335. 
|-
|valign="top"|{{Ref|Kak}}||valign="top"|  Kakizawa, Y. (2005). Bahadur exact slopes of some tests for spectral densities.  ''J. Nonparametric Stat.'',  '''17''', 745-764. 
|-
|valign="top"|{{Ref|kall}}||valign="top"|  Kallenberg, W.C.M. (1983). Intermediate efficiency, theory and examples.  ''Ann. Statist.,''  '''11''', 170--182. 
|-
|valign="top"|{{Ref|Kal}}||valign="top"|  Kallenberg, W.C.M. (1982). Chernoff efficiency and deficiency.  ''Ann. Stat.'', '''10''', 583-594. 
|-
|valign="top"|{{Ref|Niki}}||valign="top"|  Nikitin, Ya.(1995). Asymptotic Efficiency of Nonparametric Tests. Cambridge University Press. 
|-
|valign="top"|{{Ref|Otsu}}||valign="top"|  Otsu T. (2010). On Bahadur efficiency of empirical likelihood. Journ. of Econometrics, '''157''', 248-256 
|-
|valign="top"|{{Ref|Pit}}||valign="top"|  Pitman, E.J.G. (1949). Lecture Notes on Nonparametric Statistical Inference. Columbia University: Mimeographed. 
|-
|valign="top"|{{Ref|PS}}||valign="top"|  Puhalskii, A., Spokoiny, V. (1998). On large-deviation efficiency in statistical inference.  ''Bernoulli'', '''4''', 203-272. 
|-
|valign="top"|{{Ref|Serf1}}||valign="top"|  Serfling, R. (1980). Approximation Theorems of Mathematical Statistics. John Wiley and Sons, New York. 
|-
|valign="top"|{{Ref|Serf}}||valign="top"|  Serfling, R. (2010). Asymptotic relative efficiency in estimation.  ''Internatioanal Encyclopedia of Statistical Sciences'', Springer. 
|-
|valign="top"|{{Ref|Tan}}||valign="top"|  Taniguchi, M. (2001). On large deviation asymptotics of some tests in time series.  ''Journ. Stat. Plann. Inference,'' '''97''', 191-200. 
|-
|valign="top"|{{Ref|Vaart}}||valign="top"|  Van der Vaart, A.W.(1998). Asymptotic Statistics. Cambridge University Press. 
|-
|valign="top"|{{Ref|Wie}}||valign="top"|  Wieand, H.S. (1976). A condition under which the Pitman and Bahadur approaches to efficiency coincide.   ''Ann. Statist.'', '''4''', 1003--1011.
|-
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Based on an article from Lovric, Miodrag (2011), International
Encyclopedia of Statistical Science. Heidelberg: Springer Science
+Business Media, LLC.

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