University of California, Los Angeles Department of Statistics
Statistics 100B Instructor: Nicolas Christou
Noncentral χ2 and noncentral F distributions
Let Y1,Y2,…,Yn be i.i.d. random variables with Yi ∼ N(μi,σ2),i = 1,2,…,n. If each μi = 0 then n Y2
Q= i=1 i ∼χ2.Whatifeachμ ̸=0? σ2 n i
The m.g.f. of Q is given by n
Yi2
Y22
MQ(t)=E exp t
Let’s examine one of these expectations:
×E exp tσ2
i=1
σ2
=E exp tσ2
Yi2 ∞ 1 tyi2 (yi−μi)2 E exp tσ2 = σ√2πexp σ2 − 2σ2
−∞
Y12
Yn2 ×…×E exp tσ2
dyi.
2
.
Back to the expectation
Yi2tμ2i∞1 1−2tμi2
E exp tσ2 =exp σ2(1−2t) −∞ σ√2π×exp − 2σ2 yi−1−2t dyi. If we multiply and divide by √1 − 2t we have the integral of a normal pd.f. with mean μi
Evaluate the integral using:
tyi2 (yi −μi)2 yi2(1−2t) 2μiyi μ2i
σ2−2σ2 =−2σ2+2σ2−2σ2 tμ2i 1 − 2t
μi = σ2(1 − 2t) − 2σ2 yi − 1 − 2t
σ2 (and therefore it is equal to 1, to finally get 1−2t
Yi2 1 tμ2i E exp tσ2 = √1−2texp σ2(1−2t) .
n Now we can find the moment generating function of Q = i=1
−n tni=1μ2i MQ(t) = (1−2t) 2 exp σ2(1−2t) .
Yi2
.
1−2t
and variance
M (t)=(1−2t)−neθ t
follows the χ2 distribution with noncentrality parameter θ. We write Q ∼ χ2(n, θ). Therefore
σ2
In general, a random variable Q that has m.g.f. of the form Q 2 1−2t
nn Y2 μ2
Q= i=1 i ∼χ2 n, i .
σ2
Note: If the noncentrality parameter is zero then Q ∼ χ2n.
i=1
σ2