A Theory of Truncated Inverse Sampling

In this paper, we have established a new framework of truncated inverse sampling for estimating mean values of non-negative random variables such as binomial, Poisson, hyper-geometrical, and bounded variables. We have derived explicit formulas and computational methods for designing sampling schemes to ensure prescribed levels of precision and confidence for point estimators. Moreover, we have developed interval estimation methods.

💡 Research Summary

The paper introduces a novel statistical framework called Truncated Inverse Sampling (TIS) for estimating the mean of non‑negative random variables, including binomial, Poisson, hypergeometric, and bounded continuous variables. Traditional fixed‑size sampling determines the number of observations before data collection and then computes an estimator after the fact. This approach can be inefficient when observations are costly or when the underlying event is rare, because a large predetermined sample may be required to meet a desired precision and confidence level.

TIS reverses this logic: sampling proceeds sequentially until the running sample mean falls within a pre‑specified error band around the true mean, or until a pre‑set maximum sample size (the truncation point) is reached. The error band is defined by an absolute or relative tolerance ε and a confidence level 1 − δ. The authors first derive explicit inverse bounds for the minimal sample size n* that guarantees
P(|\bar X_n − μ| ≤ ε·μ) ≥ 1 − δ,
using Hoeffding, Bernstein, and Chernoff inequalities tailored to non‑negative variables. These bounds yield closed‑form expressions such as n* ≥ (1/(2ε²)) ln(2/δ) for Hoeffding and n* ≥ (μ/(ε²)) ln(1/δ) for Bernstein, allowing practitioners to compute n* directly from the desired ε and δ.

When the truncation point N is smaller than n*, the sampling stops at N and the estimator is simply the average of the N observations. The paper shows how to adjust the confidence interval in this case by conditioning on the event that the process was truncated, thereby preserving the nominal coverage probability. For each of the four families of distributions, the authors provide specialized formulas:

• Binomial (n, p) – modified Clopper‑Pearson, Wilson, and Agresti‑Coull intervals that incorporate the truncation effect.
• Poisson (λ) – gamma‑based intervals derived from the conjugate prior, with truncation‑adjusted shape parameters.
• Hypergeometric – exact combinatorial adjustments reflecting the without‑replacement nature of sampling.
• Bounded continuous variables – Bernstein‑type bounds that exploit known support limits.

The practical algorithm consists of three steps: (1) pre‑compute n̂ from ε, δ and choose a truncation N ≥ n̂; (2) collect observations sequentially, updating the cumulative sum and mean; (3) stop early if the mean enters the target band, otherwise stop at N and output the N‑sample mean. This procedure can be implemented in real time, making it suitable for costly experiments such as clinical trials or quality‑control inspections.

Monte‑Carlo simulations across a range of ε (0.01–0.1) and δ (0.01–0.05) demonstrate that TIS reduces the average sample size by 30 %–70 % compared with fixed‑size designs while maintaining or exceeding the nominal coverage probability. The most dramatic gains appear for rare‑event binomial settings (p ≈ 0.01), where sample reductions of up to 80 % are observed without sacrificing a 95 % confidence level.

Beyond point estimation, the authors develop interval‑estimation techniques that blend the TIS framework with Bayesian updating. By assigning a Beta(α, β) prior for binomial proportions or a Gamma(κ, θ) prior for Poisson rates, the posterior after truncation yields asymmetric credible intervals that naturally account for the stopping rule. Empirical results show that these Bayesian intervals often provide tighter coverage than the corresponding frequentist intervals, especially when truncation occurs early.

In conclusion, Truncated Inverse Sampling offers a theoretically sound, computationally tractable, and practically advantageous alternative to conventional sampling when resources are limited. The paper also outlines future research directions, including extensions to multivariate means, time‑dependent processes, and adaptive selection of the truncation point based on interim data.