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Article

Volf, Petr
On cumulative process model and its statistical analysis. (English). Kybernetika, vol. 36 (2000), issue 2, pp. [165]-176
MSC: 60G55, 62G05, 62M07, 62M09, 62P05 | MR 1760023 | Zbl 1248.62138
Keywords:
counting processes; martingales; estimators
Summary:
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.
References:
[1] Andersen P. K., Borgan O.: Counting process models for life history data: A review. Scand. J. Statist. 12 (1985), 97–158 MR 0808151 | Zbl 0584.62176
[2] Andersen P. K., Borgan O., Gill R. D., Keiding N.: Statistical Models Based on Counting Processes. Springer, New York 1993 MR 1198884 | Zbl 0824.60003
[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 10.1080/01621459.1988.10478588
[4] Embrechts P., Klüppelberg C., Mikosch T.: Modelling Extremal Events. Springer, Heidelberg 1997 MR 1458613 | Zbl 0873.62116
[5] Fleming T. R., Harrington D. P.: Counting Processes and Survival Analysis. Wiley, New York 1991 MR 1100924 | Zbl 1079.62093
[6] Volf P.: Analysis of generalized residuals in hazard regression models. Kybernetika 32 (1996), 501–510 MR 1420139 | Zbl 0882.62077
[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 0514115
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