| Ranked set sampling(RSS) has been shown to be a cost-effective alternative to simple random sampling(SRS). We consider three problems in this area. Much of the work done in this area have been mainly based on asymptotic theory. But for any given problem, asymptotic result may not be available, and if available, it may not be valid due to the small sample size.;In this thesis, we first consider nonparametric bootstrap as an alternative to asymptotic inference in nonparametric settings. To obtain standard error and confidence interval estimates, we discuss and compare three methods of resampling a given ranked set sample. Chen, Bai, and Sinha[11] suggest a natural method to obtain bootstrap samples from each row of a RSS. We prove that this method is consistent for a location estimator. We propose two other methods that are designed to obtain more stratified resamples from the given sample. Algorithms are provided for these methods. We recommend a method that obtains a bootstrap RSS from the observations. We prove several properties of this method, including consistency for a location parameter. We define two types of L-estimators for RSS and obtain expressions for the exact moments for one of them.;Yu and Lam[49] proposed using the RSS anolog of the SRS regression estimator to estimate the mean of Y, a variable that is difficult and expensive to measure, but a concomitant X, linearly related to Y, is available and is easy and inexpensive to rank and measure. We use semi-parametric bootstrap to obtain standard error and confidence interval estimates for the mean of Y. In particular, we consider four approaches to constructing confidence intervals for the mean of Y, one is a modification of the approach proposed by Yu and Lam[49], two that are based on resampling the residuals, and one is based on pairwise resampling.;For parametric analyses, we study situations where the parameter of a family of distributions is in the form (theta, lambda) where theta ∈ theta ⊂ R and lambda ∈ Lambda ⊂ Rq . The paramter of interest is theta while lambda is treated as a nuisance parameter. In SRS, Gong and Samaniego[16] considered replacing lambda in the likelihood equation by a consistent estimator that does not depend on theta, and give conditions under which a solution(PMLE) that maximizes the pseudo-likelihood equation exists and is asymptotically normal. We follow their approach in RSS and give analogous conditions under which a solution to the pseudo-likelihood equation exists and is asymptotically normal. As an application, we consider the bivariate normal family with parameter (0, 1, muY, s2Y , rho) where the parameter of interest is rho while (mu Y, s2Y ) is treated as a nuisance parameter. We replace (muY, s2Y ) in the likelihood equation by their corresponding moment estimates to obtain a pseudo-likelihood equation. A PMLE of rho is obtained and we show that it is consistent and asymptotically normal. |
