Distributional expansions on extremes from skew-normal distribution under power normalization (Q1706457)

Let \(X_1,X_2,\ldots\) be a sequence of independent and identically distributed random variables, each having a skew-normal distribution (with parameter \(\lambda\in\mathbb{R}\)) with probability density function given by \[ f_\lambda(x)=2\phi(x)\Phi(\lambda x)\,,\qquad x\in\mathbb{R}\,, \] where \(\phi\) and \(\Phi\) are the density and distribution functions of the standard normal random variable, respectively. The main results of this paper are expansions of the distribution and density functions of \(M_n=\max(X_1,\ldots,X_n)\) under appropriate power normalization, that is, a normalization of the form \(\left(M_n/\alpha_n\right)^{1/\beta_n}\), with appropriate choices of \(\alpha_n\) and \(\beta_n\). The authors also present numerical illustrations of their higher-order expansions, and some comparison with analogous results under a linear normalization of the maximum.