On the asymptotic normality and the construction of confidence intervals for estimators after sampling with probabilistic and deterministic stopping rules

TL;DR

This paper investigates the asymptotic normality of estimators after sequential sampling with probabilistic and deterministic stopping rules, establishing conditions for normality and analyzing confidence interval coverage.

Contribution

It provides a rigorous proof of the asymptotic normality of the sample average under most stopping rules and identifies a unique pathological case where normality fails.

Findings

Asymptotic normality holds in almost all cases except a specific deterministic stopping scenario.

The Kolmogorov distance between the distribution and the normal is 0.125 in the pathological case.

Simulations show confidence intervals maintain proper coverage despite asymptotic distribution deviations.

Abstract

A key feature of a sequential study is that the actual sample size is a random variable that typically depends on the outcomes collected. While hypothesis testing theory for sequential designs is well established, parameter and precision estimation is less well understood. Even though earlier work has established a number of ad hoc estimators to overcome alleged bias in the ordinary sample average, recent work has shown the sample average to be consistent. Building upon these results, by providing a rate of convergence for the total variation distance, it is established that the asympotic distribution of the sample average is normal, in almost all cases, except in a very specific one where the stopping rule is deterministic and the true population mean coincides with the cut-off between stopping and continuing. For this pathological case, the Kolmogorov distance with the normal is found to equal 0.125. While noticeable in the asymptotic distribution, simulations show that there fortunately are no consequences for the coverage of normally-based confidence intervals.