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| Let $X_1,\ldots,X_n$ be IID random variables, each with the standard normal distribution. Then, for any $\boldsymbol{\mu}\in\mathbb{R}^n$, the random variable $X$ defined by |
Let $X_1,\ldots,X_n$ be IID random variables, each with the standard normal distribution. Then, for any $\boldsymbol{\mu}\in\mathbb{R}^n$, the random variable $X$ defined by |
| \begin{equation*} |
\begin{equation*} |
| X=\sum_{i=1}^{n}(X_i+\mu_i)^2 |
X=\sum_{i=1}^{n}(X_i+\mu_i)^2 |
| \end{equation*} |
\end{equation*} |
| is called a \emph{non-central chi-squared random variable}. |
is called a \emph{non-central chi-squared random variable}. |
| Its distribution depends only on the number of degrees of freedom $n$ and non-centrality parameter $\lambda\equiv\Vert\boldsymbol{\mu}\Vert$. This is denoted by $\chi^2(n,\lambda)$ and has moment generating function |
Its distribution depends only on the number of degrees of freedom $n$ and non-centrality parameter $\lambda\equiv\Vert\boldsymbol{\mu}\Vert$. This is denoted by $\chi^2(n,\lambda)$ and has moment generating function |
| \begin{equation}\label{eq:1} |
\begin{equation}\label{eq:1} |
| \operatorname{M}_X(t)\equiv\mathbb{E}\left[e^{tX}\right]=\left(1-2t\right)^{-\frac{n}{2}}\exp\left(\frac{\lambda t}{1-2t}\right), |
\operatorname{M}_X(t)\equiv\mathbb{E}\left[e^{tX}\right]=\left(1-2t\right)^{-\frac{n}{2}}\exp\left(\frac{\lambda t}{1-2t}\right), |
| \end{equation} |
\end{equation} |
| which is defined for all $t\in\mathbb{C}$ with real part less than $1/2$. |
which is defined for all $t\in\mathbb{C}$ with real part less than $1/2$. |
| More generally, for any $n,\lambda\ge 0$, not necessarily integers, a random variable has the \emph{non-central chi-squared distribution}, $\chi^2(n,\lambda)$, if its moment generating function is given by (\ref{eq:1}). |
More generally, for any $n,\lambda\ge 0$, not necessarily integers, a random variable has the \emph{non-central chi-squared distribution}, $\chi^2(n,\lambda)$, if its moment generating function is given by (\ref{eq:1}). |
|
|
| A non-central chi-squared random variable for any $n,\lambda\ge 0$ can be constructed as follows. Let $Y$ be a (central) chi-squared variable with degree $n$, $Z_0,Z_1,\ldots$ be standard normals, and $N$ have the $\textrm{Poisson}(\lambda/2)$ distribution. If these are all independent then |
A non-central chi-squared random variable for any $n,\lambda\ge 0$ can be constructed as follows. Let $Y$ be a (central) chi-squared variable with degree $n$, $Z_0,Z_1,\ldots$ be standard normals, and $N$ have the $\textrm{Poisson}(\lambda/2)$ distribution. If these are all independent then |
| \begin{equation*} |
\begin{equation*} |
| X\equiv Y+\sum_{k=0}^{2N}Z_k^2. |
X\equiv Y+\sum_{k=0}^{2N}Z_k^2. |
| \end{equation*} |
\end{equation*} |
| has the $\chi^2(n,\lambda)$ distribution. Correspondingly, the probability density function for $X$ is |
has the $\chi^2(n,\lambda)$ distribution. Correspondingly, the probability density function for $X$ is |
|
\begin{equation}\label{eq:2}
|
\begin{equation}\label{eq:1}
|
| f_X(x)=\sum_{k=0}^\infty \frac{\lambda^k}{2^k k!}e^{-\lambda/2} f_{n+2k}(x), |
f_X(x)=\sum_{k=0}^\infty \frac{\lambda^k}{2^k k!}e^{-\lambda/2} f_{n+2k}(x), |
| \end{equation} |
\end{equation} |
| where $x>0$ and $f_k$ is the probability density of the $\chi^2_{(k)}$ distribution. |
where $x>0$ and $f_k$ is the probability density of the $\chi^2_{(k)}$ distribution. |
| Alternatively, this can be expressed as |
Alternatively, this can be expressed as |
| \begin{equation*} |
\begin{equation*} |
| f_X(x)=\frac{1}{2}e^{-(x+\lambda)/2}(x/\lambda)^{n/4-1/2}I_{n/2-1}\left(\sqrt{\lambda x}\right). |
f_X(x)=\frac{1}{2}e^{-(x+\lambda)/2}(x/\lambda)^{n/4-1/2}I_{n/2-1}\left(\sqrt{\lambda x}\right). |
| \end{equation*} |
\end{equation*} |
| where $I_\nu$ is a modified Bessel function of the first kind, |
where $I_\nu$ is a modified Bessel function of the first kind, |
| \begin{equation*} |
\begin{equation*} |
| I_\nu(x)=\sum_{k=0}^\infty\frac{\left(x/2\right)^{\nu+2k}}{k!\,\Gamma\left(\nu+k+1\right)}. |
I_\nu(x)=\sum_{k=0}^\infty\frac{\left(x/2\right)^{\nu+2k}}{k!\,\Gamma\left(\nu+k+1\right)}. |
| \end{equation*} |
\end{equation*} |
|
|
| \begin{figure}[H] |
\begin{figure}[H] |
| \centering |
\centering |
| \includegraphics{ncchisquared} |
\includegraphics{ncchisquared} |
| \caption{Densities of the non-central chi-squared distribution $\chi^2(n,\lambda)$.} |
\caption{Densities of the non-central chi-squared distribution $\chi^2(n,\lambda)$.} |
| \end{figure} |
\end{figure} |
|
|
| \textbf{Remarks} |
\textbf{Remarks} |
| \begin{enumerate} |
\begin{enumerate} |
| \item $\chi^2(n,\lambda)$ has mean $n+\lambda$ and variance $2n+4\lambda$. |
\item $\chi^2(n,\lambda)$ has mean $n+\lambda$ and variance $2n+4\lambda$. |
| \item $\chi^2(n,0)=\chi^2_{(n)}$. The (central) chi-squared random variable is a special case of the non-central chi-squared random variable, when the non-centrality parameter $\lambda=0$. |
\item $\chi^2(n,0)=\chi^2_{(n)}$. The (central) chi-squared random variable is a special case of the non-central chi-squared random variable, when the non-centrality parameter $\lambda=0$. |
| \item (The reproductive property of chi-squared distributions). If $Z_1,\ldots,Z_m$ are non-central chi-squared random variables such that each $Z_i\sim\chi^2(n_i,\lambda_i)$, then their total $Z=\sum Z_i$ is also a non-central chi-squared random variable with distribution $\chi^2(\sum n_i, \sum \lambda_i)$. |
\item (The reproductive property of chi-squared distributions). If $Z_1,\ldots,Z_m$ are non-central chi-squared random variables such that each $Z_i\sim\chi^2(n_i,\lambda_i)$, then their total $Z=\sum Z_i$ is also a non-central chi-squared random variable with distribution $\chi^2(\sum n_i, \sum \lambda_i)$. |
|
\item If $n>0$ then the $\chi^2(n,\lambda)$ distribution is restricted to the domain $(0,\infty)$ with probability density function (\ref{eq:2}). On the other hand, if $n=0$, then there is also an atom at $0$,
|
\item If $n>0$ then the $\chi^2(n,\lambda)$ distribution is restricted to the domain $(0,\infty)$ with probability density function (\ref{eq:1}). On the other hand, if $n=0$, then there is also an atom at $0$,
|
| \begin{equation*} |
\begin{equation*} |
| \mathbb{P}(X=0)=\lim_{t\rightarrow-\infty}\operatorname{M}_X(t)=e^{-\lambda/2}. |
\mathbb{P}(X=0)=\lim_{t\rightarrow-\infty}\operatorname{M}_X(t)=e^{-\lambda/2}. |
| \end{equation*} |
\end{equation*} |
| \item If $\boldsymbol{x}$ is a multivariate normally distributed $n$-dimensional random vector with distribution $\boldsymbol{N(\mu,V)}$ where $\boldsymbol{\mu}$ is the mean vector and $\boldsymbol{V}$ is the $n\times n$ covariance matrix. Suppose that $\boldsymbol{V}$ is singular, with $k$ = rank of $V<n$. Then $\boldsymbol{x^{\operatorname{T}}V^{-}x}$ is a non-central chi-squared random variable, where $\boldsymbol{V^{-}}$ is a generalized inverse of $\boldsymbol{V}$. Its distribution has $k$ degrees of freedom with non-centrality parameter $\lambda=\boldsymbol{\mu^{\operatorname{T}}V^{-}\mu}$. |
\item If $\boldsymbol{x}$ is a multivariate normally distributed $n$-dimensional random vector with distribution $\boldsymbol{N(\mu,V)}$ where $\boldsymbol{\mu}$ is the mean vector and $\boldsymbol{V}$ is the $n\times n$ covariance matrix. Suppose that $\boldsymbol{V}$ is singular, with $k$ = rank of $V<n$. Then $\boldsymbol{x^{\operatorname{T}}V^{-}x}$ is a non-central chi-squared random variable, where $\boldsymbol{V^{-}}$ is a generalized inverse of $\boldsymbol{V}$. Its distribution has $k$ degrees of freedom with non-centrality parameter $\lambda=\boldsymbol{\mu^{\operatorname{T}}V^{-}\mu}$. |
| \end{enumerate} |
\end{enumerate} |
