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.

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.

Infinite Parameter Estimates in Logistic Regression: Opportunities, Not Problems

2002 ◽  
Vol 27 (2) ◽  
pp. 147-161 ◽  
Author(s):  
David Rindskopf
Keyword(s):  
Logistic Regression ◽  
Maximum Likelihood ◽  
Confidence Intervals ◽  
Maximum Likelihood Estimates ◽  
Parameter Estimates ◽  
Strict Sense ◽  
Hypothesis Tests ◽  
Infinite Slope ◽  
Alternative Method

Infinite parameter estimates in logistic regression are commonly thought of as a problem. This article shows that in principle an analyst should be happy to have an infinite slope in logistic regression, because it indicates that a predictor is perfect. Using simple approaches, hypothesis tests may be performed and confidence intervals calculated even when a slope is infinite. Some functions of parameters that are infinite are still well defined, and reasonable estimates of these quantities (in particular, LD50) may be obtained even when the maximum likelihood estimates do not, in a strict sense, exist. Surprisingly, these techniques can provide more reasonable and useful results than the most popular alternative method, exact logistic regression.

scholarly journals A non-iterative alternative to ordinal Log-Linear models

2004 ◽  
Vol 8 (2) ◽  
pp. 67-86 ◽  
Author(s):  
Eric J. Beh ◽  
Pamela J. Davy
Keyword(s):  
Maximum Likelihood ◽  
Linear Models ◽  
Contingency Tables ◽  
Model Fitting ◽  
Maximum Likelihood Estimates ◽  
Order Effects ◽  
Parameter Estimates ◽  
Linear Modeling ◽  
Statistical Tool ◽  
Log Linear

Log-linear modeling is a popular statistical tool for analysing a contingency table. This presentation focuses on an alternative approach to modeling ordinal categorical data. The technique, based on orthogonal polynomials, provides a much simpler method of model fitting than the conventional approach of maximum likelihood estimation, as it does not require iterative calculations nor the fitting and re-fitting to search for the best model. Another advantage is that quadratic and higher order effects can readily be included, in contrast to conventional log-linear models which incorporate linear terms only.The focus of the discussion is the application of the new parameter estimation technique to multi-way contingency tables with at least one ordered variable. This will also be done by considering singly and doubly ordered two-way contingency tables. It will be shown by example that the resulting parameter estimates are numerically similar to corresponding maximum likelihood estimates for ordinal log-linear models.

scholarly journals A Likelihood Approach to Bornhuetter–Ferguson Analysis

Risks ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 119 ◽  
Author(s):  
Valandis Elpidorou ◽  
Carolin Margraf ◽  
María Dolores Martínez-Miranda ◽  
Bent Nielsen
Keyword(s):  
Maximum Likelihood ◽  
Cash Flow ◽  
Linear Constraints ◽  
Maximum Likelihood Estimates ◽  
Model Framework ◽  
Chain Ladder Method ◽  
Related Approach ◽  
Likelihood Approach ◽  
Chain Ladder ◽  
Constrained Maximum Likelihood

A new Bornhuetter–Ferguson method is suggested herein. This is a variant of the traditional chain ladder method. The actuary can adjust the relative ultimates using externally estimated relative ultimates. These correspond to linear constraints on the Poisson likelihood underpinning the chain ladder method. Adjusted cash flow estimates were obtained as constrained maximum likelihood estimates. The statistical derivation of the new method is provided in the generalised linear model framework. A related approach in the literature, combining unconstrained and constrained maximum likelihood estimates, is presented in the same framework and compared theoretically. A data illustration is described using a motor portfolio from a Greek insurer.

Estimating reference fishing mortality rates from noisy spawner–recruit data

2004 ◽  
Vol 61 (9) ◽  
pp. 1771-1783 ◽  
Author(s):  
A Jamie F Gibson ◽  
Ransom A Myers
Keyword(s):  
Maximum Likelihood ◽  
Mortality Rate ◽  
Mortality Rates ◽  
Maximum Sustainable Yield ◽  
Maximum Likelihood Estimates ◽  
Organizational Level ◽  
Likelihood Method ◽  
Parameter Estimates ◽  
Fishing Mortality ◽  
Over Exploitation

We review and evaluate methods of estimating reference fishing mortality rates from spawner–recruit (SR) data to obtain maximum sustainable yield. Using Monte Carlo simulations, we found that a reference fishing mortality rate derived from the maximum likelihood estimates of the SR parameters was less biased than reference fishing mortality rates obtained using the mode of the marginal probability distribution for the maximum rate that spawners produce recruits or by finding the fishing mortality rate that maximizes the expected yield. However, the maximum likelihood method produced the most variable estimates, at times leading to substantial under- or over-exploitation of the population. In contrast, the decision theoretic method of maximizing the expected yield exhibited less variability, produced higher yields, and substantially reduced the risk of overexploiting the population. We show how these methods can be extended to include information from other populations. Bayesian priors for the SR parameters, obtained through meta-analyses of population dynamics at some higher organizational level (e.g., the species), may be used to assess the plausibility of parameter estimates obtained for a single population or combined with the data for the population of interest. Reference fishing mortality rates are then estimated from the resulting joint posterior distribution.

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