[Paper Review] Estimating the Parameters of Binomial and Poisson Distributions via Multistage Sampling

Xinjia Chen|ArXiv.org|Oct 2, 2008

Fluid Dynamics and Heat Transfer3 references3 citations

TL;DR

This paper proposes multistage sampling schemes for estimating binomial and Poisson distribution parameters without prior knowledge of the parameters, using adaptive stopping rules based on cumulative binomial and Poisson tail probabilities. The method rigorously guarantees prescribed precision and confidence levels, ensuring that the estimated parameter lies within a specified absolute or relative error bound with probability at least $1 - \delta$. The key contribution is a non-asymptotic, conservative-free approach that outperforms traditional methods in efficiency and theoretical rigor.

ABSTRACT

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.

Motivation & Objective

• To address the limitations of existing parameter estimation methods for binomial and Poisson distributions, which often rely on asymptotic approximations or overly conservative bounds.
• To develop sampling schemes that require no prior knowledge of unknown parameters while guaranteeing prescribed levels of precision and confidence.
• To overcome inefficiencies and theoretical weaknesses in current approaches by introducing a non-asymptotic, multistage framework with rigorous probabilistic guarantees.
• To ensure that the estimated parameter lies within a user-specified absolute or relative error bound with high probability, regardless of the true parameter value.

Proposed method

• The method employs a multistage sampling design where sample sizes $n_1 < n_2 < \cdots < n_s$ are selected based on a recursive sequence involving $\varepsilon$, $\delta$, $\zeta$, and $\rho$, with $\tau$ determining the number of stages.
• At each stage $\ell$, the sample mean $\widehat{p}_\ell = K_\ell / n_\ell$ is computed, where $K_\ell = \sum_{i=1}^{n_\ell} X_i$.
• The stopping rule is triggered when both $S_{\mathrm{B}}(K_\ell, n_\ell, n_\ell, \widehat{p}_\ell - \varepsilon) \leq \zeta\delta$ and $S_{\mathrm{B}}(0, K_\ell, n_\ell, \widehat{p}_\ell + \varepsilon) \leq \zeta\delta$ are satisfied, ensuring the estimate is within $\varepsilon$ of the true $p$ with high confidence.
• For Poisson estimation, the method uses similar stopping rules based on $S_{\mathrm{P}}(K, \infty, n, \lambda)$ and $S_{\mathrm{P}}(0, K, n, \lambda)$, with confidence bounds derived from tail probability inequalities.
• The framework uses double-decision variables and recursive bounds on tail probabilities to ensure the overall error probability remains below $\delta$.
• Theoretical guarantees are derived using bounds on binomial and Poisson cumulative distribution functions, with key inequalities involving the gamma function and logarithmic approximations.

Experimental results

Research questions

• RQ1 Can multistage sampling schemes be designed to estimate binomial parameters with guaranteed absolute error and confidence level without prior knowledge of the parameter?
• RQ2 How can Poisson parameter estimation be achieved with both absolute and relative error bounds under the same non-asymptotic framework?
• RQ3 What is the minimal sample size required to ensure that the estimator lies within a specified error interval with probability at least $1 - \delta$?
• RQ4 How can the sampling scheme be made efficient while avoiding conservative bounds or asymptotic approximations?

Key findings

• The proposed multistage sampling scheme ensures $\Pr\{ |\widehat{p} - p| < \varepsilon \mid p \} \geq 1 - \delta$ for any $p \in (0,1)$, provided $0 < \zeta \leq \frac{1}{2(\tau + 1)}$.
• For binomial estimation, the method guarantees that the absolute error is bounded by $\varepsilon$ with confidence $1 - \delta$, using a sequence of increasing sample sizes determined by $\varepsilon$, $\delta$, and $\rho$.
• The sampling process is guaranteed to terminate almost surely because the stopping conditions are eventually satisfied due to the properties of the binomial tail probabilities.
• For Poisson estimation, the method ensures $\Pr\{ |\widehat{\lambda} - \lambda| < \varepsilon_a \text{ or } |\widehat{\lambda} - \lambda| < \varepsilon_r \lambda \} > 1 - \delta$ when $0 < \zeta < \frac{1}{2(\tau + 1)}$.
• The method avoids asymptotic approximations and conservative bounds, offering a non-asymptotic, rigorous alternative to traditional estimation techniques.
• Theoretical bounds on tail probabilities, including $S_{\mathrm{B}}$ and $S_{\mathrm{P}}$, are used to derive tight confidence intervals and ensure the error probability is bounded by $\delta$.