EDA on the asymptotic normality of the standardized sequential stopping times, Part-I: Parametric models

ABSTRACT

In sequential analysis, an experimenter may gather information regarding an unknown parameter by observing random samples in successive steps. We emphasize a number of specific parametric models under a variety of loss functions. The total number of observations collected at termination is a positive integer-valued random variable, customarily denoted by $N\equiv N_{\nu }$. The exact probability distribution of N is often hard to obtain. However, under a set of regulatory conditions, our standardized version $Q\equiv Q_{\nu }$ of the stopping variable $N_{\nu }$ from Definition 2.1 in Section 2.2 would follow a normal distribution in the asymptotic sense. In this article, we first show how these regulatory conditions build upon one another in order to conclude the asymptotic normality of such standardized stopping variable, $Q_{\nu }$. We provide exploratory data analysis (EDA) via a number of interesting illustrations obtained through large-scale simulation studies. We demonstrate the broad applicability of purely sequential methodologies included in this article along with the appropriateness of our conclusions regarding the asymptotic normality of the standardized stopping variable $Q_{\nu }$ as a practical guideline.

KEYWORDS:

• Asymptotics
• cancer research
• confidence interval
• Kolmogorov-Smirnov test
• Linex loss
• negative exponential population
• Nile example
• normal population
• point estimation
• reliability
• second-order efficiency
• SEL
• shelf life
• standardized stopping variable
• stopping time
• tumor studies

SUBJECT CLASSIFICATIONS:

• 62L12
• 62L10
• 62E17

Acknowledgments

An Associate Editor and a Referee gave valuable comments on an earlier version. This improved presentation benefited from their enthusiastic feedback. We thank both the Associate Editor and the Referee for their kind help.