A surplus process involving a compound Poisson counting process and applications

Abstract

In this paper, we introduce a surplus process involving a compound Poisson counting process, which is a generalization of the classical ruin model where the claim-counting process is a homogeneous Poisson process. The incentive is to model batch arrival of claims using a counting process that is based on a compound distribution. This reduces the difficulty of modeling claim amounts and is consistent with industrial data. Recursive formula, some properties and relevant main ruin theory results are provided. Further, we consider applications involving zero-truncated negative binomial and zero-truncated binomial batch arrivals when the claim amounts follow exponential or Erlang distribution.     

A Simple Distribution for the Sum of Correlated,Exchangeable Binary Data

This article describes a generalization of the binomial distribution. The closed form probability function for the probability of k successes out of n correlated, exchangeable Bernoulli trials depends on the number of trials and its two parameters: the common success probability and the common correlation. The distribution is derived under the assumption that the common correlation between all pairs of Bernoulli trials remains unchanged conditional on successes in all completed trials. The distribution was developed to model bond defaults but may be suited to biostatistical applications involving clusters of binary data encountered in repeated measurements or toxicity studies of families of organisms. Maximum likelihood estimates for the parameters of the distribution are found for a set of binary data from a developmental toxicity study on litters of mice.     

A generalization of the beta–binomial distribution

Summary.  The paper describes a distribution generated by the Gaussian hypergeometric function that may be seen as a generalization of the beta–binomial distribution. It is expressed as a generalized beta mixture of a binomial distribution. This new mixing distribution allows the existence of a mode and an antimode, which is very useful for fitting some data sets. Two examples illustrate the greater versatility of the new distribution compared with the beta–binomial distribution.     

Bayes estimation of the binomial parameter n

Bayes estimation of the binomial parameter n based on a general prior distribution for n is studied. As special cases improper prior and Poisson prior (which is a natural choice) are considered, and formulae for the marginal and posterior distributions are obtained. It is shown that the assumption of improper priors in both p and n leads to implausible results.     

Statistical Analysis for the Inverse Trinomial Distribution

This article considers parameter estimation, goodness of fit, likelihood ratio and score tests, and model selection by Akaike information criterion for the inverse trinomial (IT) distribution, a classical one-dimensional random walk distribution. The IT distribution has a cubic variance function of the mean and is a generalization of the negative binomial distribution. Basic distributional properties and expressions for the probability mass function, recurrence formula, moments, and score functions are also presented.     

The binomial failure rate mixture model for common cause failure data from the nuclear industry

A model for the lifetime of a system is considered in which the system is susceptible to simultaneous failures of two or more components, the failures having a common external cause. Three sets of discrete failure data from the US nuclear industry are examined to motivate and illustrate the model derivation: they are for motor-operated valves, cooling fans and emergency diesel generators. To achieve target reliabilities, these components must be placed in systems that have built-in redundancy. Consequently, multiple failures due to a common cause are critical in the risk of core meltdown. Vesely has offered a simple methodology for inference, called the binomial failure rate model: external events are assumed to be governed by a Poisson shock model in which resulting shocks kill X out of m system components, X having a binomial distribution with parameters ( m , p ), 0< p <1. In many applications the binomial failure rate model fits failure data poorly, and the model has not typically been applied to probabilistic risk assessments in the nuclear industry. We introduce a realistic generalization of the binomial failure rate model by assigning a mixing distribution to the unknown parameter p . The distribution is generally identifiable, and its unique nonparametric maximum likelihood estimator can be obtained by using a simple iterative scheme.     

Bayesian estimation and prediction for the inverse weibull distribution under general progressive censoring

Abstract

This paper deals with Bayesian estimation and prediction for the inverse Weibull distribution with shape parameter α and scale parameter λ under general progressive censoring. We prove that the posterior conditional density functions of α and λ are both log-concave based on the assumption that λ has a gamma prior distribution and α follows a prior distribution with log-concave density. Then, we present the Gibbs sampling strategy to estimate under squared-error loss any function of the unknown parameter vector (α, λ) and find credible intervals, as well as to obtain prediction intervals for future order statistics. Monte Carlo simulations are given to compare the performance of Bayesian estimators derived via Gibbs sampling with the corresponding maximum likelihood estimators, and a real data analysis is discussed in order to illustrate the proposed procedure. Finally, we extend the developed methodology to other two-parameter distributions, including the Weibull, Burr type XII, and flexible Weibull distributions, and also to general progressive hybrid censoring.     

Using Auxiliary Data for Binomial Parameter Estimation with Nonignorable Nonresponse

Nonignorable nonresponse is a nonresponse mechanism that depends on the values of the variable having nonresponse. When an observed data of a binomial distribution suffer missing values from a nonignorable nonresponse mechanism, the binomial distribution parameters become unidentifiable without any other auxiliary information or assumption. To address the problems of non identifiability, existing methods mostly based on the log-linear regression model. In this article, we focus on the model when the nonresponse is nonignorable and we consider to use the auxiliary data to improve identifiability; furthermore, we derive the maximum likelihood estimator (MLE) for the binomial proportion and its associated variance. We present results for an analysis of real-life data from the SARS study in China. Finally, the simulation study shows that the proposed method gives promising results.     

The Geometric ArcTan distribution with applications to model demand for health services

In this paper , a new discrete two–parameter distribution α ∈ ? ? {0} and 0 < θ < 1, the Geometric ArcTan (GAT) distribution is introduced. The geometric distribution is a limiting case of this model when α tends to zero. Similarly to the the latter distribution, this probabilistic family is unimodal but the mode can be located at zero or in other point of the support. Then, after deriving some of its more relevant properties , the issue of parameter investigation is investigated. Next, the GAT distribution is used to explain the demand for health services by means of a regression model. Numerical results show that this new model outperforms the negative binomial distribution.     

On the three-dimensional negative binomial distribution

We consider a particular generalization of the negative binomial distribution to the multivariate case obtained through a specification of the probability generating function as the negative power of a certain polynomial. The probability function itself has previously been derived for the two-dimensional case only, and inference in the multivariate negative binomial distribution has been restricted to the use of composite likelihood based on one- or two-dimensional marginals. In this article, we derive the three-dimensional probability function as a sum with all terms positive and study the range of possible parameter values. We illustrate the use of the three-dimensional distribution for modeling three correlated SAR images.     

The two-color urn model: recursions for the moments

For the two-color reinforcement-depletion urn model, with balancing reinforcement and depletion held constant over cycles, a recursive formula is given from which all factorial moments (for white balls, for example) can be determined. When the reinforcement of each color is positive, the stationary distribution of white balls (infinite number of cycles) turns out to be determined by three parameters. namely (i) the total number of balls in the urn, (ii) the richness of the reinforcement, or ratio of white ball reinforcement to total reinforcement, and (iii) the size of the white ball reinforcement. In addition, the distribution mimics the binomial (with less variance and skewness (√β₁:) ) and from formulas for the exact first four moments rapidly approaches normality. On the basis of the few cases studied, an approximating Gram-Charlier distribution with a binomial nucleus is only moderately successful     

Exact tests of binomial p=1/2 when one binomial event possibly has different probability

The data are n independent random binomial events, each resulting in success or failure. The event outcomes are believed to be trials from a binomial distribution with success probability p, and tests of p=1/2 are desired. However, there is the possibility that some unidentified event has a success probability different from the common value p for the other n?1 events. Then, tests of whether this common p equals 1/2 are desired. Fortunately, two-sided tests can be obtained that simultaneously are applicable for both situations. That is, the significance level for a test is same when one event has a different probability as when all events have the same probability. These tests are the usual equal-tail tests for p=1/2 (based on n independent trials from a binomial distribution).     

A Bivariate Generalization of the Noncentral Negative Binomial Distribution

This article proposes a bivariate generalization of the noncentral negative binomial distribution which arises as a model in photon and neural counting. This bivariate generalization is derived as a mixed shifted bivariate negative binomial distribution. Various properties and parameter estimation, especially by a minimum distance method based on the probability generating function, are considered. To show the practical usefulness of the bivariate distribution proposed, an application to model low-flux astronomical images is discussed and a real data set has been analyzed.     

Discrete Linnik Weibull distribution

Abstract

We introduce a new family of distributions using truncated discrete Linnik distribution. This family is a rich family of distributions which includes many important families of distributions such as Marshall–Olkin family of distributions, family of distributions generated through truncated negative binomial distribution, family of distributions generated through truncated discrete Mittag–Leffler distribution etc. Some properties of the new family of distributions are derived. A particular case of the family, a five parameter generalization of Weibull distribution, namely discrete Linnik Weibull distribution is given special attention. This distribution is a generalization of many distributions, such as extended exponentiated Weibull, exponentiated Weibull, Weibull truncated negative binomial, generalized exponential truncated negative binomial, Marshall-Olkin extended Weibull, Marshall–Olkin generalized exponential, exponential truncated negative binomial, Marshall–Olkin exponential and generalized exponential. The shape properties, moments, median, distribution of order statistics, stochastic ordering and stress–strength properties of the new generalized Weibull distribution are derived. The unknown parameters of the distribution are estimated using maximum likelihood method. The discrete Linnik Weibull distribution is fitted to a survival time data set and it is shown that the distribution is more appropriate than other competitive models.     

A COM-Poisson-type generalization of the negative binomial distribution

ABSTRACT

This paper introduces a generalization of the negative binomial (NB) distribution in analogy with the COM-Poisson distribution. Many well-known distributions are particular and limiting distributions. The proposed distribution belongs to the modified power series, generalized hypergeometric and exponential families, and also arises as weighted NB and COM-Poisson distributions. Probability and moment recurrence formulae, and probabilistic and reliability properties have been derived. With the flexibility to model under-, equi- and over-dispersion, and its various interesting properties, this NB generalization will be a useful model for count data. An application to empirical modeling is illustrated with a real data set.     

Lemuel Shattuck,Statist Founder of the American Statistical Association

Polynomial regression of degree p in one independent variable χ is considered. Numerically large sample correlations between χ^α and χ^β, α < β, a, β = 1, ···, p, may cause ill-conditioning in the matrix to be inverted in application of the method of least squares. These sample correlations are investigated. It is confirmed that centering of the independent variable to have zero sample mean removes nonessential ill-conditioning. If the sample values of χ are placed symmetrically about their mean, the sample correlation between χ^α and χ^β is reduced to zero by centering when α + β is odd, but may remain large when α + β is even. Some examples and recommendations are given.     

THE OPTIMAL TEST FOR SELECTING THE GREATER OF TWO BINOMIAL PROBABILITIES

The efficiency of a sequential test is related to the “importance” of the trials within the test. This relationship is used to find the optimal test for selecting the greater of two binomial probabilities, p_α and p_b, namely, the stopping rule is “gambler's ruin” and the optimal discipline when p_α+p_b≤ 1 (≥ 1) is play-the-winner (loser), i.e. an α-trial which results in a success is followed by an α-trial (b-trial) whereas an α-trial which results in a failure is followed by α b-trid (α-trial) and correspondingly for b-trials.     

The negative contagion reflection of the polya-eggenberger distribution

The Polya-Eggenberger distribution Involves drawing a ball from an urn containing black and white balls and, after each drawing, returning the ball together with s balls of the same color, The model represents positive contagion since the added balls are the same color as the one drawn, See Johnson and Kotz, (1977),

This paper derives and examines the probability distribution which results from the Polya-Eggenberger model with only one change namely, the s additional balls added after each drawing are of the opposite color, producing a negative contagion model.

Formulas in closed form are presented for the probability distribution function, the mean and variance, all binomial moments and, where s is greater than or equal to the number of balls in the urn at start, the mode, A formula for the mode is conjectured where s is less than the number of balls in the urn at start.

Finally, the probability of obtaining k black balls in n drawings is shown in certain instances to be equal to A_nk/n!

where A_nk are the Eulerian numbers.     

Estimation under modified Weibull distribution based on right censored generalized order statistics

In this paper, the maximum likelihood (ML) and Bayes, by using Markov chain Monte Carlo (MCMC), methods are considered to estimate the parameters of three-parameter modified Weibull distribution (MWD(β, τ, λ)) based on a right censored sample of generalized order statistics (gos). Simulation experiments are conducted to demonstrate the efficiency of the proposed methods. Some comparisons are carried out between the ML and Bayes methods by computing the mean squared errors (MSEs), Akaike's information criteria (AIC) and Bayesian information criteria (BIC) of the estimates to illustrate the paper. Three real data sets from Weibull(α, β) distribution are introduced and analyzed using the MWD(β, τ, λ) and also using the Weibull(α, β) distribution. A comparison is carried out between the mentioned models based on the corresponding Kolmogorov–Smirnov (K–S) test statistic, {AIC and BIC} to emphasize that the MWD(β, τ, λ) fits the data better than the other distribution. All parameters are estimated based on type-II censored sample, censored upper record values and progressively type-II censored sample which are generated from the real data sets.     

A three-parameter generalization of the binomial distribution

A new three-parameter distribution, a generalization of the binomial, th ebeta-binomial (BB) and the correlated binomial (CB) distributions, is derived. Improvement in fit of the new distribution over the BB and the CB distributions has been found for a set of real data