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:

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