<p>Given two populations from which independent binary observations are taken with parameters <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> respectively, estimators are proposed for the relative risk <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p_1/p_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo stretchy="false">/</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, the odds ratio <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p_1(1-p_2)/(p_2(1-p_1))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and their logarithms. The sampling strategy used by the estimators is based on two-stage sequential sampling applied to each population, where the sample sizes of the second stage depend on the results observed in the first stage. The estimators guarantee that the relative mean-square error, or the mean-square error for the logarithmic versions, is less than a target value for any <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p_1, p_2 \in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and the ratio of average sample sizes from the two populations is close to a prescribed value. The estimators can also be used with group sampling, whereby samples are taken in batches of fixed size from the two populations simultaneously, each batch containing samples from the two populations. The efficiency of the estimators with respect to the Cramér–Rao bound is good, and in particular it is close to 1 for small values of the target error.</p>

Estimation of relative risk, odds ratio and their logarithms with guaranteed accuracy and controlled sample size ratio

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2026年3月30日

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Luis Mendo

摘要

Given two populations from which independent binary observations are taken with parameters $p_1$ $p_{1}$ and $p_2$ $p_{2}$ respectively, estimators are proposed for the relative risk $p_1/p_2$ $p_{1} / p_{2}$ , the odds ratio $p_1(1-p_2)/(p_2(1-p_1))$ $p_{1} (1 - p_{2}) / (p_{2} (1 - p_{1}))$ and their logarithms. The sampling strategy used by the estimators is based on two-stage sequential sampling applied to each population, where the sample sizes of the second stage depend on the results observed in the first stage. The estimators guarantee that the relative mean-square error, or the mean-square error for the logarithmic versions, is less than a target value for any $p_1, p_2 \in (0,1)$ $p_{1}, p_{2} \in (0, 1)$ , and the ratio of average sample sizes from the two populations is close to a prescribed value. The estimators can also be used with group sampling, whereby samples are taken in batches of fixed size from the two populations simultaneously, each batch containing samples from the two populations. The efficiency of the estimators with respect to the Cramér–Rao bound is good, and in particular it is close to 1 for small values of the target error.