Robustness of Triple Sampling Inference Procedures to Underlying Distributions

Abstract

In this study, the sensitivity of the sequential normal-based triple sampling procedure for estimating
the population mean to departures from normality is discussed. We assume only that the underlying
population has finite but unknown first six moments. Two main inferential methodologies are
considered. First point estimation of the unknown population mean is investigated where a squared
error loss function with linear sampling cost is assumed to control the risk of estimating the unknown
population mean by the corresponding sample measure. We find that the behaviour of the estimators
and of the sample size depends asymptotically on both the skewness and kurtosis of the underlying
distribution and we quantify this dependence. Moreover, the asymptotic regret of using the triple
sampling inference instead of the fixed sample size approach, had the nuisance parameter been
known, is a finite but non-vanishing quantity that depends on the kurtosis of the underlying
distribution. We also supplement our findings with a simulation experiment to study the performance
of the estimators and the sample size in a range of conditions and compare the asymptotic and finite
sample results. The second part of the thesis deals with constructing a triple sampling fixed width
confidence interval for the unknown population mean with a prescribed width and coverage while
protecting the interval against Type II error. An account is given of the sensitivity of the normal-based
triple sampling sequential confidence interval for the population when the first six moments are
assumed to exist but are unknown. First, triple sampling sequential confidence intervals for the mean
are constructed using Hall’s (1981) methodology. Hence asymptotic characteristics of the constructed
interval are discussed and justified. Then an asymptotic second order approximation of a continuously
differentiable and bounded function of the stopping time is given to calculate both asymptotic
coverage based on a second order Edgeworth asymptotic expansion and the Type II error probability.
The impact of several parameters on the Type II error probability is explored for various continuous
distributions. Finally, a simulation experiment is performed to investigate the methods in finite sample
cases and to compare the finite sample and asymptotic results.

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