Noncentral chi-squared distribution


In probability theory and statistics, the noncentral chi-square distribution is a generalization of the chi-square distribution. It often arises in the power analysis of statistical tests in which the null distribution is a chi-square distribution; important examples of such tests are the likelihood-ratio tests.

Background

Let be k independent, normally distributed random variables with means and unit variances. Then the random variable
is distributed according to the noncentral chi-square distribution. It has two parameters: which specifies the number of degrees of freedom, and which is related to the mean of the random variables by:
is sometimes called the noncentrality parameter. Note that some references define in other ways, such as half of the above sum, or its square root.
This distribution arises in multivariate statistics as a derivative of the multivariate normal distribution. While the central chi-square distribution is the squared norm of a random vector with distribution, the non-central is the squared norm of a random vector with distribution. Here is a zero vector of length k, and is the identity matrix of size k.

Definition

The probability density function is given by
where is distributed as chi-square with degrees of freedom.
From this representation, the noncentral chi-square distribution is seen to be a Poisson-weighted mixture of central chi-square distributions. Suppose that a random variable J has a Poisson distribution with mean, and the conditional distribution of Z given J = i is chi-square with k + 2i degrees of freedom. Then the unconditional distribution of Z is non-central chi-square with k degrees of freedom, and non-centrality parameter.
Alternatively, the pdf can be written as
where is a modified Bessel function of the first kind given by
Using the relation between Bessel functions and hypergeometric functions, the pdf can also be written as:
Siegel discusses the case k = 0 specifically, in which case the distribution has a discrete component at zero.

Properties

Moment generating function

The moment-generating function is given by

Moments

The first few raw moments are:
The first few central moments are:
The nth cumulant is
Hence

Cumulative distribution function

Again using the relation between the central and noncentral chi-square distributions, the cumulative distribution function can be written as
where is the cumulative distribution function of the central chi-square distribution with k degrees of freedom which is given by
The Marcum Q-function can also be used to represent the cdf.

Approximation (including for quantiles)

Abdel-Aty derives a non-central Wilson-Hilferty approximation:
is approximately normally distributed, i.e.,
which is quite accurate and well adapting to the noncentrality. Also, becomes for, the chi-squared case.
Sankaran discusses a number of closed form approximations for the cumulative distribution function. In an earlier paper, he derived and states the following approximation:
where
This and other approximations are discussed in a later text book.
For a given probability, these formulas are easily inverted to provide the corresponding approximation for, to compute approximate quantiles.

Derivation of the pdf

The derivation of the probability density function is most easily done by performing the following steps:
  1. Since have unit variances, their joint distribution is spherically symmetric, up to a location shift.
  2. The spherical symmetry then implies that the distribution of depends on the means only through the squared length,. Without loss of generality, we can therefore take and.
  3. Now derive the density of . Simple transformation of random variables shows that
  4. 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-square random variables in the series above are 1 + 2i in this case.
  5. Finally, for the general case. We've assumed, without loss of generality, that are standard normal, and so has a central chi-square distribution with degrees of freedom, independent of. Using the poisson-weighted mixture representation for, and the fact that the sum of chi-square random variables is also a chi-square, completes the result. The indices in the series are + = k + 2i as required.

    Related distributions

where

Transformations

Sankaran discusses the transformations of the form
. He analyzes the expansions of the cumulants of up to the term and shows that the following choices of produce reasonable results:
Also, a simpler transformation can be used as a variance stabilizing transformation that produces a random variable with mean and variance.
Usability of these transformations may be hampered by the need to take the square roots of negative numbers.

NameStatistic
chi-square distribution
noncentral chi-square distribution
chi distribution
noncentral chi distribution

Occurrences

Use in tolerance intervals

Two-sided normal regression tolerance intervals can be obtained based on the noncentral chi-square distribution. This enables the calculation of a statistical interval within which, with some confidence level, a specified proportion of a sampled population falls.