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Uniform convergence rates of skew-normal extremes

Xiong Qian, Peng Zuoxiang, Nadarajah Saralees

Probability and Mathematical Statistics

2023

Vol. 43, Fasc. 2

165 --184

Let Mn = max(X1, . . . , Xn) denote the partial maximum of an independent and identically distributed skew-normal random sequence. In this paper, the rate of uniform convergence of skew-normal extremes is derived. It is shown that with optimal normalizing constants the convergence rate of [formula] to its ultimate extreme value distribution is proportional to [formula].

Expansions for moments of logarithmic skew-normal extremes

Liao Xin, Peng Zuoxiang, Nadarajah Saralees

Probability and Mathematical Statistics

2022

Vol. 42, Fasc. 1

41--61

Liao, Peng and Nadarajah [ J. Appl. Probab. 50 (2013), 900-907] derived asymptotic expansions for the partial maximum of a random sample from the logarithmic skew-normal distribution. Here, we derive asymptotic expansions for moments of the partial maximum using optimal norming constants. These expansions can be used to deduce convergence rates of moments of the normalized maxima to the moments of the corresponding extreme value distribution. A numerical study is made to compare the actual values of moments with their asymptotics, which shows that the convergence is exceedingly slow, and adjustment is needed whenever we use the limits to replace moments of the partial maximum.