Estimating a function of the scale parameter in a gamma distribution with bounded variance

Given a gamma population with a known shape parameter α, we develop a general theory for estimating a function g(·) of the scale parameter β with bounded variance. We begin by defining a sequential sampling procedure with g(·) satisfying some desired condition in proposing the stopping rule and show that the procedure enjoys appealing asymptotic properties. After these general considerations, we substitute g(·) 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 by two real data studies on the survival times of female patients with dementia and on the number of days that the seeds of marigold need to flower, respectively.