Abstract
This paper considers two flexible classes of omnibus goodness-of-fit tests for the inverse Gaussian distribution. The test statistics are weighted integrals over the squared modulus of some measure of deviation of the empirical distribution of given data from the family of inverse Gaussian laws, expressed by means of the empirical Laplace transform. Both classes of statistics are connected to the first nonzero component of Neyman's smooth test for the inverse Gaussian distribution. The tests, when implemented via the parametric bootstrap, maintain a nominal level of significance very closely. A large-scale simulation study shows that the new tests compare favorably with classical goodness-of-fit tests for the inverse Gaussian distribution, based on the empirical distribution function.
Similar content being viewed by others
References
Baringhaus, L., Gürtler, N. and Henze, N. (2000). Weighted integral test statistics and components of smooth tests of fit, Australian & New Zealand Journal of Statistics, 42(2), 179–192.
Baringhaus, L. and Henze, N. (1991). A class of consistent tests for exponentiality based on the empirical Laplace transform, Ann. Inst. Statist. Math., 43, 551–564.
Baringhaus, L. and Henze, N. (1992). A goodness of fit test for the Poisson distribution based on the empirical generating function, Statist. Prob. Lett., 13, 269–274.
Bhattacharyya, G. and Fries, A. (1982). Fatigue failure models—Birnbaum-Saunders vs. inverse Gaussian, IEEE Trans. Reliab., R-31, 439–441.
Billingsley, P. (1968). Convergence of Probability Measures, Wiley, New York.
Chhikara, R. Folks, J. (1977). The inverse Gaussian distribution as a lifetime model, Technometrics, 19, 461–468.
Chhikara, R. and Folks, J. (1989). The Inverse Gaussian Distribution, Theory, Methodology and Applications, Marcel, New York.
Courant, R. and Hilbert, D. (1953). Methods of Mathematical Physics, VOL I, Interscience, New York.
Edgeman, R., Scott, R. and Pavur, R. (1988). A modified Kolmogorov-Smirnov test for the inverse Gaussian density with unknown parameters, Comm. Statist. Simulation Comput., 17, 1203–1212.
Epps, T. (1995). A test of fit for lattice distributions, Comm. Statist. Theory Methods, 24(6), 1455–1479.
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed., Wiley, New York.
Gunes, H., Dietz, D., Auclair, P. and Moore, A. (1997). Modified goodness-of-fit tests for the inverse Gaussian distribution, Comput. Statist. Data Anal., 24, 63–77.
Gürtler, N. and Henze, N. (2000). Recent and classical goodness-of-fit tests for the Poisson distribution, J. Statist. Plann. Inference, 90, 207–225.
Henze, N. (1993). A new flexible class of omnibus tests for exponentiality, Comm. Statist. Theory Methods, 22, 115–133.
Henze, N. (1996). Empirical-distribution-function goodness-of-fit tests for discrete models, Canad. J. Statist., 24(1), 81–93.
Henze, N. and Meintanis, S. (2000). Tests of fit for exponentiality based on the empirical Laplace transform, Statistics (to appear).
Henze, N. and Wagner, T. (1997). A new approach to the BHEP tests for multivariate normality, J. Multivariate Anal., 62(1), 1–23.
Hougaard, P. (1984). Life table methods for heterogeneous populations: Distributions describing the heterogeneity, Biometrika, 71, 75–83.
Klar, B. (2000). Diagnostic smooth tests of fit, Metrika, 52, 237–252.
Kundu, S., Majumdar, S. and Mukherjee, K. (2000). Central limit theorems revisited, Statist. Probab. Lett., 47, 265–275.
Mergel, V. (1999). Test of goodness of fit for the inverse-gaussian distribution, Math. Commun., 4, 191–195.
Michael, J., Schucancy, W. and Haas, R. (1976). Generating random variates using transformations with multiple roots, Amer. Statist., 30(2), 88–90.
Nakamura, M. and Pérez-Abreu, V. (1993). Use of an empirical probability generating function for testing a Poisson model, Canad. J. Statist., 21(2), 149–156.
O'Reilly, F. and Rueda, R. (1992). Goodness of fit for the inverse Gaussian distribution, Canad. J. Statist., 20, 387–397.
Padgett, W. and Tsoi, S. (1986). Prediction intervals for future observations from the inverse Gaussian distribution, IEEE Trans. Reliab., R35, 406–408.
Pavur, R., Edgeman, R. and Scott, R. (1992). Quadratic statistics for the goodness-of-fit test of the inverse Gaussian distribution, IEEE Trans. Reliab., 41, 118–123.
Rueda, R. and O'Reilly, F. (1999). Tests of fit for discrete distributions based on the probability generating function, Commun. Statist. Simulation Comput, 28(1), 259–274.
Seshadri, V. (1993). The Inverse Gaussian Distribution—A Case Study in Exponential Families, Clarendon Press, Oxford.
Seshadri, V. (1999). The Inverse Gaussian Distribution—Statistical Theory and Applications, Springer, New York.
Stute, W., Manteiga, W. and Quindimil, M. (1993). Bootstrap based goodness-of-fit tests, Metrika, 40, 243–256.
Widder, D. (1959). The Laplace Transform, 5th ed., Princeton University Press, New Jersey.
Author information
Authors and Affiliations
About this article
Cite this article
Henze, N., Klar, B. Goodness-of-Fit Tests for the Inverse Gaussian Distribution Based on the Empirical Laplace Transform. Annals of the Institute of Statistical Mathematics 54, 425–444 (2002). https://doi.org/10.1023/A:1022442506681
Issue Date:
DOI: https://doi.org/10.1023/A:1022442506681