Chi-squared distribution (${\chi }_k^2$)

A ${\chi }_k^2$ variable is the sum of squares of $k$ independent standard normal variables.

For $d$ independent centered normal random variables of common variance $v$, $|\mathbf{X}{|}^2/v$ has a chi-squared distribution with $d$ degrees of freedom. Its density gives the displayed moments by a gamma integral. In three dimensions, $\mathbb{E}|\mathbf{X}{|}^{2n}=(2n+1)!!v^n$. Radial odd moments need not vanish, although any odd Cartesian product does.

A chi-bar-squared distribution is a mixture distribution ${\sum }_{j=0}^m{w}_j{\chi }_j^2$, where $w_j\ge 0$ and ${\sum }_j{w}_j=1$, and ${\chi }_0^2$ denotes a point mass at zero. It arises when a Gaussian likelihood is optimized under inequality constraints: projection onto a constraint cone selects different dimensions according to which inequalities are active. For example, if $W,Z$ are independent standard Gaussian random variables, then $Z^2+(-W{)}_{+}^2$ has distribution $\frac{1}{2}{\chi }_1^2+\frac{1}{2}{\chi }_2^2$. On $W\ge 0$ it is $Z^2$; on $W<0$ it is $Z^2+W^2$. Symmetry makes the sign of $W$ independent of $W^2$, proving the stated weights and component laws.

If $Z_1,\dots ,Z_{\nu }$ are independent standard normal random variables and $\lambda ={\sum }_j{\mu }_j^2$, then ${\sum }_j(Z_j+{\mu }_j{)}^2$ has a noncentral chi-squared distribution with $\nu$ degrees of freedom and noncentrality $\lambda$. For $\lambda =0$ it reduces to the chi-squared distribution. Squaring a shifted signed normal Wald statistic gives the case $\nu =1$.

For $Q\sim {\chi }_m^2(\lambda )$ with $m>2$, completing the square in its Gaussian representation gives $\mathbb{E}{e}^{-tQ}=(1+2t{)}^{-m/2}{e}^{-\lambda t/(1+2t)}$. Integrate the identity $1/q={\int }_0^{\infty }{e}^{-tq}dt$ and use Tonelli theorem to obtain the displayed reciprocal moment. It is strictly decreasing in $\lambda \ge 0$, since its positive integrand is strictly decreasing for every $t>0$. At $\lambda =0$ the integral equals $1/(m-2)$, proving finiteness and the sharp maximum over noncentrality parameters.

For $V\sim {\chi }_n^2$ and $x>0$, the upper tail is at most $e^{-x}$ above $n+2\sqrt{nx}+2x$, and the lower tail is at most $e^{-x}$ below $n-2\sqrt{nx}$. Indeed $\log \mathbb{E}{e}^{s(V-n)}=-ns-(n/2)\log (1-2s)\le n{s}^2/(1-2s)$ for $0<s<1/2$. The Chernoff bound with $s=\sqrt{x/n}/(1+2\sqrt{x/n})$ proves the upper estimate. For the lower estimate, $\log \mathbb{E}{e}^{-s(V-n)}\le n{s}^2$ for $s>0$, and optimization gives the result. The lower threshold can be negative; the chi-squared Chernoff lower-tail bound provides an always-positive alternative.

For $V\sim {\chi }_n^2$ and $0<a<1$, apply the Chernoff bound to $e^{-sV}$ using $\mathbb{E}{e}^{-sV}=(1+2s{)}^{-n/2}$. Optimization at $s=(a^{-1}-1)/2$ proves the displayed estimate. It remains useful when the additive lower threshold in the chi-squared concentration inequality is nonpositive. For $0<r\le 1$, choosing $a=e^{-4r}$ gives a tail at most $e^{-n{r}^2}$ because $e^{-4r}-1+4r\ge 2r^2$.

A sum of the squares of $n$ independent random variables with the standard normal distribution has the chi-squared distribution with $n$ degrees of freedom. A scaled Gaussian integral and factorization by independence give its moment-generating function $(1-2t{)}^{-n/2}$ for $t<1/2$. For positive $n$, this expectation diverges when $t\ge 1/2$.

The scaled inverse chi-squared distribution has densityIt is an inverse-gamma distribution under a change of parameters.

If $r<s$, coupling by extra independent squared normals gives ${\chi }_s^2={\chi }_r^2+W$ with $W\ge 0$, so every fixed upper quantile increases with the degrees of freedom.

Many quadratic and likelihood-ratio statistics converge under a regular null hypothesis to a chi-squared distribution. Using its quantiles for a finite sample is accurate when the sample is large relative to the model dimension and expected category counts are not too small.