10.6.∗Adaptive Procedures 621μ 1 ,μ 2 ,andσ^2 , such as the statistics
∑n^11(Xi−X)^2∑n^21(Yi−Y)^2,∑n^11|Xi−median(Xi)|∑n^21|Yi−median(Yi)|,
range(X 1 ,X 2 ,...,Xn 1 )
range(Y 1 ,Y 2 ,...,Yn 2 ).Thus, in general, we would hope to be able to find a selectorQthat is a function
of the complete sufficient statistics for the parameters, underH 0 ,sothatitis
independent of the test statistic.
It is particularly interesting to note that it is relatively easy to use this technique
innonparametricmethods by using the independence result based upon complete
sufficient statistics forparameters. For the situations here, we must find complete
sufficient statistics for a cdf,F, of the continuous type. In Chapter 7, it is shown
that the order statisticsY 1 <Y 2 <···<Ynof a random sample of sizenfrom a
distribution of the continuous type with pdfF′(x)=f(x) are sufficient statistics
for the “parameter”f(orF). Moreover, if the family of distributions contains all
probability density functions of the continuous type, the family of joint probability
density functions ofY 1 ,Y 2 ,...,Ynis also complete. That is, the order statistics
Y 1 ,Y 2 ,...,Ynare complete sufficient statistics for the parametersf(orF).
Accordingly, our selectorQis based upon those complete sufficient statistics, the
order statistics underH 0. This allows us to independently choose a distribution-
free test appropriate for this type of underlying distribution, and thus increase the
power of our test.
A statistical test that maintains the significance level close to a desired signif-
icance levelαfor a wide variety of underlying distributions with good (not neces-
sarily the best for any one type of distribution) power for all these distributions is
described as beingrobust. As an illustration, the pooledt-test (Student’st)usedto
test the equality of the means of two normal distributions is quite robustprovided
that the underlying distributions are rather close to normal ones with common vari-
ance. However, if the class of distributions includes those that are not too close to
normal ones, such as contaminated normal distributions, the test based upontis
notrobust; the significance level is not maintained and the power of thet-test can
be quite low for heavy-tailed distributions. As a matter of fact, the test based on
the Mann–Whitney–Wilcoxon statistic (Section 10.4) is a much more robust test
than that based upontif the class of distributions includes those with heavy tails.
In the following example, we illustrate a robust, adaptive, distribution-free pro-
cedure in the setting of the two-sample problem.
Example 10.6.1.LetX 1 ,X 2 ,...,Xn 1 be a random sample from a continuous-
type distribution with cdfF(x)andletY 1 ,Y 2 ,...,Yn 2 be a random sample from a
distribution with cdfF(x−Δ). Letn=n 1 +n 2 denote the combined sample size.
We test
H 0 :Δ=0versusH 1 :Δ> 0 ,
by using one of four distribution-free statistics, one being the Wilcoxon and the
other three being modifications of the Wilcoxon. In particular, the test statistics