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A randomized UMP test for a binomial model

Example

Assume AC. Let $X = (X_{1}, X_{2}, X_{3}) \in {0, 1}^{3}$ have mass function $P_{p} (X = x) = p^{T (x)} (1 - p)^{3 - T (x)}$ for $0 < p < 1$ , where $T (x) = x_{1} + x_{2} + x_{3}$ . For $H_{0} : p \leq \frac{1}{2}$ against $H_{1} : p > \frac{1}{2}$ , the randomized test with rejection function $ϕ (x) = 1_{{T (x) > 2}} + \frac{1}{3} 1_{{T (x) = 2}}$ is UMP of level $α = \frac{1}{4}$ .

Verification

Given: The finite experiment and hypotheses in the statement.

[F1] In a family with MLR in $T$ , the upper boundary-randomized threshold test calibrated at $p_{0}$ is UMP for $p \leq p_{0}$ versus $p > p_{0}$ . (The Karlin–Rubin theorem for one-sided tests)

[F2] A randomized test is a binary-action probability kernel; its test function is a measurable rejection probability in $[0, 1]$ . (Randomized tests)

[F3] MLR means the likelihood ratio for every larger parameter is nondecreasing in $T$ , including the extended-ratio support convention. (Families with a monotone likelihood ratio)

[F4] The size of a test is the supremum of its null rejection probabilities, and level $α$ means that size is at most $α$ . (Test size and power)

[F5] A probability kernel assigns a probability measure to each observation and has measurable event-evaluation functions. (Measure kernel and probability kernel)

1.1givenalgebra

The sample space has eight points, and $\sum_{x \in {0, 1}^{3}} p^{T (x)} (1 - p)^{3 - T (x)} = (p + (1 - p))^{3} = 1$ , so the masses define a probability law; the statistic $T$ is measurable on the finite power-set space.

1.2givenF3algebra

For $0 < p_{1} < p_{2} < 1$ , the likelihood ratio is $\frac{p_{2}^{T (x)} (1 - p_{2})^{3 - T (x)}}{p_{1}^{T (x)} (1 - p_{1})^{3 - T (x)}} = {(\frac{1 - p_{2}}{1 - p_{1}})}^{3} {(\frac{p_{2} (1 - p_{1})}{p_{1} (1 - p_{2})})}^{T (x)}$ . The second base exceeds one since $p_{2} (1 - p_{1}) - p_{1} (1 - p_{2}) = p_{2} - p_{1} > 0$ , so the ratio increases with $T$ and [F3] applies.

1.3givenF2F5algebraconstruct

At $p_{0} = 1 / 2$ , each point has mass $1 / 8$ , with one point at $T = 3$ and three at $T = 2$ ; hence $P_{1 / 2} (T > 2) = 1 / 8$ and $P_{1 / 2} (T = 2) = 3 / 8$ . The threshold $c = 2, γ = 1 / 3$ therefore has endpoint-null rejection probability $1 / 8 + (1 / 3) (3 / 8) = 1 / 4$ . Define $K (x, B) = ϕ (x) 1_{{1 \in B}} + (1 - ϕ (x)) 1_{{0 \in B}}$ for $B \subseteq {0, 1}$ . The two weights are nonnegative and sum to one, and the formula is countably additive because each point belongs to at most one event in a disjoint sequence, so each section is a probability measure. Every event evaluation is measurable because the sample space is finite with its full sigma-algebra. Thus [F5] gives a probability kernel and [F2] makes it a randomized test with rejection function $ϕ$ .

2.1F1F4step 1.2step 1.3∎

By step 1.2 the family has MLR in $T$ , and step 1.3 calibrates the upper boundary-randomized threshold at $p_{0} = 1 / 2$ to $α = 1 / 4$ . The Karlin–Rubin result [F1] therefore makes this test UMP of level $1 / 4$ for $H_{0} : p \leq 1 / 2$ versus $H_{1} : p > 1 / 2$ , with size and level interpreted by [F4].

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Sources

Jun Shao, Stat 610 Mathematical Statistics, Lecture 11, Theorem 8.3.17, PDF pp. 9–10 (standard reference, not scraped)