Confidence intervals and point estimators for a normal mean under purely sequential strategies involving Gini’s mean difference and mean absolute deviation

Abstract

ABSTRACTWe have developed purely sequential methodologies for problems associated with both fixed-width confidence interval estimation and minimum risk point estimation for the normal mean μ when the variance σ2 is assumed unknown. New stopping rules are constructed by replacing the sample variance with appropriate multiples of Gini’s mean difference (GMD) and mean absolute deviation (MAD) in defining the conditions for boundary crossing. A number of asymptotic first-order consistency, efficiency, and risk efficiency properties associated with these new estimation strategies have been investigated. These are followed by summaries obtained from extensive sets of simulations by drawing samples from (i) normal universes or (ii) mixture-normal universes where samples may be reasonably treated as observations from a normal universe in a large majority of simulations. We also include illustrations using sales data and horticulture data. Overall, we empirically feel confident that our newly developed GMD-based or MAD-based methodologies are more robust for practical purposes when we compare them with the sample variance–based methodologies respectively, especially when up to 20% suspect outliers may be expected.