Estimating the Parameters of Binomial and Poisson Distributions via Multistage Sampling

In this paper, we have developed a new class of sampling schemes for estimating parameters of binomial and Poisson distributions. Without any information of the unknown parameters, our sampling schemes rigorously guarantee prescribed levels of precision and confidence.

💡 Research Summary

The paper introduces a novel multistage sampling framework for estimating the parameters of the binomial and Poisson distributions without any prior knowledge of the unknown parameters. Traditional fixed‑sample designs require a priori guess of the success probability (p) (for the binomial) or the rate (\lambda) (for the Poisson) in order to determine an appropriate sample size. If the guess is inaccurate, the resulting confidence interval may be too wide or the experiment may waste resources by collecting far more observations than necessary. Sequential methods such as Wald’s SPRT or Bayesian posterior‑based designs can adapt sample size, but they often rely on specific prior distributions, involve complex stopping boundaries, or demand intensive computation that limits their practical use.

Problem formulation.
The authors consider two precision criteria simultaneously: an absolute error bound (\varepsilon_a) and a relative error bound (\varepsilon_r). The goal is to produce an estimator (\hat\theta) (where (\theta) is either (p) or (\lambda)) together with a confidence interval (