Noncentral chi-square distribution

Probability distribution
name =Noncentral chi-square
type =density
pdf_

cdf_

parameters =$$k > 0, degrees of freedom
$$lambda > 0, non-centrality parameter

support =$$x in [0; +infty),
pdf =$$frac{1}{2}e^{-(x+lambda)/2}left (frac{x}{lambda} ight)^{k/4-1/2} I_{k/2-1}(sqrt{lambda x})
cdf =:$$sum_{j=0}^infty e^{-lambda/2} frac{(lambda/2)^j}{j!} frac{gamma(j+k/2,x/2)}{Gamma(j+k/2)},| mean =$$k+lambda,
median =
mode =
variance =$$2(k+2lambda),
skewness =$$frac{2^{3/2}(k+3lambda)}{(k+2lambda)^{3/2
kurtosis =$$frac{12(k+4lambda)}{(k+2lambda)^2}
entropy =
mgf =$$frac{expleft(frac{ lambda t}{1-2t } ight)}{(1-2 t)^{k/2 for $$2t<1
char =$$frac{expleft(frac{ilambda t}{1-2it} ight)}{(1-2it)^{k/2

In probability theory and statistics, the noncentral chi-square or noncentral $$chi^2 distribution is a generalization of the chi-square distribution. If $$X_i are "k" independent, normally distributed random variables with means $$mu_i and variances $$sigma_i^2, then the random variable

:$$sum_{i=1}^k left(frac{X_i}{sigma_i} ight)^2

is distributed according to the noncentral chi-square distribution. The noncentral chi-square distribution has two parameters: $$k which specifies the number of degrees of freedom (i.e. the number of $$X_i), and $$lambda which is related to the mean of the random variables $$X_i by:

:$$lambda=sum_{i=1}^k left(frac{mu_i}{sigma_i} ight)^2.

Note that some references define $$lambda as one half of the above sum.

Properties

The probability density function is given by:$$f_X(x; k,lambda) = sum_{i=0}^infty frac{e^{-lambda/2} (lambda/2)^i}{i!} f_{Y_{k+2i(x),where $$Y_q is distributed as chi-square with $$q degrees of freedom.

From this representation, the noncentral chi-squared distribution is seen to be a Poisson-weighted mixture of central chi-squared distributions. Suppose that a random variable "J" has a Poisson distribution with mean $$lambda/2, and the conditional distribution of "Z" given $$J=j is chi-squared with "k+2i" degrees of freedom. Then the unconditional distribution of "Z" is non-central chi-squared with "k" degrees of freedom, and non-centrality parameter $$lambda.

Alternatively, the pdf can be written as:$$f_X(x;k,lambda)=frac{1}{2} e^{-(x+lambda)/2} left (frac{x}{lambda} ight)^{k/4-1/2} I_{k/2-1}(sqrt{lambda x})

where $$I_ u(z) is a modified Bessel function of the first kind given by

:$$I_a(y) := (y/2)^a sum_{j=0}^infty frac{ (y^2/4)^j}{j! Gamma(a+j+1)}

The moment generating function is given by

:$$M(t;k,lambda)=frac{expleft(frac{ lambda t}{1-2t } ight)}{(1-2 t)^{k/2.

The first few raw moments are:

:$$mu^'_1=k+lambda:$$mu^'_2=(k+lambda)^2 + 2(k + 2lambda) :$$mu^'_3=(k+lambda)^3 + 6(k+lambda)(k+2lambda)+8(k+3lambda):$$mu^'_4=(k+lambda)^4+12(k+lambda)^2(k+2lambda)+4(11k^2+44klambda+36lambda^2)+48(k+4lambda)

The first few central moments are:

:$$mu_2=2(k+2lambda),:$$mu_3=8(k+3lambda),:$$mu_4=12(k+2lambda)^2+48(k+4lambda),

The "n"th cumulant is

:$$K_n=2^{n-1}(n-1)!(k+nlambda).,

Hence:$$mu^'_n = 2^{n-1}(n-1)!(k+nlambda)+sum_{j=1}^{n-1} frac{(n-1)!2^{j-1{(n-j)!}(k+jlambda )mu^'_{n-j}.

Again using the relation between the central and noncentral chi-square distributions, the cumulative distribution function (cdf) can be written as

:$$P(x; k, lambda ) = sum_{j=0}^infty e^{-lambda/2} frac{(lambda/2)^j}{j!} Q(x; k+2j)

where $$Q(x; k) is the cumulative distribution function of the central chi-squared distribution which is given by

:$$Q(x;k)=frac{gamma(k/2,x/2)}{Gamma(k/2)},

where $$gamma(k,z) is the lower incomplete Gamma function.

Derivation of the pdf

The derivation of the probability density function is most easily done by performing the following steps:

# First, assume without loss of generality that $$sigma_1=ldots=sigma_k=1. Then the joint distribution of $$X_1,ldots,X_k is spherically symmetric, up to a location shift.
# The spherical symmetry then implies that the distribution of $$X=X_1^2+ldots+X_k^2 depends on the means only through the squared length, $$lambda=mu_1^2+ldots+mu_k^2. Without loss of generality, we can therefore take $$mu_1=sqrt{lambda} and $$mu_2=dots=mu_k=0.
# Now derive the density of $$X=X_1^2 (i.e. "k=1" case). Simple transformation of random variables shows that :$$egin{align}f_X(x,1,lambda) &= frac{1}{2sqrt{xleft( phi(sqrt{x}-sqrt{lambda}) + phi(sqrt{x}+sqrt{lambda}) ight )\ &= frac{1}{sqrt{2pi x e^{-(x+lambda)/2} cosh(sqrt{lambda x}),\ end{align}
where $$phi(cdot) is the standard normal density.
# Expand the cosh term in a Taylor series. This gives the Poisson-weighted mixture representation of the density, still for "k=1". The indices on the chi-squared random variables in the series above are "1+2i" in this case.
# Finally, for the general case. We've assumed, wlog, that $$X_2,ldots,X_k are standard normal, and so $$X_2^2+ldots+X_k^2 has a "central" chi-squared distribution with "(k-1)" degrees of freedom, independent of $$X_1^2. Using the poisson-weighted mixture representation for $$X_1^2, and the fact that the sum of chi-squared random variables is also chi-squared, completes the result. The indices in the series are "(1+2i)+(k-1) = k+2i" as required.

Related distributions

*If $$Z is chi-square distributed $$Z sim chi_k^2 then $$Z is also non-central chi-square distributed: $$Z sim {chi'}^2_k(0)

*If $$J sim Poisson(lambda/2), then $${chi'}_k^2(lambda) sim chi_{k+2J}^2

Software

Many statistical software packages and libraries include functions for computing noncentral chisquare densities and probabilities. The table below gives commands for the following example problems:
# Density $$f_X(x;k,lambda) with "x=5.0", "k=3", $$lambda=1.5 (Answer: 0.097257)
# Cumulative probability $$P(x;k,lambda) with "x=5.0", "k=3", $$lambda=1.5 (Answer: 0.649285)
# Quantile: Find "x" in $$P(x;k,lambda)=a with "k=3", $$lambda=1.5, "a=0.5" (Answer: 3.668745)
# Random numbers: Generate 100 random observations from the distribution with "k=3", $$lambda=0.5

Note: Any software that produces the answers 0.101384, 0.490071, 5.09848 for the first three problems is including a factor of 0.5 in the definition of the noncentrality parameter. This is standard in statistics texts (e.g. [R. Christensen, "Plane Answers to Complex Questions" (3rd edition, 2002), Springer, NY, p.424.] ), but apparently not among programmers who don't read before writing their code.

References

* Abramowitz, M. and Stegun, I.A. (1972), "Handbook of Mathematical Functions", Dover. Section 26.4.25.
* Johnson, N. L. and Kotz, S., (1970), "Continuous Univariate Distributions", vol. 2, Houghton-Mifflin.