non-central chi-squared random variable
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∑i=1(Xi+μi)2 |
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≡∥𝝁∥. This is denoted by χ2(n,λ) and has moment generating function
MX(t)≡𝔼[etX]=(1-2t)-n2exp(λt1-2t), | (1) |
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 (1).
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∑k=1Z2k. |
has the χ2(n,λ) distribution. Correspondingly, the probability density function for X is
fX(x)=∞∑k=0λk2kk!e-λ/2fn+2k(x), | (2) |
where x>0 and fk is the probability density of the χ2(k) distribution. Alternatively, this can be expressed as
fX(x)=12e-(x+λ)/2(x/λ)n/4-1/2In/2-1(√λx). |
where Iν is a modified Bessel function of the first kind,
Iν(x)=∞∑k=0(x/2)ν+2kk!Γ(ν+k+1). |
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 non-centrality 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), then their total Z=∑Zi is also a 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 (2). On the other hand, if n=0, then there is also an atom at 0,
ℙ(X=0)=lim -
5.
If is a multivariate normally distributed -dimensional random vector with distribution where is the mean vector and is the covariance matrix
. Suppose that is singular
, with = rank of . Then is a non-central chi-squared random variable, where is a generalized inverse of . Its distribution has degrees of freedom with non-centrality parameter .
Title | non-central chi-squared random variable |
---|---|
Canonical name | NoncentralChisquaredRandomVariable |
Date of creation | 2013-03-22 14:56:16 |
Last modified on | 2013-03-22 14:56:16 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 11 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 62E99 |
Classification | msc 60E05 |
Synonym | non-central chi-squared distribution |
Defines | non-centrality parameter |