Poisson size-biased Lindley distribution and its applications

2021 ◽  
Author(s):  
Showkat Ahmad Dar ◽  
Anwar Hassan ◽  
Peer Bilal Ahmad
Keyword(s):  
Parameter Estimation ◽  
Count Data ◽  
Hazard Rate ◽  
Risk Model ◽  
Probability Generating Function ◽  
Likelihood Estimation ◽  
Lindley Distribution ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Erlang Distributions

In this paper, a new model for count data is introduced by compounding the Poisson distribution with size-biased three-parameter Lindley distribution. Statistical properties, such as reliability, hazard rate, reverse hazard rate, Mills ratio, moments, shewness, kurtosis, moment genrating function, probability generating function and order statistics, have been discussed. Moreover, the collective risk model is discussed by considering the proposed distrubution as the primary distribution and the expoential and Erlang distributions as the secondary ones. Parameter estimation is done using maximum likelihood estimation (MLE). Finally a real dataset is discussed to demonstrate the suitability and applicability of the proposed distribution in modeling count dataset.

scholarly journals A New Generalized Poisson-Lindley Distribution: Applications and Properties

2015 ◽  
Vol 44 (4) ◽  
pp. 35-51 ◽  
Author(s):  
Deepesh Bhati ◽  
DVS Sastry ◽  
PZ Maha Qadri
Keyword(s):  
Risk Model ◽  
Cumulative Distribution ◽  
Likelihood Method ◽  
Lindley Distribution ◽  
Generalized Poisson ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Competing Models ◽  
Secondary Distribution ◽  
Better Than

A new generalized Poisson Lindley distribution is obtained by compounding Poissondistribution with two parameter generalised Lindley distribution. The new distribution isshown to be unimodal and over dispersed. This distribution has a tendency to accommodate right tail as well as for particular values of parameter the tail tends to zero at a faster rate. Various properties like cumulative distribution function, generating function, Moments etc. are derived. Knowledge about the parameters is obtained through Method of Moments, Maximum Likelihood Method and EM Algorithm. Moreover, an actuarial application in collective risk model is shown by considering the proposed distribution as primary and Exponential and Erlang as secondary distribution. The model is applied to real dataset and found to perform better than competing models.

Matrix-form Recursive Evaluation of the Aggregate Claims Distribution Revisited

2011 ◽  
Vol 5 (2) ◽  
pp. 163-179 ◽  
Author(s):  
Kok Keng Siaw ◽  
Xueyuan Wu ◽  
David Pitt ◽  
Yan Wang
Keyword(s):  
Risk Model ◽  
Likelihood Estimation ◽  
Real Data ◽  
Recursive Formula ◽  
Matrix Form ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Potential Applications ◽  
Aggregate Claims ◽  
Recursive Evaluation

AbstractThis paper aims to evaluate the aggregate claims distribution under the collective risk model when the number of claims follows a so-called generalised (a, b, 1) family distribution. The definition of the generalised (a, b, 1) family of distributions is given first, then a simple matrix-form recursion for the compound generalised (a, b, 1) distributions is derived to calculate the aggregate claims distribution with discrete non-negative individual claims. Continuous individual claims are discussed as well and an integral equation of the aggregate claims distribution is developed. Moreover, a recursive formula for calculating the moments of aggregate claims is also obtained in this paper. With the recursive calculation framework being established, members that belong to the generalised (a, b, 1) family are discussed. As an illustration of potential applications of the proposed generalised (a, b, 1) distribution family on modelling insurance claim numbers, two numerical examples are given. The first example illustrates the calculation of the aggregate claims distribution using a matrix-form Poisson for claim frequency with logarithmic claim sizes. The second example is based on real data and illustrates maximum likelihood estimation for a set of distributions in the generalised (a, b, 1) family.

scholarly journals A Multi-Year Microlevel Collective Risk Model

2021 ◽  
Author(s):  
Rosy Oh ◽  
Himchan Jeong ◽  
Jae Youn Ahn ◽  
Emiliano A. Valdez
Keyword(s):  
Risk Model ◽  
Collective Risk Model ◽  
Collective Risk

scholarly journals Comparing of Car-Bym, Generalized Poisson dan Negative Binomial on Tuberculosis Data in Banyumas Districs

2021 ◽  
Vol 5 (1) ◽  
pp. 130-140
Author(s):  
Jajang Jajang ◽  
Budi Pratikno ◽  
Mashuri Mashuri
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Count Data ◽  
Degrees Of Freedom ◽  
Negative Binomial ◽  
Likelihood Estimation ◽  
Poisson Model ◽  
Generalized Poisson ◽  
Central Java ◽  
Area Data

In 2019 the number of people with TB (Tuberculosis) in Banyumas, Central Java, is high (1,910 people have been detected with TB). The number of people infected Tuberculosis (TB) in Banyumas is the count data and it is also the area data. In modeling, the parameter estimation and characteristic of the data need to be considered. Here, we studied comparing Generalized Poisson (GP), negative binomial (NB), and Poisson and CAR.BYM model for TB cases in Banyumas. Here, we use two methods for parameter estimation, maximum likelihood estimation (MLE) and Bayes. The MLE is used for GP and NB models, whereas Bayes is used for Poisson and CAR-BYM. The results showed that Poisson model detected overdispersion where deviance value is 67.38 for 22 degrees of freedom. Therefore, ratio of deviance to degrees of freedom is 3.06 (>1). This indicates that there was overdispersion. The folowing GP, NB, Poisson-Bayes and CAR-BYM are used to modeling TB data in Banyumas and we compare their RMSE. With refer to RMES criteria, we found that CAR-BYM is the best model for modeling TB in Banyumas because its RMSE is smallest.

scholarly journals A zero truncated discrete distribution: Theory and applications to count data

2020 ◽  
pp. 167-190
Author(s):  
Tassaddaq Hussain Kiani
Keyword(s):  
Count Data ◽  
Negative Binomial ◽  
Residual Life ◽  
Probability Generating Function ◽  
Strength Parameter ◽  
Mean Residual Life ◽  
Factorial Moments ◽  
Residual Life Function ◽  
Collective Risk Model ◽  
Life Function

The analysis and modeling of zero truncated count data is of primary interest in many elds such as engineering, public health, sociology, psychology, epidemiology. Therefore, in this article we have proposed a new and simple structure model, named a zero truncated discrete Lindley distribution. Thedistribution contains some submodels and represents a two-component mixture of a zero truncated geometric distribution and a zero truncated negative binomial distribution with certain parameters. Several properties of the distribution are obtained such as mean residual life function, probability generating function, factorial moments, negative moments, moments of residual life function, Bonferroni and Lorenz curves, estimation of parameters, Shannon and Renyi entropies, order statistics with the asymptotic distribution of their extremes and range, a characterization, stochastic ordering and stress-strength parameter. Moreover, the collective risk model is discussed by considering theproposed distribution as primary distribution and exponential and Erlang distributions as secondary ones. Test and evaluation statistics as well as three real data applications are considered to assess the peformance of the distribution among the most frequently zero truncated discrete probability models.

scholarly journals Posterior Regret Γ-Minimax Estimation of Insurance Premium in Collective Risk Model

2008 ◽  
Vol 38 (1) ◽  
pp. 277-291 ◽  
Author(s):  
Agata Boratyńska
Keyword(s):  
Loss Function ◽  
Unknown Parameter ◽  
Risk Model ◽  
Insurance Premium ◽  
Loss Functions ◽  
Expected Number ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Asymmetric Loss Function ◽  
Robust Procedures

The collective risk model for the insurance claims is considered. The objective is to estimate a premium which is defined as a functional H specified up to an unknown parameter θ (the expected number of claims). Four principles of calculating a premium are applied. The Bayesian methodology, which combines the prior knowledge about a parameter θ with the knowledge in the form of a random sample is adopted. Two loss functions (the square-error loss function and the asymmetric loss function LINEX) are considered. Some uncertainty about a prior is assumed by introducing classes of priors. Considering one of the concepts of robust procedures the posterior regret Γ-minimax premiums are calculated, as an optimal robust premiums. A numerical example is presented.

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