Noncentral chi distribution

If $X\_i$ are k independent, normally distributed random variables with means $\backslash mu\_i$ and variances $\backslash sigma\_i^2$, then the statistic

:$Z\; =\; \backslash sqrt\{\backslash sum\_\{i=1\}^k\; \backslash left(\backslash frac\{X\_i\}\{\backslash sigma\_i\}\backslash right)^2\}$

is distributed according to the noncentral chi distribution. The noncentral chi distribution has two parameters: $k$ which specifies the number of degrees of freedom (i.e. the number of $X\_i$), and $\backslash lambda$ which is related to the mean of the random variables $X\_i$ by:

:$\backslash lambda=\backslash sqrt\{\backslash sum\_\{i=1\}^k\; \backslash left(\backslash frac\{\backslash mu\_i\}\{\backslash sigma\_i\}\backslash right)^2\}$

The probability density function (pdf) is

:$f(x;k,\backslash lambda)=\backslash frac\{e^\{-(x^2+\backslash lambda^2)/2\}x^k\backslash lambda\}$

{(\lambda x)^{k/2}} I_{k/2-1}(\lambda x)

where $I\_\backslash nu(z)$ is a modified Bessel function of the first kind.

The first few raw moments are:

:$\backslash mu^\text{'}\_1=\backslash sqrt\{\backslash frac\{\backslash pi\}\{2\}\}L\_\{1/2\}^\{(k/2-1)\}\backslash left(\backslash frac\{-\backslash lambda^2\}\{2\}\backslash right)$

:$\backslash mu^\text{'}\_2=k+\backslash lambda^2$

:$\backslash mu^\text{'}\_3=3\backslash sqrt\{\backslash frac\{\backslash pi\}\{2\}\}L\_\{3/2\}^\{(k/2-1)\}\backslash left(\backslash frac\{-\backslash lambda^2\}\{2\}\backslash right)$

:$\backslash mu^\text{'}\_4=(k+\backslash lambda^2)^2+2(k+2\backslash lambda^2)$

where $L\_n^\{(a)\}(z)$ is a Laguerre function. Note that the 2$n$th moment is the same as the $n$th moment of the noncentral chi-squared distribution with $\backslash lambda$ being replaced by $\backslash lambda^2$.

Let $X\_j\; =\; (X\_\{1j\},\; X\_\{2j\}),\; j\; =\; 1,\; 2,\; \backslash dots\; n$, be a set of n independent and identically distributed bivariate normal random vectors with marginal distributions $N(\backslash mu\_i,\backslash sigma\_i^2),\; i=1,2$, correlation $\backslash rho$, and mean vector and covariance matrix

:$E(X\_j)=\; \backslash mu=(\backslash mu\_1,\; \backslash mu\_2)^T,\; \backslash qquad$

\Sigma =

\begin{bmatrix}

\sigma_{11} & \sigma_{12} \\

\sigma_{21} & \sigma_{22}

\end{bmatrix}

= \begin{bmatrix}

\sigma_1^2 & \rho \sigma_1 \sigma_2 \\

\rho \sigma_1 \sigma_2 & \sigma_2^2

\end{bmatrix},

with $\backslash Sigma$ positive definite. Define

:$$

U = \left[ \sum_{j=1}^n \frac{X_{1j}^2}{\sigma_1^2} \right]^{1/2}, \qquad

V = \left[ \sum_{j=1}^n \frac{X_{2j}^2}{\sigma_2^2} \right]^{1/2}.

Then the joint distribution of U, V is central or noncentral bivariate chi distribution with n degrees of freedom.{{cite journal|author=Marakatha Krishnan|title= The Noncentral Bivariate Chi Distribution|journal= SIAM Review |volume=9|issue=4|year=1967|pages=708–714|doi=10.1137/1009111|bibcode= 1967SIAMR...9..708K}}{{cite journal|author=P. R. Krishnaiah, P. Hagis, Jr. and L. Steinberg |title= A note on the bivariate chi distribution|journal=SIAM Review|volume= 5|year=1963|issue= 2|pages= 140–144|jstor=2027477|doi=10.1137/1005034|bibcode= 1963SIAMR...5..140K}}

If either or both $\backslash mu\_1\; \backslash neq\; 0$ or $\backslash mu\_2\; \backslash neq\; 0$ the distribution is a noncentral bivariate chi distribution.

• If $X$ is a random variable with the non-central chi distribution, the random variable $X^2$ will have the noncentral chi-squared distribution. Other related distributions may be seen there.
• If $X$ is chi distributed: $X\; \backslash sim\; \backslash chi\_k$ then $X$ is also non-central chi distributed: $X\; \backslash sim\; NC\backslash chi\_k(0)$. In other words, the chi distribution is a special case of the non-central chi distribution (i.e., with a non-centrality parameter of zero).
• A noncentral chi distribution with 2 degrees of freedom is equivalent to a Rice distribution with $\backslash sigma=1$.
• If X follows a noncentral chi distribution with 1 degree of freedom and noncentrality parameter λ, then σX follows a folded normal distribution whose parameters are equal to σλ and σ² for any value of σ.

{{Reflist}}

{{ProbDistributions|continuous-semi-infinite}}

{{DEFAULTSORT:Noncentral Chi Distribution}}

Category:Continuous distributions

Category:Noncentral distributions