Applications of Empirical Likelihood to quantile estimation and longitudinal data
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As a non-parametric method, Empirical Likelihood (EL) has been attracting serious attention from researchers in statistics, econometrics, engineering and biostatistics. By defining the estimation equations in EL appropriately, we can extend EL to various data settings, especially those in which parametric likelihoods are absent. In this dissertation, two applications of empirical likelihood are explored: quantile estimation and longitudinal data analysis. Quantile estimation for discrete data analysis has not been well studied. For a given 0 < p < 1, the commonly used sample quantile may or may not be consistent for the pth quantile, depending on whether or not the underlying distribution has a plateau at the level of p. I propose an EL-based categorization procedure which not only helps determine the shape of the true distribution at level p, but also provides a way of formulating a new estimator that is consistent in any case. For non-Gaussian longitudinal data, generalized estimating equations (GEE) are a popular class of marginal models. While the GEE estimator is consistent and robust, it may suffer significant loss of efficiency if the working correlation structure is misspecified. I consider the use of EL to select working correlations for GEE models, for which parametric likelihoods are absent and quasi-likelihoods are difficult to construct.