Back
 AM  Vol.12 No.4 , April 2021
Kolmogorov-Smirnov APF Test for Inhomogeneous Poisson Processes with Shift Parameter
Abstract: In this article, we study the Kolmogorov-Smirnov type goodness-of-fit test for the inhomogeneous Poisson process with the unknown translation parameter as multidimensional parameter. The basic hypothesis and the alternative are composite and carry to the intensity measure of inhomogeneous Poisson process and the intensity function is regular. For this model of shift parameter, we propose test which is asymptotically partially distribution free and consistent. We show that under null hypothesis the limit distribution of this statistic does not depend on unknown parameter.
Cite this paper: Tanguep, E. and Njomen, D. (2021) Kolmogorov-Smirnov APF Test for Inhomogeneous Poisson Processes with Shift Parameter. Applied Mathematics, 12, 322-335. doi: 10.4236/am.2021.124023.
References

[1]   Lehmann, E.L. and Romano, J.P. (2005) Testing Statistical Hypothesis. 3rd Edition, Springer-Verlag, New York.

[2]   Durbin, J. (1973) Distribution Theory for Tests Based on the Sample Distribution Function. SIAM, Philadelphia.
https://doi.org/10.1137/1.9781611970586

[3]   Mann, H.B. and Wald, A. (1942) On the Choice of the Number of Class Intervals in the Application of Chi-Square Test. Annals of Mathematical Statistics, 13, 306-317.
https://doi.org/10.1214/aoms/1177731569

[4]   Ingster, Yu.I. and Suslina, I.A. (2003) Nonparametric Goodness-of-Fit Testing Under Gaussian Models. Springer-Verlag, New York.
https://doi.org/10.1007/978-0-387-21580-8

[5]   Ingster, Yu.I. and Kutoyants, Yu.A. (2007) Nonparametric Hypothesis Testing for an Intensity of Poisson Process. Mathematical Methods of Statistics, 16, 217-245.
https://doi.org/10.3103/S1066530707030039

[6]   Dachian, S. and Kutoyants, Yu.A. (2007) On the Goodness-of-Fit Tests for Some Continuous Time Processes. In: Vonta, F., Nikulin, M., Limnios, N. and Huber-Carol, C., Eds., Statistical Models and Methods for Biomedical and Technical Systems, Birkhäuser, Boston, 395-413.
https://doi.org/10.1007/978-0-8176-4619-6_27

[7]   Dabye, A.S. (2013) On the Cramér-von Mises Test with Parametric Hypothesis for Poisson Processes. Statistical Inference for Stochastic Processes, 16, 1-13.
https://doi.org/10.1007/s11203-013-9077-y

[8]   Dabye, A.S., Tanguep, W.E.D. and Top, A. (2016) On the Cramér-von Mises Test for Poisson Processes with Scale Parameter. Far East Journal of Theoretical Statistics, 52, 419-441.

[9]   Rao, C.R. (1965) Linear Statistical Inference and Its Applications. Wiley, New York.

[10]   Khmaladze, E. (1981) Martingale Approach in the Theory of Goodness-of-Fit Tests. Theory of Probability and Its Applications, 26, 240-257. (Translated by A.B. Aries)
https://doi.org/10.1137/1126027

[11]   Bai, J. (2002) Testing Parametric Conditional Distributions of Dynamic Models. Boston College, Chestnut Hill, MA.

[12]   Koenker, R. and Xiao, Z. (2002) Inference on the Quantile Regression Process. Econometrica, 70, 1583-1612.
https://doi.org/10.1111/1468-0262.00342

[13]   Darling, D.A. (1958) The Cramér-Smirnov Test in the Parametric Case. Annals of Mathematical Statistics, 26, 1-20.
https://doi.org/10.1214/aoms/1177728589

[14]   Kutoyants, Yu.A. (1998) Statistical Inference for Spatial Poisson Processes. Lecture Notes in Statistics, 134, Springer-Verlag, New York.
https://doi.org/10.1007/978-1-4612-1706-0

[15]   Gihman, I.I. and Skorohod, A.V. (1974) The Theory of Stochastic Processes I. Springer-Verlag, New-York.

 
 
Top