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non-central chi-squared random variable


Let X1,,Xn be IID random variablesMathworldPlanetmath, each with the standard normal distributionMathworldPlanetmath. Then, for any 𝝁n, the random variable X defined by

X=ni=1(Xi+μi)2

is called a non-central chi-squared random variable. Its distributionPlanetmathPlanetmath depends only on the number of degrees of freedom n and non-centrality parameter λ𝝁. 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 with real part less than 1/2. More generally, for any n,λ0, 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,λ0 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 independentPlanetmathPlanetmath then

XY+2Nk=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).
Figure 1: DensitiesPlanetmathPlanetmath of the non-central chi-squared distribution χ2(n,λ).

Remarks

  1. 1.

    χ2(n,λ) has mean n+λ and varianceMathworldPlanetmath 2n+4λ.

  2. 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. 3.

    (The reproductive property of chi-squared distributions). If Z1,,Zm are non-central chi-squared random variables such that each Ziχ2(ni,λi), then their total Z=Zi is also a non-central chi-squared random variable with distribution χ2(ni,λi).

  4. 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,

    (X=0)=lim
  5. 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 matrixMathworldPlanetmath. Suppose that 𝑽 is singularPlanetmathPlanetmath, with k = rank of V<n. Then 𝒙𝐓𝑽-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=𝝁𝐓𝑽-𝝁.

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