Chapter 12 Nonparametric Hypothesis Tests............................
12.1 Introduction
In this chapter, we shall develop some hypothesis tests in situations where the data come
from a probability distribution whose underlying form is not specified. That is, it will not
be assumed that the underlying distribution is normal, or exponential, or any other given
type. Because no particular parametric form for the underlying distribution is assumed,
such tests are callednonparametric.
The strength of a nonparametric test resides in the fact that it can be applied without any
assumption on the form of the underlying distribution. Of course, if there is justification
for assuming a particular parametric form, such as the normal, then the relevant parametric
test should be employed.
In Section 12.2, we consider hypotheses concerning the median of a continuous dis-
tribution and show how thesign testcan be used in their study. In Section 12.3, we
consider thesigned rank test, which is used to test the hypothesis that a continuous popu-
lation distribution is symmetric about a specified value. In Section 12.4, we consider the
two-sample problem, where one wants to use data from two separate continuous distribu-
tions to test the hypothesis that the distributions are equal, and present therank sum test.
Finally, in Section 12.5 we study theruns test, which can be used to test the hypothesis that
a sequence of 0’s and 1’s constitutes a random sequence that does not follow any specified
pattern.
12.2 The Sign Test
LetX 1 ,...,Xndenote a sample from a continuous distributionFand suppose that we
are interested in testing the hypothesis that the median ofF, call itm, is equal to a
specified valuem 0. That is, consider a test of
H 0 :m=m 0 versus H 1 :m=m 0wheremis such thatF(m)=.5.
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