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Ordered Random Variables have attracted several authors. The basic
building block of Ordered Random Variables is Order Statistics
which has several applications in extreme value theory and ordered
estimation. The general model for ordered random variables, known
as Generalized Order Statistics has been introduced relatively
recently by Kamps (1995).
This book intended to cover the overview of unequal probability
sampling and Generalization of Murthy's (1957) estimator. The
researcher in the field of survey sampling can get benefit from
this book. This book covers an overview of the unequal probability
sampling, work done for Horvitz and Thompson (1952), Raj(1956) and
Murthy (1957) estimators. Simulation and Empirical studies are
conducted to discuss the performance of various available
estimators for unequal probability sampling.
The Murthy (1957) estimator has been extensively studied by various
authors. The design based variance of the estimator has been
studied by the various survey statisticians. The modifications of
the Murthy (1957) estimator has been proposed by Shahbaz (2004)
alongside the design based study. The model based study of the
estimator has been presented in this monograph. We have also
studied the stability of the variance estimator of the modified
Murthy (1957) estimator under the linear super-population model. It
has been observed that the anticipated variance of the modified
Murthy (1957) estimator achieves the Godambe-Joshi (1965) lower
bound and the variance estimator is stable.
As we all know that sampling with unequal probability plays most
important role in large scale sample surveys, therefore in this
monograph some recent developments regarding this topic has been
discussed. Moreover since joint probability is difficult calculate
therefore some approximations of the joint probability have been
derived. Special attention has been given to general theory of
arbitrary probability. For this theory sampling with replacement
and without replacement are the special cases of this general
theory. Generalized Murthy estimator has also been discussed in
details.
Unequal Probability Sampling has been efficiently used for
estimation of population characteristics. Several developments have
been made from time to time to increase the precision of estimates.
In this book we present some new methods of estimation in unequal
probability sampling. Some selection procedures have been discussed
for use with the Horvitz and Thompson (1952) estimator. We have
also proposed a general selection procedure which yields several
selection procedures as special case. We have also discussed the
method of developing the approximations for variance formulae for
Horvitz and Thompson (1952) estimator. Some new estimators have
been discussed alongside their design and model based study. The
empirical comparison of proposed techniques has also been given
with some well known estimation techniques of unequal probability
sampling.
The development of estimators of population parameters based on
two-phase sampling schemes has seen a dramatic increase in the past
decade. Various authors have developed estimators of population
mean of study variables using either single or two auxiliary
variables. The present volume is a comprehensive collection of
estimators available in single and two phase sampling. The book
covers estimators which utilize information on single, two and
multiple auxiliary variables of both quantitative and qualitative
nature. The estimators discussed in the monograph are based upon
different mechanisms of availability of auxiliary information,
termed as Full, Partial and No Information. Multivariate Estimators
in survey sampling are also discussed in the book. Two-Phase
Sampling will prove an invaluable point of reference for
researchers working in the field of survey sampling in general and
in the field of two-phase sampling in particular.
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