标题： 逆二项抽样下在归一化线性-线性和逆线性损失下的概率估计 显示英文标题

标题： Estimation of a probability in inverse binomial sampling under normalized linear-linear and inverse-linear loss

摘要： 在逆二项抽样中考虑了成功概率$p$的顺序估计。 对于任何估计量$\hat p$，其质量通过与线性-线性或逆线性形式的归一化损失函数相关的风险来衡量。 这些函数可能是不对称的，具有任意斜率参数$a$和$b$，分别对应于$\hat p<p$和$\hat p>p$。 对这些函数的兴趣是由于它们的重要性和潜在用途，这些用途将简要讨论。 估计量给出的损失在$p$趋向于$0$时具有渐近值，并且保证对于任何在$(0,1)$中的$p$，损失都低于其渐近值。 这允许选择所需的成功次数，$r$，以满足规定的质量，而不管未知的$p$。 此外，所提出的估计量被证明当 $a/b$ 不偏离 $1$太远时是近似最小最大值的，并且当 $r$趋向于无穷大时，当 $a=b$ 时是渐近最小最大值的。 显示英文摘要

摘要： Sequential estimation of the success probability $p$ in inverse binomial sampling is considered in this paper. For any estimator $\hat p$, its quality is measured by the risk associated with normalized loss functions of linear-linear or inverse-linear form. These functions are possibly asymmetric, with arbitrary slope parameters $a$ and $b$ for $\hat p<p$ and $\hat p>p$ respectively. Interest in these functions is motivated by their significance and potential uses, which are briefly discussed. Estimators are given for which the risk has an asymptotic value as $p$ tends to $0$, and which guarantee that, for any $p$ in $(0,1)$, the risk is lower than its asymptotic value. This allows selecting the required number of successes, $r$, to meet a prescribed quality irrespective of the unknown $p$. In addition, the proposed estimators are shown to be approximately minimax when $a/b$ does not deviate too much from $1$, and asymptotically minimax as $r$ tends to infinity when $a=b$.