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Families of distributions arising from distributions of order statistics

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Abstract

Consider starting from a symmetric distributionF on ℜ 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 withF 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 ℜ+ are discussed, and links are forged between the distributions underlying the skewt and logF distributions and Tadikamalla and Johnson'sL U family.

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References

  • Arnold, B. C., Balakrishnan, N., andNagaraja, H. N. (1992).A First Course in Order Statistics. Wiley, New York.

    MATH  Google Scholar 

  • Arnold, B. C. andBeaver, R. J. (2000). Hidden truncation models.Sankhya, 62: 23–35.

    MATH  MathSciNet  Google Scholar 

  • Arnold, B. C. andBeaver, R. J. (2002). Skewed multivariate models related to hidden truncation and/or selective reporting.Test, 11: 7–54.

    Article  MATH  MathSciNet  Google Scholar 

  • Arnold, B. C. andGroeneveld, R. A. (1995). Measuring skewness with respect to the mode.The American Statistician, 49: 34–38.

    Article  MathSciNet  Google Scholar 

  • Aroian, L. A. (1941). A study of R. A. Fisher'sz distribution and the relatedf distribution.Annals of Mathematical Statistics, 12: 429–448.

    MathSciNet  MATH  Google Scholar 

  • Azzalini, A. (1985). A class of distributions which includes the normal ones.Scandinavian Journal of Statistics, 12: 171–178.

    MathSciNet  MATH  Google Scholar 

  • Azzalini, A. andDalla Valle, A. (1996). The multivariate skew-normal distribution.Biometrika, 83: 715–726.

    Article  MATH  MathSciNet  Google Scholar 

  • Balakrishnan, N. (2002). Discussion of “skewed multivariate models related to hidden truncation and/or selective reporting”.Test, 11: 37–39.

    Google Scholar 

  • Balakrishnan, N. andAkhundov, I. S. (2003). A characterization by linearity of the regression function based on order statistics.Statistics and Probability Letters, 63: 435–440.

    Article  MATH  MathSciNet  Google Scholar 

  • Barlow, R. E. andProschan, F. (1966). Inequalities for linear combinations of order statistics from restricted families.Annals of Mathematical Statistics 37: 1574–1592.

    MathSciNet  MATH  Google Scholar 

  • Barlow, R. E. andProschan, F. (1975).Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rhinehart and Winston, New York.

    Google Scholar 

  • Barndorff-Nielsen, O., Kent, J., andSørensen, M. (1982). Normal variance-mean mixtures andz distributions.International Statistical Review, 50: 145–159.

    Article  MATH  MathSciNet  Google Scholar 

  • Birnbaum, A. andDudman, J. (1963). Logistic order statistics.Annals of Mathematical Statistics, 34: 658–663.

    MathSciNet  MATH  Google Scholar 

  • Box, G. E. P. andTiao, G. C. (1973).Bayesian Inference in Statistical Analysis. Addison-Wesley, Reading, MA

    MATH  Google Scholar 

  • Brown, B. W., Spears, F. M., andLevy, L. B. (2002). The logF: a distribution for all seasons.Computional Statistics, 17: 47–58.

    Article  MATH  MathSciNet  Google Scholar 

  • David, H. A. andNagaraja, H. N. (2003).Order Statistics. Wiley, Hoboken, NJ, 3rd ed.

    Book  MATH  Google Scholar 

  • Devroye, L. (1986).Non-Uniform Random Variate Generation, Springer, New York.

    MATH  Google Scholar 

  • Duncan, A. J. (1986).Quality Control and Industrial Statistics, Irwin, Homewood, Illinois, 5th ed.

    Google Scholar 

  • Durrans, S. R. (1992). Distributions of fractional order statistics in hydrology.Water Resources Research, 28: 1649–1655.

    Article  Google Scholar 

  • Eugene, N. (2001).A generalized normal distribution: properties, estimation and applications. Ph.D. thesis, Central Michigan University.

  • Eugene, N., Lee, C., andFamoye, F. (2002). Beta-normal distribution and its applications.Communications in Statistics—Theory and Methods, 31: 497–512.

    Article  MATH  MathSciNet  Google Scholar 

  • Fang, H. B., Fang, K. T. andKotz, S. (2002). The meta-elliptical distributions with given marginals.Journal of Multivariate Analysis, 82: 1–16.

    Article  MATH  MathSciNet  Google Scholar 

  • Ferreira, J. T. andSteel, M. F. J. (2003). Bayesian multivariate regression analysis with a new class of skewed distributions. Technical Report UKC/IMS/03/28, University of Kent.

  • Ferreira, J. T. A. S. andSteel, M. F. J. (2004a). A constructive representation of univariate skewed distributions. Statistics Research Report 422, University of Warwick.

  • Ferreira, J. T. A. S. andSteel, M. F. J. (2004b). Model comparison of coordinate-free multivariate skewed distributions with an application to stochastic frontiers. Statistics Research Report 423, University of Warwick.

  • Fisher, R. A. (1924). On a distribution yielding the error functions of several well known statistics. InProceedings of the International Mathematical Congress, vol. 2, pp. 805–813. Toronto.

  • Genton, M. G., He, L., andLiu, X. (2001). Moments of skew-normal random vectors and their quadratic forms.Statistics and Probability Letters, 51: 319–325.

    Article  MATH  MathSciNet  Google Scholar 

  • Greenland, S. (2001). Putting background information about relative risks into conjugate prior distributions.Biometrics, 57: 663–670.

    Article  MathSciNet  Google Scholar 

  • Greenwood, J. A., Landwehr, J. M., Matalas, N. C., andWallis, J. R. (1979). Probability weighted moments: Definition and relation to parameters of several distributions expressible in inverse form.Water Resources Research, 15: 1049–1054.

    Article  Google Scholar 

  • Gupta, R. D. andKundu, D. (1999). Generalized exponential distributions.Australian and New Zealand Journal of Statistics, 41: 173–188.

    Article  MATH  MathSciNet  Google Scholar 

  • Gupta, R. D. andKundu, D. (2001a). Exponentiated exponential family: An alternative to gamma and Weibull distributions.Biometrical Journal, 43:117–130.

    Article  MATH  MathSciNet  Google Scholar 

  • Gupta, R. D. andKundu, D. (2001b). Generalized exponential distribution: Different method of estimations.Journal of Statistical Computation and Simulation, 69:315–337.

    MATH  MathSciNet  Google Scholar 

  • Gupta, R. D. andKundu, D. (2003a). Closeness of gamma and generalized exponential distribution.Communications in Statistics—Theory and Methods, 32:705–721.

    Article  MATH  MathSciNet  Google Scholar 

  • Gupta, R. D. andKundu, D. (2003b). Discriminating between Weibull and generalized exponential distributions.Computational Statistics and Data Analysis, 43:179–196.

    MathSciNet  MATH  Google Scholar 

  • Gupta, R. D. andShah, B. K. (1965). Exact moments and percentage points of the order statistics and the distribution of the range from the logistic distribution.Annals of Mathematical Statistics, 36:907–920.

    MathSciNet  MATH  Google Scholar 

  • Huillard d'Aignaux, J. N., Cousens, S. N., andSmith, P. G. (2003). The predictability of the epidemic of variant Creutzfeldt-Jacob disease by back-calculation methods.Statistical Methods in Medical Research, 12:203–220.

    Article  MathSciNet  MATH  Google Scholar 

  • Johnson, N. L. andKotz, S. (1972).Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York.

    MATH  Google Scholar 

  • Johnson, N. L. andKotz, S. (1973). Extended and multivariate Tukey lambda distributions.Biometrika, 60:655–661.

    Article  MATH  MathSciNet  Google Scholar 

  • Johnson, N. L., Kotz, S., andBalakrishnan, N. (1995).Continuous Univariate Distributions, vol. 2. Wiley, New York, 2nd ed.

    MATH  Google Scholar 

  • Jones, M. C. (2001a). Multivariatet and beta distributions associated with the multivariateF distribution.Metrika, 54:215–231.

    Article  MathSciNet  Google Scholar 

  • Jones, M. C. (2001b). A skewt distribution. In C. A. Charalambides, M. V. Koutras, and N. Balakrishnan, eds.,Probability and Statistical Models with Applications, pp. 269–278. Chapman and Hall/CRC, London.

    Google Scholar 

  • Jones, M. C. (2002a). Changing marginals of multivariate densities by multiplication, with application to skewing spherically symmetric distributions.Journal of Multivariate Analysis, 81:85–99.

    Article  MATH  MathSciNet  Google Scholar 

  • Jones, M. C. (2002b). The complementary beta distribution.Journal of Statistical Planning and Inference, 104:329–337.

    Article  MATH  MathSciNet  Google Scholar 

  • Jones, M. C. (2002c). On fractional uniform order statistics.Statistics and Probability Letters, 58:93–96.

    Article  MATH  MathSciNet  Google Scholar 

  • Jones, M. C. (2002d). Student's simplest distribution.Journal of the Royal Statistical Society, Series D, 51:41–49.

    Article  Google Scholar 

  • Jones, M. C. andFaddy, M. J. (2003). A skew extension of thet distribution, with applications.Journal of the Royal Statistical Society, Series B, 65:159–174.

    Article  MATH  MathSciNet  Google Scholar 

  • Jones, M. C. andLarsen, P. V. (2004). Multivariate distributions with support above the diagonal.Biometrika. To appear.

  • Kamps, U. (1991). A general recurrence relation for moments of order statistics in a class of probability distributions and characterizations.Metrika, 38:215–225.

    MathSciNet  MATH  Google Scholar 

  • Kent, J. T. (1978). Some probabilistic properties of Bessel functions.Annals of Probability, 6:760–770.

    MATH  MathSciNet  Google Scholar 

  • Kent, J. T. andTyler, D. E. (1991). Redescendingm-estimates of multivariate location and scatter.Annals of Statistics, 19:2102–2119.

    MATH  MathSciNet  Google Scholar 

  • Kent, J. T., Tyler, D. E. andVardi, Y. (1994). A curious likelihood identity for the multivariatet distribution.Communications in Statistics: Simulation and Computation, 23:441–453.

    MATH  MathSciNet  Google Scholar 

  • Lange, K. L., Little, R. J. A. andTaylor, J. M. G. (1989). Robust statistical modelling using thet distribution.Journal of the American Statistical Association, 84:881–896.

    Article  MathSciNet  Google Scholar 

  • Lehmann, E. L. (1953). The power of rank tests.Annals of Mathematical Statistics, 24:23–43.

    MathSciNet  MATH  Google Scholar 

  • Numerical Algorithms Group (2003). NAG FORTRAN libraries.

  • Olkin, I. andLiu, R. (2003). A bivariate beta distribution.Statistics and Probability Letters, 62:407–412.

    Article  MATH  MathSciNet  Google Scholar 

  • Papadatos, N. (1995). Intermediate order statistics with applications to nonparametric estimation.Statistics and Probability Letters, 22:231–238.

    Article  MATH  MathSciNet  Google Scholar 

  • Patel, J. K. andRead, C. B. (1982).Handbook of the Normal Distribution. Dekker, New York.

    MATH  Google Scholar 

  • Peng, Y., Dear, K. B. G., andDenham, J. W. (1998). A generalizedF mixture model for cure rate estimation.Statistics in Medicine, 17:813–830.

    Article  Google Scholar 

  • Prentice, R. L. (1975). Discrimination among some parametric models.Biometrika, 62:607–614.

    Article  MATH  MathSciNet  Google Scholar 

  • Ratnaparkhi, M. V. andMosimann, J. E. (1990). On the normality of transformed beta and unit-gamma random-variables.Communications in Statistics—Theory and Methods, 19:3833–3854.

    Google Scholar 

  • Reiss, R. D. (1989).Approximate Distributions of Order Statistics; with Applications to Nonparametric Statistics. Springer, New York.

    MATH  Google Scholar 

  • Rohatgi, V. K. andSaleh, A. K. M. E. (1988). A class of distributions connected to order statistics with nonintegral sample size.Communications in Statistics—Theory and Methods, 17:2005–2012.

    MATH  MathSciNet  Google Scholar 

  • Rydberg, T. H. (2000). Realistic statistical modelling of financial data.International Statistical Review, 68:233–258.

    MATH  Google Scholar 

  • Samorodnitsky, G. andTaqqu, M. S. (1994).Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman and Hall/CRC, Boca Raton.

    MATH  Google Scholar 

  • Smyth, G. K. (1994). A note on modelling cross-correlations: Hyperbolic secant regression.Biometrika, 81:396–402.

    Article  MATH  Google Scholar 

  • Stigler, S. M. (1977). Fractional order statistics, with applications.Journal of the American Statistical Association, 72:544–550.

    Article  MATH  MathSciNet  Google Scholar 

  • Tadikamalla, P. K. (1980). Random sampling from the exponential power distribution.Journal of the American Statistical Association, 75:683–686.

    Article  MATH  MathSciNet  Google Scholar 

  • Tadikamalla, P. K. andJohnson, N. L. (1982). Systems of frequency curves generated by transformations of logistic variables.Biometrika, 69:461–465.

    Article  MathSciNet  Google Scholar 

  • Vaughan, D. C. (1992). Expected values, variances and covariances of order statistics for Student'st-distribution with two degrees of freedom.Communications in Statistics—Simulation and Computation, 21:391–404.

    MATH  MathSciNet  Google Scholar 

  • Wand, M. P. andJones, M. C., (1995).Kernel Smoothing. Chapman and Hall, London.

    MATH  Google Scholar 

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  1. Department of Statistics, The Open University, Walton Hall, MK7 6AA, Milton Keynes, U.K.

    M. C. Jones

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Jones, M.C. Families of distributions arising from distributions of order statistics. Test 13, 1–43 (2004). https://doi.org/10.1007/BF02602999

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