Noncentral chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral 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. Let be k independent, normally distributed random variables with means and unit variances. Then the random variable is distributed according to the noncentral chi-squared distribution. It has two parameters: which specifies the number of degrees of freedom (i.e. the number of ), 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-squared distribution is the squared norm of a random vector with distribution (i.e., the squared distance from the origin to a point taken at random from that 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. The probability density function (pdf) is given by where is distributed as chi-squared with 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 , 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 .