BAYESIAN CHAIN LADDER MODELS

2014 ◽  
Vol 45 (1) ◽  
pp. 75-99 ◽  
Author(s):  
Greg Taylor
Keyword(s):  
Credibility Estimator ◽  
Implicit Equations ◽  
Special Cases ◽  
Recursive Formulas ◽  
Error Terms ◽  
Map Estimator ◽  
The Cross ◽  
Chain Ladder ◽  
Ladder Models ◽  
Linear Bayes

AbstractThe literature on Bayesian chain ladder models is surveyed. Both Mack and cross-classified forms of the chain ladder are considered. Both cases are examined in the context of error terms distributed according to a member of the exponential dispersion family. Tweedie and over-dispersed Poisson errors follow as special cases. Bayesian cross-classified chain ladder models may randomise row, column or diagonal parameters. Column and diagonal randomisation has been largely absent from the literature until recently. The present paper allows randomisation of row and column parameters. The Bayes estimator, the linear Bayes (credibility) estimator, and the MAP estimator are shown to be identical in the Mack case, and in the cross-classified case provided that the error terms are Tweedie distributed. In the Mack case the variance structure is generalised considerably from the existing literature. In the cross-classified case the model structure differs somewhat from the existing literature, and a comparison is made between the two. MAP estimators for the cross-classified case are often given by implicit equations that require numerical solution. Recursive formulas are given for these in the general case of error terms from the exponential dispersion family. The connection between the cross-classified case and Bornhuetter-Ferguson prediction is explored.

EXISTENCE AND UNIQUENESS OF CHAIN LADDER SOLUTIONS

2016 ◽  
Vol 47 (1) ◽  
pp. 1-41 ◽  
Author(s):  
Greg Taylor
Keyword(s):  
Maximum Likelihood ◽  
Existence And Uniqueness ◽  
Maximum Likelihood Estimates ◽  
Parameter Estimates ◽  
The Cross ◽  
Chain Ladder ◽  
Ladder Models

AbstractThe cross-classified chain ladder has a number of versions, depending on the distribution to which observations are subject. The simplest case is that of Poisson distributed observations, and then maximum likelihood estimates of parameters are explicit. Most other cases, however, including Bayesian chain ladder models, lead to implicit MAP (Bayesian) or MLE (non-Bayesian) solutions for these parameter estimates, raising questions as to their existence and uniqueness. The present paper investigates these questions in the case where observations are distributed according to some member of the exponential dispersion family.

scholarly journals CROSS-MOMENTS COMPUTATION FOR STOCHASTIC CONTEXT-FREE GRAMMARS

2018 ◽  
Vol 33 (1) ◽  
pp. 041
Author(s):  
Velimir M. Ilić ◽  
Miroslav D. Ćirić ◽  
Miomir S. Stanković
Keyword(s):  
Random Variable ◽  
Sample Space ◽  
Efficient Computation ◽  
Context Free Grammar ◽  
Special Cases ◽  
The Cross ◽  
Stochastic Context Free Grammars ◽  
Context Free ◽  
New Algorithms ◽  
Vector Random Variable

In this paper we consider the problem of efficient computation of cross-moments of a vector random variable represented by a stochastic context-free grammar. Two types of cross-moments are discussed. The sample space for the first one is the set of all derivations of the context-free grammar, and the sample space for the second one is the set of all derivations which generate a string belonging to the language of the grammar. In the past, this problem was widely studied, but mainly for the cross-moments of scalar variables and up to the second order. This paper presents new algorithms for computing the cross-moments of an arbitrary order, while the previously developed ones are derived as special cases.

scholarly journals An Individual Claims Reserving Model

2007 ◽  
Vol 37 (01) ◽  
pp. 113-132 ◽  
Author(s):  
Keyword(s):  
Stochastic Models ◽  
Seasonal Effects ◽  
Claim Size ◽  
Unrealistic Assumption ◽  
Chain Ladder ◽  
The Individual ◽  
Ladder Models

Traditional Chain Ladder models are based on a few cells in an upper triangle and often give inaccurate projections of the reserve. Traditional stochastic models are based on the same few summaries and in addition are based on the often unrealistic assumption of independence between the aggregate incremental values. In this paper a set of stochastic models with weaker assumptions based on the individual claims development are described. These models can include information about settlement and can handle seasonal effects, changes in mix of business and claim types as well as changes in mix of claim size. It is demonstrated how the distribution of the process can be specified and especially how the distribution of the reserve can be determined. The method is illustrated with an example.

scholarly journals Parameter reduction in log-normal chain-ladder models

2015 ◽  
Vol 5 (2) ◽  
pp. 355-380 ◽  
Author(s):  
Richard J. Verrall ◽  
Mario V. Wüthrich
Keyword(s):  
Parameter Reduction ◽  
Chain Ladder ◽  
Log Normal ◽  
Ladder Models

Bounds on the Maximum Thermoelastic Stress and Deflection in a Beam and Plate

1966 ◽  
Vol 33 (4) ◽  
pp. 881-887 ◽  
Author(s):  
Bruno A. Boley
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Practical Interest ◽  
Thermoelastic Stress ◽  
Cross Sectional ◽  
Special Cases ◽  
End Conditions ◽  
Instantaneous Temperature ◽  
The Cross ◽  
Sectional Shape

It is shown in this paper that the thermal stress in a beam or plate cannot exceed the value kαEΔT, where ΔT is the maximum instantaneous temperature excursion in a cross section, and k is a coefficient dependent on the shape of the cross section. A simple general formula for k is found, and results for several special cases of practical interest are given. For rectangular beams (suitably oriented) and for plates, for example, k = 4/3. For any section, k = 1 if the thermal moment is zero; simplifications also occur if the thermal force is zero. The corresponding results for beam deflections are also carried out: The maximum deflection cannot exceed the value kδ kδ′αLΔT, where kδ and kδ′ are coefficients depending respectively on the cross-sectional shape and on the end conditions. For example, for rectangular cross sections, kδ = 3/4; and for a simply supported beam, kδ′ = 1/8.

CHAIN LADDER AND ERROR PROPAGATION

2016 ◽  
Vol 46 (2) ◽  
pp. 293-330 ◽  
Author(s):  
Ancus Röhr
Keyword(s):  
Prediction Error ◽  
Error Propagation ◽  
Classical Case ◽  
Algebraic Form ◽  
Development Factor ◽  
Process Error ◽  
Special Cases ◽  
Squared Prediction Error ◽  
Chain Ladder ◽  
Development Result

AbstractWe show how estimators for the chain ladder prediction error in Mack's (1993) distribution-free stochastic model can be derived using the error propagation formula. Our method allows for the treatment of the general case of the prediction error of the loss development result between two arbitrary future horizons. In the well-known special cases considered previously by Mack (1993) and Merz and Wüthrich (2008), our estimators coincide with theirs. However, the algebraic form in which we cast them is new, considerably more compact and more intuitive to understand. For example, in the classical case treated by Mack (1993), we show that the mean squared prediction error divided by the squared estimated ultimate loss can be written as ∑jû2j, where ûj measures the (relative) uncertainty around the jth development factor and the proportion of the estimated ultimate loss that it affects. The error propagation method also provides a natural split into process error and parameter error. Our proofs identify and exploit symmetries of “chain ladder processes” in a novel way. For the sake of wider practical applicability of the formulae derived, we allow for incomplete historical data and the exclusion of outliers in the triangles.

Response of Beams and Plates to Random Loads

1957 ◽  
Vol 24 (1) ◽  
pp. 46-52
Author(s):  
A. C. Eringen
Keyword(s):  
Cross Correlation ◽  
External Pressure ◽  
Rectangular Plates ◽  
Circular Plates ◽  
Probable Number ◽  
Simply Supported ◽  
Random Loads ◽  
Stochastic Loading ◽  
Special Cases ◽  
The Cross

Abstract With the use of generalized harmonic analysis the problem of vibrating damped beams and plates under stochastic loading is solved. The resulting equations give the cross-correlation functions for displacements, stresses, moments, and so on, in terms of the cross-correlation function of external pressure. Mean square values of these functions are special cases of these results. Using a method due to Rice, we also calculate the probable number of times per unit time the random displacements or stresses will exceed a given value. The case of simply supported bars, cantilever bars, clamped circular plates, and simply supported rectangular plates are worked out in detail.

scholarly journals An itertive algorithm with error terms for solving a system of implicit n-variational inclusions

2020 ◽  
Vol 8 (1) ◽  
pp. 242-253
Author(s):  
Zubair Khan ◽  
Syed Shakaib Irfan ◽  
M. Firdosh Khan ◽  
P. Shukla
Keyword(s):  
Iterative Algorithm ◽  
Approximate Solutions ◽  
Variational Inclusions ◽  
Special Cases ◽  
Error Terms ◽  
New System

A new system of implicit n-variational inclusions is considered. We propose a new algorithm with error terms for computing the approximate solutions of our system. The convergence of the iterative sequences generated by the iterative algorithm is also discussed. Some special cases are also discussed.

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