Higher-order Expansions For Distributions Of Extremes From Two Kinds Of Distributions

For a sequence of independent and identically distributed random variables with marginal general error distribution and skew-t distribution, the higher-order expansions and application of extremes are studied in the thesis, which are organized as two parts.For the first part of this thesis, the higher-order expansions of density and moments of normalized maximum with general error distribution are established. Firstly the higher-order expansions of density are obtained under two couple of norming constants. Subsequently, the results are applied to derive the higher-order expansions of the moments of extremes, which is consistent with the previous re-search results.In the second part, asymptotic behaviors of the extremes from the skew-t distri-bution are considered. The study is two folded. The first derives the Mills’equality Mills’ratio and tail behavior of skew-t distribution, which tells the extreme val-ue distribution of the normalized maximum. The second establishes higher order expansions of the probability density and cumulative distribution functions for the maximum under suitable norming constants.