Journal of Statistical Theory and Practice

, Volume 12, Issue 1, pp 111–135

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

Article

## 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 σ2 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, ω. 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 σ2, 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

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