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non-central chi-squared random variable∗
CWoo†
2013-03-21 18:31:48
Let X1 , . . . , Xn be IID random variables, each with the standard normal
distribution. Then, for any µ ∈ Rn , the random variable X defined by
X=
n
X
(Xi + µi )2
i=1
is called a non-central chi-squared random variable. Its distribution depends
only on the number of degrees of freedom n and non-centrality parameter λ ≡
kµk. This is denoted by χ2 (n, λ) and has moment generating function
tX λt
−n
2
,
(1)
exp
MX (t) ≡ E e
= (1 − 2t)
1 − 2t
which is defined for all t ∈ C with real part less than 1/2. More generally, for
any n, λ ≥ 0, not necessarily integers, a random variable has the non-central
chi-squared distribution, χ2 (n, λ), if its moment generating function is given by
(??).
A non-central chi-squared random variable for any n, λ ≥ 0 can be constructed as follows. Let Y be a (central) chi-squared variable with degree n,
Z1 , Z2 , . . . be standard normals, and N have the Poisson(λ/2) distribution. If
these are all independent then
X≡Y +
2N
X
Zk2 .
k=1
has the χ2 (n, λ) distribution. Correspondingly, the probability density function
for X is
∞
X
λk −λ/2
fX (x) =
e
fn+2k (x),
(2)
2k k!
k=0
∗ hNoncentralChisquaredRandomVariablei
created: h2013-03-21i by: hCWooi version:
h36628i Privacy setting: h1i hDefinitioni h62E99i h60E05i
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where x > 0 and fk is the probability density of the χ2(k) distribution. Alternatively, this can be expressed as
fX (x) =
√ 1 −(x+λ)/2
λx .
e
(x/λ)n/4−1/2 In/2−1
2
where Iν is a modified Bessel function of the first kind,
Iν (x) =
∞
X
k=0
ν+2k
(x/2)
.
k! Γ (ν + k + 1)
Figure 1: Densities of the non-central chi-squared distribution χ2 (n, λ).
Remarks
1. χ2 (n, λ) has mean n + λ and variance 2n + 4λ.
2. χ2 (n, 0) = χ2(n) . The (central) chi-squared random variable is a special case of the non-central chi-squared random variable, when the noncentrality parameter λ = 0.
3. (The reproductive property of chi-squared distributions). If Z1 , . . . , Zm
are non-central chi-squared
random variables such that each Zi ∼ χ2 (ni , λi ),
P
then their total Z =
ZiP
is also
Pa non-central chi-squared random variable with distribution χ2 ( ni , λi ).
4. If n > 0 then the χ2 (n, λ) distribution is restricted to the domain (0, ∞)
with probability density function (??). On the other hand, if n = 0, then
there is also an atom at 0,
P(X = 0) = lim MX (t) = e−λ/2 .
t→−∞
5. If x is a multivariate normally distributed n-dimensional random vector
with distribution N (µ, V ) where µ is the mean vector and V is the n × n
covariance matrix. Suppose that V is singular, with k = rank of V < n.
Then xT V − x is a non-central chi-squared random variable, where V − is
a generalized inverse of V . Its distribution has k degrees of freedom with
non-centrality parameter λ = µT V − µ.
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