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Families of distributions arising from distributions of order statistics. (English) Zbl 1110.62012

Summary: Consider starting from a symmetric distribution $F$ on $\Re$ and generating a family of distributions from it by employing two parameters whose role is to introduce skewness and to vary tail weight. The proposal in this paper is a simple generalisation of the use of the collection of order statistics distributions associated with $F$ for this purpose; an alternative derivation of this family of distributions is as the result of applying the inverse probability integral transformation to the beta distribution.

General properties of the proposed family of distributions are explored. It is argued that two particular special cases are especially attractive because they appear to provide the most tractable instances of families with power and exponential tails; these are the skewt distribution and the $logF$ distribution, respectively. Limited experience with fitting the distributions to data in their four-parameter form, with location and scale parameters added, is described, and hopes for their incorporation into complex modelling situations expressed. Extensions to the multivariate case and to ${\Re }^{+}$ are discussed, and links are forged between the distributions underlying the skewt and $logF$ distributions and P.R. Tadikamalla and N.L. Johnson’s ${L}_{U}$ family [see Commun. Stat., Simulation Comput. 11, 249–271 (1982; Zbl 0496.62099)].

##### MSC:
 62E10 Characterization and structure theory of statistical distributions 62G30 Order statistics; empirical distribution functions 60E05 General theory of probability distributions
KernSmooth; nag
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