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Hypothesis testing

where F is given by the variance ratio

2
N 1 s /(N 1 − 1) u 2
F = 1 ≡ (3.15)
2
N 2 s /(N 2 − 1) v 2
2
and s 1 and s 2 are the standard deviations of the two samples. On plotting λ as a function of F, it
is apparent that the rejection region λ < λ crit corresponds to a two-tailed test on F. Nevertheless,
as will shall see below, by defining the fraction (3.15) appropriately, it is customary to make a one-
tailed test on F. The distribution of F may be obtained in a reasonably straightforward manner by
2
making use of the distribution of the sample variance s given in (3.14). Under our null hypothesis
2
H 0 , the two Gaussian distributions share a common variance, which we denote by σ . Changing
2
2
2
the variable in (3.14) from s to u we find that u has the sampling distribution
( ) (N−1)/2 [ 2 ]
N − 1 1 (N − 1)u
2 (N−3)/2
2
P(u |H 0 ) = (u ) exp − .
1
2σ 2 Γ( (N − 1)) 2σ 2
2
2
2
Since u and v are independent, their joint distribution is simply the product of their individual
distributions and is given by
[ 2 2 ]
(N 1 − 1)u + (N 2 − 1)v
2
2
2 (N 1 −3)/2
2 (N 2 −3)/2
P(u |H 0 )P)v |H 0 ) = A(u ) (v ) exp − ,
2σ 2
where the constant A is given by
(N 1 − 1) (N 1 −1)/2 (N 2 − 1) (N 2 −1)/2
A = . (3.16)
1
1
σ
2 (N 1 +N 2 −2)/2 (N 1 +N 2 −2) Γ( (N 1 − 1))Γ( (N 2 − 1))
2 2
2
2
2
2
Now, for fixed v we have u = Fv and d(u ) = v dF. Thus, the joint sampling distribution
2
P(v , F|H 0 ) is obtained by requiring that
2
2
2
2
2
2
P(v , F|H 0 )d(v )dF = P(u |H 0 )P(v |H 0 )d(u )d(v ). (3.17)
2
2
In order to find the distribution of F alone, we now integrate P(v , F|H 0 ) with respect to v from
0 to ∞, from which we obtain
( ) (N 1 −1)/2 ( ) −(N 1 +N 2 −2)/2
N 1 − 1 1 N 1 − 1
P(F|H 0 ) = F (N 1 −3)/2 1 + F ,
1
1
N 2 − 1 B( (N 1 − 1), (N 2 − 1)) N 2 − 1
2 2
(3.18)
( )
1
where B 1 (N 1 − 1), (N 2 − 1) is the beta function defined below
2 2
1
∫
B(x, y) = t x−1 (1 − t) y−1 dt.
0
P(F|H 0 ) is called the F-distribution (or occasionally the Fisher distribution) with (N 1 −1, N 2 −1)
degrees of freedom.
IfarandomvariableX hasanF-distributionwithparametersd 1 andd 2 ,wewriteX ∼F(d 1 , d 2 ).
Then the probability density function (pdf) for X is given by
√
d
(d 1 x) 1 d d 2 ) d 1 d 1 +d 2
2
d
(d 1 x+d 2 ) 1 +d 2 1 ( d 1 2 d 1 ( d 1 ) − 2
f(x; d 1 , d 2 ) = ( ) = ( ) x 2 −1 1 + x
x B d 1 , d 2 B d 1 , d 2 d 2 d 2
2 2 2 2
for real x ≥ 0. In many applications, the parameters d 1 and d 2 are positive integers, but the
distribution is well-defined for positive real values of these parameters.
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