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**On empirical Bayes selection rules for sampling inspection.**
*(English)*
Zbl 0797.62004

Summary: We study the problem of acceptance sampling for \(k\) independent hypergeometric populations, say \(\pi_ i = \pi (M_ i,m_ i,d_ i)\), \(i=1,\dots,k\), where \(M_ i\) is the number of items in population \(\pi_ i\), \(d_ i\) is the number of defectives in \(\pi_ i\) and \(m_ i\) is the number of items taken from \(\pi_ i\) for inspection. Let \(d_{i0}\) be an integer such that \(0<d_{i0}<M_ i\), which is used to evaluate the quality of population \(\pi_ i\). Population \(\pi_ i\) is said to be good and is acceptable if \(d_ i<d_{i0}\) and to be bad otherwise. Our goal is to select all good populations and to exclude all bad populations.

This selection problem is formulated through the empirical Bayes approach. Two empirical Bayes selection rules, called as parametric empirical Bayes and hierarchical empirical Bayes, are studied. The asymptotic optimality of the proposed empirical Bayes selection rules are investigated. It is shown that for each empirical Bayes selection rule, the corresponding Bayes risk tends to the minimum Bayes risk with a rate of convergence of order \(O(\exp (-ck + \ln k))\) for some positive constant \(c\), where the value of \(c\) varies depending on the rule. A simulation study is also carried out to investigate the performance of the empirical Bayes selection rules for small to moderate \(k\) case.

This selection problem is formulated through the empirical Bayes approach. Two empirical Bayes selection rules, called as parametric empirical Bayes and hierarchical empirical Bayes, are studied. The asymptotic optimality of the proposed empirical Bayes selection rules are investigated. It is shown that for each empirical Bayes selection rule, the corresponding Bayes risk tends to the minimum Bayes risk with a rate of convergence of order \(O(\exp (-ck + \ln k))\) for some positive constant \(c\), where the value of \(c\) varies depending on the rule. A simulation study is also carried out to investigate the performance of the empirical Bayes selection rules for small to moderate \(k\) case.

### MSC:

62C12 | Empirical decision procedures; empirical Bayes procedures |

62F07 | Statistical ranking and selection procedures |

62P30 | Applications of statistics in engineering and industry; control charts |

62D05 | Sampling theory, sample surveys |

### Keywords:

sampling inspection; acceptance sampling; independent hypergeometric populations; empirical Bayes selection rules; parametric empirical Bayes; hierarchical empirical Bayes; asymptotic optimality; minimum Bayes risk; rate of convergence; simulation study
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\textit{S. S. Gupta} and \textit{T. Liang}, J. Stat. Plann. Inference 38, No. 1, 43--64 (1994; Zbl 0797.62004)

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### References:

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