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The log-likelihood in the normal linear model, up to an additive constant, isFull column rank makes $X^{T}X$ positive-definite, so minimizing the residual sum of squares gives the normal equations $X^{T}X\hat{\beta }=X^{T}Y$. Maximizing over ${\sigma }^2$ afterward gives the maximum-likelihood estimatorsThe residual sum of squares is positive almost surely because $n-p>0$ and ${\sigma }^2>0$, so this maximization has an interior variance solution almost surely. The first estimator is a linear Gaussian statistic with mean $\beta$ and covariance ${\sigma }^2(X^{T}X{)}^{-1}$, giving

Let $P_X=X(X^{T}X{)}^{-1}{X}^T$ be the orthogonal projection onto the design column space, and let $A=I-P_X$. This is a symmetric idempotent matrix of rank $n-p$. Choose an orthonormal basis for its range and collect its columns in $L$, so $A=L{L}^T$, $L^{T}L=I_{n-p}$ and $L^{T}X=0$. ThenThe definition of the chi-squared distribution therefore givesThe maximum-likelihood denominator is $n$; the unbiased residual-variance estimator uses $n-p$.

Finally put $B=(X^{T}X{)}^{-1}{X}^T$. It has rank $p$ and $BA=0$, because $X^T(I-P_X)=0$. Part (a) implies that $BY=\hat{\beta }$ is independent of $Y^{T}AY=n{\hat{\sigma }}^2$. Multiplication by a fixed constant preserves independence, so the normal linear model maximum-likelihood sampling distributions satisfy