96 research outputs found
Evaluation of Tweedie exponential dispersion model densities by Fourier inversion
The Tweedie family of distributions is a family of exponential dispersion models with power variance functions V (Ī¼) = Ī¼^p for p not between (0, 1). These distributions do not generally have density functions that can be written in closed form. However, they have simple moment generating functions, so the densities can be evaluated numerically by Fourier inversion of the characteristic functions. This paper develops numerical methods to make this inversion fast and accurate. Acceleration techniques are used to handle oscillating integrands. A range of analytic results are used to ensure convergent computations and to reduce the complexity of the parameter space. The Fourier inversion method is compared to a series evaluation method and the two methods are found to be complementary in that they perform well in different regions of the parameter
space
Flexible Tweedie regression models for continuous data
Tweedie regression models provide a flexible family of distributions to deal
with non-negative highly right-skewed data as well as symmetric and heavy
tailed data and can handle continuous data with probability mass at zero. The
estimation and inference of Tweedie regression models based on the maximum
likelihood method are challenged by the presence of an infinity sum in the
probability function and non-trivial restrictions on the power parameter space.
In this paper, we propose two approaches for fitting Tweedie regression models,
namely, quasi- and pseudo-likelihood. We discuss the asymptotic properties of
the two approaches and perform simulation studies to compare our methods with
the maximum likelihood method. In particular, we show that the quasi-likelihood
method provides asymptotically efficient estimation for regression parameters.
The computational implementation of the alternative methods is faster and
easier than the orthodox maximum likelihood, relying on a simple Newton scoring
algorithm. Simulation studies showed that the quasi- and pseudo-likelihood
approaches present estimates, standard errors and coverage rates similar to the
maximum likelihood method. Furthermore, the second-moment assumptions required
by the quasi- and pseudo-likelihood methods enables us to extend the Tweedie
regression models to the class of quasi-Tweedie regression models in the
Wedderburn's style. Moreover, it allows to eliminate the non-trivial
restriction on the power parameter space, and thus provides a flexible
regression model to deal with continuous data. We provide \texttt{R}
implementation and illustrate the application of Tweedie regression models
using three data sets.Comment: 34 pages, 8 figure
Monte Carlo Methods for Insurance Risk Computation
In this paper we consider the problem of computing tail probabilities of the
distribution of a random sum of positive random variables. We assume that the
individual variables follow a reproducible natural exponential family (NEF)
distribution, and that the random number has a NEF counting distribution with a
cubic variance function. This specific modelling is supported by data of the
aggregated claim distribution of an insurance company. Large tail probabilities
are important as they reflect the risk of large losses, however, analytic or
numerical expressions are not available. We propose several simulation
algorithms which are based on an asymptotic analysis of the distribution of the
counting variable and on the reproducibility property of the claim
distribution. The aggregated sum is simulated efficiently by importancesampling
using an exponential cahnge of measure. We conclude by numerical experiments of
these algorithms.Comment: 26 pages, 4 figure
The Tweedie Index Parameter and Its Estimator: An Introduction with Applications to Actuarial Ratemaking
87 pagesTweedie random variables are exponential dispersion models that have power unit
variance functions, are infnitely divisible, and are closed under translations and scale
transformations. Notably, a Tweedie random variable has an indexing/power param-
eter that is key in describing its distribution. Actuaries typically set this parameter to
a default value, whereas R's tweedie package provides tools to estimate the Tweedie
power via maximum likelihood estimation. This estimation is tested on simulations
and applied to an auto severity dataset and a home loss cost dataset. Models built
with an estimated Tweedie power observe lower Akaike Information Criterion rela-
tive to models built with default Tweedie powers. However, this parameter tuning
only marginally changes regression coefficients and model predictions. Given time
constraints, we recommend actuaries use default Tweedie powers and consider alter-
native feature engineering
Excess zeros under GAM: Tweedie or two-part?
Positive, right-skewed data with excess zeros are encountered in many real-life situations. Two possible techniques to analyze this type of data are: Two-part models and Tweedie models. The two-part models assume existence of a separate zero generating process, while the Tweedie models are based on distributions that allow mass at zero. The paper aims to present a simulation study to investigate the performance of Generalized Additive Models (GAM) under the distribution of Tweedie and two-part models for such data with excess zero by using MSE (Mean Square Error) and relative bias to compare the performance of both methods. We found that under different practical scenarios, the two-part model has a better performance than the Tweedie
Spatial risk estimation in Tweedie compound Poisson double generalized linear models
Tweedie exponential dispersion family constitutes a fairly rich sub-class of
the celebrated exponential family. In particular, a member, compound Poisson
gamma (CP-g) model has seen extensive use over the past decade for modeling
mixed response featuring exact zeros with a continuous response from a gamma
distribution. This paper proposes a framework to perform residual analysis on
CP-g double generalized linear models for spatial uncertainty quantification.
Approximations are introduced to proposed framework making the procedure
scalable, without compromise in accuracy of estimation and model complexity;
accompanied by sensitivity analysis to model mis-specification. Proposed
framework is applied to modeling spatial uncertainty in insurance loss costs
arising from automobile collision coverage. Scalability is demonstrated by
choosing sizable spatial reference domains comprised of groups of states within
the United States of America.Comment: 34 pages, 10 figures and 12 table
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