Fitting procedure for the two-state Batch Markov modulated Poisson process

Yera, Yoel G.; Lillo, Rosa E.; Ramírez‐Cobo, Pepa

doi:10.1016/j.ejor.2019.04.018

Cited by 5 publications

(7 citation statements)

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“…Also, it is of interest to obtain probabilities of the counting process N T (t) (N K (k)), number of failures up to t(k), as in Neuts and Li (1997). A third theoretical problem is the extension of the process proposed in this paper to its batch counterpart, where failures can occur in simultaneous way (see Yera et al (2018Yera et al ( , 2019 for results concerning the batch counterpart of the MMPP 2 ). Also, characterizing the bivariate MMPP 2 by a set of moments is a challenging goal from the inference point of view.…”

Section: Discussionmentioning

confidence: 99%

A bivariate two-state Markov modulated Poisson process for failure modeling

Yera

Lillo

Nielsen

et al. 2021

Reliability Engineering & System Safety

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This dissertation is mainly motivated by the problem of statistical modeling via a specific point process, namely, the (Batch) Markov Modulated Poisson process. Point processes arise in a wide range of situations in physics, biology, engineering, or economics. In general, the occurrence of events is defined depending on the context, but in many areas are defined by the occurrence of an event at a specific time. Sometimes, in order to simplify the models and obtain closed form expressions for the quantities of interest, the exponentiality and/or independence of the inter-event times is assumed. However, the independence and exponentiability assumptions become unrealistic and restrictive in practice, and therefore, there is a need of more realistic models to fit the data.

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Section: Discussionmentioning

confidence: 99%

A bivariate two-state Markov modulated Poisson process for failure modeling

Yera

Lillo

Nielsen

et al. 2021

Reliability Engineering & System Safety

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“…A number of papers has considered estimation for different classes of BMAP s, the method of moments and the EM algorithm being the most studied approaches, see [5,8,11,54,66,78] . Given a real trace of inter-loss times from the OpRisk context, we explore in this paper the estimation of the MAP 2 via a direct maximization of the likelihood function in the same spirit as in [10] .…”

Section: Statistical Estimation Of the Mapmentioning

confidence: 99%

“…Taking into account that convergence is achieved after 50 0 0 0 iterations, the Bayesian method turns out significantly slower. Recently, [78] compared the EM algorithm with a moments matching method in the context of Batch Markovian arrival processes, resulting the last one considerably faster. The estimation of the MAP 2 is not the main goal of this paper, and therefore we refer the reader to [10] , where the technical details regarding the previous approach as well as several numerical illustrations are discussed more in depth.…”

Section: Statistical Estimation Of the Mapmentioning

confidence: 99%

“…In a BMAP , the times between events (losses, arrivals...) are dependent and phase-type (PH) distributed, and as will be described later, the mean and variance of the counting process do not coincide. All these properties have made the BMAP suitable for modeling a variety of real life contexts as queuing theory, reliability, teletraffic, and climatology, see [14,38,42,62,66,78] , just to cite a few. In particular, the MAP has been also widely considered in the insurance literature, see [1,4,19,21,32,52,79] .…”

Section: Introductionmentioning

confidence: 99%

“…A number of papers have considered statistical inference for different classes of BMAP s. For example, [5,11,28,66,74] and [78] investigate moments matching approaches. Other authors adopt a Bayesian viewpoint, as [29] or [60] , where exact Gibbs samplers are proposed to obtain the posterior distribution of the model parameters.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Analysis of an aggregate loss model in a Markov renewal regime

Ramírez‐Cobo

Carrizosa

Lillo

2021

Applied Mathematics and Computation

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In this article we consider an aggregate loss model with dependent losses. The loss occurrence process is governed by a two-state Markovian arrival process ( MAP 2 ), a Markov renewal process that allows for (1) correlated inter-loss times, (2) non-exponentially distributed inter-loss times and, (3) overdisperse loss counts. Some quantities of interest to measure persistence in the loss occurrence process are obtained. Given a real OpRisk database, the aggregate loss model is estimated by fitting separately the inter-loss times and severities. The MAP 2 is estimated via direct maximization of the likelihood function, and severities are modeled by the heavy-tailed, double-Pareto Lognormal distribution. In comparison with the fit provided by the Poisson process, the results point out that taking into account the dependence and overdispersion in the inter-loss times distribution leads to higher capital charges.

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