Abstract
The inverse Gaussian (IG) family is strikingly analogous to the Gaussian family in terms of having simple inference solutions, which use the familiar χ2, t and F distributions, for a variety of basic problems. Hence, the IG family, consisting of asymmetric distributions is widely used for modelling and analyzing nonnegative skew data. However, the process lacks measures of model appropriateness corresponding to \(\sqrt {\beta _1 } \) and β2, routinely employed in statistical analyses. We use known similarities between the two families to define a concept termed IG-symmetry, an analogue of the symmetry, and to develop IG-analogues δ1 and δ2 of \(\sqrt {\beta _1 } \) and β2, respectively. Interestingly, the asymptotic null distributions of the sample versions d 1, d 2 of δ1, δ2 are exactly the same as those of their normal counterparts \(\sqrt {b_1 } \) and b 2. Some applications are discussed, and the analogies between the two families, enhanced during this study are tabulated.
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References
Ang, A. H.-S. and Tang, W. H. (1975). Probability Concepts in Engineering Planning and Design, Vol. I, Wiley, New York.
Balanda, K. P. and MacGillivray, H. L. (1988). Kurtosis: A critical review, Amer. Statist., 42, 111–119.
Bartlett, M. S. (1937). Properties of sufficiency and statistical tests, Proceedings in Royal Statistical Society, A, 160, 268–282.
Bickel, P. J. and Lehmann, E. L. (1975). Descriptive statistics for nonparametric models. II: Location, Ann. Statist., 3, 1045–1069.
Bowman, K. O. and Shenton, L. R. (1975). Omnibus test contours for departures from normality based on √b1 and b 2, Biometrika, 62, 243–250.
Box, G. E. P. (1953). Non normality and tests on variances, Biometrika, 40, 318–335.
Chhikara, R. S. and Folks, J. L. (1989). The Inverse Gaussian Distribution, Marcel Dekker, New York.
Cramér, H. (1946). Mathematical Methods of Statistics, Princeton University Press, Princeton, New Jersey.
D'Agostino, R. B. and Stephens, M. A. (1986). Goodness-of-fit Techniques, Marcel Dekker, New York.
Dugué, D. (1941). Sur un nouveau type de courbe de fréquence, Comptes Rendes de l'Academie des Sciences, 213, 634–635.
Efron, B. (1982). The jackknife, the bootstrap and other resampling plans, CBMS Regional Conf. Ser. in Appl. Math., 38.
Efron, B. and Olshen, R. A. (1978). How broad is the class of normal scale mixtures?, Ann. Statist., 6, 1159–1164.
Elderton, W. P. and Johnson, N. L. (1969). Systems of Frequency Curves, Cambridge University Press, London-New York.
Folks, J. L. and Chhikara, R. S. (1978). The Inverse Gaussian distribution and its Statistical Application-A Review, J. Roy. Statist. Soc. Ser. B, 40, 263–289.
Freimer, M., Mudholkar, G. S., Kollia, G. and Lin, C. T. (1988). A study of the generalized Tukey lambda family, Comm. Statist. Theory Methods, 17(10), 3547–3567.
Heyde, C. C. (1963). On a property of the lognormal distribution, J. Roy. Statist. Soc., Ser. B, 25, 392–393.
Huberman, B. A., Pirolli, P. L. T., Pitkow, J. E. and Lukose R. M. (1998). Strong regularities in World Wide Web Surfing, Science, 280, 95–97.
Iyengar, S. and Patwardhan, G. (1988). Recent developments in the inverse Gaussian distribution, Handbook of Statist. (eds. P. R. Krishnaiah and C. R. Rao), 7, 479–480.
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Wiley, New York.
Kagan, A. M., Linnik, Y. and Rao, C. R. (1973). Characterization Problems in Mathematical Statistics, Wiley, New York.
Keilson, J. and Steutel, F. W. (1974). Mixtures of distributions: Moment inequality measures of exponentiality and normality, Ann. Probab., 2, 112–130.
Kendall, M. G. and Stuart, A. (1969). Kendall's Advanced Theory of Statistics, Vol. I, Carles Griffin, London.
Khatri, C. G. (1962). A characterization of the inverse Gaussian distribution, Ann. Math. Statist., 33, 800–803.
Lin, C. C. and Mudholkar, G. S. (1980). A simple test for normality against asymmetric alternatives, Biometrika, 67, 455–461.
Lukacs, E. and Laha, R. G. (1964). Applications of Characteristic Functions, Hafner, New York.
MacGillivray, H. L. (1986). Skewness and asymmetry: Measures and ordering, Ann. Statist., 14, 994–1011.
Mooley, D. A. (1973). Gamma distribution probability model for Asian summer monsoon monthly rainfall, Monthly Weather Review, 101, 160–176.
Mudholkar, G. S. and Hutson, A. (1996). The exponentiated Weibull family: Some properties and a flood data application, Comm. Statist.-Theory Methods, 25(12), 3059–3083.
Mudholkar, G. S. and Hutson, A. (1998). LQ-moments: Analogs of L-moments, J. Statist. Plann. Inference, 71, 191–208.
Mudholkar, A. Mudholkar, G. S. and Srivastava, D. K. (1991). A construction and appraisal of pooled trimmed-t statistics, Comm. Statist. Ser. A, 20, 1345–1359.
Mudholkar, G. S., Marchetti, C., and Lin, C. T. (2001a). Independence characterization and testing normality against restricted skewness and kurtosis alternatives, J. Statist. Plann. Inference (to appear).
Mudholkar, G. S., Natarajan, R. and Chaubey, Y. P. (2001b). Independence characterization and inverse Gaussian goodness-of-fit composite hypothesis, Sankhyā, Ser. B (to appear).
Ord, J. K. (1972). Families of Frequency Distributions, Hafner, New York.
Pearson, K. R. (1905). Skew variation, a rejoinder, Biometrika, 4, 169–212.
Perricchi, L. R. and Rodriguez-Iturbe, I. (1985). On the statistical analysis of floods, The ISI Centenary volume (eds. A. C. Atkinson and S. E. Feinberg), 511–541, Springer, New York.
Schrödinger, E. (1915). Zur theorie der fall-und steigversuche an teilchenn mit Brownsche bewegung, Physikalische Zeitschrift, 16, 289–295.
Serfling, R. J. (1980). Approximation Thorems of Mathematical Statistics, Wiley, New York.
Seshadri, V. (1993). The Inverse Gaussian distribution: A case study in exponential families, Clarendon Press, Oxford.
Seshadri, V. (1997). Halphen's Law in Encyclopedia of Statistical Science, Update Vol. 1, 302–306, Wiley, New York.
Seshadri, V. (1999). The Inverse Gaussian distribution-Statistical Theory and Applications, Lecture Notes in Statist., No. 137, Springer, New York.
Shohat, J. A. and Tamarkin, J. D. (1943). The Problem of Moments, American Mathematical Society, New York.
Smithsonian Institution, Miscellaneous Collections (1927). World Weather Records, 79, Washington, D. C., p. 323.
Smithsonian Institution, Miscellaneous Collections (1934). World Weather Records, 1921–1930, 90, Washington, D. C., p. 146.
Smithsonian Institution, Miscellaneous Collections (1947). World Weather Records, 1931–1940, 105, Washington, D. C., p. 145.
Smoluchowsky, M. V. (1915). Notizüber die Berechning der Brownshen Molkular-bewegung bei des Ehrenhaft-millikanchen Versuchsanordnung, Physikalische Zeitschrift, 16, 318–321.
Stieltjes, T. J. (1894). Recherches sur les fractions continues, Annales de la Faculté des Sciences de Toulouse, 8, 1–122.
Tweedie, M. C. K. (1945). Inverse statistical variates, Nature, 155, p. 453.
U. S. Department of Commerce, Weather Bureau (1959). World Weather Records, 1941–1950, Washington D. C., p. 398.
U. S. Department of Commerce, Weather Bureau (1967). World Weather Records, 1951–1960, 4, Asia, Washington D. C., p. 345.
U. S. Water Resources Council, Washington, Hydrology Comittee (1977). Guidelines for determining flood flow frequency, United States. Water Resources Council, Washington D. C.
Wald, A. (1947). Sequential Analysis, Wiley, New York.
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Mudholkar, G.S., Natarajan, R. The Inverse Gaussian Models: Analogues of Symmetry, Skewness and Kurtosis. Annals of the Institute of Statistical Mathematics 54, 138–154 (2002). https://doi.org/10.1023/A:1016173923461
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DOI: https://doi.org/10.1023/A:1016173923461