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The likelihood-ratio rule is a measurable test of exact size

Statement

In the setup of Likelihood-ratio tests with boundary randomization, for every $α \in [0, 1]$ the candidate rejection function $φ_{α}^{*}$ is measurable into $[0, 1]$ , defines a randomized test, and has size exactly $α$ for the simple null $P_{0}$ .

Facts & Assumptions

Given: The common-dominator two-law experiment, specified density representatives $p_{0}, p_{1}$ , and $α \in [0, 1]$ . The proof uses only these fixed representatives.

[F1]

Likelihood-ratio tests with boundary randomization and Likelihood densities: $p_{0}, p_{1}$ are finite, nonnegative measurable densities; the candidate is $1_{U}$ at $α = 0$ and $1_{U} + 1_{A} + γ 1_{B}$ at $α > 0$ , where $U = {p_{0} = 0 < p_{1}}$ , $A = {p_{0} > 0, R > c}$ , and $B = {p_{0} > 0, R = c}$ ; in the latter case $c \geq 0$ , $0 \leq γ \leq 1$ , and $P_{0} (R > c) + γ P_{0} (R = c) = α$ .

[F2]

The common-support likelihood ratio is measurable: $R$ is finite, nonnegative, and measurable.

[F3]

A measurable function between measurable spaces: preimages of Borel sets under $p_{0}, p_{1}, R$ are measurable.

[F4]

Closure properties of measurable functions used by the integral: finite sums and real scalar multiples of measurable functions are measurable.

[F5]

Every measurable rejection function defines a randomized test: every measurable $[0, 1]$ -valued rejection function defines a binary randomized test.

[F6]

The Radon-Nikodym derivative as an almost-everywhere equivalence class and A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density: for every measurable $C$ , $P_{0} (C) = \int_{C} p_{0} d μ$ .

[F7]

Nonnegative simple measurable functions, The integral of a nonnegative simple function, and The nonnegative integral agrees with the simple integral on simple functions: the integral of a finite-valued nonnegative simple function is the sum of its values times the measures of their disjoint level sets.

[F8]

Measures are monotone: a measurable subset of a measurable $P_{0}$ -null set is $P_{0}$ -null.

[F9]

Expectation of a nonnegative or integrable random variable and Probability measures and probability spaces: $E_{P_{0}}$ is integration under the probability measure $P_{0}$ .

[F10]

Test size and power: for the simple null ${0}$ , size is the expectation of the rejection function under $P_{0}$ .

Proof

technique · direct

1.1F1F2F3F4

The sets $U, A, B$ in [F1] are measurable by [F2] and [F3]. For $α > 0$ they are pairwise disjoint, so [F4] makes $φ_{α}^{*} = 1_{U} + 1_{A} + γ 1_{B}$ measurable; for $α = 0$ , [F4] makes $1_{U}$ measurable.

1.2F1F6F7F8algebra

Let $N = {p_{0} = 0}$ . By [F1], [F6], and the zero-function case of [F7], $P_{0} (N) = \int_{N} p_{0} d μ = 0$ , since $p_{0}$ vanishes on $N$ . Thus $P_{0} (U) = 0$ by [F8].

2.1F1F5step 1.1algebra

The disjointness and $0 \leq γ \leq 1$ show that every value of $φ_{α}^{*}$ lies in $[0, 1]$ . Hence [F5] makes it a randomized test.

2.2F1F7F9F10step 1.2

If $α = 0$ , [F1] gives $φ_{0}^{*} = 1_{U}$ . The simple integral identity [F7] and [F9] give $E_{P_{0}} [φ_{0}^{*}] = P_{0} (U) = 0$ , so [F10] makes its size $0 = α$ .

3.1F1F7F9F10step 1.2algebra∎

Suppose $α > 0$ . Since $R = 0$ on $N$ and $c \geq 0$ , $A = {R > c}$ . Also $B = {R = c}$ when $c > 0$ ; when $c = 0$ , the two sets differ only by $N$ . Step 1.2 makes that difference $P_{0}$ -null. The three disjoint indicator terms in [F1] therefore have expectation $P_{0} (R > c) + γ P_{0} (R = c) = α$ by [F7], [F9], and the threshold equation in [F1]. By [F10], the test has size $α$ .

Depends on

Used by

A continuous likelihood-ratio test Example
Randomize at a discrete likelihood-ratio boundary Example
Neyman–Pearson lemma with singular alternative mass Theorem

Dependency tree · two levels

49 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

George Deligiannidis, SB2.1 Foundations of Statistical Inference, Chapter 13, §13.1.2, PDF p. 77 (standard reference, not scraped)
Anna Mikusheva, MIT 14.381 Lecture 9: Uniformly Most Powerful Tests, PDF pp. 2–3 (standard reference, not scraped)