Lecture notes. Written from the material of Berkeley Stat 210A, licensed CC BY 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.

48. Testing with one real parameter¶

What this covers¶

The lecture's own outline for this session has three parts — one-sided tests, two-sided tests, and UMP unbiased tests — but the surviving material only reaches the first, and even that breaks off mid-example. This chapter is therefore short on purpose: it covers the single question the notes actually answer, which is what to do when testing a one-sided hypothesis about a real parameter and no uniformly most powerful (UMP) test exists. It assumes the Neyman–Pearson framework from earlier in the course — power functions, test size, and what "UMP" means for a simple-versus-simple test — and builds the score test as the natural substitute.

Why a one-sided hypothesis can defeat UMP¶

Take a family $\mathcal{P}=\{P_\theta:\theta\in\Theta\subseteq\mathbb{R}\}$ indexed by a single real parameter, fix $\theta_0\in\Theta$, and consider the one-sided hypothesis $$ H_0:\theta\le\theta_0 \qquad\text{vs.}\qquad H_1:\theta>\theta_0 . $$ Both sides are composite: $H_0$ covers a whole ray of parameter values, and so does $H_1$.

For a single alternative $\theta_1>\theta_0$, Neyman–Pearson gives the most powerful test of $\theta_0$ against $\theta_1$: reject when the likelihood ratio $p_{\theta_1}(X)/p_{\theta_0}(X)$ is large. The trouble is that this rejection region can change shape as $\theta_1$ moves — the notes record this plainly as "LRT may vary for different $\theta_1$ values." If the region that is best against one alternative is not the region that is best against another, no single test can be most powerful against every point of $H_1$ at once, and in general no UMP test exists for the one-sided problem.

Local alternatives and the score test¶

The fix the notes take is to stop trying to be optimal against a fixed, distant alternative and instead ask what happens against an alternative that sits just barely inside $H_1$: for $n$ large, prioritize $\theta_1=\theta_0+\varepsilon$ and let $\varepsilon\downarrow0$.

Write $\ell(\theta;x)=\log p_\theta(x)$ for the log-likelihood and $\dot\ell(\theta;x)=\partial_\theta\,\ell(\theta;x)$ for its derivative in $\theta$ — the score. The log likelihood ratio between the local alternative and the null point is, by a first-order Taylor expansion in $\varepsilon$ around $\varepsilon=0$, $$ \log LR(X)=\log\frac{p_{\theta_0+\varepsilon}(X)}{p_{\theta_0}(X)}=\ell(\theta_0+\varepsilon;X)-\ell(\theta_0;X)\approx \varepsilon\,\dot\ell(\theta_0;X). $$ Since $\varepsilon>0$ is fixed and small, ranking outcomes $X$ by $\log LR(X)$ agrees, to first order, with ranking them by $\dot\ell(\theta_0;X)$ alone — the $\varepsilon$ is just a positive scale factor. So the Neyman–Pearson rejection region for the local alternative, "reject for large $\log LR$", collapses in the limit $\varepsilon\downarrow0$ to "reject for large score at $\theta_0$." That is the argument for using the score at $\theta_0$ as the test statistic regardless of which nearby $\theta_1$ one had in mind — this is the construction usually called the score test: $$ \phi(X)=\mathbf{1}{\dot\ell(\theta_0;X)\ge c_\alpha}. $$

The local-alternative argument: instead of fixing one point of $H_1$, push the alternative $\theta_0+\varepsilon$ toward the boundary $\theta_0$ and see what the optimal rejection region converges to.

Checking the size against a composite null¶

Because $H_0:\theta\le\theta_0$ is composite, it is not enough to tune $c_\alpha$ so that $P_{\theta_0}(\phi(X)=1)=\alpha$ at the boundary point alone. Validity as a level-$\alpha$ test requires the power function $$ \beta_\phi(\theta)=P_\theta(\phi(X)=1) $$ to satisfy $\beta_\phi(\theta)\le\alpha$ for every $\theta\le\theta_0$, not just at $\theta_0$ itself. The notes flag this explicitly as a step that has to be checked in any application of the score test — they state the requirement but do not carry out the check.

The Laplace example, as far as the source goes¶

The notes open a worked example and then stop: let $X_1,\dots,X_n\overset{iid}{\sim}\frac12 e^{-|x-\theta|}$, the Laplace (double-exponential) distribution with location parameter $\theta$, and consider testing $H_0:\theta\le\theta_0$ against $H_1:\theta>\theta_0$ — presumably to work out the score $\dot\ell(\theta_0;X)$ for this family and see what the resulting test looks like. The source breaks off mid-sentence at this point, so no score function, test statistic, or size calculation for the Laplace case survives in the material available here.

Sources¶

All of the above is from the handwritten lecture note statistics/berkeley/stat210a/fall-2024/handwritten/lecture15-testonepar.md (converted from handwritten/lecture15-testonepar.pdf, licensed CC BY 4.0). The note itself was reconstructed by a model from a PDF with no text layer, and its own banner marks the prose as paraphrase and every equation as unverified — treat the equations here with the same caution.
The note's own outline lists three parts: one-sided tests in general, two-sided tests, and UMP unbiased tests. Only the first is present in the material supplied for this chapter; two-sided tests and UMP unbiased tests are named in the outline but their content is not in the source and is not covered here.
The worked example (testing a location parameter for the Laplace distribution) is stated in the source but not completed — the note cuts off after posing the hypotheses, before deriving the score or the test.
Background taken as already known — the Neyman–Pearson lemma, power functions, size, and what a UMP test is for a simple hypothesis — was not supplied with this chapter and is assumed from earlier in the course.

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