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A location-scale family generated by a given symmetric distribution and the sample variance. (English) Zbl 1068.62011

Summary: Let \(X_i\), \(i=1,\dots,n\), \(n\geq 2\), be a random sample from a distribution \(F\) of a location scale family of distribution \(F_0((x-\mu)/ \sigma)\), \(\mu\) is some real number, \(\sigma>0\) for some given distributions \(F_0\). Assume that \(F_0\) is a symmetric distribution with finite moments of all orders and is determined by its moments. Two characterizations of \(F_0((x-\mu)/\sigma)\) based on the distribution function \(G_n\) of \(\sum^n_{i=1}(X_i-\overline X)^2/\sigma^2\), are given.

MSC:

62E10 Characterization and structure theory of statistical distributions
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62N05 Reliability and life testing
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