Modified linex two-stage and purely sequential estimation of the variance in a normal distribution with illustrations using horticultural data

ABSTRACT

In a normal distribution with its mean unknown, we have developed Stein type two-stage and Chow and Robbins type purely sequential strategies to estimate the unknown variance under a modified Linex loss function. We control the associated risk function per unit cost by bounding it from above with a fixed preassigned positive number, . Under both proposed estimation strategies, we have emphasized (i) exact calculations of the distributions and moments of the stopping times as well as the biases and risks associated with our terminal estimators of , along with (ii) selected asymptotic properties. In developing asymptotic second-order properties under the purely sequential estimation methodology, we have relied upon nonlinear renewal theory. We report extensive data analysis carried out via (i) exact calculations as well as (ii) simulations when requisite sample sizes range from small to moderate to large. Both estimation methodologies have been implemented and illustrated with the help of real data sets recorded by Mukhopadhyay et al. from designed experiments in the field of horticulture.

KEYWORDS:

• Asymptotics
• bounded risk
• exact calculations
• first-order properties
• Linex loss
• modified Linex loss
• purely sequential methodology
• real data
• risk
• risk per unit cost
• scale parameter
• second order properties
• simulations
• two-stage methodology

AMS SUBJECT CLASSIFICATION:

• 62L12
• 62L05
• 62G20
• 62F10
• 62P10

Acknowledgments

Two reviewers and an associate editor shared with us a number of helpful comments. We sincerely thank them all.