Solution
The Most Powerful Test with a Leftover Budget
Show the problem again
A single observation $X\in\{1,2,3,4,5\}$ has one of two distributions: | x | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---| | $P_0(x)$ | 0.40 | 0.30 | 0.15 | 0.10 | 0.05 | | $P_1(x)$ | 0.10 | 0.20 | 0.30 | 0.15 | 0.25 | Among all tests of $H_0$ (distribution $P_0$) against $H_1$ (distribution $P_1$), including randomized tests, whose size is at most 0.12, what is the largest possible power? Give a decimal or reduced fraction.
Worked solution
By the Neyman-Pearson lemma the optimal test rejects for large likelihood ratio $P_1(x)/P_0(x)$. The ratios are $1/4, 2/3, 2, 3/2, 5$ for $x=1,\dots,5$, so the order of priority is $x=5,3,4,2,1$ (note it is not the natural order of $x$). Spend the size budget 0.12 in that order: $x=5$ costs 0.05 and buys power 0.25. The remaining 0.07 goes to $x=3$, which costs 0.15 for power 0.30; reject there with probability $0.07/0.15=7/15$, gaining $0.30\cdot7/15=0.14$. Total power $0.25+0.14=0.39=39/100$. Any nonrandomized test within budget does worse (for instance $\{5\}$ alone gives 0.25, $\{4,5\}$ gives 0.40 at size 0.15, over budget).
Source: Neyman-Pearson lemma (1933). Numbers and framing written for AxiomIQ. Statement written for AxiomIQ.