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Characterization Theorems and Goodness-of-Fit Tests

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Goodness-of-Fit Tests and Model Validity

Part of the book series: Statistics for Industry and Technology ((SIT))

Abstract

Karl Pearson’s chi-square goodness-of-fit test of 1900 is considered an epochal contribution to the science in general and statistics in particular. Regarded as the first objective criterion for agreement between a theory and reality, and suggested as “beginning the prelude to the modern era in statistics,” it stimulated a broadband enquiry into the basics of statistics and led to numerous concepts and ideas which are now common fare in statistical science. Over the decades of the twentieth century the goodness-of-fit has become a substantial field of statistical science of both theoretical and applied importance, and has led to development of a variety of statistical tools. The characterization theorems in probability and statistics, the other topic of our focus, are widely appreciated for their role in clarifying the structure of the families of probability distributions. The purpose of this paper is twofold. The first is to demonstrate that characterization theorems can be natural, logical and effective starting points for constructing goodness-of-fit tests. Towards this end, several entropy and independence characterizations of the normal and the inverse gaussian (IG) distributions, which have resulted in goodness-of-fit tests, are used. The second goal of this paper is to show that the interplay between distributional characterizations and goodness-of-fit assessment continues to be a stimulus for new discoveries and ideas. The point is illustrated using the new concepts of IG symmetry, IG skewness and IG kurtosis, which resulted from goodness-of-fit investigations and have substantially expanded our understanding of the striking and intriguing analogies between the IG and normal families.

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References

  1. Barlow, R. E., Bartholomew, D. J., Bremner, J. M., and Brunk, H. D. (1971).Statistical inference under order restrictionsNew York: John Wiley & Sons.

    Google Scholar 

  2. Bartholomew, D. J. (1959a). A test of homogeneity for ordered alternativesBiometrika,46, 36–38.

    MathSciNet  MATH  Google Scholar 

  3. Bartholomew, D. J. (1959b). A test of homogeneity for ordered alternatives IIBiometrika, 46,328–335.

    MathSciNet  MATH  Google Scholar 

  4. Beirlant, J., Dudewicz, E. J., Györfi, L., and van der Meulen, E. C. (1997). Nonparametric entropy: An overviewInternational Journal of Mathematical and Statistical Sciences,6, 17–39.

    MathSciNet  MATH  Google Scholar 

  5. Bingham, N. H. (2000). Studies in the history of probability and statistics XLVI. Measure into probability: From Lebesgue to KolmogorovBiometrika,87, 145–156.

    Article  MathSciNet  MATH  Google Scholar 

  6. Blum, J. R., Kiefer, J., and Rosenblatt, M. (1961). Distribution free tests of independence based on the sample distribution functionAnnals of Mathematical Statistics, 32, 485–498.

    Article  MathSciNet  MATH  Google Scholar 

  7. Box, G. E. P. and Cox, D. R. (1964). An analysis of transformationsJournal of the Royal Statistical Society Series B 26211–252.

    MathSciNet  MATH  Google Scholar 

  8. Chernoff, H. (1954). Testing homogeneity against ordered alternativesAnnals of Mathematical Statistics, 34, 945–956.

    Google Scholar 

  9. Cramér, H. (1936). Über eine Eigenschaft der normalen VerteilungsfunktionMath. Zeitschrift,41, 405–411.

    Article  Google Scholar 

  10. Cressie, N. (1976). On the logarithms of high-order spacingsBiometrika, 63, 343–355.

    Article  MathSciNet  MATH  Google Scholar 

  11. Csiszar, I. (1975). I-divergence geometry of probability distributions an minimization problemsAnnals of Probability,3, 146–158.

    Article  MathSciNet  MATH  Google Scholar 

  12. D’Agostino, R. B. and Stephens, M. A. (Eds.) (1986).Goodness-of-fit TechniquesMarcel Dekker: New York.

    MATH  Google Scholar 

  13. Darling, D. A. (1953). On a class of problems related to the random division of an intervalAnnals of Mathematical Statistics, 24, 239–253.

    Article  MathSciNet  MATH  Google Scholar 

  14. Dawid, A. P. (1978). Comments on “The inverse Gaussian distribution and its statistical application: A review,”Journal of the Royal Statistical Society Series B, 40, 280.

    Google Scholar 

  15. Dudewicz, E. and van der Meulen E. C. (1981). Entropy-based tests of uniformityJournal of the American Statistical Association, 76, 967–974.

    Article  MathSciNet  MATH  Google Scholar 

  16. Dugué, D. (1941), Sur un nouveau type de courbe de frequenceComptes Rendus de l’Academie des Sciences Paris, 213, 634–635.

    Google Scholar 

  17. Ebrahimi, N., Habibullah, M., and Soofi, E. S. (1992). Testing for exponentiality based on Kullback-Leiber informationJournal of the Royal Statistical Society Series B, 54, 739–748.

    MathSciNet  MATH  Google Scholar 

  18. Fisher, R. A. (1932).Statistical Methods for Research WorkersLondon: Oliver and Boyd,.

    MATH  Google Scholar 

  19. Folks, J. L. and Chhikara, R. S. (1978). The inverse gaussian distribution and its statistical application - a reviewJournal of the Royal Statistical Society Series B, 40, 263–289.

    MathSciNet  MATH  Google Scholar 

  20. Freimer, M., Kollia, G., Mudholkar, G. S., and Lin, C. T. (1989). Extremes, extreme spacings and outliers in the Tukey and Weibull familiesCommunications in Statistics-Theory and Methods, 18, 4261–4274.

    Article  MathSciNet  MATH  Google Scholar 

  21. Fujino, Y. (1979). Tests for the homogeneity of a set of variances against ordered alternativesBiometrika, 66, 133–140.

    Article  MathSciNet  MATH  Google Scholar 

  22. Geary, R. C. (1936). The distribution of ‘Student’s’ ratio for non-normal samplesSupplement to the Journal of the Royal Statistical Society, 3, 178–184.

    Article  MATH  Google Scholar 

  23. Gokhale, D. V. (1983). On entropy-based goodness-of-fit testsComputational Statistics and Data Analysis, 1, 157–165.

    Article  MATH  Google Scholar 

  24. Greenwood, M. (1946). The statistical study of infectious diseaseJournal of the Royal Statistical Society Series B, 109, 85–110.

    MathSciNet  Google Scholar 

  25. HackingI.(1984). Trial by numbersScience-8.4, 5, 69–73.

    Google Scholar 

  26. HallP.(1984). Limit theorems for sums of general functions of m-spacingsMathematical Statistics and Data Analysis, 1, 517–532.

    Google Scholar 

  27. Hall, P. (1986). On powerful distributional tests on sample spacingsJournal of Multivariate Analysis, 19, 201–255.

    Article  MathSciNet  MATH  Google Scholar 

  28. Höeffding, W. (1948). A nonparametric test of independenceAnnals of Mathematical Statistics, 19, 546.

    Article  MathSciNet  Google Scholar 

  29. Hwang, T-Y. and Hu, C-Y. (1999). On a characterization of the gamma distribution: The independence of the sample mean and the sample coefficient of variationAnnals of the Institute of Statistical Mathematics, 51, 749–753.

    Article  MathSciNet  MATH  Google Scholar 

  30. Iyengar, S. and Patwardhan, G. (1988). Recent developments in the inverse Gaussian distributionHandbook of Statistics, Volume 7479–490.

    Article  Google Scholar 

  31. Kagan, A. M., Linnik, Y. V., and Rao, B. (1973).Characterization Problems in Mathematical StatisticsNew York: John Wiley & Sons.

    MATH  Google Scholar 

  32. Kashimov, S. A. (1989). Asymptotic properties of functions of spacingsTheory of Probability and its Applications, 34, 298–307.

    Article  MathSciNet  Google Scholar 

  33. Khatri, C. G. (1962). A characterization of the inverse gaussian distributionAnnals of Mathematical Statistics, 33, 800–803.

    Article  MathSciNet  MATH  Google Scholar 

  34. Kolmogorov, A. (1933). Sulla determinazione empirica di una legge di distribuzioneGior. Ist. Ital. Attuari, 4, 83–91.

    MATH  Google Scholar 

  35. Kullback, S. (1959).Information Theory and Statistics p.15, New York: John Wiley & Sons.

    Google Scholar 

  36. Kullback, S. and Leibler, R. A. (1951). On information and sufficiencyAnnals of Mathematical Statistics, 22, 79–86.

    Article  MathSciNet  MATH  Google Scholar 

  37. Lin, C. C. and Mudholkar, G. S. (1980). A simple test for normality against asymmetric alternativesBiometrika, 67, 455–461.

    Article  MathSciNet  MATH  Google Scholar 

  38. Lukacs, E. (1942). A characterization of the normal distributionAnnals of Mathematical Statistics, 13, 91–93.

    Article  MathSciNet  MATH  Google Scholar 

  39. Lukacs, E. and Laha, R. G. (1964).Applications of CharacterizationsNew York: Hafner.

    Google Scholar 

  40. McDermott, M. P. and Mudholkar, G. S. (1993). A simple approach to testing homogeneity of order-constrained meansJournal of the American Statistical Association 881371–1379.

    Article  MathSciNet  MATH  Google Scholar 

  41. Moran, P. A. P. (1951). The random division of an interval - Part IIJournal of the Royal Statistical Society Series B, 9, 92–98.

    Google Scholar 

  42. Mudholkar, G. S. and Lin, C. T. (1984). On two applications of characterization theorems to goodness of fitColloquia Mathematica Societatis Janos Bolyai, 45, 395–414.

    MathSciNet  Google Scholar 

  43. Mudholkar, G. S., Marchetti, C. E., and Lin, C. T. (2000). Independence characterizations and testing normalityJournal of Statistical Planning and Inference(to appear).

    Google Scholar 

  44. Mudholkar, G. S. and McDermott, M. P. (1989). A class of tests for equality of ordered meansBiometrika, 76, 161–168.

    Article  MathSciNet  MATH  Google Scholar 

  45. Mudholkar, G. S., McDermott, M. P., and Aumont, J. (1993). Testing homogeneity of ordered variancesMetrika, 40, 271–281.

    Article  MathSciNet  MATH  Google Scholar 

  46. Mudholkar, G. S., McDermott, M. P., and Mudholkar, A. (1995). Robust finite-intersection tests for homogeneity of ordered variancesJournal of Statistical Planning and Inference, 43, 185–195.

    Article  MathSciNet  MATH  Google Scholar 

  47. Mudholkar, G. S., McDermott, M. P., and Srivastava, D. K. (1992). A test of p-variate normalityBiometrika, 79, 850–854.

    Article  MathSciNet  MATH  Google Scholar 

  48. Mudholkar, G. S. and Natarajan, R. (1999). The inverse gaussian analogs of symmetry, skewness and kurtosisAnnals of the Institute of Statistical Mathematics(to appear).

    Google Scholar 

  49. Mudholkar, G. S., Natarajan, R., and Chaubey, Y. P. (2000), Independence characterization and inverse gaussian goodness of fit composite hypothesis, Sankhyá (to appear).

    Google Scholar 

  50. Mudholkar, G. S. and Tian, L. (2000). An entropy characterization of the inverse gaussian distribution and related goodness of fit testTechnical ReportUniversity of Rochester, Rochester, NY. Submitted for publication.

    Google Scholar 

  51. Natarajan, R. (1998). An investigation of the inverse Gaussian distribution with an emphasis on Gaussian analogiesPh.D. ThesisUniversity of Rochester, Rochester, NY.

    Google Scholar 

  52. Pearson, K. (1892).The Grammar of ScienceLondon: W. Scott.

    Book  MATH  Google Scholar 

  53. Pearson, K. (1900). On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can reasonably be supposed to have risen from random samplingPhil. Mag., 5, 157–175.

    Google Scholar 

  54. Pearson, K. (1933). On a method of determining whether a sample of given size n supposed to have been drawn from a parent population having a known probability integral has probably been drawn at randomBiometrika, 25, 379–410.

    Google Scholar 

  55. Pearson, K. and Filon, L. N. G. (1898). xxxPhilosophical Transactions of the Royal Society of London, 191, 229–311.

    MATH  Google Scholar 

  56. Rao, C. R. and Shanbhag, D. N. (1986). Recent results on characterization of probability distributions: A unified approach through extensions of Deny’s theoremAdvances in Applied Probability 18,660–678.

    Article  MathSciNet  MATH  Google Scholar 

  57. Robertson, T., Wright, F. T., and Dykstra, R. L. (1988).Order Restricted Statistical InferenceNew York: John Wiley & Sons.

    MATH  Google Scholar 

  58. Schrödinger, E. (1915). Zur Theorie der Fall-und-Steigversuche an Teilchenn mit Brownsche BewegungPhysikalische Zeitschrift,16, 289–295.

    Google Scholar 

  59. Seshadri, V. (1993).The Inverse Gaussian Distribution: A Case Study in Exponential FamiliesOxford: Clarendon Press.

    Google Scholar 

  60. Seshadri, V. (1999).The Inverse Gaussian Distribution: Statistical Theory and ApplicationsNew York: Springer-Verlag.

    Book  MATH  Google Scholar 

  61. Shannon, C. E. (1949).The Mathematical Theory of Communication, p. 55, University of Illinois Press.

    Google Scholar 

  62. Shapiro, S. S. and Wilk, M. B. (1965). An analysis of variance test for normality (complete samples)Biometrika, 52, 591–611.

    MathSciNet  MATH  Google Scholar 

  63. Smoluchovsky, M. V. (1915). Notiz uber die berechnung der Browschen molekular-bewegung bei der ehrenhaft-millikanschen versuchsanordnungPhy. Z., 16, 318–321.

    Google Scholar 

  64. Soofi, E. S., Ebrahimi, N., and Habibullah, M. (1995). Information distinguishability with application to analysis of failure dataJournal of the American Statistical Association, 90, 657–668.

    Article  MathSciNet  MATH  Google Scholar 

  65. Tweedie, M. C. K. (1945). Inverse statistical varianceNature, 155, 453.

    Article  MathSciNet  MATH  Google Scholar 

  66. van Es, B. (1992). Estimating functionals related to a density by a class of statistics based on spacingsScandinavian Journal of Statistics, 19, 61–72.

    MATH  Google Scholar 

  67. Vasicek, O. (1976). A test for normality based on the sample entropyJournal of the Royal Statistical Society Series B, 38, 54–59.

    MathSciNet  MATH  Google Scholar 

  68. Vincze, I. (1984).Colloquia Mathematica Societatis Janos Bolyai, 45, 395–414.

    Google Scholar 

  69. Wald, A. (1947).Sequential AnalysisNew York: John Wiley&Sons.

    MATH  Google Scholar 

  70. Wilson, E. B. and Hilferty, M. M. (1931). The distribution of chi-squareProceedings of the National Academy of Sciences, 17, 684–688.

    Article  Google Scholar 

  71. Zinger, A. A. (1951). On independent samples from normal populationsUspeki Mat. Nauk, 6, 172–175.

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Rochester Institute of Technology, Rochester, New York, USA

    Carol E. Marchetti & Govind S. Mudholkar

  2. University of Rochester, Rochester, New York, USA

    Carol E. Marchetti & Govind S. Mudholkar

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  1. Carol E. Marchetti
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  2. Govind S. Mudholkar
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Editor information

Editors and Affiliations

  1. Laboratoire de Statistique Médicale, Université Rene Descartes-Paris 5, Paris, 75006, France

    C. Huber-Carol

  2. Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada

    N. Balakrishnan

  3. Laboratoire Statistique Mathématique, Université Bordeaux 2, Bordeaux Cedex, 33076, France

    M. S. Nikulin

  4. Laboratory of Statistical Methods, V. Steklov Mathematical Institute, St. Petersburg, 191011, Russia

    M. S. Nikulin

  5. Laboratoire de Statistique Appliquée, Université de Bretagne Sud, Vannes, 56000, France

    M. Mesbah

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Marchetti, C.E., Mudholkar, G.S. (2002). Characterization Theorems and Goodness-of-Fit Tests. In: Huber-Carol, C., Balakrishnan, N., Nikulin, M.S., Mesbah, M. (eds) Goodness-of-Fit Tests and Model Validity. Statistics for Industry and Technology. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0103-8_10

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  • DOI: https://doi.org/10.1007/978-1-4612-0103-8_10

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6613-6

  • Online ISBN: 978-1-4612-0103-8

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