Asymptotic expansions for moments of skew-normal extremes

Abstract

Skew normal distribution, an extension of normal distribution, has been widely used in applied statistics, engineer, meteorology and financial time series modeling. We have considered the Mills type inequalities and Mills type ratios of skew normal distribution which are applied to establish the asymptotic expansions of distributions of extremes. In this short note, we focus on the limiting behaviors of moments for normalized partial maxima of skew-normal samples. Under optimal norming constants, asymptotic expansions for moments of maxima of skew-normal samples are derived. These expansions are used to deduce convergence rates of moments of the normalized maxima to the moments of the corresponding extreme value distribution.

Moment convergence of powered normal extremes

“…Nair (1981) derived asymptotic expansions for the moments of standard normal extremes. Further, Liao et al (2013) and Jia et al (2015) extended the results to that of skew-normal distribution and general error distribution, respectively. For other work related to moments convergence on extremes, see, e.g.…”

Expansions for moments of logarithmic skew-normal extremes

“…Pickands (1968) showed that moments of normalized extremes converge to corresponding moments of the extreme value distribution provided that the moments are finite for sufficiently large n and F is in the domain of attraction of an extreme value distribution. For more studies related to the asymptotic expansions for moments of M n , we refer to Nair (1981) for the normal distribution, Liao et al (2014) for the skew-normal distribution, and Pu et al (2015) for the general error distribution.…”

Competition Mathematics as a Complement and Enhancement for In-Class Instruction

“…For λ = 0, the Mills' ratios, the distributional tail representation and the higher-order expansions of the extremes of SN(λ) were studied by Liao et al (2014b). The higher-order expansions of moments of the extremes of SN(λ), λ = 0 were studied by Liao et al (2013a). Liao et al (2013bLiao et al ( , 2014a) also considered the tail behaviors and higher-order expansions of the distribution of extremes for the log-skew-normal distribution.…”