Statistical Inference Problems Under Non-simple Random Sampling

The most common sampling approach for collecting data from a population with the goal of making inferences about unknown features of the population is a simple random sample.However,we have to use other methods sometimes when it can not be carried out or costs much time or money.In this paper,we discussed two non-simple random sampling method:instruction sampling and ranked set sampling.We discuss the problem of estimating the expectation and variance of a population under instruction sampling in the thesis first. It is unsuitable for us to infer a population only by sample from instruction sampling because it has no representativeness. In the thesis, relying on observed information, we have supplemented sample information which has not received instruction of investigation by probability and statistics method. Then we can infer the population using observed and supplementary information. Particularly, we have obtained iterative formula about estimation of population mean and variance and its application in investigation of economy crime and epidemic.Next,we consider the problems of ranked set sampling.This method is applicable whenever ranking of a set of sample units can be done easily by a judgement method of the interest variable or of the concomitant variable.When estimating the expectation of the interest variable Y of a bivariate variable (X,Y),people used the method of balanced ranked set sampling based on the concomitant X in previous paper.We give the method of unbalanced ranked set sampling based on a concomitant variable.Suppose (X,Y) follows morgenstern type distribution,we prove that sample mean under balanced ranked set sam-pling based on a concomitant variable is good than the sample mean under simple random sampling if they are of the same size.When (X,Y) follows morgenstern type bivariate ex-ponential distribution,we give the optimal allocation for estimating the mean of Y when the correlation coefficient of X and Y is known,using the method of unbalanced ranked set sampling based on a concomitant variable.Furthermore, we give the optimal alloca-tion when (X,Y) follows morgenstern type distribution.In the case of unknown correlation coefficient,we discuss the optimality of the allocation under several criteria and prove the allocation given in this paper is Bayes, admissible and minimax.In the following content,we discuss the optimal allocation for unbiased estimators of the correlation coefficientρ. We first give a class of unbiased estimators of p when the meanρof the interest variable Y is known and obtains a complete subclass of this class. Further, the optimal allocation of the unbiased estimators is found in this subclass and is proved to be Bayes, admissible and minimax. Lastly, the unbiased estimator ofρunder the optimal allocation in the case of knownθis reformed for estimatingρin the case of unknownθ, and the reformed estimator is shown to be strong consistent.In the last part of the paper,we discuss the exact distribu-tion and asymptotic distribution of sample sum under balanced ranked set sampling based on a concomitant variable when (X,Y) follows morgenstern type distribution.