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'non-central chi-squared random variable'
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| Title of object: |
non-central chi-squared random variable |
| Canonical Name: |
NonCentralChiSquaredRandomVariable |
| Type: |
Definition |
| Created on: |
2005-01-07 16:53:14 |
| Modified on: |
2005-01-07 16:55:41 |
| Classification: |
msc:60E05, msc:62E99 |
| Defines: |
non-centrality parameter |
| Synonyms: |
non-central chi-squared random variable=non-central chi-squared distribution |
Preamble:
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Content:
Let $X_1,\ldots,X_n$ be iid random variables, each with the distribution $N(0,1)$, the standard normal distribution. Define new iid random variables $Y_i$, $i=1,\ldots,n$ by $$Y_i=X_i+c_i,$$ where $c_i\in\mathbb{R}$. Then the random variable $Z$, defined by
$$Z:=\sum_{i=1}^{n}Y_i^2=\sum_{i=1}^{n}X_i^2+2\sum_{i=1}^{n}X_ic_i+\sum_{i=1}^{n}c_i^2$$ is called a \emph{non-central chi-squared random variable}, whose distribution, called the \emph{non-central chi-squared distribution}, is denoted by $\chi^2(n,c)$, where $n$ is the degrees of freedom of the random variable, and $c$, called the \emph{non-centrality parameter}, is the sum of squares of the $c_i$'s, or the last term of the rightmost expression above.
\textbf{Remarks}
\begin{enumerate}
\item $\chi^2(n,c)$ has mean $n+c$ and variance $2n+4c$.
\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 $c=0$.
\item (The reproductive property of chi-squared distributions). If $Z_1,\ldots,Z_m$ are non-central chi-squared random variables such that each $Z_i\sim\chi^2(n_i,c_i)$, then their total $Z=\sum Z_i$ is also a non-central chi-squared random variable with distribution $\chi^2(\sum n_i, \sum c_i)$.
\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 $c=\boldsymbol{\mu^{\operatorname{T}}V^{-}\mu}$.
\end{enumerate} |
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