| The thesis is composed of two parts. In Part I we extend the class of non-negative, asymmetric kernel density estimators and propose Birnbaum-Saunders (BS) and lognormal (LN) kernel density functions. The density functions have bounded support on (0,infinity). Both BS and LN kernel estimators are free of boundary bias, non-negative, with natural varying shave and achieve the optimal rate of convergence for the mean integrated squared error. We apply BS and LN kernel density estimators to both high frequency time duration data and simulated data. The comparisons are made on several nonparametric kernel density estimators. BS and LN kernels perform better near the boundary in terms of bias reduction. In Part II we deal with the estimation of the extreme value index (EVI) gamma for i.i.d. data from heavy-tailed distributions (i.e. gamma > 0). We study in detail the doman of attraction (DOA) approach. First, we show the equivalence of multiple versions of first and second order conditions in the regular variation framework. Second, we give an overview of the existing EVI estimators and compare common EVI estimators by minimal AMSE criterion. Our region plot enables one to choose the best estimator within the approach. Third, we perform extensive simulations to choose optimal number k of largest order statistics in the sample. Finally, we estimate the EVI gamma and extreme quantile delta p from both DOA and parametric approaches. Our real data analysis shows that the DOA approach provides good quality of estimates. |
