Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

however, one wishes to use the data to give a ‘yes’ or ‘no’ answer to a particular

question. For example, one might wish to know whether some assumed model

does, in fact, provide a good fit to the data, or whether two parameters have the

same value.

31.7.1 Simple and composite hypotheses

In order to use data to answer questions of this sort, the question must be

posed precisely. This is done by first asserting that somehypothesisis true.

The hypothesis under consideration is traditionally called thenull hypothesis

and is denoted byH 0. In particular, this usually specifies some formP(x|H 0 )

for the probability density function from which the dataxare drawn. If the

hypothesis determines the PDF uniquely, then it is said to be asimple hypothesis.

If, however, the hypothesis determines the functional form of the PDF but not the

values of certain parametersaon which it depends then it is called acomposite

hypothesis.

One decides whether toacceptorrejectthe null hypothesisH 0 by performing

somestatistical test, as described below in subsection 31.7.2. In fact, formally

one uses a statistical test to decide between the null hypothesisH 0 and the

alternative hypothesisH 1. We define the latter to be the complementH 0 of the

null hypothesiswithin some restricted hypothesis space known (or assumed) in

advance. Hence, rejection ofH 0 implies acceptance ofH 1 , and vice versa.

As an example, let us consider the case in which a samplexis drawn from a

Gaussian distribution with a known varianceσ^2 but with an unknown meanμ.

If one adopts the null hypothesisH 0 thatμ= 0, which we write asH 0 :μ=0,

then the corresponding alternative hypothesis must beH 1 :μ= 0. Note that,

in this case,H 0 is a simple hypothesis whereasH 1 is a composite hypothesis.

If, however, one adopted the null hypothesisH 0 :μ<0 then the alternative

hypothesis would beH 1 :μ≥0, so that bothH 0 andH 1 would be composite

hypotheses. Very occasionally bothH 0 andH 1 will be simple hypotheses. In our

illustration, this would occur, for example, if one knew in advance that the mean

μof the Gaussian distribution were equal to either zero or unity. In this case, if

one adopted the null hypothesisH 0 :μ= 0 then the alternative hypothesis would

beH 1 :μ=1.

31.7.2 Statistical tests

In our discussion of hypothesis testing we will restrict our attention to cases in

which the null hypothesisH 0 issimple(see above). We begin by constructing a

test statistict(x) from the data sample. Although, in general, the test statistic need

not be just a (scalar) number, and could be a multi-dimensional (vector) quantity,

we will restrict our attention to the former case. Like any statistic,t(x) will be a