Distribution

This book presents a study of statistical inferences based on the kernel-type estimators of distribution functions. The inferences involve matters such as quantile estimation, nonparametric tests, and mean residual life expectation, to name just some. Convergence rates for the kernel estimators of density functions are slower than ordinary parametric estimators, which have root-n consistency. If the appropriate kernel function is used, the kernel estimators of the distribution functions recover the root-n consistency, and the inferences based on kernel distribution estimators have root-n consistency. Further, the kernel-type estimator produces smooth estimation results. The estimators based on the empirical distribution function have discrete distribution, and the normal approximation cannot be improved—that is, the validity of the Edgeworth expansion cannot be proved. If the support of the population density function is bounded, there is a boundary problem, namely the estimator does not have consistency near the boundary. The book also contains a study of the mean squared errors of the estimators and the Edgeworth expansion for quantile estimators.

This book provides the statistical basis for quantitative risk management by presenting and explaining the most important distributions. Distributions describe the occurrence and impact of a risk. They are a prerequisite for risk aggregation, risk analysis and risk assessment as required by the German revision standards IDW PS 340, StaRUG and FISG.

Distribution systems analysis employs a set of techniques to simulate, analyse, and optimise power distribution systems. Combined with automation, these techniques underpin the concept of the smart grid. In recent years, distribution systems have been facing growing challenges, due to increasing demand as well as the rising shares of distributed and volatile renewable energy sources.