Multivariate distributions with support above the diagonal

A general family of distributions for the empirical modelling of ordered multivariate data is proposed. The family is based on, but greatly extends, the joint distribution of order statistics from an independent and identically distributed univariate sample. General properties, including marginal and conditional distributions, bivariate dependence, limiting distributions and links to the Dirichlet distribution are described. Univariate and bivariate special cases of the multivariate distributions, the latter including an equivalent rotated version, are considered. Two particular tractable special cases are stressed. The models are successfully and usefully fitted, by maximum likelihood, to meteorological data. The models are also applicable to data in which one variable is unconstrained and the other are all nonnegative. Copyright 2004, Oxford University Press.

[1]  R. Fisher.  036: On a Distribution Yielding the Error Functions of Several Well Known Statistics. , 1924 .

[2]  A. T. McKay,et al.  Sampling from Batches , 1934 .

[3]  Leo A. Aroian,et al.  A Study of R. A. Fisher's $z$ Distribution and the Related F Distribution , 1941 .

[4]  P. Bickel.  Some contributions to the theory of order statistics , 1967 .

[5]  Ross L. Prentice,et al.  Discrimination among some parametric models , 1975 .

[6]  Stephen M. Stigler,et al.  Fractional Order Statistics, with Applications , 1977 .

[7]  J. D. T. Oliveira,et al.  The Asymptotic Theory of Extreme Order Statistics , 1979 .

[8]  Wayne S. Smith,et al.  Interactive Elicitation of Opinion for a Normal Linear Model , 1980 .

[9]  P. Holland,et al.  Dependence function for continuous bivariate densities , 1987 .

[10]  R. Reiss.  Approximate Distributions of Order Statistics , 1989 .

[11]  B. Arnold,et al.  A first course in order statistics , 1994 .

[12]  Nickos Papadatos,et al.  Intermediate order statistics with applications to nonparametric estimation , 1995 .

[13]  Bounded Bivariate Distributions with Nearly Normal Marginals , 1996 .

[14]  K. Joag-dev,et al.  Bivariate Dependence Properties of Order Statistics , 1996 .

[15]  A. Subbotin,et al.  New data on the morphological differences of anchovy eggs (Engraulis encrasicolus L) in the Black Sea , 1997 .

[16]  H. Joe.  Multivariate models and dependence concepts , 1998 .

[17]  S. Nadarajah,et al.  Ordered multivariate extremes , 1998 .

[18]  W. Walls,et al.  Uncertainty in the Movie Industry: Does Star Power Reduce the Terror of the Box Office? , 1999 .

[19]  M. Parlange,et al.  An Extended Version of the Richardson Model for Simulating Daily Weather Variables , 2000 .

[20]  Chris Jones,et al.  A skew t distribution , 2000 .

[21]  S. Coles,et al.  An Introduction to Statistical Modeling of Extreme Values , 2001 .

[22]  G. L’italien,et al.  Distribution of blood pressure and hypertension in Canada and the United States. , 2001, American journal of hypertension.

[23]  M. C. Jones.  On fractional uniform order statistics , 2002 .

[24]  M. C. Jones.  Student's simplest distribution , 2002 .

[25]  C. W. Richardson,et al.  Evaluating the Adequacy of Simulating Maximum and Minimum Daily Air Temperature with the Normal Distribution , 2002 .

[26]  Lawrence B. Levy,et al.  The log F: A Distribution for All Seasons , 2002, Comput. Stat..

[27]  M. C. Jones,et al.  A skew extension of the t‐distribution, with applications , 2003 .

[28]  M. C. Jones.  Families of distributions arising from distributions of order statistics , 2004 .