Modified linex two-stage and purely sequential estimation of the variance in a normal distribution with illustrations using horticultural data
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.
KeywordsAsymptotics 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 Classification62L12 62L05 62G20 62F10 62P10
Unable to display preview. Download preview PDF.
- Ghosh, M., and N. Mukhopadhyay. 1975. Asymptotic normality of stopping times in sequential analysis. Unpublished report. Indian Statistical Institute, Calcutta.Google Scholar
- Mukhopadhyay, N., and S. R Bapat. 2016a. Multistage point estimation methodologies for a negative exponential location under a modified Linex loss function: Illustrations with infant mortality and bone marrow data. Sequential Analysis 35:175–206. https://doi.org/10.1080/07474946.2016.1165532.MathSciNetCrossRefGoogle Scholar
- Mukhopadhyay, N., and S. R Bapat. 2016b. Multistage estimation of the difference of locations of two negative exponential populations under a modified linex loss function: real data illustrations from cancer studies and reliability analysis. Sequential Analysis 35:387–412. https://doi.org/10.1080/07474946.2016.1206386.MathSciNetCrossRefGoogle Scholar
- Robbins, H. 1959. Sequential estimation of the mean of a normal population. In Probability and statistics, Vol. ed. H. Cramer, ed. U. Grenander, 235–45, Uppsala Sweden: Almquist and Wiksell.Google Scholar
- Stein, C. 1949. Some problems in sequential estimation (abstract). Econometrica 17:77–78.Google Scholar
- Varían, Η. R. 1975. A Bayesian approach to real estate assessment. In Studies in Bayesian econometrics and statistics, ed. L. J. Savage, S. E. Fienberg, and A. Zellner, 195–208. Amsterdam, The Netherlands: North Holland, Elsevier.Google Scholar