Page 333 - Fundamentals of Probability and Statistics for Engineers

316 Fundamentals of Probability and Statistics for Engineers
q
q
mass function (pmf) p(x; ), where may be specified or unspecified. We denote
by hypothesis H the hypothesis that the sample represents n values of a random
q
q
variable with pdf f (x; ) or p(x; ). This hypothesis is called a simple hypothesis
when the underlying distribution is completely specified; that is, the parameter
values are specified together with the functional form of the pdf or the pmf;

otherwise, it is a composite hypothesis. To construct a criterion for hypotheses
testing, it is necessary that an alternative hypothesis be established against
which hypothesis H can be tested. An example of an alternative hypothesis is
simply another hypothesized distribution, or, as another example, hypothesis
H can be tested against the alternative hypothesis that hypothesis H is not true.
In our applications, the latter choice is considered more practical and we shall
in general deal with the task of either accepting or rejecting hypothesis H on
the basis of a sample from the population.

10.1.1 TYPE-I AND TYPE-II ERRORS

As in parameter estimation, errors or risks are inherent in deciding whether a
hypothesis H should be accepted or rejected on the basis of sample information.
Tests for hypotheses testing are therefore generally compared in terms of the
probabilities of errors that might be committed. There are basically two types
of errors that are likely to be made – namely, reject H when in fact H is true or,
alternatively, accept H when in fact H is false. We formalize the above with
Definition 10.1.
Definition 10.1. in testing hypothesis H, a Type-I error is committed when H

is rejected when in fact H is true; a Type-II error is committed when H is
accepted when in fact H is false.
In hypotheses testing, an important consideration in constructing statistical
tests is thus to control, insofar as possible, the probabilities of making these
errors. Let us note that, for a given test, an evaluation of Type-I errors can be
made when hypothesis H is given, that is, when a hypothesized distribution is
specified. In contrast, the specification of an alternative hypothesis dictates
Type-II error probabilities. In our problem, the alternative hypothesis is simply
that hypothesis H is not true. The fact that the class of alternatives is so large
makes it difficult to use Type-II errors as a criterion. In what follows, methods
of hypotheses testing are discussed based on Type-I errors only.

10.2 CHI-SQUARED GOODNESS-OF-FIT TEST

As mentioned above, the problem to be addressed is one of testing hypothesis H
that specifies the probability distribution for a population X compared with the

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