Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.7 HYPOTHESIS TESTING

0

0

0. 05

0. 10

10 20 30 40

λ(u)

u

λcrit

ab

Figure 31.12 The sampling distributionP(u|H 0 )forN= 10; this is a chi-

squared distribution forN−1 degrees of freedom.

distribution with unknownμandσ, and we wish to distinguish between the two

hypotheses

H 0 :σ^2 =σ^20 , −∞<μ<∞ and H 1 :σ^2 =σ^20 , −∞<μ<∞,

whereσ^20 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 lineσ^2 =σ^20 ,−∞<μ<∞.

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

]

.

The maximum ofLinAoccurs atμ= ̄xandσ^2 =s^2 , whereas the maximum of

LinSis atμ= ̄xandσ^2 =σ 02. Thus, the generalised likelihood ratio is given by

λ(x)=

L(x;x, σ ̄ 02 )

L(x;x, s ̄^2 )

=

(u

N

)N/ 2

exp

[

−^12 (u−N)

]

,

where we have introduced the variable

u=

Ns^2

σ^20

=

∑

i(xi− ̄x)

2

σ^20

. (31.121)

An example of this distribution is plotted in figure 31.12 forN= 10. From

the figure, we see that the rejection regionλ<λcritcorresponds to a two-tailed

rejection region onugiven by

0 <u<a and b<u<∞,

whereaandbare such thatλcrit(a)=λcrit(b), as shown in figure 31.12. In practice,