Compound Negative Binomial Distribution Research Articles

Compound negative binomial shared frailty model with random probability of susceptibility

The shared frailty models are now popular for modelling heterogeneity in survival analysis. It assumes that the same frailty is shared by all individual members within the families. Also, it is believed that all the individuals in the population are susceptible to the event of interest and will eventually experience the event. This may not always be the situation in reality. There may be a certain fraction of the population which is non-susceptible for an event and hence may not experience the event under study. Non-susceptibility is modelled by frailty models with compound frailty distribution. Further, susceptibility may be different for different families. This can be attained by randomizing the parameter of frailty distribution. This paper incorporates both the things, non-susceptibility and different susceptibility for different families by considering compound negative binomial distribution with random probability of susceptibility as frailty distribution. The inferential problem is solved in a Bayesian framework using Markov Chain Monte Carlo methods. The proposed model is then applied to a real-life data set.

The Infinite Divisibility of Compound Negative Binomial Distribution as the Sum of Laplace Distribution

The infinite divisibility of compound negative binomial distribution especially as the sum of Laplace distribution has important roles in governing the mathematical model based on its characteristic function. In order to show the property of characteristic function of this compound negative binomial distribution, it is used Fourier-Stieltjes transform to have characteristic function and then governed the property of continuity and quadratic form by using analytical approaches. The infinite divisibility property is obtained by introducing a function satisfied the criteria to be a characteristic function such that its convolution has the characteristic function of compound negative binomial distribution. Then it is concluded that the characteristic function of compound negative binomial distribution as the sum of Laplace distribution satisfies the property of continuity, quadratic form and infinite divisibility.

Compound negative binomial multivariate correlated frailty model for long-term survivors

In the correlated frailty models, it is believed that all the individuals in the population will eventually experience the event of interest. This may not always be the situation in reality. There may be a certain fraction of the population which is immune to the event and hence may not experience the event under study. For such individuals, frailty may not be active i.e., zero. Hence, frailty distribution should provide a positive mass at zero. In this article, a new correlated frailty model with compound negative binomial distribution as frailty distribution, is introduced to incorporate the non-susceptibility of individuals. The inferential problem is solved in a Bayesian framework using Markov Chain Monte Carlo methods. The proposed model is then applied to a real life data set.

The Remarks on Seven New Publications Based on Sub-Independence Concept

Chesneau and Palacios considered the infinite decomposability of the Geometric (Chesneau and Palacios(2021b), (paper 1)), and Gamma, Laplace and n-Laplace (Chesneau and Palacios(2021a), (paper 2)) of two (as well as n) independent random variables. They obtained, very nicely, certain important results on the decomposability concept. Also, George Yanev published a paper entitled” Exponential and Hyper exponential Distributions: Some Characterizations” (Yanev G.(2020), (paper 3)) and reported a paper entitled ”On Arnold-Villasenor Conjectures for Characterizing Exponential Distribution Based on Sample of Size Three” (Yanev(2020), (paper 4)). In both papers, George Yanev considered the distribution of the sum or a linear combination of the independent random variables. Yanev obtained certain nice results in these two papers under the assumption of independence of the summands. Roozegar and Bazyani published a paper entitled ”Exact Distribution of Random Weighted Convolution of Some Beta Distributions Through an Integral Transform” (Roozegar and Bazyari(2017), (paper 5)), in which they considered the exact distribution of the weighted average of n independent beta random variables and provided a new integral transformation with some of its mathematical properties. Ahmad et al.(2021) considered ”Compound Negative Binomial Distribution as the Sum of Independent Laplace Variates” (paper 6) and discussed infinite divisibility of the underlining distribution. Furthermore, Marques et al.(2015) considered the distribution of the linear combinations of independent Gumbel random variables and obtain, very nicely, certain important results (paper 7). In this short note, we like to show that the very strong assumption of ”independence” can be replaced with a much weaker assumption of ”sub-independence” in all aforementioned papers. This short paper may be helpful to other investigators dealing with the random variables which are not necessary independent, but could be sub-independent.

Refinements of bounds for tails of compound distributions and ruin probabilities

In this paper we derive lower bounds for right-tails of compound geometric distributions and ruin probabilities in the classical compound Poisson risk model in the heavy-tailed cases using the truncated Lundberg condition, which improve all the corresponding known lower bounds. Some upper bounds are also derived. Examples are given and numerical comparison for ruin probabilities when the adjustment coefficient does not exist are also considered, illustrating the effectiveness of the proposed new bounds. In addition, several bounds for tails of negative binomial distributions are obtained in terms of the tail of compound geometric distributions as well as bounds in the heavy-tailed cases. Also, some bounds for the stop-loss premium associated with compound negative binomial distributions are given. Using Chernoff's upper bounds we derive two-sided bounds for tails of compound Poisson distributions. Finally, two-sided bounds are given for tails of compound logarithmic distributions in the heavy-tailed cases.

The Characterization of Infinite Divisibility of Compound Negative Binomial Distribution as the Sum of Exponential Distribution

The sum of random variables that are identical and independent from an exponential distribution creates the compound distribution. It is called compound negative binomial distribution as the sum of exponential distribution when the number of random variables added follows the negative binomial distribution. This compound distribution’s characteristic function is established by using mathematical analysis methods, included its uniform continuity property. The characteristic function's parametric curves never disappear from the complex plane, which means it is a positively defined function. Another characteristic function's property shows that this compound distribution is one of infinitely divisible distribution.

Generalized Negative Binomial Distribution: Compound Negative Binomial Distribution as the Sum of Laplace Variates

Generalized Negative Binomial Distribution: Compound Negative Binomial Distribution as the Sum of Laplace Variates

COUNTING DISTRIBUTIONS IN RISK THEORY

In this paper we introduce some significant counting distributionsin risk theory. The first one is the I-Delaporte distribution. It is ageneralization of the Non-central negative binomial distribution. Thesecond distribution is the Non-central P´olya-Aeppli distribution. It isa sum of two independent random variables, one that is a Poisson andanother one, a P´olya-Aeppli distributed. The P´olya-Aeppli-Lindley,the compound P´olya and compound binomial distributions are alsoconsidered. They are mixed P´olya-Aeppli distribution with Lindleymixing distribution, compound negative binomial and compound bi-nomial distribution with geometric compounding distribution. Themain application of these distributions is that they can be used ascorresponding counting processes’ distributions in risk models

COUNTING DISTRIBUTIONS IN RISK THEORY

In this paper we introduce some significant counting distributionsin risk theory. The first one is the I-Delaporte distribution. It is ageneralization of the Non-central negative binomial distribution. Thesecond distribution is the Non-central P´olya-Aeppli distribution. It isa sum of two independent random variables, one that is a Poisson andanother one, a P´olya-Aeppli distributed. The P´olya-Aeppli-Lindley,the compound P´olya and compound binomial distributions are alsoconsidered. They are mixed P´olya-Aeppli distribution with Lindleymixing distribution, compound negative binomial and compound bi-nomial distribution with geometric compounding distribution. Themain application of these distributions is that they can be used ascorresponding counting processes’ distributions in risk models

A compound negative binomial distribution with mutative termination conditions based on a change point

In this paper, an extended negative binomial distribution called NBM is proposed by considering mutative termination conditions based on a change point. If the condition A is satisfied before the occurrence of the change point, the trial is terminated according to condition A. Otherwise, if the condition A does not happen and the change point is satisfied, the termination condition of the trial changes from condition A to condition B. We consider the conditions under which the resulting distribution can be degenerated to the existing negative binomial distributions and other new negative binomial distributions. The finite Markov chain imbedding approach is employed to derive the new negative binomial distribution (NBM) and to obtain the related probabilistic indexes. Furthermore, we study the distribution of the compound negative binomial distribution with mutative termination conditions based on a change point (CNBM) by means of phase-type representations.

Expectation Thinning Operators Based on Linear Fractional Probability Generating Functions

We introduce a two-parameter expectation thinning operator based on a linear fractional probability generating function. The operator is then used to define a first-order integer-valued autoregressive INAR (1) process. Distributional properties of the INAR (1) process are described. We revisit the Bernoulli-geometric INAR (1) process of Bourguignon and Weis (Test 26(4):847–868, 2017. https://doi.org/10.1007/s11749-017-0536-4 ) and we introduce a new stationary INAR (1) process with a compound negative binomial distribution. Lastly, we show how a proper randomization of our operator leads to a generalized notion of monotonicity for distributions on $$\mathbf{Z}_+$$ .

A Bivariate Model based on Compound Negative Binomial Distribution

A new bivariate model is introduced by compounding negative binomial and geometric distributions. Distributional properties, including joint, marginal and conditional distributions are discussed. Expressions for the product moments, covariance and correlation coefficient are obtained. Some properties such as ordering, unimodality, monotonicity and self-decomposability are studied. Parameter estimators using the method of moments and maximum likelihood are derived. Applications to traffic accidents data are illustrated.

CMPH: a multivariate phase-type aggregate loss distribution

Abstract We introduce a compound multivariate distribution designed for modeling insurance losses arising from different risk sources in insurance companies. The distribution is based on a discrete-time Markov Chain and generalizes the multivariate compound negative binomial distribution, which is widely used for modeling insurance losses.We derive fundamental properties of the distribution and discuss computational aspects facilitating calculations of risk measures of the aggregate loss, as well as allocations of the aggregate loss to individual types of risk sources. Explicit formulas for the joint moment generating function and the joint moments of different loss types are derived, and recursive formulas for calculating the joint distributions given. Several special cases of particular interest are analyzed. An illustrative numerical example is provided.

The Tempered Discrete Linnik distribution

We introduce a new family of integer-valued distributions by considering a tempered version of the Discrete Linnik law. The proposal is actually a generalization of the well-known Poisson–Tweedie law. The suggested family is extremely flexible since it contains a wide spectrum of distributions ranging from light-tailed laws (such as the Binomial) to heavy-tailed laws (such as the Discrete Linnik). The main theoretical features of the Tempered Discrete Linnik distribution are explored by providing a series of identities in law, which describe its genesis in terms of mixture Poisson distribution and compound Negative Binomial distribution—as well as in terms of mixture Poisson–Tweedie distribution. Moreover, we give a manageable expression and a suitable recursive relationship for the corresponding probability function. Finally, an application to scientometric data—which deals with the scientific output of the researchers of the University of Siena—is considered.

A new inverse regression model applied to radiation biodosimetry.

Biological dosimetry based on chromosome aberration scoring in peripheral blood lymphocytes enables timely assessment of the ionizing radiation dose absorbed by an individual. Here, new Bayesian-type count data inverse regression methods are introduced for situations where responses are Poisson or two-parameter compound Poisson distributed. Our Poisson models are calculated in a closed form, by means of Hermite and negative binomial (NB) distributions. For compound Poisson responses, complete and simplified models are provided. The simplified models are also expressible in a closed form and involve the use of compound Hermite and compound NB distributions. Three examples of applications are given that demonstrate the usefulness of these methodologies in cytogenetic radiation biodosimetry and in radiotherapy. We provide R and SAS codes which reproduce these examples.

Compound Poisson Approximation to Convolutions of Compound Negative Binomial Variables

In this paper, the problem of compound Poisson approximation to the convolution of compound negative binomial distributions, under total variation distance, is considered. First, we obtain an error bound using the method of exponents and it is compared with existing ones. It is known that Kerstan’s method is more powerful in compound approximation problems. We employ Kerstan’s method to obtain better estimates, using higher-order approximations. These bounds are of higher-order accuracy and improve upon some of the known results in the literature. Finally, an interesting application to risk theory is discussed.

The destructive negative binomial cure rate model with a latent activation scheme

A new flexible cure rate survival model is developed where the initial number of competing causes of the event of interest (say lesions or altered cells) follows a compound negative binomial (NB) distribution. This model provides a realistic interpretation of the biological mechanism of the event of interest, as it models a destructive process of the initial competing risk factors and records only the damaged portion of the original number of risk factors. Besides, it also accounts for the underlying mechanisms that lead to cure through various latent activation schemes. Our method of estimation exploits maximum likelihood (ML) tools. The methodology is illustrated on a real data set on malignant melanoma, and the finite sample behavior of parameter estimates are explored through simulation studies.

Further Review of Panjer's Recursion for Evaluation of Compound Negative Binomial Distribution Using R

Panjer's recursion formula is used for the evaluation of compound distributions. The use of this algorithm has become a widespread standard technique for life and general insurance problems. This study reviews and extends Panjer's recursion formula for evaluation of compound negative binomial distributions. For gamma and a mixture of gamma distributions, the theory has been developed so that the software R can be directly applicable. The accuracy of the method used in this study is better and the computation time is faster.

On the Sums of Compound Negative Binomial and Gamma Random Variables

We study the convolution of compound negative binomial distributions with arbitrary parameters. The exact expression and also a random parameter representation are obtained. These results generalize some recent results in the literature. An application of these results to insurance mathematics is discussed. The sums of certain dependent compound Poisson variables are also studied. Using the connection between negative binomial and gamma distributions, we obtain a simple random parameter representation for the convolution of independent and weighted gamma variables with arbitrary parameters. Applications to the reliability ofm-out-of-n:G systems and to the shortest path problem in graph theory are also discussed.

On the Sums of Compound Negative Binomial and Gamma Random Variables

We study the convolution of compound negative binomial distributions with arbitrary parameters. The exact expression and also a random parameter representation are obtained. These results generalize some recent results in the literature. An application of these results to insurance mathematics is discussed. The sums of certain dependent compound Poisson variables are also studied. Using the connection between negative binomial and gamma distributions, we obtain a simple random parameter representation for the convolution of independent and weighted gamma variables with arbitrary parameters. Applications to the reliability of m-out-of-n:G systems and to the shortest path problem in graph theory are also discussed.