Estimation of the scale parameter of a family of distributions using a newly derived minimal sufficient statistic

Abstract

A new class of statistics obtained by ordering the absolute values of the observations arising from absolutely continuous distributions which are symmetrically distributed about zero is introduced in this paper. The statistics generated by the above method are named as absolved order statistics (AOS) of the given sample. The association of the distribution of these statistics with the distribution of order statistics arising from the folded form of the parental density about zero is outlined. The vector of AOS is proved to be a minimal sufficient statistic for the class ${\mathcal{F}}_{\theta }^{(1)}$ of all absolutely continuous distributions which are symmetrically distributed about zero. A method of estimation of the scale parameter of any distribution belonging to ${\mathcal{F}}_{\theta }^{(1)}$ using AOS is described. Illustration on the advantage of the above method of estimation is described for the distributions such as (i) logistic, (ii) normal, and (iii) double Weibull. A more realistic censoring scheme involving AOS as well is discussed in this paper. We have derived the U-statistic estimator based on AOS for the scale parameter σ of any distribution $f(x,\sigma )\in {\mathcal{F}}_{\theta }^{(1)}$ using the best linear unbiased estimate (BLUE) based on AOS of a preliminary sample as kernel. We have illustrated the performance of this estimator with an U-statistic generated from BLUE based on order statistics for each of (i) logistic (ii) normal and (iii) double Weibull distributions.

Keywords:

• Absolved order statistics
• symmetric distributions
• minimal sufficiency
• scale parameter
• best linear unbiased estimate
• maximum likelihood estimate
• censored samples
• U-statistics

MATHEMATICAL SUBJECT CLASSIFICATION:

• 62G30
• 62B05
• 62F10

Acknowledgments

The anonymous reviewer is very much thanked for many of his valuable comments.