Basic Statistics and Data Analysis

Lecture notes, MCQS of Statistics

Non Central Chi Squared Distribution

The Non Central Chi Squared Distribution is a generalization of the Chi Squared Distribution.
If $Y_{1} ,Y_{2} ,\cdots ,Y_{n} \sim N(0,1)$ i.e. $(Y_{i} \sim N(0,1)) \Rightarrow y_{i}^{2} \sim \psi _{i}^{2}$ and $\sum y_{i}^{2}  \sim \psi _{(n)}^{2} $

If mean ($\mu $) is non-zero then $y_{i} \sim N(\mu _{i} ,1)$ i.e each $y_{i} $ has different mean
\begin{align*}
\Rightarrow  & \qquad y_i^2 \sim \psi_{1,\frac{\mu_i^2}{2}} \\
\Rightarrow  & \qquad \sum y_i^2 \sim \psi_{(n,\frac{\sum \mu_i^2}{2})} =\psi_{(n,\lambda )}^{2}
\end{align*}

Note that if $\lambda =0$ then we have central $\psi ^{2} $. If $\lambda \ne 0$ then it is non central chi squared distribution because it has no central mean (as distribution is not standard normal).

Central Chi-Square Distribution $f(x)=\frac{1}{2^{\frac{n}{2}} \left|\! {\overline{\frac{n}{2} }}  \right. } \chi ^{\frac{n}{2} -1} e^{-\frac{x}{2} }; \qquad 0<x<\infty $

Theorem:

If $Y_{1} ,Y_{2} ,\cdots ,Y_{n} $ are independent normal random variables with $E(y_{i} )=\mu _{i} $ and $V(y_{i} )=1$ then $w=\sum y_{i}^{2}  $ is distributed as non central chi square with $n$ degree of freedom and non-central parameter $\lambda $, where $\lambda =\frac{\sum \mu _{i}^{2}  }{2} $ and has pdf

\begin{align*}
f(w)=e^{-\lambda } \sum _{i=0}^{\infty }\left[\frac{\lambda ^{i} w^{\frac{n+2i}{2} -1} e^{-\frac{w}{2} } }{i!\, 2^{\frac{n+2i}{2} } \left|\! {\overline{\frac{n+2i}{2} }}  \right. } \right]\qquad 0\le w\le \infty
\end{align*}

Proof:

Consider the moment generating function of $w=\sum y_{i}^{2}  $

\begin{align*}
M_{w} (t)=E(e^{wt} )=E(e^{t\sum y_{i}^{2}  } ); \qquad \text{ where } y_{i} \sim N(\mu \_{i} ,1)
\end{align*}

By definition
\begin{align*}
M_{w} (t) &= \int \cdots \int e^{t\sum y_{i}^{2} } .f(y_{i} )dy_{i} \\
&= K_{1} \int \cdots \int e^{-\frac{1}{2} (1-2t)\left[\sum y_{i}^{2} -\frac{2\sum y_{i} \mu _{i} }{1-2t} \right]}   dy_{1} .dy_{2} \cdots dy_{n} \\
&\text{By completing square}\\
& =K_{1} \int \cdots \int e^{\frac{1}{2} (1-2t)\sum \left[\left[y_{i} -\frac{\mu _{i} }{1-2t} \right]^{2} -\frac{\mu _{i}^{2} }{(1-2t)^{2} } \right]}   dy_{1} .dy_{2} \cdots dy_{n} \\
&= e^{-\frac{\sum \mu_{i}^{2} }{2} \left(1-\frac{1}{1-2t} \right)} \int \cdots \int \left(\frac{1}{\sqrt{2\pi } } \right)^{n} \frac{\frac{1}{\left(\sqrt{1-2t} \right)^{n} } }{\frac{1}{\left(\sqrt{1-2t} \right)^{n} } }  \, e^{-\frac{1}{2.\frac{1}{1-2t} } .\sum \left(y_{i} -\frac{\mu _{i} }{1-2t} \right)^{2} }  dy_{1} .dy_{2} \cdots dy_{n}\\
&=e^{-\frac{\sum \mu _{i}^{2} }{2} \left(1-\frac{1}{1-2t} \right)} .\frac{1}{\left(\sqrt{1-2t} \right)^{n} } \int \cdots \int \left(\frac{1}{\sqrt{2\pi } } \right)^{n}  \frac{1}{\left(\sqrt{\frac{1} {1-2t}} \right)^n} e^{-\, \frac{1}{2.\frac{1}{1-2t} } .\sum \left(y_{i} -\frac{\mu_i}{1-2t}\right)^{2} } dy_{1} .dy_{2} \cdots dy_{n}\\
\end{align*}

where

\[\int_{-\infty}^{\infty } \cdots \int _{-\infty }^{\infty }\left(\frac{1}{\sqrt{2\pi}} \right)^{n} \frac{1}{\left(\frac{1}{1-2t} \right)^{\frac{n}{2}}} e^{-{\frac{1}{2}.\frac{1}{1-2t} }} .\sum \left(y_{i} -\frac{\mu _{i} }{1-2t} \right)^{2} dy_{1} .dy_{2} \cdots dy_{n}\]
is integral of complete density

\begin{align*}
M_{w}(t)&=e^{-\frac{\sum \mu_i^2}{2} \left(1-\frac{1}{1-2t}\right)} .\left(\frac{1}{\sqrt{1-2t} } \right)^{n} \\
&=\left(\frac{1}{\sqrt{1-2t}}\right)^{n} e^{-\lambda \left(1-\frac{1}{1-2t} \right)} \\
&=e^{-\lambda }.e^{\frac{\lambda}{1-2t}} \frac{1}{(1-2t)^{\frac{n}{2}}}\\
&\text{Using Taylor series about zero}\\
&=e^{-\lambda } \sum _{i=0}^{\infty }\frac{\lambda ^{i} }{i!(1-2t)^{i} (1-2t)^{n/2} }\\
M_{w=y_{i}^{2} } (t)&=e^{-\lambda } \sum _{i=0}^{\infty }\frac{\lambda ^{i} }{i!(1-2t)^{\frac{n+2i}{2} } }\tag{A}
\end{align*}

Now Moment Generating Function (MGF) for non-central distribution for a given density function is
\begin{align*}
M_{\omega} (t) & = E(e^{\omega t} )\\
&=\int _{0}^{\infty }e^{\omega \lambda } e^{-\lambda } \sum _{i=0}^{\infty }\frac{\lambda ^{i} \omega ^{\frac{n+2i}{2} -1} e^{-\frac{\omega }{2} } }{i!2^{\frac{n+2i}{2} } \left|\! {\overline{\frac{n+2i}{2} }}  \right. } d\omega\\
&=e^{-\lambda } \sum _{i=0}^{\infty }\frac{\lambda ^{i} }{i!2^{\frac{n+2i}{2} } \left|\! {\overline{\frac{n+2i}{2} }}  \right. }  \int _{0}^{\infty }e^{\frac{\omega }{2} (1-2t)}  \omega ^{\frac{n+2i}{2} -1} d\omega
\end{align*}
Let
\begin{align*}
\frac{\omega }{2} (1-2t)&=P\\
\Rightarrow \omega & =\frac{2P}{1-2t} \\
\Rightarrow d\omega &=\frac{2dp}{1-2t}
\end{align*}

\begin{align*}
&=e^{-\lambda } \sum\limits_{i=0}^{\infty }\frac{\lambda ^{i} }{i!2^{\frac{n+2i}{2} } \left|\! {\overline{\frac{n+2i}{2} }}  \right. }  \int _{0}^{\infty }e^{-P} \left(\frac{2P}{1-2t} \right)^{\frac{n+2i}{2} -1} \frac{2dP}{1-2t}  \\
&=e^{-\lambda } \sum _{i=0}^{\infty }\frac{\lambda ^{i} 2^{\frac{n+2i}{2} } }{i!2^{\frac{n+2i}{2} } \left|\! {\overline{\frac{n+2i}{2} }}  \right. (1-2t)^{\frac{n+2i}{2} -1} } \int _{0}^{\infty }e^{-P} P^{\frac{n+2i}{2} -1}  dP \\
&=e^{-\lambda } \sum _{i=0}^{\infty }\frac{\lambda ^{i} }{i!\left|\! {\overline{\frac{n+2i}{2} }}  \right. (1-2t)^{\frac{n+2i}{2} } } \left|\! {\overline{\frac{n+2i}{2} }}  \right.
\end{align*}

as \[\int\limits _{0}^{\infty }e^{-P} P^{\frac{n+2i}{2} -1}  dP=\left|\! {\overline{\frac{n+2i}{2} }}  \right. \]

\[M_{\omega } (t)=e^{-\lambda } \sum _{i=0}^{\infty }\frac{\lambda ^{i} }{i!(1-2t)^{\frac{n+2i}{2} } }  \tag{B}\]

Comparing ($A$) and ($B$)
\[M_{w=\sum y_{i}^{2} } (t)=M_{\omega } (t)\]

By Uniqueness theorem

\[f_{w} (w)=f_{\omega } (\omega )\]
\begin{align*}
\Rightarrow \qquad f_{w} (t)&=f(\psi ^{2} )\\
&=e^{-\lambda } \sum _{i=0}^{\infty }\frac{\lambda ^{i} w^{\frac{n+2i}{2} -1} e^{-\frac{w}{2} } }{i!2^{\frac{n+2i}{2} } \left|\! {\overline{\frac{n+2i}{2} }}  \right. };  \qquad o\le w\le \infty
\end{align*}
is the pdf of non central chi square with n df and $\lambda =\frac{\sum \mu _{i}^{2} }{2} $ is the non-centrality parameter. Non central chi squared distribution is also Additive as central chi square distribution.

Reference:

Download pdf file:

 

The Author

Muhammad Imdadullah

Student and Instructor of Statistics and business mathematics. Currently Ph.D. Scholar (Statistics), Bahauddin Zakariya University Multan.Like Applied Statistics and Mathematics and Statistical Computing. Statistical and Mathematical software used are: SAS, STATA, GRETL, EVIEWS, R, SPSS, VBA in MS-Excel.Like to use type-setting LaTeX for composing Articles, thesis etc.

Leave a Reply

Your email address will not be published. Required fields are marked *

Copy Right © 2011 ITFEATURE.COM
error: Content is protected !!