In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral χ 2 {\displaystyle \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 ( X 1 , X 2 , … , X i , … , X k ) {\displaystyle (X_{1},X_{2},\ldots ,X_{i},\ldots ,X_{k})} be k independent, normally distributed random variables with means μ i {\displaystyle \mu _{i}} and unit variances. Then the random variable
∑ i = 1 k X i 2 {\displaystyle \sum _{i=1}^{k}X_{i}^{2}}
is distributed according to the noncentral chi-squared distribution. It has two parameters: k {\displaystyle k} which specifies the number of degrees of freedom (i.e. the number of X i {\displaystyle X_{i}} ), and λ {\displaystyle \lambda } which is related to the mean of the random variables X i {\displaystyle X_{i}} by:
λ = ∑ i = 1 k μ i 2 . {\displaystyle \lambda =\sum _{i=1}^{k}\mu _{i}^{2}.}
λ {\displaystyle \lambda } is sometimes called the noncentrality parameter. Note that some references define λ {\displaystyle \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 ) {\displaystyle 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 χ 2 {\displaystyle \chi ^{2}} is the squared norm of a random vector with N ( μ , I k ) {\displaystyle N(\mu ,I_{k})} distribution. Here 0 k {\displaystyle 0_{k}} is a zero vector of length k, μ = ( μ 1 , … , μ k ) {\displaystyle \mu =(\mu _{1},\ldots ,\mu _{k})} and I k {\displaystyle I_{k}} is the identity matrix of size k.
Density The probability density function (pdf) is given by
f X ( x ; k , λ ) = ∑ i = 0 ∞ e − λ / 2 ( λ / 2 ) i i ! f Y k + 2 i ( x ) , {\displaystyle 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 {\displaystyle Y_{q}} is distributed as chi-squared with q {\displaystyle 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 λ / 2 {\displaystyle \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 λ {\displaystyle \lambda } . Alternatively, the pdf can be written as
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