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，我们发展了一套具有有界方差的尺度参数$\beta$的函数$g(\cdot)$估计的一般理论。我们首先定义了一个序贯抽样程序，其中$g(\cdot)$在提出停止规则时满足某些期望条件，并证明该程序具有良好的渐近性质。在这些一般条件之后，我们将$g(\cdot)$替换为具体函数，包括伽马均值、伽马方差、伽马率参数和伽马生存概率，作为四种可能的示例。对于每个示例，通过蒙特卡洛模拟验证了我们提出的序贯程序的显著性能。这一结论进一步通过一项关于新生儿体重的真实数据研究得到证实。