Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

containingRof theM parameters.) IfH 0 is true then it follows from our

discussion in subsection 31.5.6 (although we shall not prove it) that, when the

sample sizeNis large, the quantity−2lntfollows approximately achi-squared

distribution of orderR.

31.7.5 Student’st-test

Student’st-test is just a special case of the generalised likelihood ratio test applied

to a samplex 1 ,x 2 ,...,xNdrawn independently from a Gaussian distribution for

whichboththe meanμand varianceσ^2 are unknown, and for which one wishes

to distinguish between the hypotheses

H 0 :μ=μ 0 , 0 <σ^2 <∞, and H 1 :μ=μ 0 , 0 <σ^2 <∞,

whereμ 0 is a given number. Here, the parameter spaceAis the half-plane

−∞<μ<∞,0<σ^2 <∞, whereas the subspaceScharacterised by the null

hypothesisH 0 is the lineμ=μ 0 ,0<σ^2 <∞.

The likelihood function for this situation is given by

L(x;μ, σ^2 )=

1

(2πσ^2 )N/^2

exp

[

−

∑

i(xi−μ)

2

2 σ^2

]

.

On the one hand, as shown in subsection 31.5.1, the values ofμandσ^2 that

maximiseLinAareμ=x ̄andσ^2 =s^2 ,wherex ̄is the sample mean ands^2 is

the sample variance. On the other hand, to maximiseLin the subspaceSwe set

μ=μ 0 , and the only remaining parameter isσ^2 ; the value ofσ^2 that maximises

Lis then easily found to be

σ̂^2 =^1

N

∑N

i=1

(xi−μ 0 )^2.

To retain, in due course, the standard notation for Student’st-test, in this section

we will denote the generalised likelihood ratio byλ(rather thant); it is thus

given by

λ(x)=

L(x;μ 0 ,σ̂^2 )

L(x;x, s ̄^2 )

=

[(2π/N)

∑

i(xi−μ^0 )

(^2) ]−N/ (^2) exp(−N/2)
[(2π/N)
∑
i(xi− ̄x)
(^2) ]−N/ (^2) exp(−N/2)=
[∑
i(xi− ̄x)
2
∑
i(xi−μ^0 )^2
]N/ 2

. (31.112)

Normally, our next step would be to find the sampling distribution ofλunder

the assumption thatH 0 were true. It is more conventional, however, to work in

terms of a related test statistict, which was first devised by William Gossett, who

wrote under the pen name of ‘Student’.