Robustness of Triple Sampling Inference Procedures to Underlying Distributions

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

Name dc:type.qualificationname

Ph.D.

Level dc:type.qualificationlevel

doctoral

Grantor dc:publisher.institution

University of Southampton

Year dc:date.issued

2010

Author dc:creator

• Yousef, Ali Saleh Ali

Advisor dc:contributor.advisor

• Kimber, Alan