Abstract A purely sequential minimum risk point estimation (MRPE) methodology with associated stopping time N is designed to come up with a useful estimation strategy. We work under an appropriately formulated weighted squared error loss (SEL) due to estimation of a function of μ, with plus linear cost of sampling from a population having both parameters unknown. A series of important first-order and second-order asymptotic (as c, the cost per unit sample, ) results is laid out including the first-order and second-order efficiency properties. Then, accurate sequential risk calculations are launched, which are then followed by two main results: (i) Theorem 4.1 shows an asymptotic risk efficiency property, and (ii) Theorem 5.1 shows an asymptotic second-order regret expansion associated with the proposed purely sequential MRPE strategy assuming suitable conditions on g(.). We also provide a bias-corrected version of the terminal estimator, We follow up with a number of interesting illustrations where Theorems 4.1–5.1 are readily exploited to conclude an asymptotic risk efficiency property and second-order regret expansion, respectively. A number of other interesting illustrations are highlighted where it is possible to verify the conclusions from Theorems 4.1–5.1 more directly with less stringent assumptions on the pilot sample size.

中文翻译：

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正态分布中均值函数的纯序贯最小风险点估计 (MRPE) 的一般理论

摘要 设计了一种具有相关停止时间 N 的纯顺序最小风险点估计 (MRPE) 方法来提出有用的估计策略。由于对 μ 函数的估计，我们在适当公式化的加权平方误差损失 (SEL) 下工作，加上从具有两个参数未知的总体抽样的线性成本。列出了一系列重要的一阶和二阶渐近（如 c，单位样本成本，）结果，包括一阶和二阶效率属性。然后，启动准确的顺序风险计算，然后得到两个主要结果：（i）定理 4.1 显示了渐近风险效率属性，以及（ii）定理 5。图 1 显示了与提议的纯顺序 MRPE 策略相关的渐近二阶遗憾扩展，假设 g(.) 上有合适的条件。我们还提供了终端估计器的偏差校正版本，我们跟进了许多有趣的插图，其中很容易利用定理 4.1-5.1 分别得出渐近风险效率属性和二阶后悔扩展。突出显示了许多其他有趣的插图，其中可以更直接地验证定理 4.1-5.1 的结论，而对试点样本量的假设不太严格。1 很容易分别用于得出渐近风险效率属性和二阶后悔扩展。突出显示了许多其他有趣的插图，其中可以更直接地验证定理 4.1-5.1 的结论，而对试点样本量的假设不太严格。1 很容易分别用于得出渐近风险效率属性和二阶后悔扩展。突出显示了许多其他有趣的插图，其中可以更直接地验证定理 4.1-5.1 的结论，而对试点样本量的假设不太严格。