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The skew-normal distribution and related multivariate families. (English) Zbl 1091.62046
If \(f_0\) is a \(d\)-dimensional density with \(f_0(x)=f_0(-x)\), \(G\) is a one-dimensional CDF with symmetric about 0 PDF and \(w: R^d\to R\) is any function with \(w(-x)=-w(x)\), then \(f(z)=2f_0(z)G(w(z))\) is a density function on \(R^d\). With \(f_0\sim N(0,\Omega)\), \(G\sim N(0,1)\), and a linear function \(w\) one obtains \(f\) being a skew-normal (SN) multivariate distribution. An overview of properties of univariate and multivariate SN distributions is given. Statistical inference for i.i.d. SN observations based on the maximum likelihood is described. Skew-elliptical distributions with elliptical \(f_0\) (specially skew \(t\)-distribution) and distributions with polynomial \(w\) are discussed. Such distributions arise in selective sampling problems, stochastic frontier models and some models of financial markets. In the discussion some semiparametric and nonparametric problems related to such distributions are described.

MSC:
62H10 Multivariate distribution of statistics
62H12 Estimation in multivariate analysis
62E15 Exact distribution theory in statistics
62F10 Point estimation
Software:
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