Expansions On Extremes From Logarithmic General Error Distribution

Logarithmic general error distribution is an extension of the log-normal dis-tribution. Fv(x) and fv(x) are the cumulative distribution function and probabil-ity density function of logarithmic general error distribution, v≥ 1. This paper attempts to discuss the higher-order expansions of Fvn(anx+bn) under linear nor-malizing constants an and bn. and the higher-order expansions of distributions of maxima and its density under power normalization.In the first part, we study the higher-order expansions on extremes under lin-ear normalization for a sequence of independent and identically distributed ran-dom variables with marginal distribution Fv(x). Firstly, we derive two facts that Fv ∈ D1(∧),v>1, F1∈(φ(?)) and optimal norming constants an, bn from the tail representation of the logarithmic general error distribution. Secondly, with pre-cise distributional tail representation we establish the higher-order expansions of F?(anx+bn).In the secondly part, under power normalization we study the higher-order expansions of density and distribution of extremes form by logarithmic general error distribution. The fact that Fv ∈ Dp(φ1), v> 1 under power normalization can be found by the domain of attraction of Fv(x) under linear normalization. Subsequently, we obtain the optimal norming constants αn, βn and the higher-order expansions of distribution and density of extremes under power normalization.