Parametric Estimations For A Class Of Continuous Families Under Optimal Ranked Set Sampling

Ranked set sampling(RSS)was introduced by McIntyre(1952)for estimating the pasture yields.It is appropriate for situations where quantification of sampling units is either costly or difficult,but ranking the units in a small set is easy and inexpensive.Research on parametric methods to RSS were mainly focused on the minimum variance unbiased estimator(MVUE)in a class of some unbiased estimators.Lesitha et al.(2013)and Sarikavanij et al.(2014)have given some recent results about the MVUE.Those earlier results about the MVUE have been summarized by Chuiv et al.(1998).Notwithstanding the large number of papers devoted to MVUEs of parameters for various distributions in RSS.However,their results are only the MVUE in a class of some unbiased estimators.Hence it is meaningful to find the optimal RSS according to the character of a population and obtain the MVUE in the class of all unbiased estimators.On the other hand,researches on parametric estimation in existing literatures were mainly focused on the maximum likelihood estimators(MLEs).For example,Abu-Dayyeh et al.(2013)studied the MLE of the shape parameter for Pareto dis-tribution under the balanced RSS and the RSS based on maximizing the Fisher information,respectively.However,their results do not directly apply other popu-lations besides Pareto distribution.Hence it is meaningful to find an optimal RSS for other distributions and obtain the MLE of the parameter in these distributions under this optimal RSS.Based on the above discussions,in this thesis,parameter(s)estimators for a class of continuous families with scale ?,lower truncation parameter ? and upper truncation parameter v under optimal RSS are studied.The thesis consists of six chapters:In Chapter One,we summarize the introduction of the related problems and state the main results of the present thesis.We also simply introduce models,methods and some preliminary results in this thesis.In Chapter Two,we study parameters estimator(s)for a continuous truncated parameter family with unknown parameters A and/or v in the present of known parameter ?.Firstly we find the optimal ranked set sampling(RSS)according to the character of this family,viz,arrange RSS based on complete and sufficient statistics of independent and identically distributed(iid)samples.Then under this RSS,the minimum variance unbiased estimator(MVUE)of the parameter in the family is found.Numerical simulations for some usual distributions in this family fully support the result from the above two step optimizations.In Chapter Three,we study estimation of ? for a continuous one-parameter exponential family with unknown parameter ? in the present of known parameters? and v.We firstly find the optimal RSS according to the character of this family,viz,arrange the RSS based on quasi complete and sufficient statistic of independent and identically distributed(iid)samples.Then under this RSS,a maximum likelihood estimator(MLE)of ? for the family is studied.Some sufficient conditions for the existence and uniqueness of MLE,which are easily used in practice,are obtained.Using these conditions,the existence and uniqueness of the MLEs of the parameters for some usual distributions in this family are proved.Numerical simulations for these distributions fully support the result from the above two step optimizations of the sampling and the estimation method.In Chapter Four,we study estimation of ? or v and ? for a continuous expo-nential family with unknown parameter ? or v in the present of unknown parameter?.We firstly find the optimal RSS according to the character of this family,viz,arrange the RSS based on quasi complete and sufficient statistic of iid samples.Then under this RSS,MLEs of A or v and B for the family are studied.A sufficient condition for the existence of the MLEs,which is easily used in practice,is obtained.Using the condition,the existence for the MLEs of the parameters from some usual distributions in this family are proved.In Chapter Five,we study estimation of ?,v and ? for a continuous exponential family with unknown parameters ? and v in the present of unknown parameter ?.We firstly find the optimal RSS according to the character of this family,viz,arrange the RSS based on quasi complete and sufficient statistic of iid samples.Then under this RSS,MLEs of ?,v and ? for the family are studied.A sufficient condition for the existence of the MLEs,which is easily used in practice,is obtained.In Chapter Six,we study a modified MLE of scale parameter under moving extremes RSS.