A family of consistent normally distributed tests for Poissonity. (English) Zbl 1539.62122
Summary: A family of consistent tests, derived from a characterization of the probability generating function, is proposed for assessing Poissonity against a wide class of count distributions, which includes some of the most frequently adopted alternatives to the Poisson distribution. Actually, the family of test statistics is based on the difference between the plug-in estimator of the Poisson cumulative distribution function and the empirical cumulative distribution function. The test statistics have an intuitive and simple form and are asymptotically normally distributed, allowing a straightforward implementation of the test. The finite sample properties of the test are investigated by means of an extensive simulation study. The test shows satisfactory behaviour compared to other tests with known limit distribution.
MSC:
| 62G10 | Nonparametric hypothesis testing |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62E10 | Characterization and structure theory of statistical distributions |
Keywords:
asymptotics; consistent test; normal distribution; Poissonity; probability generating functionSoftware:
RReferences:
| [1] | Ainsbury, EA; Vinnikov, VA; Maznyk, NA, A comparison of six statistical distributions for analysis of chromosome aberration data for radiation biodosimetry, Radiat. Prot. Dosimetry, 155, 3, 253-267, 2013 · doi:10.1093/rpd/ncs335 |
| [2] | Baringhaus, L.; Henze, N., A goodness of fit test for the Poisson distribution based on the empirical generating function, Stat. Probab. Lett., 13, 269-274, 1992 · Zbl 0741.62043 · doi:10.1016/0167-7152(92)90033-2 |
| [3] | Gürtler, N.; Henze, N., Recent and classical goodness-of-fit tests for the Poisson distribution, J. Stat. Plan. Inference, 90, 207-225, 2000 · Zbl 0958.62043 · doi:10.1016/S0378-3758(00)00114-2 |
| [4] | Henze, N., Empirical-distribution-function goodness-of-fit tests for discrete models, Can. J. Stat., 24, 81-93, 1996 · Zbl 0846.62037 · doi:10.2307/3315691 |
| [5] | Inglot, T., Data driven efficient score tests for poissonity, Probab. Math. Stat., 39, 115-126, 2019 · Zbl 1422.62162 · doi:10.19195/0208-4147.39.1.8 |
| [6] | Janssen, A., Global power functions of goodness of fit tests, Ann. Stat., 28, 1, 239-253, 2000 · Zbl 1106.62329 · doi:10.1214/aos/1016120371 |
| [7] | Johnson, NL; Kotz, S.; Kemp, AW, Univariate discrete distributions, 2005, United States: John Wiley & Sons, United States · Zbl 1092.62010 · doi:10.1002/0471715816 |
| [8] | Kocherlakota, S.; Kocherlakota, K., Goodness of fit tests for discrete distributions, Commun. Stat. Theor. Method., 15, 815-829, 1986 · Zbl 0618.62049 · doi:10.1080/03610928608829153 |
| [9] | Le Cam L.: Asymptotic methods in statistical decision theory. Springer Science & Business Media (2012) |
| [10] | Meintanis, S.; Bassiakos, Y., Goodness-of-fit tests for additively closed count models with an application to the generalized Hermite distribution, Sankhyā: Indian J. Stat., 67, 538-552, 2005 · Zbl 1193.62081 |
| [11] | Meintanis, S.; Nikitin, YY, A class of count models and a new consistent test for the poisson distribution, J. Stat. Plan. Inference, 138, 3722-3732, 2008 · Zbl 1146.62324 · doi:10.1016/j.jspi.2007.12.011 |
| [12] | Mijburgh, P.; Visagie, I., An overview of goodness-of-fit tests for the poisson distribution, S. Afr. Stat. J., 54, 2, 207-230, 2020 · Zbl 1488.62060 · doi:10.37920/sasj.2020.54.2.6 |
| [13] | Nakamura, M.; Pérez-Abreu, V., Use of an empirical probability generating function for testing a poisson model, Can. J. Stat., 21, 149-156, 1993 · Zbl 0779.62039 · doi:10.2307/3315808 |
| [14] | Puig, P.; Weiß, CH, Some goodness-of-fit tests for the poisson distribution with applications in biodosimetry, Comput. Stat. Data Anal., 144, 106, 878, 2020 · Zbl 1504.62022 · doi:10.1016/j.csda.2019.106878 |
| [15] | R Core Team: R: A Language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, (2021). https://www.R-project.org/ |
| [16] | Rémillard, B.; Theodorescu, R., Inference based on the empirical probability generating function for mixtures of poisson distributions, Stat. Decis., 18, 349-366, 2000 · Zbl 1179.62046 · doi:10.1524/strm.2000.18.4.349 |
| [17] | Rueda, R.; O’Reilly, F., Tests of fit for discrete distributions based on the probability generating function, Commun. Stat. Simul. Comput., 28, 259-274, 1999 · Zbl 1054.62546 · doi:10.1080/03610919908813547 |
| [18] | Weiß, CH; Homburg, A.; Puig, P., Testing for zero inflation and overdispersion in inar (1) models, Stat. Pap., 60, 823-848, 2019 · Zbl 1420.62401 · doi:10.1007/s00362-016-0851-y |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
