Given a gamma population with known shape parameter $\alpha$, we develop a general theory for estimating a function $g(\cdot)$ of the scale parameter $\beta$ with bounded variance. We begin by defining a sequential sampling procedure with $g(\cdot)$ satisfying some desired condition in proposing the stopping rule, and show the procedure enjoys appealing asymptotic properties. After these general conditions, we substitute $g(\cdot)$ with specific functions including the gamma mean, the gamma variance, the gamma rate parameter, and a gamma survival probability as four possible illustrations. For each illustration, Monte Carlo simulations are carried out to justify the remarkable performance of our proposed sequential procedure. This is further substantiated with a real data study on weights of newly born babies.

翻译：给定一个形状参数$\alpha$已知的伽马总体，我们建立了一套用于估计尺度参数$\beta$的函数$g(\cdot)$且保证方差有界的一般性理论。我们首先定义了一个序贯抽样程序，其中$g(\cdot)$需满足提出停止规则时所要求的某些条件，并证明了该程序具有优良的渐近性质。在这些一般性条件之后，我们将$g(\cdot)$替换为具体函数进行说明，包括伽马均值、伽马方差、伽马比率参数以及伽马生存概率四种情形。针对每种情形，均进行了蒙特卡洛模拟以验证我们所提出的序贯程序的卓越性能。一项关于新生儿体重的实际数据研究进一步证实了该方法的有效性。