Essays on inference from multi -stage samples with applications to inequality measurement and on estimation of monotone index model

The first chapter of this thesis develops a GMM-based framework for asymptotic inference to analyze data from surveys whose designs involve stratification and clustering. I set up the estimation problem, derive the appropriate asymptotic distribution theory as the number of clusters tends to infinity and compute asymptotic standard errors that are robust to sample-design effects. The analysis is then extended to nonparametric regression and to semiparametric estimation based on U-processes. I also describe how to apply these methods in situations involving multiple levels of stratification and clustering.;The second chapter extends the methods of the first chapter to continuous stochastic processes as is essential for the estimation of Lorenz curves and the Gini coefficient of economic inequality. It then develops a consistent test for inequality dominance using the asymptotic distribution theory obtained above. These methods are then applied to test for inequality dominance in terms of monthly per capita expenditure using Indian household consumption data from 1987--8 and 1993--4. Ignoring the survey design implies qualitatively different conclusions in several cases.;The third and final chapter proposes a new technique for estimating the parameters (up to location and scale) of a monotone-index model, based on sorting the data. The key observation guiding the estimation procedure is that the sum of distances between pairs of adjacent observations is minimized (over all possible permutations) when the observations are sorted by their values. The method does not require subjective bandwidth choice and involves minimization of a U-process. I derive n -consistency and asymptotic normality of the estimator and a method for estimating the covariance matrix consistently when the data come from simple random samples. I then extend the analysis to cover monotone-index panel data models where every individual is observed over at least two periods. The results are then extended to the situation where the sampling design involves stratification and clustering.