Background Let (X_1,X_2, \ldots, X_i, \ldots,X_k) be
k independent,
normally distributed random variables with means \mu_i and unit variances. Then the random variable : \sum_{i=1}^k X_i^2 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=\sum_{i=1}^k \mu_i^2. \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, \mu = (\mu_1, \ldots, \mu_k) and I_k is the
identity matrix of size
k.
Density The
probability density function (pdf) is given by : 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 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 + 2
i 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 : f_X(x;k,\lambda)=\frac 1 2 e^{-(x+\lambda)/2} \left (\frac x \lambda \right)^{k/4-1/2} I_{k/2-1}(\sqrt{\lambda x}) where I_\nu(y) is a modified
Bessel function of the first kind given by : I_\nu(y) = (y/2)^\nu \sum_{j=0}^\infty \frac{ (y^2/4)^j}{j! \Gamma(\nu+j+1)}. Using the relation between
Bessel functions and
hypergeometric functions, the pdf can also be written as: :f_X(x;k,\lambda)={{\rm e}^{-\lambda/2}} _0F_1(;k/2;\lambda x/4)\frac 1 {2^{k/2} \Gamma(k/2)} {\rm e}^{-x/2} x^{k/2-1}. 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: • Since X_1,\ldots,X_k have unit variances, their joint distribution is spherically symmetric, up to a location shift. • The spherical symmetry then implies that the distribution of X=X_1^2+\cdots+X_k^2 depends on the means only through the squared length, \lambda=\mu_1^2+\cdots+\mu_k^2. Without loss of generality, we can therefore take \mu_1=\sqrt{\lambda} and \mu_2=\cdots=\mu_k=0. • Now derive the density of X=X_1^2 (i.e. the
k = 1 case). Simple transformation of random variables shows that :::\begin{align}f_X(x,1,\lambda) &= \frac{1}{2\sqrt{x}}\left( \phi(\sqrt{x}-\sqrt{\lambda}) + \phi(\sqrt{x}+\sqrt{\lambda}) \right )\\ &= \frac{1}{\sqrt{2\pi x}} e^{-(x+\lambda)/2} \cosh(\sqrt{\lambda x}), \end{align} ::where \phi(\cdot) 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 + 2
i in this case. • Finally, for the general case. We've assumed, without loss of generality, that X_2,\ldots,X_k are standard normal, and so X_2^2+\cdots+X_k^2 has a
central chi-squared distribution with (
k − 1) degrees of freedom, independent of X_1^2. Using the poisson-weighted mixture representation for X_1^2, 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 + 2
i) + (
k − 1) =
k + 2
i as required. == Properties ==