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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.

MSC:
62M09 Non-Markovian processes: estimation
62M07 Non-Markovian processes: hypothesis testing
62P05 Applications of statistics to actuarial sciences and financial mathematics
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