Families of distributions arising from distributions of order statistics

Jones, M.C. (2004). Families of distributions arising from distributions of order statistics. TEST, 13(1) pp. 1–43.

DOI: https://doi.org/10.1007/bf02602999

URL: http://www.seio.es/test/Archivos/t131/Test131jon.p...

Abstract

Consider starting from a symmetric distribution F on IR 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 statistic 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 skew t distribution and the log F 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 IR+ are discussed, and links are forged between the distributions underlying the skew t and log F distributions and Tadikamalla and Johnson's LU family.

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