Noncentral F- and beta-distribution.

12 Let Y1,..., Yr; Ys+1,..., Yn be independently normally distributed with common variance σ2 and means E(Yi) =

ηi (i = 1,...,r); E(Yi) = 0 (i = s + 1,..., n).

(i) The probability density of W = r i=1 Y 2 i /

n i=s+1 Y 2 i is given by (7.6). The distribution of the constant multiple (n − s)W/r of W is the noncentral F -

distribution.

(ii) The distribution of the statistic B = r i=1 Y 2 i /(r i=1 Y 2 i + n i=s+1 Y 2 i ) is the noncentral beta-distribution, which has probability density

∞

k=0 Pk (ψ)g1 2 r+k, 1 2 (n−s)(b),

where gp,q

(b) = (p + q)
(p)(q)
bp−1 (1 − b)
q−1 , 0 ≤ b ≤ 1 (7.62)
is the probability density of the (central) beta-distribution.