A Statistical Analysis of Intensities Estimation on the Modeling of Non-Life Insurance Claim Counting Process

This study presents an estimation approach to non-life insurance claim counts relating to a specified time. The objective of this study is to estimate the parameters in non-life insurance claim counting process, including the homogeneous Poisson process (HPP) and the non-homogeneous Poisson process (NHPP) with a bell-shaped intensity. We use the estimating function, the zero mean martingale (ZMM) as a procedure of parameter estimation in the insurance claim counting process. Then, Λ (t), the compensator of N (t) is proposed for the number of claims in the time interval (0, t]. We present situations through a simulation study of both processes on the time interval (0, t]. Some examples of the situations in the simulation study are depicted by a sample path relating N (t) to its compensator Λ (t). In addition, an example of the claim counting process illustrates the result of the compensator estimate misspecification.


Introduction
Nowadays, insurance is a common way of managing risks and the insurance industry has grown rapidly over time.Insurance industry owners, especially, consider the components of risk management, such as the premiums which are the main income of insurance businesses, reserves, underwriting, investment planning, reinsurance planning, etc.Also, estimating claims play an important part in each component in the non-life insurance field.In the past four decades, a few researchers have studied the claim counts model for non-life insurance.Klugman et al. [1] and Denuit et al. [2] were interested in studying the frequency distribution of insurance claims, including the parameter estimation methods.Bühlmann [3,4] presented the credibility approach in the form of a linear function to estimate and predict the expected claim counts in upcoming periods, using past experience of claims as a risk class or related risk classes.Bühlmann's credibility approach is interesting and can be extended to other approaches, such as the Bühlmann-Straub model, Jewell's model or the Exact credibility approach, etc., (see Klugman et al. [1]).Calculating the expected claim counts using the credibility approach only depends on the information from prior experience of claim counts, and does not consider the occurrence behavior of claim counts over time.Some authors have found an alternative approach to claim counts relating to a specified time or their behavior over time, for example, Mikosch [5] viewed the claim counting process as a homogeneous Poisson process (HPP) in the Cramér-Lundberg model, one of the most popular and useful risk models in non-life insurance, and Matsui and Mikosch [6] also considered a Poisson cluster model for the modeling of a total claims amount by a point of claim counts as an HPP with a constant rate of occurrence called the constant intensity.For some non-life insurance portfolios, the claim counts during a time period are caused by periodic phenomena or seasonality.These claim counts are modeled in terms of a non-homogeneous Poisson process (NHPP) with a period time-dependent intensity rate.Morales [7] presented the periodic risk model consisting of the claim counting process with a bell-shaped intensity function (called the Gaussian intensity) of the form , where s is an initial season, s = 0, 1, 2, •••, σ and  are parameters, and  is the standard normal distribution function.He estimated the unknown parameters of the periodic model intensity by using the maximum likelihood estimation (MLE), and he also considered evaluating the ruin probability through a simulation study.Furthermore, Lu and Garrido [8] explored the periodic NHPP model with a Beta-shaped intensity function.
The precision of claim count estimation is a key to running the insurance business successfully.In this study, we will present an estimation approach to non-life insurance claim counts related to a specification of the two different claim counting processes, i.e., HPP, and NHPP with a bell-shaped intensity function, through a simulation study.Our purpose is to estimate the parameters in the non-life insurance claim counting process.The parameters in the insurance claim counting process, intensity function ( ) in terms of mean value function , makes a complicated distribution function of insurance claim counts.An estimating function, such as the zero mean martingale (ZMM), is used here as a procedure of parameter estimation of an insurance claim counts model, and the parameters of model intensity are estimated by the MLE method.

A Definition of the Non-Life Insurance Claim Counting Process
We define the insurance claim counting process ; , and the insurance claim counts which have occurred in the time interval (0,t] where is a claim arrival time and i is independent and identically distributed (iid) Exponential with the parameter , called the intensity rate, is a counting process which is non-decreasing, can be written as where is an increment of is called the multiplicative intensity, where and k t are defined as the intensity rate and the exposure risk, respectively.We con-sider as a non-decreasing right continuous step function 0 at time t = 0 and jumps of size 1, and In this study we consider the insurance claim counting process which are the HPP with   t    , a constant intensity, and the NHPP with a bell-shaped intensity function as an initial season, s = 0 [7], where *  (an average number of claims over a period) and  are the parameters, *  , 0   .

Parameter Estimation in the Non-Life Insurance Claim Counting Process
In this section, we introduce the methods which are useful for parameter estimation in the non-life insurance claim counting process, including the estimating function, the martingale method, and the MLE.

Estimating Function
On a probability space   In t study, the parameter estim As a result of the ma of the is useful for con for a ZMM [11,12].ethod -study martingale m structing an estimating function for a parameter estima-tion in the insurance claim counting process.The process takes place over a small time interval ( , d ]  and as a res e meaning he martingale can be written as ) which is a martingale-difference.Then, the following maror it is rewritten in the form of ult of th

A Maximum Likelihood Estimation of the Model Intensity
In order to get the estimate of the compensator of  [5,13] is given by Thus, is an estimating equation for th n the insurance claim counting process.Also, as a result of the parameter estimate in the process, this can be interpreted as an , and this estimate is useful for predicting the times currence of insurance claim counts [12].We can depict the systematic part of the process of insurance claim counts,     The second simulation study of the insurance claim counting process, in which we consider the NHPP with a bell-shaped intensity function, or as the general form of mean value function . So, in this study, the th n claim arrival time, ( ) for al , s generated by where

Conclusion
This simulation study of the non-life insurance claim counting process, of both the HPP and the NHPP with a bell-shaped intensity, demonstrates that the fitting of the

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ation in the insurance claim counting process is provided by the martingale method.The martingales are random proc On a probability space   , which is the available data at the time t .The process and (b), respectively of 15 independent random times of claims occurrence in the NHPP with an intensity of , b a sam ased on

Figure 1 .
Figure 1.In a sample of 15 independent random times of claims occurrence with the intensity                   2 1 48exp 10 2  t

10  10  5   in Figure 3 .
as the observation number is 15 and 20 (slightly larger than the intensity The compensa r   with to  t fits   N t , as the o vation number is nd 20 (slightly larger than the parameter of model intensity * 10  bser 15 a  ).
Figure 2. and its compensator   N t
on the modeling of the non-life insurance claim counting process, both the HPP and NHPP, the parameters of the intensity function are estimated by the MLE method.Given   , MLE estimator of the model intensity, which is a comdure, i.e. the Newton-Raphson algorithm, to solve these

Table 2 . MSE of the comp tor es ensa timate
N t is illustrated by a sample path of the NHPP so that the MSE of fitting the compensator estimate misspecification   N t .