A new goodness-of-fit test for the inverse Gaussian distribution

Abstract

The Inverse Gaussian (IG) distribution is widely used in practice and therefore an important issue is to develop a powerful goodness-of-fit test (GOF) for this distribution. In this article, we propose and examine a new GOF test for the IG distribution based on a new estimate of Kullback-Leibler (KL) information. The properties of the test statistic are presented. In order to compute the proposed test statistic, parameters of the IG distribution are estimated by maximum likelihood estimators, which are simple explicit estimators. Critical values and the actual sizes of the proposed test are obtained. Through a simulation study, power values of the proposed test are compared with some prominent existing tests. Finally, two illustrative examples are presented and analyzed.

Keywords

• Inverse Gaussian distribution
• Kullback-Leibler information
• Monte Carlo simulation
• goodness-of-fit tests
• critical points
• test power

References

1. [1] T.W. Anderson and D.A. Darling, A test of goodness of fit, J. Amer. Statist. Assoc. 49 (268), 765-769, 1954.
2. [2] I. Arizono and H. Ohta, A test for normality based on Kullback-Leibler information, Amer. Statist. 43 (1), 20-22, 1989.
3. [3] N. Balakrishnan, A. Habibi Rad and N.R. Arghami, Testing exponentiality based on Kullback-Leibler information with progressively type-II censored data, IEEE Trans. Rel. 56 (2), 301-307, 2007.
4. [4] N. Balakrishnan H.K.T. Ng and N. Kannan, Goodness-of-fit tests based on spacings for progressively type-II censored data from a general location-scale distribution, IEEE Trans. Rel. 53 (3), 349-356, 2004.
5. [5] W.E. Bardsley, Note on the use of the inverse Gaussian distribution for wind energy applications, J. Appl. Meteorol. 19 (9), 1126-1130, 1980.
6. [6] O.E. Barndorff-Nielsen, A note on electrical networks and the inverse Gaussian distribution, Adv. in Appl. Probab. 26 (1), 63-67, 1994.
7. [7] G. Chen and N. Balakrishnan, A general purpose approximate goodness-of-fit test, J. Qual. Technol. 27 (2), 154-161, 1995.
8. [8] Z. Chen, A new two-parameter lifetime distribution with bathtub shape or increasing failure note function, Statist. Probab. Lett. 49 (2), 155-161, 2000.

9. [9] R.S. Chhikara and J.L. Folks, The inverse Gaussian distribution as a lifetime model, Technometrics 19 (4), 461-468, 1977.
10. [10] B. Choi and K. Kim, Testing goodness-of-fit for Laplace distribution based on maximum entropy, Statistics 40 (6), 517-531, 2006.
11. [11] S. Chouia and N. Seddik-Ameur, Different EDF goodness-of-fit tests for competing risks models, Comm. Statist. Simulation Comput. 52 (8), 3491-3501, 2023.
12. [12] J.C. Correa, A new estimator of entropy, Comm. Statist. Theory Methods 24 (10), 2439-2449, 1995.
13. [13] R.B. D’Agostino and M.A. Stephens, Goodness-of-Fit Techniques, Marcel Dekker, New York, 1986.
14. [14] B.S. Dhillon, Life distributions, IEEE Trans. Rel. 30 (5), 457-459, 1981.
15. [15] E.J. Dudewicz and E.C. van der Meulen, Entropy-based tests of uniformity, J. Amer. Statist. Assoc. 76 (376), 967-974, 1981.
16. [16] N. Ebrahimi, M. Habibullah and E. Soofi, Testing exponentiality based on Kullback- Leibler information, J. R. Stat. Soc. Ser. B. Stat. Methodol. 54 (3), 739-748, 1992.
17. [17] N. Ebrahimi, K. Pflughoeft and E. Soofi, Two measures of sample entropy, Statist. Probab. Lett. 20 (3), 225-234, 1994.
18. [18] J.L. Folks and R.S. Chhikara, The inverse Gaussian distribution and its statistical application-a review, J. R. Stat. Soc. Ser. B. Stat. Methodol. 40 (3), 263-289, 1978.
19. [19] J.L. Folks and R.S. Chhikara, The Inverse Gaussian Distribution, Theory, Methodology and Applications, Marcel Dekker, New York, 1989.
20. [20] D.V. Gokhale, On entropy-based goodness-of-fit test, Comput. Statist. Data Anal. 1, 157-165, 1983.
21. [21] E. González-Estrada and J.A. Villaseñor, An R package for testing goodness of fit: goft, J. Stat. Comput. Simul. 88 (4), 726-751, 2018.
22. [22] P. Hall and S.C. Morton, On the estimation of entropy, Ann. Inst. Statist. Math. 45 (1), 69-88, 1993.
23. [23] C. Huber-Carol, N. Balakrishnan, M.S. Nikulin and M. Mesbah, Goodness-of-Fit Tests and Model Validity, Birkhauser, Boston, 2002.
24. [24] N.L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, 1 and 2, Wiley, New York, 1994.
25. [25] A.N. Kolmogorov, Sulla determinazione empirica di une legge di distribuzione, Giornale dell’Intituto Italiano degli Attuari 4, 83-91, 1933.
26. [26] N.H. Kuiper, Tests concerning random points on a circle, Proc. Koninkl. Nederl. Akad. van Wetenschappen, Ser. A 63, 34-47, 1960.
27. [27] S. Lee, I. Vonta and A. Karagrigoriou, A maximum entropy type test of fit, Comput. Statist. Data Anal. 55 (9), 2635-2643, 2011.
28. [28] C-T. Lin, Y-L. Huang and N. Balakrishnan, A new method for goodness-of-fit testing based on type-II right censored samples, IEEE Trans. Rel. 57 (4), 633-642, 2008.
29. [29] G.S. Mudholkar and L. Tian An entropy characterization of the inverse Gaussian distribution and related goodness-of-fit test, J. Statist. Plann. Inference 102 (2), 211- 221, 2002.
30. [30] H.A. Noughabi, A new estimator of entropy and its application in testing normality, J. Stat. Comput. Simul. 80 (10), 1151-1162, 2010.
31. [31] H.A. Noughabi, A new estimator of Kullback-Leibler information and its application in goodness of fit tests, J. Stat. Comput. Simul. 89 (10), 1914-1934, 2019.
32. [32] H.A. Noughabi, A new goodness of fit test for the logistic distribution, Sankhya B 84, 303-319, 2022.
33. [33] H.A. Noughabi, Cumulative residual entropy applied to testing uniformity, Comm. Statist. Theory Methods 51 (12), 4151-4161, 2022.
34. [34] H.A. Noughabi and N.R. Arghami, General treatment of goodness of fit tests based on Kullback-Leibler information, J. Stat. Comput. Simul. 83 (8), 1556-1569, 2013.
35. [35] H.A. Noughabi and N. Balakrishnan, Goodness of fit using a new estimate of Kullback- Leibler information based on type II censored data, IEEE Trans. Rel. 64 (2), 627-635, 2015.
36. [36] R. Pakyari, Goodness-of-fit testing based on Gini index of spacings for progressively type-II censoring, Comm. Statist. Simulation Comput. 52 (7), 3223-3232, 2023.
37. [37] R. Pakyari and A. Baklizi, On goodness-of-fit testing for Burr type X distribution under progressively type-II censoring, Comput. Statist. 37 (5), 2249-2265, 2022.
38. [38] R. Pakyari and N. Balakrishnan, A general purpose approximate goodness-of-fit test for progressively type-II censored data, IEEE Trans. Rel. 61 (1), 238-244, 2012.
39. [39] R. Pakyari and N. Balakrishnan, Goodness-of-fit tests for progressively type-II censored data from location-scale distributions, J. Stat. Comput. Simul. 83 (1), 167-178, 2013.
40. [40] G. Qiu and K. Jia, Extropy estimators with applications in testing uniformity, J. Nonparametr. Stat. 30 (1), 182-196, 2018.
41. [41] A.H. Rad, F. Yousefzadeh and N. Balakrishnan, Goodness-of-fit test based on Kullback-Leibler information for progressively type-II censored data, IEEE Trans. Rel. 60 (3), 570-579, 2011.
42. [42] V. Seshadri, The Inverse Gaussian Distribution: Statistical Theory and Applications, Springer, New York, 1999.
43. [43] C.E. Shannon, A mathematical theory of communications, Bell Syst. Tech. J. 27 (3), 379-423, 1948.
44. [44] B. van Es, Estimating functionals related to a density by class of statistics based on spacings, Scand. J. Stat. 19 (1), 61-72, 1992.
45. [45] O. Vasicek, A test for normality based on sample entropy, J. R. Stat. Soc. Ser. B. Stat. Methodol. 38 (1), 54-59, 1976.
46. [46] J.A. Villasenor and E. Gonzalez-Estrada, Tests of fit for inverse Gaussian distributions, Statist. Probab. Lett. 105, 189-194, 2015.
47. [47] J.A. Villasenor, E. Gonzalez-Estrada and A. Ochoa, On testing the inverse Gaussian distribution hypothesis, Sankhya B 81 (1), 60-74, 2019.
48. [48] R. von Mises, Wahrscheinlichkeitsrechnung und ihre Anwendung in der Statistik und theoretischen Physik, Leipzig and Vienna: Deuticke, 1931.
49. [49] G.S. Watson, Goodness of fit tests on a circle, Biometrika 48 (1-2), 109-114, 1961.
50. [50] R. Wieczorkowski and P. Grzegorzewsky, Entropy estimators improvements and comparisons, Comm. Statist. Simulation Comput. 28 (2), 541-567, 1999.