Expansions On Extremes From Skew-normal Distribution

In this thesis, with optimal normalizing constants the asymptotic expansions of maxima of independent and identically distributed random variables with skew-normal distribution were considered under linear normalization and power normal-ization respectively. There are two parts of this thesis.For the first part of this thesis, firstly by using the higher-order expansions of distribution function of maximum from skew-normal distribution under linear normalization and the relationship between distribution function and probability density function, we get its higher-order expansions of probability density function of maximum. Subsequently, by using the higher-order expansions of density and the dominated convergence theorem, we derive the higher-order expansions of linear normalized moments of maximum.The second part of this thesis studied the higher-order expansions of distri-bution function and probability density function of maximum from skew-normal distribution under power normalization. Firstly we deduce the type of extreme value distribution under power normalization and norming constants through the relationship between linear normalized extreme value theorem and the power nor-malized extreme value theorem. Then we establish the higher-order expansions of distribution function and probability density function of maximum from skew-normal distribution under power normalization. Furthermore, numerical studies are presented to illustrate the accuracy of higher-order expansions of the cumulative distribution function from skew-normal distribution under two different normaliza-tion, we have the following findings:i) The asymptotics under linear normalization are more closer to its actual values, ii) All asymptotics are more closer to their actual values as λ becomes larger except some special cases, iii) The third-order asymptotics are closest to the actual values as λ≥0, contrary to the case of λ< 0, which shows that the first-order asymptotics may be better.