Yousef, Ali Saleh Ali
(2010)
Robustness of Triple Sampling Inference Procedures to
Underlying Distributions.
*University of Southampton, School of Mathematics, Doctoral Thesis*, 233pp.

## 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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