ON SOME PROPERTIES OF A CLASS OF MULTIVARIATE ERLANG MIXTURES WITH INSURANCE APPLICATIONS

2014 ◽  
Vol 45 (1) ◽  
pp. 151-173 ◽  
Author(s):  
Gordon E. Willmot ◽  
Jae-Kyung Woo
Keyword(s):  
Residual Lifetime ◽  
Risk Allocation ◽  
Insurance Risk ◽  
Lifetime Distributions ◽  
Scale Parameters ◽  
Distributional Properties ◽  
Premium Calculation ◽  
Equilibrium Distributions ◽  
Stop Loss ◽  
Erlang Distributions

AbstractWe discuss some properties of a class of multivariate mixed Erlang distributions with different scale parameters and describes various distributional properties related to applications in insurance risk theory. Some representations involving scale mixtures, generalized Esscher transformations, higher-order equilibrium distributions, and residual lifetime distributions are derived. These results allows for the study of stop-loss moments, premium calculation, and the risk allocation problem. Finally, some results concerning minimum and maximum variables are derived and applied to pricing joint life and last survivor policies.

scholarly journals Excess of Loss Reinsurance with Reinstatements Revisited

2005 ◽  
Vol 35 (01) ◽  
pp. 211-238 ◽  
Author(s):  
Werner Hürlimann
Keyword(s):  
Incomplete Information ◽  
Size Distribution ◽  
Claim Size ◽  
Inverse Gaussian ◽  
Insurance Risk ◽  
Distribution Free ◽  
Analytical Approximations ◽  
Claim Size Distribution ◽  
Aggregate Claims ◽  
Stop Loss

The classical evaluation of pure premiums for excess of loss reinsurance with reinstatements requires the knowldege of the claim size distribution of the insurance risk. In the situation of incomplete information, where only a few characteristics of the aggregate claims to an excess of loss layer can be estimated, the method of stop-loss ordered bounds yields a simple analytical distribution-free approximation to pure premiums of excess of loss reinsurance with reinstatements. It is shown that the obtained approximation is enough accurate for practical purposes and improves the analytical approximations obtained using either a gamma, translated gamma, translated inverse Gaussian or a mixture of the last two distributions.

scholarly journals A Dynamic Failure Time Degradation-Based Model

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1532
Author(s):  
Abdulhakim A. Albabtain ◽  
Mansour Shrahili ◽  
Lolwa Alshagrawi ◽  
Mohamed Kayid
Keyword(s):  
Hazard Rate ◽  
Failure Time ◽  
Degradation Process ◽  
Residual Lifetime ◽  
Time To Failure ◽  
Lifetime Distributions ◽  
Mean Residual Lifetime ◽  
Aging Properties ◽  
Degradation Models ◽  
Characterization Property

A novel methodology for modelling time to failure of systems under a degradation process is proposed. Considering the method degradation may have influenced the failure of the system under the setup of the model several implied lifetime distributions are outlined. Hazard rate and mean residual lifetime of the model are obtained and a numerical situation is delineated to calculate their amounts. The problem of modelling the amount of degradation at the failure time is also considered. Two monotonic aging properties of the model is secured and a characterization property of the symmetric degradation models is established.

Entropy-based measure of uncertainty in past lifetime distributions

2002 ◽  
Vol 39 (02) ◽  
pp. 434-440 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Maria Longobardi
Keyword(s):  
Shannon Entropy ◽  
Lifetime Distribution ◽  
Residual Entropy ◽  
Residual Lifetime ◽  
Lifetime Distributions ◽  
The Past ◽  
Life Distributions ◽  
Measure Of Uncertainty

As proposed by Ebrahimi, uncertainty in the residual lifetime distribution can be measured by means of the Shannon entropy. In this paper, we analyse a dual characterization of life distributions that is based on entropy applied to the past lifetime. Various aspects of this measure of uncertainty are considered, including its connection with the residual entropy, the relation between its increasing nature and the DRFR property, and the effect of monotonic transformations on it.

On higher-order properties of compound geometric distributions

2002 ◽  
Vol 39 (02) ◽  
pp. 324-340 ◽  
Author(s):  
Gordon E. Willmot
Keyword(s):  
Equilibrium Distribution ◽  
Risk Model ◽  
Higher Order ◽  
Residual Lifetime ◽  
Classical Risk Model ◽  
Time Of Ruin ◽  
The Laplace Transform ◽  
The Time Of Ruin ◽  
Stop Loss ◽  
The Classical Risk Model

An explicit convolution representation for the equilibrium residual lifetime distribution of compound zero-modified geometric distributions is derived. Second-order reliability properties are seen to be essentially preserved under geometric compounding, and complement results of Brown (1990) and Cai and Kalashnikov (2000). The convolution representation is then extended to thenth-order equilibrium distribution. This higher-order convolution representation is used to evaluate the stop-loss premium and higher stop-loss moments of the compound zero-modified geometric distribution, as well as the Laplace transform of thekth moment of the time of ruin in the classical risk model.

scholarly journals SOME DISTRIBUTIONAL PROPERTIES OF A CLASS OF COUNTING DISTRIBUTIONS WITH CLAIMS ANALYSIS APPLICATIONS

2013 ◽  
Vol 43 (2) ◽  
pp. 189-212 ◽  
Author(s):  
Gordon E. Willmot ◽  
Jae-Kyung Woo
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Negative Binomial ◽  
Claims Analysis ◽  
Explicit Formulas ◽  
Distributional Properties ◽  
Special Cases ◽  
Aggregate Claims ◽  
Stop Loss

AbstractWe discuss a class of counting distributions motivated by a problem in discrete surplus analysis, and special cases of which have applications in stop-loss, discrete Tail value at risk (TVaR) and claim count modelling. Explicit formulas are developed, and the mixed Poisson case is considered in some detail. Simplifications occur for some underlying negative binomial and related models, where in some cases compound geometric distributions arise naturally. Applications to claim count and aggregate claims models are then given.

A GENERALIZED ENTROPY-BASED RESIDUAL LIFETIME DISTRIBUTIONS

2011 ◽  
Vol 04 (02) ◽  
pp. 171-184 ◽  
Author(s):  
VIKAS KUMAR ◽  
H. C. TANEJA
Keyword(s):  
Residual Life ◽  
Survival Function ◽  
Monotonicity Property ◽  
Entropy Measure ◽  
Residual Lifetime ◽  
Mean Residual Life ◽  
Present Communication ◽  
Dynamic Information ◽  
Lifetime Distributions ◽  
Life Function

The present communication considers Havrda and Charvat entropy measure to propose a generalized dynamic information measure. It is shown that the proposed measure determines the survival function uniquely. The residual lifetime distributions have been characterized. A bound for the dynamic entropy measure in terms of mean residual life function has been derived, and its monotonicity property is studied.

On higher-order properties of compound geometric distributions

2002 ◽  
Vol 39 (2) ◽  
pp. 324-340 ◽  
Author(s):  
Gordon E. Willmot
Keyword(s):  
Equilibrium Distribution ◽  
Risk Model ◽  
Higher Order ◽  
Residual Lifetime ◽  
Classical Risk Model ◽  
Time Of Ruin ◽  
The Laplace Transform ◽  
The Time Of Ruin ◽  
Stop Loss ◽  
The Classical Risk Model

An explicit convolution representation for the equilibrium residual lifetime distribution of compound zero-modified geometric distributions is derived. Second-order reliability properties are seen to be essentially preserved under geometric compounding, and complement results of Brown (1990) and Cai and Kalashnikov (2000). The convolution representation is then extended to the nth-order equilibrium distribution. This higher-order convolution representation is used to evaluate the stop-loss premium and higher stop-loss moments of the compound zero-modified geometric distribution, as well as the Laplace transform of the kth moment of the time of ruin in the classical risk model.

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