初始编码及最大序列号均未知时总体容量的估计  

Estimation of the Population Size with Unknown Minimum and Maximum Serial Numbers

作　　者：李亚男 王昱泉 陈国蓝 胡跃清 LI Ya'nan;WANG Yuquan;CHEN Guolan;HU Yueqing(Deparement of Biostatistics and Computational Biology,School of Life Sciences,Fudan University,Shanghai 200433 China)

机构地区：[1]复旦大学生命科学学院生物统计学与计算生物学系

出　　处：《复旦学报（自然科学版）》2019年第5期549-559,共11页Journal of Fudan University：Natural Science

基　　金：国家自然科学基金(11571082)

摘　　要：实际生活中,很多产品的序列号是连续的,估计出最大序列号就知道产品的总产量.例如二战中德军坦克数量的估计就是初始编码为1时最大序列号的估计.但在很多商业活动中,需要估计某产品在某个时间段的产量.本文研究这种初始编码及最大序列号均未知时总体容量的估计问题,从不同角度提出6个估计量:有放回抽样时极大似然估计量以及它的Jackknife法、近似法、迭代法及区间长度法得出的相应估计量及无放回抽样时的最小方差无偏估计量,然后通过大量随机模拟来评价这6个估计量的无偏性和均方根误差的相对大小.模拟结果表明:极大似然估计在小样本时表现欠佳,Jackknife法确实可以改进偏倚,后4个估计量的表现不论是有放回还是无放回抽样均相差不大,最小方差无偏估计量在无放回抽样时表现最佳.As we all know,the serial numbers of many products are continuous.The classic German tank problem in statistics,namely estimating the size of a discrete uniformly distributed population with a known minimum based on sampling without replacement,raised the studies of serial number analysis.Sometimes,it is necessary to estimate the output of a product in a certain period of time in many commercial activities.Then the problem’s scenario is extended to situation with an also unknown minimum.This study proposed six estimators about the size of such a population from sampling with replacement by different strategies.Bias and root mean squared error obtained from a series of simulations were used to compare and evaluate these estimators.The results showed that maximum likelihood estimator has a large bias in small sample and Jackknife method can reduce the bias.Besides,the iterative estimator,the approximate estimator,the gap estimator and the minimum variance unbiased estimator under sampling without replacement have the best overall performance.

关 键 词：序列号估计 极大似然估计 迭代估计 最小方差无偏估计 

分 类 号：O212.2[理学—概率论与数理统计]