In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 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

Let be k independent, normally distributed random variables with means \mu_i and unit variances. Then the random variable is distributed according to the noncentral chi-squared distribution. It 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 is sometimes called the noncentrality parameter. Note that some references define \lambda 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 with N(0_k,I_k) distribution (i.e., the squared distance from the origin to a point taken at random from that distribution), the non-central \chi^2 is the squared norm of a random vector with N(\mu,I_k) distribution. Here 0_k is a zero vector of length k, and I_k is the identity matrix of size k.

Density

The probability density function (pdf) is given by where Y_q is distributed as chi-squared 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 = i 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 where I_\nu(y) 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:

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 where Q(x; k), is the cumulative distribution function of the central chi-squared distribution with k degrees of freedom which is given by The Marcum Q-function Q_M(a,b) 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., which is quite accurate and well adapting to the noncentrality. Also, becomes f = k for \lambda=0, 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 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 For a given probability, these formulas are easily inverted to provide the corresponding approximation for x, to compute approximate quantiles.

Related distributions

Transformations

Sankaran (1963) discusses the transformations of the form . He analyzes the expansions of the cumulants of z up to the term and shows that the following choices of b 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.

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.