zbMATH — the first resource for mathematics

On cumulative process model and its statistical analysis. (English) Zbl 1248.62138
Summary: The notion of counting processes is recalled and the idea of ‘cumulative’ processes is presented. While the counting process describes a 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 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, a goodness-of-fit test and a test of homogeneity are proposed. We also give an example of applications to analysis of financial transactions.

62M09 Non-Markovian processes: estimation
62M07 Non-Markovian processes: hypothesis testing
62P05 Applications of statistics to actuarial sciences and financial mathematics
Full Text: Link EuDML
[1] Andersen P. K., Borgan O.: Counting process models for life history data: A review. Scand. J. Statist. 12 (1985), 97-158 · Zbl 0584.62176
[2] Andersen P. K., Borgan O., Gill R. D., Keiding N.: Statistical Models Based on Counting Processes. Springer, New York 1993 · 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 · Zbl 0873.62116
[5] Fleming T. R., Harrington D. P.: Counting Processes and Survival Analysis. Wiley, New York 1991 · Zbl 1079.62093
[6] Volf P.: Analysis of generalized residuals in hazard regression models. Kybernetika 32 (1996), 501-510 · Zbl 0882.62077 · www.kybernetika.cz · eudml:27971
[7] Volf P.: On counting process with random increments. 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 · Zbl 0499.60035
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.