Estimation of aggregate losses of secondary cancer cases using PH Panjer class (a,b,1) distributions,

This research in-cooperates phase type distributions of Panjer class $(a,b,1)$ in estimation of aggregate loss probabilities of secondary cancer. Matrices of the phase type distributions are derived using Chapman-Kolmogorov equation and transition probabilities estimated using modified Kaplan-Meir and consequently the transition intensities and transition probabilities. Stationary probabilities of the Markov chains represents $\vec{\gamma}$. Claim amount are modeled using OPPL, TPPL, Pareto, Generalized Pareto and Wei-bull distributions. PH ZT Poisson with Generalized Pareto distribution provided the best fit.

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TRANSIENT ANALYSIS OF A MULTI-COMPONENT SYSTEM MODELED BY A GENERAL MARKOV PROCESS

Asia Pacific Journal of Operational Research ◽

10.1142/s0217595906000954 ◽

2006 ◽

Vol 23 (03) ◽

pp. 311-327 ◽

Cited By ~ 9

Author(s):

RAFAEL PÉREZ-OCÓN ◽

DELIA MONTORO-CAZORLA ◽

JUAN ELOY RUIZ-CASTRO

Keyword(s):

Markov Process ◽

Performance Measures ◽

Transient Analysis ◽

Transition Probabilities ◽

Dynamic Environment ◽

General Interest ◽

System A ◽

Phase Type ◽

Phase Type Distributions ◽

Reliability Systems

An M-unit system in dynamic environment with operational and repair times following phase-type distributions and incorporating geometrical processes is considered. A general Markov process with vectorial states is the appropriate structure for modeling this system. A transient analysis is performed for this complex system and the transition probabilities are calculated. Some performance measures of general interest in the study of systems are obtained using an algorithmic approach, and applied to G-out-of-M systems. A numerical example is presented and the transient performance measures are calculated and compared with the stationary ones. This paper extends previous reliability systems, that can be considered as particular cases of this one. Throughout the paper, the mathematical expressions are given by algorithmic methods, that emphasized the utility of phase-type distributions in the analysis of lifetime data.

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Matrix-Form Recursions for a Family of Compound Distributions

Astin Bulletin ◽

10.2143/ast.40.1.2049233 ◽

2010 ◽

Vol 40 (1) ◽

pp. 351-368 ◽

Cited By ~ 2

Author(s):

Xueyuan Wu ◽

Shuanming Li

Keyword(s):

Risk Model ◽

Discrete Distribution ◽

Recursive Formula ◽

Matrix Form ◽

Claim Amount ◽

Collective Risk Model ◽

Collective Risk ◽

Compound Distribution ◽

Phase Type ◽

Phase Type Distributions

AbstractIn this paper, we aim to evaluate the distribution of the aggregate claims in the collective risk model. The claim count distribution is firstly assumed to belong to a generalised (a, b, 0) family. A matrix form recursive formula is then derived to evaluate the related compound distribution when individual claim amounts follow a discrete distribution on non-negative integers. The corresponding formula is also given for continuous individual claim amounts. Secondly, we pay particular attention to the recursive formula for compound phase-type distributions, since only certain types of discrete phase-type distributions belong to the generalised (a, b, 0) family. Similar recursive formulae are obtained for discrete and continuous individual claim amount distributions. Finally, numerical examples are presented for three counting distributions.

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Comparison of Tail Performance of the Champernowne Transformed Kernel Density Estimator, the Generalized Pareto Distribution and the G-and-H Distribution

SSRN Electronic Journal ◽

10.2139/ssrn.1395682 ◽

2009 ◽

Cited By ~ 2

Author(s):

Tine Buch-Kromann

Keyword(s):

Pareto Distribution ◽

Generalized Pareto Distribution ◽

Kernel Density ◽

Kernel Density Estimator ◽

Density Estimator ◽

Generalized Pareto

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Inference on P(Y

Calcutta Statistical Association Bulletin ◽

10.1177/0008068320974472 ◽

2020 ◽

Vol 72 (2) ◽

pp. 89-110

Author(s):

Manoj Chacko ◽

Shiny Mathew

Keyword(s):

Maximum Likelihood ◽

Pareto Distribution ◽

Generalized Pareto Distribution ◽

Maximum Likelihood Estimators ◽

Record Values ◽

Asymptotic Distributions ◽

Bayes Estimators ◽

Generalized Pareto ◽

Pareto Distributions ◽

Generalized Pareto Distributions

In this article, the estimation of [Formula: see text] is considered when [Formula: see text] and [Formula: see text] are two independent generalized Pareto distributions. The maximum likelihood estimators and Bayes estimators of [Formula: see text] are obtained based on record values. The Asymptotic distributions are also obtained together with the corresponding confidence interval of [Formula: see text]. AMS 2000 subject classification: 90B25

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Phase Type Distributions with Finite Support

Stochastic Models ◽

10.1080/15326349.2014.956226 ◽

2014 ◽

Vol 30 (4) ◽

pp. 576-597 ◽

Cited By ~ 3

Author(s):

V. Ramaswami ◽

N. C. Viswanath

Keyword(s):

Finite Support ◽

Phase Type ◽

Phase Type Distributions

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Analysis of R-out-of-N repairable systems: the case of phase-type distributions

Advances in Applied Probability ◽

10.1239/aap/1077134467 ◽

2004 ◽

Vol 36 (1) ◽

pp. 116-138 ◽

Cited By ~ 8

Author(s):

Yonit Barron ◽

Esther Frostig ◽

Benny Levikson

Keyword(s):

Repairable System ◽

Repairable Systems ◽

Regenerative Processes ◽

Numerical Examples ◽

Independent Components ◽

Duality Results ◽

Phase Type ◽

Phase Type Distributions ◽

Two Systems ◽

Repair Facility

An R-out-of-N repairable system, consisting of N independent components, is operating if at least R components are functioning. The system fails whenever the number of good components decreases from R to R-1. A failed component is sent to a repair facility. After a failed component has been repaired it is as good as new. Formulae for the availability of the system using Markov renewal and semi-regenerative processes are derived. We assume that either the repair times of the components are generally distributed and the components' lifetimes are phase-type distributed or vice versa. Some duality results between the two systems are obtained. Numerical examples are given for several distributions of lifetimes and of repair times.

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The Algebraic Degree of Phase-Type Distributions

Journal of Applied Probability ◽

10.1017/s0021900200006963 ◽

2010 ◽

Vol 47 (03) ◽

pp. 611-629

Author(s):

Mark Fackrell ◽

Qi-Ming He ◽

Peter Taylor ◽

Hanqin Zhang

Keyword(s):

Lower Bound ◽

Upper Bound ◽

Polynomial Algorithm ◽

Algebraic Degree ◽

Nonnegative Matrices ◽

Stieltjes Transform ◽

Special Cases ◽

Phase Type ◽

Phase Type Distributions ◽

The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

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The Gamma Generalized Pareto Distribution with Applications in Survival Analysis

International Journal of Statistics and Probability ◽

10.5539/ijsp.v6n3p141 ◽

2017 ◽

Vol 6 (3) ◽

pp. 141 ◽

Cited By ~ 2

Author(s):

Thiago A. N. De Andrade ◽

Luz Milena Zea Fernandez ◽

Frank Gomes-Silva ◽

Gauss M. Cordeiro

Keyword(s):

Monte Carlo Simulation ◽

Pareto Distribution ◽

Generalized Pareto Distribution ◽

Real Data ◽

Model Parameters ◽

Monte Carlo Simulation Study ◽

Lorenz Curves ◽

Generalized Pareto ◽

Mathematical Properties ◽

Pareto Model

We study a three-parameter model named the gamma generalized Pareto distribution. This distribution extends the generalized Pareto model, which has many applications in areas such as insurance, reliability, finance and many others. We derive some of its characterizations and mathematical properties including explicit expressions for the density and quantile functions, ordinary and incomplete moments, mean deviations, Bonferroni and Lorenz curves, generating function, R\'enyi entropy and order statistics. We discuss the estimation of the model parameters by maximum likelihood. A small Monte Carlo simulation study and two applications to real data are presented. We hope that this distribution may be useful for modeling survival and reliability data.

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