Geodesic
On cumulative process model and its statistical analysis
Kybernetika, Tome 36 (2000) no. 2, pp. 165-176
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The notion of the counting process is recalled and the idea of the ‘cumulative’ process is presented. While the counting process describes the sequence of events, by the cumulative process we understand a stochastic process which cumulates random increments at random moments. It is described by an intensity of the random (counting) process of these moments and by a distribution of increments. We derive the martingale – compensator decomposition of the process and then we study the estimator of the cumulative rate of the process. We prove the uniform consistency of the estimator and the asymptotic normality of the process of residuals. On this basis, the goodness- of-fit test and the test of homogeneity are proposed. We also give an example of application to analysis of financial transactions.
The notion of the counting process is recalled and the idea of the ‘cumulative’ process is presented. While the counting process describes the sequence of events, by the cumulative process we understand a stochastic process which cumulates random increments at random moments. It is described by an intensity of the random (counting) process of these moments and by a distribution of increments. We derive the martingale – compensator decomposition of the process and then we study the estimator of the cumulative rate of the process. We prove the uniform consistency of the estimator and the asymptotic normality of the process of residuals. On this basis, the goodness- of-fit test and the test of homogeneity are proposed. We also give an example of application to analysis of financial transactions.
Classification : 60G55, 62G05, 62M07, 62M09, 62P05
Keywords: counting processes; martingales; estimators 
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     title = {On cumulative process model and its statistical analysis},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2000_36_2_a1/}
}
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Volf, Petr. On cumulative process model and its statistical analysis. Kybernetika, Tome 36 (2000) no. 2, pp. 165-176. http://geodesic.mathdoc.fr/item/KYB_2000_36_2_a1/

[1] Andersen P. K., Borgan O.: Counting process models for life history data: A review. Scand. J. Statist. 12 (1985), 97–158 | MR | Zbl

[2] Andersen P. K., Borgan O., Gill R. D., Keiding N.: Statistical Models Based on Counting Processes. Springer, New York 1993 | MR | Zbl

[3] Arjas E.: A graphical method for assessing goodness of fit in Cox’s proportional hazards model. J. Amer. Statist. Assoc. 83 (1988), 204–212 | DOI

[4] Embrechts P., Klüppelberg C., Mikosch T.: Modelling Extremal Events. Springer, Heidelberg 1997 | MR | Zbl

[5] Fleming T. R., Harrington D. P.: Counting Processes and Survival Analysis. Wiley, New York 1991 | MR | Zbl

[6] Volf P.: Analysis of generalized residuals in hazard regression models. Kybernetika 32 (1996), 501–510 | MR | Zbl

[7] Volf P.: On counting process with random increments. In: Proceedings of Prague Stochastics’98, Union of Czech Math. Phys., Prague 1998, pp. 587–590

[8] Winter B. B., Földes A., Rejtö L.: Glivenko–Cantelli theorems for the product limit estimate. Problems Control Inform. Theory 7 (1978), 213–225 | MR