Unordering of Estimators in Sampling Theory: Revisited
- PDF / 883,616 Bytes
- 12 Pages / 439.37 x 666.142 pts Page_size
- 44 Downloads / 175 Views
Unordering of Estimators in Sampling Theory: Revisited T. J. Rao1,2 Accepted: 11 October 2020 © Grace Scientific Publishing 2020
Abstract Rao–Blackwellization, a term credited to C. R. Rao based on his 1945 ‘breakthrough’ paper published in the Bulletin of the Calcutta Mathematical Society, besides providing improved estimators in conventional, adaptive, link-tracing, sizebiased sampling theories, found applications in post simulation improvement of Monte Carlo methods, cross validation and non parametric boot strapping, particle filtering, stereology, data compression, Rao-Blackwellized Field Goal percentage estimator (RB-FG %), Rao-Blackwellized Gaussian Smoothing, Rao–Blackwellized Parts-Constellation Tracker, Rao-Blackwellized Tempered Sampling (RTS) and a host of others. In this paper, we shall consider applications related to improving of estimators in finite population sampling theory. Taking cue from Basu (1958) wherein he showed that the ‘order statistic ‘(sample units in ascending order of their labels) is a sufficient statistic, Pathak (Sankhya A 23:409–414, 1961) in the context of sampling from finite populations,first noticed that ‘any estimator which is not a function of the order statistic’, can be uniformly improved by the use of Rao– Blackwellization technique. See also Sinha and Sen (Sankhya [B] 51: 65–83, 1989) who go beyond variance comparisons and generalize to convex loss functions. In this paper, we shall revisit some of the estimators in Probability Proportional to Size sampling With Out Replacement (PPSWOR) schemes and show how Rao–Blackwellization provides improved estimators. Keywords Probability proportional to size sampling without replacement · Ordered and unordered estimators · Des Raj estimator · Das estimator · Brewer’s and Durbin’s sampling schemes · Rao-Blackwellization
This article is part of the topical collection “Celebrating the Centenary of Professor C. R. Rao” guest edited by, Ravi Khattree, Sreenivasa Rao Jammalamadaka, and M. B. Rao. * T. J. Rao [email protected] 1
Indian Statistical Institute, Kolkata, India
2
306 Radha Beach Residency, R K Beach Road, Maharanipeta, Visakhapatnam 530002, India
13
Vol.:(0123456789)
12
Page 2 of 12
Journal of Statistical Theory and Practice
(2021) 15:12
1 Introduction We shall first revisit one of the earliest estimators, considered by Das, for selecting units from a finite population with probability proportional to size and without replacement. We comment that this estimator did not receive as much attention as others and briefly discuss some of its useful properties. Its unordered version was not properly considered in the literature. Here we shall derive its unordered (symmetrized) version in alternative forms. Recently, there has been renewed interest in symmetrized estimators. Observing that Brewer’s and Durbin’s methods are also based on ordered selection of units, we shall obtain symmetrized (unordered) Brewer and symmetrized Durbin estimators for PPSWOR scheme which were not studied earlier. An illustrative exampl
Data Loading...
