Applications of asymptotic expansions to some statistical problems

The dissertation is composed of four research papers. In all the papers asymptotic methods and techniques are the main tools used to reach conclusions. [1] A Berry-Esseen theorem for Hypergeometric probabilities under minimal conditions. [2] Normal Approximation to the Hypergeometric distribution in nonstandard cases and a sub-Gaussian Berry-Esseen Theorem. [3] Asymptotic properties of sample quantiles from a finite population. [4] Edgeworth expansions for Spectral density estimates.; The papers [1]-[3] consider the problem of quantile estimation in a finite population setup and related asymptotic results regarding Normal approximation to Hypergeometric random variables. The asymptotic properties of the sample quantile are derived under a superpopulation model. It is shown that the sample quantile is asymptotically normal and the scaled sample variance of the sample quantile converges to the asymptotic variance under a slight moment condition. The performance of the bootstrap is also investigated. We show that the usual bootstrap suggested by Gross (1980) fails in this case. A suitably modified version of the bootstrapped sample quantile converges in distribution to the same asymptotic normal distribution as the sample quantile. Consistency of the modified bootstrap variance estimate is proved under the same moment conditions.; The paper [4] considers the problem of Edgeworth expansion of spectral density estimators of a stationary time series. The spectral density estimate at each frequency is based on tapered periodograms of overlapping blocks of observations. We give conditions for the validity of a general order Edgeworth expansion under an approximate strong mixing condition on the random variables, and also establish a moderate deviation inequality. We also verify the conditions explicitly for linear time series, which are satisfied under mild and easy-to-check conditions on the innovation variables and on their nonrandom co-efficients. The work makes use of the results of Lahiri (2007) where general order Edgeworth expansions for functions of blocks of weakly dependent random variables are derived. In our work we relax the assumption of Gaussianity, which was used by Velasco and Robinson (2001) to obtain similar Edgeworth expansions.