Fitting Mixtures of Erlangs to Censored and Truncated Data Using the Em Algorithm

Verbelen, Roel; Gong, Liutang; Antonio, Katrien; Badescu, Andrei L.; Lin, Sheldon S.

doi:10.1017/asb.2015.15

Cited by 70 publications

(23 citation statements)

References 29 publications

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“…More precisely, by observing that the terms involving the truncation P U (τ − t r(l) i ) from the second and third parts of (5.2) cancel each other out, in the first step we maximize the likelihood of the observed reported delay densities k l=1 n r l i=1 p U r (l) i (du r(l) i ). As, according to (5.1), the reporting delays are interval-censored, we use an EM algorithm for fitting a censored Erlang mixture (Verbelen et al 2015).…”

Section: Estimation Resultsmentioning

confidence: 99%

“…In this section, we will present an EM algorithm for fitting our class of Pascal-HMMs, which includes the three discretely observed processes discussed above. Variations of the EM algorithm have been used to fit Erlang-based mixture distributions to data in Lee and Lin (2010), Badescu et al (2015) and Verbelen et al (2015). As we are dealing with time series data and fitting the proposed stochastic model to the data in this paper, the proposed EM algorithm in this section, although it has some similarity to the aforementioned EM algorithms due to the use of the Erlang distribution, is very different.…”

Section: Parameter Estimation: An Em Algorithmmentioning

confidence: 99%

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A Marked Cox Model for the Number of Ibnr Claims: Estimation and Application

et al. 2019

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Incurred but not reported (IBNR) loss reserving is of great importance for Property & Casualty (P&C) insurers. However, the temporal dependence exhibited in the claim arrival process is not reflected in many current loss reserving models, which might affect the accuracy of the IBNR reserve predictions. To overcome this shortcoming, we proposed a marked Cox process and showed its many desirable properties in Badescu et al. (2016).In this paper, we consider the model estimation and applications. We first present an expectation–maximization (EM) algorithm which guarantees the efficiency of the estimators unlike the moment estimation methods widely used in estimating Cox processes. In addition, the proposed fitting algorithm can be implemented at a reasonable computational cost. We examine the performance of the proposed algorithm through simulation studies. The applicability of the proposed model is tested by fitting it to a real insurance claim data set. Through out-of-sample tests, we find that the proposed model can provide realistic predictive distributions.

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Section: Estimation Resultsmentioning

confidence: 99%

Section: Parameter Estimation: An Em Algorithmmentioning

confidence: 99%

A Marked Cox Model for the Number of Ibnr Claims: Estimation and Application

et al. 2019

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“…In risk theory, using the mixed Erlang distribution as a claim size model, an analytical form for the finite time ruin probability has been derived by Dickson and Willmot [6] and Dickson [5]. Recently, using the EM algorithm, mixed Erlang distribution has been fitted to catastrophic loss data in the United States by Lee and Lin [11] and also to censored and truncated data by Verbelen et al [21] . Moreover, Lee and Lin [12], Willmot and Woo [25] have developed the multivariate mixed Erlang distribution to overcome some drawbacks of the copula approach while Badescu et al [1] have used multivariate mixed Poisson distribution with mixed Erlang claim sizes to model operational risks.…”

Section: Mixed Erlang Marginalsmentioning

confidence: 99%

On mixed Erlang reinsurance risk: aggregation, capital allocation and default risk

Ratovomirija

2016

Eur. Actuar. J.

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In this paper, we address the aggregation of dependent stop loss reinsurance risks where the dependence among the ceding insurer(s) risks is governed by the Sarmanov distribution and each individual risk belongs to the class of Erlang mixtures. We investigate the effects of the ceding insurer(s) risk dependencies on the reinsurer risk profile by deriving a closed formula for the distribution function of the aggregated stop loss reinsurance risk. Furthermore, diversification effects from aggregating reinsurance risks are examined by deriving a closed expression for the risk capital needed for the whole portfolio of the reinsurer and also the allocated risk capital for each business unit under the TVaR capital allocation principle. Moreover, given the risk capital that the reinsurer holds, we express the default probability of the reinsurer analytically. In case the reinsurer is in default, we determine analytical expressions for the amount of the aggregate reinsured unpaid losses and the unpaid losses of each reinsured line of business of the ceding insurer(s). These results are illustrated by numerical examples.

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“…For example, Bernardi et al (2012) proposes finite mixture of skew normal distributions in the framework of Bayesian analysis. Verbelen et al (2015) develops finite mixtures of Erlang distributions and adopt the Expectation-Maximization (EM) algorithm to estimate the model. Gómez-Déniz et al (2013) propose a gamma mixture with the generalized inverse Gaussian distribution to model a mainland US hurricane damage data set.…”

Section: Introductionmentioning

confidence: 99%

Generalizing the Log-Moyal Distribution and Regression Models for Heavy-Tailed Loss Data

Li¹,

Beirlant²,

Meng³

2020

ASTIN Bull.

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Catastrophic loss data are known to be heavy-tailed. Practitioners then need models that are able to capture both tail and modal parts of claim data. To this purpose, a new parametric family of loss distributions is proposed as a gamma mixture of the generalized log-Moyal distribution from Bhati and Ravi (2018), termed the generalized log-Moyal gamma (GLMGA) distribution. While the GLMGA distribution is a special case of the GB2 distribution, we show that this simpler model is effective in regression modeling of large and modal loss data. Regression modeling and applications to risk measurement are illustrated using a detailed analysis of a Chinese earthquake loss data set, comparing with the results of competing models from the literature. To this end, we discuss the probabilistic characteristics of the GLMGA and statistical estimation of the parameters through maximum likelihood. Further illustrations of the applicability of the new class of distributions are provided with the fire claim data set reported in Cummins et al. (1990) and a Norwegian fire losses data set discussed recently in Bhati and Ravi (2018).

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Fitting Mixtures of Erlangs to Censored and Truncated Data Using the Em Algorithm

Cited by 70 publications

References 29 publications

A Marked Cox Model for the Number of Ibnr Claims: Estimation and Application

A Marked Cox Model for the Number of Ibnr Claims: Estimation and Application

On mixed Erlang reinsurance risk: aggregation, capital allocation and default risk

Generalizing the Log-Moyal Distribution and Regression Models for Heavy-Tailed Loss Data

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