Noncentral chi-squared distribution

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.

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]

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

Siegel (1979) discusses the case k = 0 specifically (zero degrees of freedom), in which case the distribution has a discrete component at zero.

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.

The moment-generating function is given by

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

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

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

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

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

Abdel-Aty[5] derives (as "first approx.") a non-central Wilson–Hilferty transformation:

${\displaystyle \left({\frac {\chi '^{2}}{k+\lambda }}\right)^{\frac {1}{3}}}$ is approximately normally distributed, ${\displaystyle \sim {\mathcal {N}}\left(1-{\frac {2}{9f}},{\frac {2}{9f}}\right),}$ i.e.,

${\displaystyle P(x;k,\lambda )\approx \Phi \left\{{\frac {\left({\frac {x}{k+\lambda }}\right)^{1/3}-\left(1-{\frac {2}{9f}}\right)}{\sqrt {\frac {2}{9f}}}}\right\},{\text{where }}\ f:={\frac {(k+\lambda )^{2}}{k+2\lambda }}=k+{\frac {\lambda ^{2}}{k+2\lambda }},}$

which is quite accurate and well adapting to the noncentrality. Also, ${\displaystyle f=f(k,\lambda )}$ becomes ${\displaystyle f=k}$ for ${\displaystyle \lambda =0}$, the (central) chi-squared case.

Sankaran[6] discusses a number of closed form approximations for the cumulative distribution function. In an earlier paper,[7] he derived and states the following approximation:

${\displaystyle P(x;k,\lambda )\approx \Phi \left\{{\frac {({\frac {x}{k+\lambda }})^{h}-(1+hp(h-1-0.5(2-h)mp))}{h{\sqrt {2p}}(1+0.5mp)}}\right\}}$

where

${\displaystyle \Phi \lbrace \cdot \rbrace \,}$ denotes the cumulative distribution function of the standard normal distribution;

${\displaystyle h=1-{\frac {2}{3}}{\frac {(k+\lambda )(k+3\lambda )}{(k+2\lambda )^{2}}}\,;}$

${\displaystyle p={\frac {k+2\lambda }{(k+\lambda )^{2}}};}$

${\displaystyle m=(h-1)(1-3h)\,.}$

This and other approximations are discussed in a later text book.[8]

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

${\displaystyle P(x;2n,\lambda )\approx {\frac {1}{2}}\left[P(x;2n-1,\lambda )+P(x;2n+1,\lambda )\right].}$

Another approximation that also serves as an upper bound is given by

${\displaystyle P(x;2n,\lambda )\approx 1-\left[(1-P(x;2n-1,\lambda ))(1-P(x;2n+1,\lambda ))\right]^{1/2}.}$

For a given probability, these formulas are easily inverted to provide the corresponding approximation for ${\displaystyle x}$, to compute approximate quantiles.

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	${\displaystyle {\sqrt {\sum _{1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}}}$

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.