Build one test for a one-sided family

Try

Test a Bernoulli probability against increases above p₀. In the previous lesson, every fixed larger probability p₁ produced the same ordering: larger success totals have larger likelihood ratios. This raises a stronger possibility. Could one upper-tail rule be most powerful against every p₁>p₀? The answer is yes for this one-sided binomial family when the rule’s size is calibrated at the null boundary and the whole null is controlled. The common ordering is useful precisely because it does not depend on which larger alternative is selected.

Predict

Predict where the largest rejection probability occurs under H₀:p≤p₀ for an upper-tail rule. Intuitively it is at p₀, but intuition must be converted into a stochastic-ordering argument. Merely checking a few smaller parameter values cannot prove a supremum statement over a continuum. We will use a coupling that generates Bernoulli observations for every p on one probability space and makes their monotonicity visible observation by observation.

Experiment

Take four fixed uniform values 0.1,0.3,0.6,0.9 as an illustrative record. For each p, set Xᵢ(p)=1 when the corresponding value is at most p. These are teaching values for inspecting the mechanism; random independent uniforms supply the actual coupling proof.

Probability p	Success vector	Total
0.2	1000	1
0.5	1100	2
0.8	1110	3

Increasing p never changes a success back to failure. Therefore any nondecreasing rejection function of the total also never decreases on each coupled outcome. Taking expectations proves monotonicity of the rejection probability, including rules that randomize at a boundary with an intermediate rejection probability.

Understand

A monotone likelihood-ratio family has ratios fθ₁/fθ₀ nondecreasing in a statistic T whenever θ₁>θ₀, under the relevant common-support conventions. In regular one-parameter settings this supports upper-tail tests. Rather than invoke the broad theorem without its details, the binomial case can be proved directly: its likelihood ratio is a constant times a base greater than one raised to S. Neyman–Pearson therefore makes an upper-tail size-α rule at p₀ most powerful against each fixed larger p₁. The coupling argument proves that no smaller null probability has larger rejection probability.

Combining those two facts gives uniform most-powerful, or UMP, status against the one-sided alternative. A competing level-α test for the composite null also has size at most α at p₀, so the simple-null optimality comparison applies at every larger p₁. “Uniform” refers to all alternative parameter values, not to equal power or equal rejection probability across them.

Apply

For two Bernoulli trials, reject only if both are successes. Its rejection probability is p². Under p≤1/2, size is 1/4; under p=3/4, power is 9/16. Derive both from the same function and state the different roles of the two parameter values. If a smaller target size is needed, randomizing at the high-total boundary can adjust it while preserving monotonicity.

For normal observations with known variance, the likelihood ratio for a larger mean is increasing in the sample sum. A mean threshold calibrated at μ₀ controls all μ≤μ₀ because shifting the mean down shifts the distribution left. With unknown variance, a nuisance parameter changes the problem: validity of a t test does not by itself prove unrestricted UMP optimality. Additional restrictions or different optimality criteria are needed.

Continue

Do not extend a one-sided UMP result to a two-sided alternative by attaching a second tail. Opposite alternatives generally prefer opposite error allocations. Continue to Likelihood-ratio tests for a general nested-model construction. Its appeal is broad applicability, while its calibration and optimality must be argued separately. This distinction prevents a useful testing recipe from being presented as a universal best test.