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

Definitions

Background

be k independent, normally distributed and unit variances. Then the random variable

which specifies the number of degrees of freedom by:

is sometimes called the noncentrality parameter 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-squared distribution is the squared norm of a random vector is a zero vector of length k is the identity matrix of size k.

Density

The probability density function (pdf) is given by

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, and the conditional distribution of Z given J=i is chi-squared with k+2i degrees of freedom. Then the unconditional distribution of Z is non-central chi-squared with k.

Alternatively, the pdf can be written as

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:

The case k=0 (zero degrees of freedom), in which case the distribution has a discrete component at zero, is discussed by Torgersen (1972) and further by Siegel (1979).

Derivation of the pdf

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

1. have unit variances, their joint distribution is spherically symmetric, up to a location shift.

2. .

3. (i.e. the k=1 case). Simple transformation of random variables shows that

   1. 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.

4. has a central chi-squared distribution with (k−, and the fact that the sum of chi-squared random variables is also a chi-square, completes the result. The indices in the series are (1+2i)+(k−1) =k+2i as required.

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-squared distributions, the cumulative distribution function (cdf) can be written as

is the cumulative distribution function of the central chi-squared distribution with k degrees of freedom which is given by

1. is the lower incomplete gamma function.

The Marcum Q-function can also be used to represent the cdf.

When the degrees of freedom k is positive odd integer, we have a closed form expression for the complementary cumulative distribution function given by

where n is non-negative integer, Q is the Gaussian Q-function, and I is the modified Bessel function of first kind with half-integer order. The modified Bessel function of first kind with half-integer order in itself can be represented as a finite sum in terms of hyperbolic functions.

In particular, for k = 1, we have

Also, for k = 3, we have

Approximation (including for quantiles)

Abdel-Aty derives (as "first approx.") a non-central Wilson–Hilferty transformation:

is approximately normally distributed i.e.,

, the (central) 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

1. denotes the cumulative distribution function of the standard normal distribution;

This and other approximations are discussed in a later text book.

More recently, since the CDF of non-central chi-squared distribution with odd degree of freedom can be exactly computed, the CDF for even degree of freedom can be approximated by exploiting the monotonicity and log-concavity properties of Marcum-Q function as

Another approximation that also serves as an upper bound is given by

, to compute approximate quantiles.

Related distributions

• is chi-square

• , is generalized chi-square distributed.

• then a noncentral F-distributed

• takes the Rice distribution.

• Normal approximation:.

• .

• random variables. It remains to plug in the MGF for the non-central chi square distributions into the product and compute the new MGF– this is left as an exercise. Alternatively it can be seen via the interpretation in the background section above as sums of squares of independent normally distributed random variables with variances of 1 and the specified means.

• The complex noncentral chi-squared distribution has applications in radio communication and radar systems. be independent scalar complex random variables with twice the degrees of freedom and twice the noncentrality parameter:

  1. ,

  .

Transformations

. He analyzes the expansions of the cumulants produce reasonable results:

can be used as a variance stabilizing transformation.

Usability of these transformations may be hampered by the need to take the square roots of negative numbers.

Occurrence and applications

Use in tolerance intervals

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

Notes

References

• Abramowitz, M. and Stegun, I. A. (1972), Handbook of Mathematical Functions, Dover.

• Johnson, N. L., Kotz, S., Balakrishnan, N. (1995), Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley.

• Muirhead, R. (2005) Aspects of Multivariate Statistical Theory (2nd Edition). Wiley.