= Chi-bar-squared distribution
{title2=$\sum_jw_j\chi_j^2$}
A chi-bar-squared distribution is a <mixture distribution> $\sum_{j=0}^m w_j\chi_j^2$, where $w_j\geq0$ and $\sum_jw_j=1$, 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 $\tfrac12\chi_1^2+\tfrac12\chi_2^2$. On $W\geq0$ 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.
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