non-central chi-squared random variable

Let $X_1,\mathrm{\dots },X_n$ be IID random variables, each with the standard normal distribution. Then, for any $𝝁\in {ℝ}^n$, the random variable $X$ defined by

$$X=\sum _{i=1}^n{(X_i+{\mu }_i)}^2$$

is called a non-central chi-squared random variable.Its distribution depends only on the number of degrees of freedom $n$ and non-centrality parameter $\lambda \equiv \parallel 𝝁\parallel$. This is denoted by ${\chi }^2(n,\lambda )$ and has moment generating function

$${\mathrm{M}}_X(t)\equiv 𝔼\left[e^{tX}\right]={\left(1-2t\right)}^{-\frac{n}{2}}\exp \left(\frac{\lambda t}{1-2t}\right),$$(1)

which is defined for all $t\in ℂ$ with real part less than $1/2$.More generally, for any $n,\lambda \ge 0$, not necessarily integers, a random variable has the non-central chi-squared distribution, ${\chi }^2(n,\lambda )$, if its moment generating function is given by (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 $n$, $Z_1,Z_2,\mathrm{\dots }$ be standard normals, and $N$ have the $\text{Poisson}(\lambda /2)$ distribution. If these are all independent then

$$X\equiv Y+\sum _{k=1}^{2N}{Z}_k^2.$$

has the ${\chi }^2(n,\lambda )$ distribution. Correspondingly, the probability density function for $X$ is

$$f_X(x)=\sum _{k=0}^{\mathrm{\infty }}\frac{{\lambda }^k}{2^{k}k!}{e}^{-\lambda /2}{f}_{n+2k}(x),$$(2)

where $x>0$ and $f_k$ is the probability density of the ${\chi }_{(k)}^2$ distribution.Alternatively, this can be expressed as

$$f_X(x)=\frac{1}{2}{e}^{-(x+\lambda )/2}{(x/\lambda )}^{n/4-1/2}{I}_{n/2-1}\left(\sqrt{\lambda x}\right).$$

where $I_{\nu }$ is a modified Bessel function of the first kind,

$$I_{\nu }(x)=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(x/2\right)}^{\nu +2k}}{k!\mathrm{\Gamma }\left(\nu +k+1\right)}.$$

Figure 1: Densities of the non-central chi-squared distribution ${\chi }^2(n,\lambda )$.

Remarks

1. ${\chi }^2(n,\lambda )$ has mean $n+\lambda$ and variance $2n+4\lambda$.

2. ${\chi }^2(n,0)={\chi }_{(n)}^2$. The (central) chi-squared random variable is a special case of the non-central chi-squared random variable, when the non-centrality parameter $\lambda =0$.

3. (The reproductive property of chi-squared distributions). If $Z_1,\mathrm{\dots },Z_m$ 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 ${\chi }^2(\sum n_i,\sum {\lambda }_i)$.

4. If $n>0$ then the ${\chi }^2(n,\lambda )$ distribution is restricted to the domain $(0,\mathrm{\infty })$ with probability density function (2). On the other hand, if $n=0$, then there is also an atom at $0$,

   $$\mathbb{P}(X=0)=\underset{t\to -\mathrm{\infty }}{lim}{\mathrm{M}}_X(t)=e^{-\lambda /2}.$$

5. If $𝒙$ is a multivariate normally distributed $n$-dimensional random vector with distribution $𝑵\mathbf{(}𝝁\mathbf{,}𝑽\mathbf{)}$ where $𝝁$ is the mean vector and $𝑽$ is the $n\times n$ covariance matrix. Suppose that $𝑽$ is singular, with $k$ = rank of . Then ${𝒙}^{𝐓}{𝑽}^{\mathbf{-}}𝒙$ is a non-central chi-squared random variable, where ${𝑽}^{\mathbf{-}}$ is a generalized inverse of $𝑽$. Its distribution has $k$ degrees of freedom with non-centrality parameter $\lambda ={𝝁}^{𝐓}{𝑽}^{\mathbf{-}}𝝁$.