Making Tweedie’s compound Poisson model more accessible

The most commonly used regression model in general insurance pricing is the compound Poisson model with gamma claim sizes. There are two different parametrizations for this model: the Poisson-gamma parametrization and Tweedie’s compound Poisson parametrization. Insurance industry typically prefers the Poisson-gamma parametrization. We review both parametrizations, provide new results that help to lower computational costs for Tweedie’s compound Poisson parameter estimation within generalized linear models, and we provide evidence supporting the industry preference for the Poisson-gamma parametrization.

Computational Simulations of Similar Probabilistic Distributions to the Binomial and Poisson Distributions

This study has developed a Matlab application for simulating statistical models project (SMp) probabilistic distributions that are similar to binomial and Poisson, which were created by mathematical procedures. The simulated distributions are graphically compared with these popular distributions. The application allows to obtain many probabilistic distributions, and shows the trend (τ ) for n trials with success probability p, i.e. the maximum probability as τ=np. While the Poisson distribution PD(x;µ) is a unique probabilistic distribution, where PD=0 in x=+∞, the application simulates many SMp(x;µ,Xmax) distributions, where µ is the Poisson parameter and value of x with generally the maximum probability, and Xmax is the upper limit of x with SMp(x;µ,Xmax) ≥ 0 and limit of the stochastic region of a random discrete variable. It is shown that by simulation via, one can get many and better probabilistic distributions than by mathematical one.

Computational Simulations of Similar Probabilistic Distributions to the Binomial and Poisson Distributions

This study has developed a Matlab application for simulating statistical models project (SMp) probabilistic distributions that are similar to binomial and Poisson, which were created by mathematical procedures. The simulated distributions are graphically compared with these legendary distributions. The application allows to obtain many probabilistic distributions, and shows the trend (τ ) for n trials with success probability p, i.e. the maximum probability as τ=np. While the Poisson distribution PD(x;µ) is a unique probabilistic distribution, where PD=0 in x=+∞, the application simulates many SMp(x;µ,Xmax) distributions, where µ is the Poisson parameter and value of x with generally the maximum probability, and Xmax is upper limit of x with SMp(x;µ,Xmax) ≥ 0 and limit of the stochastic region of the random discrete variable X. It is shown that by simulation via, one can get many and better probabilistic distributions than by mathematical one.

Random forests for homogeneous and non-homogeneous Poisson processes with excess zeros

We propose a general hurdle methodology to model a response from a homogeneous or a non-homogeneous Poisson process with excess zeros, based on two forests. The first forest in the two parts model is used to estimate the probability of having a zero. The second forest is used to estimate the Poisson parameter(s), using only the observations with at least one event. To build the trees in the second forest, we propose specialized splitting criteria derived from the zero truncated homogeneous and non-homogeneous Poisson likelihood. The particular case of a homogeneous process is investigated in details to stress out the advantages of the proposed method over the existing ones. Simulation studies show that the proposed methods perform well in hurdle (zero-altered) and zero-inflated settings, for both homogeneous and non-homogeneous processes. We illustrate the use of the new method with real data on the demand for medical care by the elderly.

Tracking Poisson Parameter for Non-Stationary Discontinuous Time Series with Taylor’s Abnormal Fluctuation Scaling

We propose a parameter estimation method for non-stationary Poisson time series with the abnormal fluctuation scaling, known as Taylor’s law. By introducing the effect of Taylor’s fluctuation scaling into the State Space Model with the Particle Filter, the underlying Poisson parameter’s time evolution is estimated correctly from given non-stationary time series data with abnormally large fluctuations. We also developed a discontinuity detection method which enables tracking the Poisson parameter even for time series including sudden discontinuous jumps. As an example of application of this new general method, we analyzed Point-of-Sales data in convenience stores to estimate change of probability of purchase of commodities under fluctuating number of potential customers. The effectiveness of our method for Poisson time series with non-stationarity, large discontinuities and Taylor’s fluctuation scaling is verified by artificial and actual time series.

Estimation of the Poisson Parameter with Moment Generating Method

A new estimator of the Poisson parameter is proposed using the moment generating function. Some statistical properties of the proposed estimator are studied. The performance of the new estimator is compared with the maximum likelihood estimator (MLE) via examples and simulation in terms of goodness of fit and relative efficiency. Simulation and examples to real-life data suggest that the new estimator has higher relative efficiency compared to the MLE, while both are comparable in goodness of fit. The R program utilized in all computation and simulation is incorporated to facilitate the implementation of the new estimator in computation.