Derivation — Taylor Expansion Proof of Wilks' Theorem

The previous note stated Wilks' theorem and previewed, in one paragraph, why a Taylor expansion of the log-likelihood is the engine behind it. This note carries that argument out in full, in two stages. First, the scalar case — testing a single fully-specified $\theta_0$ against an unrestricted one-dimensional $\theta$ — reduces to a short, self-contained calculation: expand $\ell(\theta_0;x)$ around the unrestricted MLE $\hat\theta$, use the fact that the score vanishes at $\hat\theta$ to kill the first-order term, and recognise the surviving quadratic term as (a constant times) the square of the already-known asymptotically Normal quantity $\hat\theta-\theta_0$. Second, the general multiparameter case extends this by the same spectral-decomposition machinery Cochran's theorem already supplied for splitting a sum of squares into independent chi-squared pieces — here applied not to $\bar X$ and $S^2$, but to the restricted and unrestricted maximum-likelihood fits.

Learning objectives. By the end of this note you should be able to: