LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Binary zero-one loss risks are the testing error probabilities

Statement

Let $Θ_{0}, Θ_{1}$ be disjoint subsets of a parameter set $Θ$ in a statistical experiment. Let $K$ be a randomized test on the binary action space ${0, 1}$ , with $1$ meaning reject, and write $φ (x) = K (x, {1})$ . Define the zero-one error loss by $L (θ, a) = 1_{Θ_{0}} (θ) a + 1_{Θ_{1}} (θ) (1 - a),$ so $L$ is zero when $θ \notin Θ_{0} \cup Θ_{1}$ . For $β_{φ} (θ) = E_{θ} [φ (X)]$ and the risk $R (θ, K)$ of Loss functions and risk, $R (θ, K) = β_{φ} (θ) (θ \in Θ_{0}), R (θ, K) = 1 - β_{φ} (θ) (θ \in Θ_{1}) .$

Facts & Assumptions

Given: A statistical experiment, disjoint hypotheses $Θ_{0}, Θ_{1}$ , and a binary randomized test $K$ with test function $φ$ .

[F1]

Randomized tests: the action space is ${0, 1}$ with its full sigma-algebra, each $K (x, \cdot)$ is a probability measure, and $φ (x) = K (x, {1})$ .

[F2]

Loss functions and risk: if each action section of $L$ is measurable and nonnegative, then $R (θ, K) = \int [\int L (θ, a) K (x, d a)] P_{θ} (d x)$ .

[F3]

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 an indicator of a measurable singleton under $K (x, \cdot)$ is the $K (x, \cdot)$ -measure of that singleton.

[F4]

Expectation of a nonnegative or integrable random variable: expectation under $P_{θ}$ is integration with respect to $P_{θ}$ .

[F5]

Monotonicity and nonnegative homogeneity of the nonnegative integral: if $0 \leq f \leq g$ , then $\int f d P_{θ} \leq \int g d P_{θ}$ .

[F6]

Integrable real and complex functions, and their integrals: a real measurable function is integrable when the integral of its absolute value is finite.

[F7]

The Lebesgue integral is linear on $L^{1} (μ)$ : the integral is linear on integrable functions.

[F8]

Statistical experiments and Probability measures and probability spaces: each $P_{θ}$ is a probability measure with $P_{θ} (X) = 1$ .

Proof

technique · direct

1.1givenF1F2

For fixed $θ$ , the action section $a \mapsto L (θ, a)$ is one of $a$ , $1 - a$ , or $0$ . Each is nonnegative and measurable on the full sigma-algebra of ${0, 1}$ , so [F2] defines its risk.

2.1F1F3step 1.1algebra

On $Θ_{0}$ , the section is $a = 1_{{1}} (a)$ , whose integral under $K (x, \cdot)$ is $K (x, {1}) = φ (x)$ by [F1] and [F3]. On $Θ_{1}$ , the section is $1 - a = 1_{{0}} (a)$ , whose integral is $K (x, {0}) = 1 - φ (x)$ because $K (x, \cdot)$ is a probability measure.

3.1F2F4step 2.1

Substituting these inner integrals into [F2] and using [F4] gives $R (θ, K) = E_{θ} φ (X)$ on $Θ_{0}$ and $R (θ, K) = E_{θ} [1 - φ (X)]$ on $Θ_{1}$ .

4.1F1F5F6F7F8step 3.1algebra∎

Since $0 \leq φ \leq 1$ and $0 \leq 1 - φ \leq 1$ , [F5] and [F8] show that each has integral at most $1$ . Thus their absolute-value integrals are finite, so they are integrable by [F6]. Linearity [F7] and [F8] give $E_{θ} [1 - φ] = 1 - E_{θ} [φ]$ . The two risk identities now have the asserted form.

Depends on

Used by

Test size and power Definition

Dependency tree · two levels

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Sources

Ryan Martin, Lecture Notes on Advanced Statistical Theory, §5.1, PDF pp. 100–102 (standard reference, not scraped)
Peter Hoff, Statistical Decision Problems, §§4.1–4.2, PDF pp. 7–8 (standard reference, not scraped)