Background

Let ${\displaystyle (X_{1},X_{2},\ldots ,X_{i},\ldots ,X_{k})}$ be k independent, normally distributed random variables with means ${\displaystyle \mu _{i}}$ and unit variances. Then the random variable

${\displaystyle \sum _{i=1}^{k}X_{i}^{2}}$

is distributed according to the noncentral chi-squared distribution. It has two parameters: ${\displaystyle k}$ which specifies the number of degrees of freedom (i.e. the number of ${\displaystyle X_{i}}$), and ${\displaystyle \lambda }$ which is related to the mean of the random variables ${\displaystyle X_{i}}$ by:

${\displaystyle \lambda =\sum _{i=1}^{k}\mu _{i}^{2}.}$

${\displaystyle \lambda }$ is sometimes called the noncentrality parameter. Note that some references define ${\displaystyle \lambda }$ 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 ${\displaystyle N(0_{k},I_{k})}$ distribution (i.e., the squared distance from the origin to a point taken at random from that distribution), the non-central ${\displaystyle \chi ^{2}}$ is the squared norm of a random vector with ${\displaystyle N(\mu ,I_{k})}$ distribution. Here ${\displaystyle 0_{k}}$ is a zero vector of length k, ${\displaystyle \mu =(\mu _{1},\ldots ,\mu _{k})}$ and ${\displaystyle I_{k}}$ is the identity matrix of size k.

Density

The probability density function (pdf) is given by

${\displaystyle f_{X}(x;k,\lambda )=\sum _{i=0}^{\infty }{\frac {e^{-\lambda /2}(\lambda /2)^{i}}{i!}}f_{Y_{k+2i}}(x),}$

where ${\displaystyle Y_{q}}$ is distributed as chi-squared with ${\displaystyle q}$ 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 ${\displaystyle \lambda /2}$, 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 ${\displaystyle \lambda }$.

Alternatively, the pdf can be written as

${\displaystyle f_{X}(x;k,\lambda )={\frac {1}{2}}e^{-(x+\lambda )/2}\left({\frac {x}{\lambda }}\right)^{k/4-1/2}I_{k/2-1}({\sqrt {\lambda x}})}$

where ${\displaystyle I_{\nu }(y)}$ is a modified Bessel function of the first kind given by

${\displaystyle I_{\nu }(y)=(y/2)^{\nu }\sum _{j=0}^{\infty }{\frac {(y^{2}/4)^{j}}{j!\Gamma (\nu +j+1)}}.}$

Using the relation between Bessel functions and hypergeometric functions, the pdf can also be written as:^[2]

${\displaystyle f_{X}(x;k,\lambda )={{\rm {e}}^{-\lambda /2}}_{0}F_{1}(;k/2;\lambda x/4){\frac {1}{2^{k/2}\Gamma (k/2)}}{\rm {e}}^{-x/2}x^{k/2-1}.}$

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:

1. Since ${\displaystyle X_{1},\ldots ,X_{k}}$ have unit variances, their joint distribution is spherically symmetric, up to a location shift.
2. The spherical symmetry then implies that the distribution of ${\displaystyle X=X_{1}^{2}+\cdots +X_{k}^{2}}$ depends on the means only through the squared length, ${\displaystyle \lambda =\mu _{1}^{2}+\cdots +\mu _{k}^{2}}$. Without loss of generality, we can therefore take ${\displaystyle \mu _{1}={\sqrt {\lambda }}}$ and ${\displaystyle \mu _{2}=\cdots =\mu _{k}=0}$.
3. Now derive the density of ${\displaystyle X=X_{1}^{2}}$ (i.e. the k = 1 case). Simple transformation of random variables shows that

${\displaystyle {\begin{aligned}f_{X}(x,1,\lambda )&={\frac {1}{2{\sqrt {x}}}}\left(\phi ({\sqrt {x}}-{\sqrt {\lambda }})+\phi ({\sqrt {x}}+{\sqrt {\lambda }})\right)\\&={\frac {1}{\sqrt {2\pi x}}}e^{-(x+\lambda )/2}\cosh({\sqrt {\lambda x}}),\end{aligned}}}$

where ${\displaystyle \phi (\cdot )}$ is the standard normal density.

1. 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.
2. Finally, for the general case. We've assumed, without loss of generality, that ${\displaystyle X_{2},\ldots ,X_{k}}$ are standard normal, and so ${\displaystyle X_{2}^{2}+\cdots +X_{k}^{2}}$ has a central chi-squared distribution with (k − 1) degrees of freedom, independent of ${\displaystyle X_{1}^{2}}$. Using the poisson-weighted mixture representation for ${\displaystyle X_{1}^{2}}$, 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 + 2i) + (k − 1) = k + 2i as required.

Moment generating function

The moment-generating function is given by

${\displaystyle M(t;k,\lambda )={\frac {\exp \left({\frac {\lambda t}{1-2t}}\right)}{(1-2t)^{k/2}}}.}$

Moments

The first few raw moments are:

${\displaystyle \mu '_{1}=k+\lambda }$

${\displaystyle \mu '_{2}=(k+\lambda )^{2}+2(k+2\lambda )}$

${\displaystyle \mu '_{3}=(k+\lambda )^{3}+6(k+\lambda )(k+2\lambda )+8(k+3\lambda )}$

${\displaystyle \mu '_{4}=(k+\lambda )^{4}+12(k+\lambda )^{2}(k+2\lambda )+4(11k^{2}+44k\lambda +36\lambda ^{2})+48(k+4\lambda ).}$

The first few central moments are:

${\displaystyle \mu _{2}=2(k+2\lambda )\,}$

${\displaystyle \mu _{3}=8(k+3\lambda )\,}$

${\displaystyle \mu _{4}=12(k+2\lambda )^{2}+48(k+4\lambda )\,}$

The nth cumulant is

${\displaystyle \kappa _{n}=2^{n-1}(n-1)!(k+n\lambda ).\,}$

Hence

${\displaystyle \mu '_{n}=2^{n-1}(n-1)!(k+n\lambda )+\sum _{j=1}^{n-1}{\frac {(n-1)!2^{j-1}}{(n-j)!}}(k+j\lambda )\mu '_{n-j}.}$

Cumulative distribution function

Again using the relation between the central and noncentral chi-squared distributions, the cumulative distribution function (cdf) can be written as

${\displaystyle P(x;k,\lambda )=e^{-\lambda /2}\;\sum _{j=0}^{\infty }{\frac {(\lambda /2)^{j}}{j!}}Q(x;k+2j)}$

where ${\displaystyle Q(x;k)\,}$ is the cumulative distribution function of the central chi-squared distribution with k degrees of freedom which is given by

${\displaystyle Q(x;k)={\frac {\gamma (k/2,x/2)}{\Gamma (k/2)}}\,}$

and where ${\displaystyle \gamma (k,z)\,}$ is the lower incomplete gamma function.

The Marcum Q-function ${\displaystyle Q_{M}(a,b)}$ can also be used to represent the cdf.^[3]

${\displaystyle P(x;k,\lambda )=1-Q_{\frac {k}{2}}\left({\sqrt {\lambda }},{\sqrt {x}}\right)}$

When the degrees of freedom k is positive odd integer, we have a closed form expression for the complementary cumulative distribution function given by^[4]

${\displaystyle {\begin{aligned}P(x;2n+1,\lambda )&=1-Q_{n+1/2}({\sqrt {\lambda }},{\sqrt {x}})\\&=1-\left[Q({\sqrt {x}}-{\sqrt {\lambda }})+Q({\sqrt {x}}+{\sqrt {\lambda }})+e^{-(x+\lambda )/2}\sum _{m=1}^{n}\left({\frac {x}{\lambda }}\right)^{m/2-1/4}I_{m-1/2}({\sqrt {\lambda x}})\right],\end{aligned}}}$

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

${\displaystyle P(x;1,\lambda )=1-\left[Q({\sqrt {x}}-{\sqrt {\lambda }})+Q({\sqrt {x}}+{\sqrt {\lambda }})\right].}$

Also, for k = 3, we have

${\displaystyle P(x;3,\lambda )=1-\left[Q({\sqrt {x}}-{\sqrt {\lambda }})+Q({\sqrt {x}}+{\sqrt {\lambda }})+{\sqrt {\frac {2}{\pi }}}{\frac {\sinh({\sqrt {\lambda x}})}{\sqrt {\lambda }}}e^{-(x+\lambda )/2}\right].}$

Transformations

Sankaran (1963) discusses the transformations of the form ${\displaystyle z=[(X-b)/(k+\lambda )]^{1/2}}$. He analyzes the expansions of the cumulants of ${\displaystyle z}$ up to the term ${\displaystyle O((k+\lambda )^{-4})}$ and shows that the following choices of ${\displaystyle b}$ produce reasonable results:

• ${\displaystyle b=(k-1)/2}$ makes the second cumulant of ${\displaystyle z}$ approximately independent of ${\displaystyle \lambda }$
• ${\displaystyle b=(k-1)/3}$ makes the third cumulant of ${\displaystyle z}$ approximately independent of ${\displaystyle \lambda }$
• ${\displaystyle b=(k-1)/4}$ makes the fourth cumulant of ${\displaystyle z}$ approximately independent of ${\displaystyle \lambda }$

Also, a simpler transformation ${\displaystyle z_{1}=(X-(k-1)/2)^{1/2}}$ can be used as a variance stabilizing transformation that produces a random variable with mean ${\displaystyle (\lambda +(k-1)/2)^{1/2}}$ and variance ${\displaystyle O((k+\lambda )^{-2})}$.

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

Various chi and chi-squared distributions

Name	Statistic
chi-squared distribution	${\displaystyle \sum _{1}^{k}\left({\frac {X_{i}-\mu _{i}}{\sigma _{i}}}\right)^{2}}$
noncentral chi-squared distribution	${\displaystyle \sum _{1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}$
chi distribution	${\displaystyle {\sqrt {\sum _{1}^{k}\left({\frac {X_{i}-\mu _{i}}{\sigma _{i}}}\right)^{2}}}}$
noncentral chi distribution	${\displaystyle {\sqrt {\sum _{1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}}}$

Use in tolerance intervals

Two-sided normal regression tolerance intervals can be obtained based on the noncentral chi-squared distribution.^[10] This enables the calculation of a statistical interval within which, with some confidence level, a specified proportion of a sampled population falls.