494 Optimal Tests of HypotheseswhereZhas a standard normal distribution. Hence the asymptotic test would use
the critical regionC 2 ={|Tn|≥zα/ 2 }. By (8.3.7) the critical regionC 2 would have
approximate sizeα. In practice, we would useC 1 , becausetcritical values are gen-
erally larger thanzcritical values and, hence, the use ofC 1 would be conservative;
i.e., the size ofC 1 would be slightly smaller than that ofC 2. As Exercise 8.3.4
shows, the two-samplet-test is also asymptotically correct, provided the underlying
distributions have thesamevariance.
For nonnormal situations where the distribution is “close” to the normal distri-
bution, thet-test is essentially valid; i.e., the true level of significance is close to the
nominalα. In terms of robustness, we would say that for these situations thet-test
possessesrobustness of validity.Butthet-test may not possessrobustness of
power. For nonnormal situations, there are more powerful tests than thet-test;
see Chapter 10 for discussion.
For finite sample sizes and for distributions that are decidedly not normal, very
skewed for instance, the validity of thet-test may also be questionable, as we illus-
trate in the following simulation study.
Example 8.3.4(Skewed Contaminated Normal Family of Distributions).Consider
the random variableXgiven by
X=(1−I )Z+I Y, (8.3.8)whereZhas aN(0,1) distribution,Yhas aN(μc,σc^2 ) distribution,I has abin(1, )
distribution, andZ,Y,andI are mutually independent. Assume that < 0. 5
andσc>1, so thatY is the contaminating random variable in the mixture. If
μc =0,thenX has the contaminated normal distribution discussed in Section
3.4.1, which is symmetrically distributed about 0. Forμc = 0, the distribution ofX,
(8.3.8), is skewed and we call it theskewed contaminated normal distribution,
SCN( , σc,μC). Note thatE(X)= μcand in Exercise 8.3.18 the cdf and pdf ofX
are derived. The R functionrscngenerates random variates from this distribution.
In this example, we show the results of a small simulation study on the validity
of thet-test for random samples from the distribution ofX. Consider the one-sided
hypotheses
H 0 : μ=μXversusH 0 : μ<μX.
LetX 1 ,X 2 ,...,Xnbe a random sample from the distribution ofX.Asatest
statistic we consider thet-test discussed in Example 4.5.4, which is also given in
expression (8.3.6); that is, the test statistic isTn=(X−μX)/(Sn/
√
n), whereX
andSnare the sample mean and standard deviation ofX 1 ,X 2 ,...,Xn, respectively.
We set the level of significance atα=0.05 and used the decision rule: RejectH 0
ifTn≤t 0. 05 ,n− 1. For the study, we setn= 30, =0.20, andσc= 25. Forμc,we
selected the five values of 0, 5 , 10 ,15, and 20, as shown in Table 8.3.1. For each of
these five situations, we ran 10,000 simulations and recordedα̂, which is the number
of rejections ofH 0 divided by the number of simulations, i.e., the empiricalαlevel.
For the test to be valid,α̂should be close to the nominal value of 0.05. As
Table 8.3.1 shows, though, for all cases other thanμc=0,thet-test is quite liberal;
that is, its empirical significance level far exceeds the nominal 0.05 level (as Exercise