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.     

Fiducial prediction intervals

This paper presents an approach for constructing prediction intervals for any given distribution. The approach is based on the principle of fiducial inference. We use several examples, including the normal, binomial, exponential, gamma, and Weibull distributions, to illustrate the proposed procedure.     

On Generalized Binomial and Negative Binomial Distributions for Dependent Bernoulli Variables

We study the distributions of the random variables S_n and V_r related to a sequence of dependent Bernoulli variables, where S_n denotes the number of successes in n trials and V_r the number of trials necessary to obtain r successes. The purpose of this article is twofold: (1) Generalizing some results on the “nature” of the binomial and negative binomial distributions we show that S_n and V_r can follow any prescribed discrete distribution. The corresponding joint distributions of the Bernoulli variables are characterized as the solutions of systems of linear equations. (2) We consider a specific type of dependence of the Bernoulli variables, where the probability of a success depends only on the number of previous successes. We develop some theory based on new closed-form representations for the probability mass functions of S_n and V_r which enable direct computations of the probabilities.     

A sequential design for binomial clinical trials

We investigate a sequential procedure for comparing two treatments in a binomial clinical trial. The procedure uses play-the-winner sampling with termination as soon as the absolute difference in the number of successes of the two treatments reaches a critical value. The important aspect of our procedure is that the critical value is modified as the experiment progresses. Numerical results are given which show that this procedure is preferred to all other existing procedures on the basis of the sample size on the poorer treatment and also on the basis of total sample size.     

A generalization of the logistic linear'model

Consider the logistic linear model, with some explanatory variables overlooked. Those explanatory variables may be quantitative or qualitative. In either case, the resulting true response variable is not a binomial or a beta-binomial but a sum of binomials. Hence, standard computer packages for logistic regression can be inappropriate even if an overdispersion factor is incorporated. Therefore, a discrete exponential family assumption is considered to broaden the class of sampling models. Likelihood and Bayesian analyses are discussed. Bayesian computation techniques such as Laplacian approximations and Markov chain simulations are used to compute posterior densities and moments. Approximate conditional distributions are derived and are shown to be accurate. The Markov chain simulations are performed effectively to calculate posterior moments by using the approximate conditional distributions. The methodology is applied to Keeler's hardness of winter wheat data for checking binomial assumptions and to Matsumura's Accounting exams data for detailed likelihood and Bayesian analyses.     

Generalized Markov-Polya urn models with predetermined strategies

By using combinatorial methods involving lattice path combinatorics, three generalized probability models dependent on predetermined strategies have been obtained with the help of urn models.The models have been developed with the help of a sampling scheme which unifies both, the binomial and the inverse binomial sampling schemes. These models generate a number of important discrete probability distributions both as particular cases and as limiting cases. Recurrence relations among the moments of the models have also been obtained.     

Discrete associated kernels method and extensions

Discrete kernel estimation of a probability mass function (p.m.f.), often mentioned in the literature, has been far less investigated in comparison with continuous kernel estimation of a probability density function (p.d.f.). In this paper, we are concerned with a general methodology of discrete kernels for smoothing a p.m.f. f. We give a basic of mathematical tools for further investigations. First, we point out a generalizable notion of discrete associated kernel which is defined at each point of the support of f and built from any parametric discrete probability distribution. Then, some properties of the corresponding estimators are shown, in particular pointwise and global (asymptotical) properties. Other discrete kernels are constructed from usual discrete probability distributions such as Poisson, binomial and negative binomial. For small samples sizes, underdispersed discrete kernel estimators are more interesting than the empirical estimator; thus, an importance of discrete kernels is illustrated. The choice of smoothing bandwidth is classically investigated according to cross-validation and, novelly, to excess of zeros methods. Finally, a unification way of this method concerning the general probability function is discussed.     

The monotone boundary property and the full coverage property of confidence intervals for a binomial proportion

The methodology for deriving the exact confidence coefficient of some confidence intervals for a binomial proportion is proposed in Wang [2007. Exact confidence coefficients of confidence intervals for a binomial proportion. Statist. Sinica 17, 361–368]. The methodology requires two conditions of confidence intervals: the monotone boundary property and the full coverage property. In this paper, we show that for some confidence intervals of a binomial proportion, the two properties hold for any sample size. Based on results presented in this paper, the procedure in Wang [2007. Exact confidence coefficients of confidence intervals for a binomial proportion. Statist. Sinica 17, 361–368] can be directly used to calculate the exact confidence coefficients of these confidence intervals for any fixed sample size.     

Bayesian Estimation of the Size of a Closed Population Using Photo-ID Data with Part of the Population Uncatchable

Abstract

We develop a Bayesian statistical model for estimating bowhead whale population size from photo-identification data when most of the population is uncatchable. The proposed conditional likelihood function is a product of Darroch's model, formulated as a function of the number of good photos, and a binomial distribution of captured whales given the total number of good photos at each occasion. The full Bayesian model is implemented via adaptive rejection sampling for log concave densities. We apply the model to data from 1985 and 1986 bowhead whale photographic studies and the results compare favorably with the ones obtained in the literature. Also, a comparison with the maximum likelihood procedure with bootstrap simulation is considered using different vague priors for the capture probabilities.     

Omnitab-a simple language for statistical computations

The OMNITAB system of computer programming provides a simple, yet extremely versatile means of communicating with the computer using simple English words and phrases, It is written in FORTRAN and is available for implementation on UNIVAC 1108, IBM 360, Burroughs 5500 and CDC 6600 equipment. It is based on a worksheet, stored in the computer, which is a two dimensional array easy to visualize even by beginners. The system has particular use for students of statistics, of all ages, in that it allows them to interact with the data without requiring that they master any diffiicult and esoteric (for statisticians) computer skills. The programming system is described together with several applications to statistical problems. Full program documentation is available from the authors.     

A generalized negative binomial and applications

A generalized negative binomial distribution is derived from the Markov Bernoulli sequence of successes and failures. We study the properties and applications of this distribution. The properties are illustrated by two examples of discrete time queueing systems. The distribution is then fitted to two data sets, the eruption record of Mt. Sangay, and a record of computer disk failure accesses. In the first case there is a strong serial dependence in the data and the generalized negative binomial provides a good fit, while in the second case, although there is a significant serial dependence, it is insufficient to justify the additional parameter of the distribution. We conclude by demonstrating the usefulness of the distribution in the field of statistical quality control.     

Another Approach to Incomplete Integrals

One usually writes the incomplete integrals as the sums of discrete probabilities by repeating the procedure of integration by parts. This work provides another approach by employing the binomial expansion.     

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.     

Reference priors for constrained rate models of count data

We derive reference priors for constrained rate models of count data using the sequential algorithm of Berger and Bernardo (1992b). The event counts for various groups of subjects are modeled as discrete random variables (Poisson, binomial, or negative binomial) with group specific rates. We consider situations in which the groups can be completely ordered according to one covariate. The priors enforce monotonicity (or monotonicity and convexity) of the rates with respect to the ordering. We use the priors to model a data set on mortality rates for men in different age groups assuming that the mortality rates increase with respect to age. We also consider the situation in which the parameter space is augmented to include rates corresponding to unobserved age groups, and the case of a random upper bound on the mortality rates. In addition, we provide an evaluation of the out-of-sample predictive performance of the proposed methods.     

Resolving the conflict over fisher's exact test

Fisher's exact test for two-by-two contingency tables has repeatedly been criticized as being too conservative. These criticisms arise most frequently in the context of a planned experiment for which the numbers of successes in each of two experimental groups are assumed to be binomially distributed. It is argued here that the binomial model is often unrealistic, and that the departures from the binomial assumptions reduce the conservatism in Fisher's exact test. Further discussion supports a recent claim of Barnard (1989) that the residual conservatism is attributable, not to any additional information used by the competing method, but to the discrete nature of the test, and can be drastically reduced through the use of Lancaster's mid-p-value. The binomial model is not recommended in that it depends on extra, questionable assumptions.     

Inference for a binomial proportion in the presence of ties

We suppose a case is to be compared with controls on the basis of a test that gives a single discrete score. The score of the case may tie with the scores of one or more controls. However, scores relate to an underlying quantity of interest that is continuous and so an observed score can be treated as the rounded value of an underlying continuous score. This makes it reasonable to break ties. This paper addresses the problem of forming a confidence interval for the proportion of controls that have a lower underlying score than the case. In the absence of ties, this is the standard task of making inferences about a binomial proportion and many methods for forming confidence intervals have been proposed. We give a general procedure to extend these methods to handle ties, under the assumption that ties may be broken at random. Properties of the procedure are given and an example examines its performance when it is used to extend several methods. A real example shows that an estimated confidence interval can be much too small if the uncertainty associated with ties is not taken into account. Software implementing the procedure is freely available.     

Asymptotic Normality in Mixtures of Power Series Distributions

Abstract.  The problem of estimating the individual probabilities of a discrete distribution is considered. The true distribution of the independent observations is a mixture of a family of power series distributions. First, we ensure identifiability of the mixing distribution assuming mild conditions. Next, the mixing distribution is estimated by non-parametric maximum likelihood and an estimator for individual probabilities is obtained from the corresponding marginal mixture density. We establish asymptotic normality for the estimator of individual probabilities by showing that, under certain conditions, the difference between this estimator and the empirical proportions is asymptotically negligible. Our framework includes Poisson, negative binomial and logarithmic series as well as binomial mixture models. Simulations highlight the benefit in achieving normality when using the proposed marginal mixture density approach instead of the empirical one, especially for small sample sizes and/or when interest is in the tail areas. A real data example is given to illustrate the use of the methodology.     

On the Alias Method for Generating Random Variables from a Discrete Distribution

The alias method of Walker is a clever, new, fast method for generating random variables from an arbitrary, specified discrete distribution. A simple probabilistic proof is given, in terms of mixtures, that the method works for any discrete distribution with a finite number of outcomes. A more efficient version of the table-generating portion of the method is described. Finally, a brief discussion on efficiency of the method is given. We believe that the generality, speed, and simplicity of the method make it attractive for use in generating discrete random variables.     

Mixture models for capture-recapture count data

The contribution investigates the problem of estimating the size of a population, also known as the missing cases problem. Suppose a registration system is targeting to identify all cases having a certain characteristic such as a specific disease (cancer, heart disease, ...), disease related condition (HIV, heroin use, ...) or a specific behavior (driving a car without license). Every case in such a registration system has a certain notification history in that it might have been identified several times (at least once) which can be understood as a particular capture-recapture situation. Typically, cases are left out which have never been listed at any occasion, and it is this frequency one wants to estimate. In this paper modelling is concentrating on the counting distribution, e.g. the distribution of the variable that counts how often a given case has been identified by the registration system. Besides very simple models like the binomial or Poisson distribution, finite (nonparametric) mixtures of these are considered providing rather flexible modelling tools. Estimation is done using maximum likelihood by means of the EM algorithm. A case study on heroin users in Bangkok in the year 2001 is completing the contribution.     

On a discrete interaction risk model with delayed claims and stochastic incomes under random discount rates

In this paper, we study a discrete interaction risk model with delayed claims and stochastic incomes in the framework of the compound binomial model. A generalized Gerber-Shiu discounted penalty function is proposed to analyse this risk model in which the interest rates follow a Markov chain with finite state space. We derive an explicit expression for the generating function of this Gerber-Shiu discounted penalty function. Furthermore, we derive a recursive formula and a defective renewal equation for the original Gerber-Shiu discounted penalty function. As an application, the joint distributions of the surplus one period prior to ruin and the deficit at ruin, as well as the probabilities of ruin are obtained. Finally, some numerical illustrations from a specific example are also given.