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.     

Selected singular-generalized-hyperbolic distributions with applications to order statistics and reliability

In this paper, the truncated version of the selected multivariate generalized-hyperbolic distributions is introduced. Considering special truncations, the joint distribution of the consecutive order statistics from the multivariate generalized-hyperbolic (GH) distribution is derived. It is shown that this joint distribution can be expressed as mixtures of the truncated selected-GH distributions. All of these truncated distributions are expressed as the selected singular-GH distributions. These results are used to obtain some expressions for the reliability measures such as mean residual life time, mean inactivity time and regression mean residual life for k-out-of-n systems.     

Two-Sided Generalized Exponential Distribution

We derive a generalization of the exponential distribution by making log transformation of the standard two-sided power distribution. We show that this new generalization is in fact a mixture of a truncated exponential distribution and truncated generalized exponential distribution introduced by Gupta and Kundu [Generalized exponential distributions. Aust. N. Z. J. Stat. 41(1999):173–188]. The newly defined distribution is more flexible for modeling data than the ordinary exponential distribution. We study its properties, estimate the parameters, and demonstrate it on some well-known real data sets comparing other existing methods.     

Moments of order statistics from doubly truncated continuous distributions and characterizations

In this paper, recurrence relations from a general class of doubly truncated continuous distributions which are satisfied by single as well as product moments of order statistics are obtained. Recurrence relations from doubly truncated generalized Weibull, exponential, Raleigh and logistic distributions have been derived as special cases of our result, Some previous results for doubly truncated Weibull, standard exponential, power function and Burr type XII distributions are obtained as special cases. The general recurrence relation of single moments has been used in the case of the left and right truncation to characterize the Weibull, Burr type XII and Pareto distributions.     

Attribute control chart for some popular distributions

The binomial distribution is often used to display attribute control data. In this paper, a statistical model is settled for attribute control chart under truncated life test. By Burr X & XII, inverse Gaussian (IG), and exponential lifetime-truncated distributions, a Shewhart-type attribute control chart is built to display the data. The performance of attributed control chart constructed on truncated life test is evaluated by average run length, which compares the performance of all distributions. Our study arranges that IG is better distribution among all.     

Moments of order statistics from a non-overlapping mixture model with applications to truncated laplace distribution

We employ two different approaches to derive single and product moments of order statistics from a truncated Laplace distribution. A direct evaluation method establishes recurrence relations whereas the more general non-overlapping mixture model incorporates the truncated Laplace distribution as a special case. The results are thereafter applied to estimate location and scale parameters of such distributions.     

Some results on the truncated multivariate t distribution

The use of truncated distributions arises often in a wide variety of scientific problems. In the literature, there are a lot of sampling schemes and proposals developed for various specific truncated distributions. So far, however, the study of the truncated multivariate t (TMVT) distribution is rarely discussed. In this paper, we first present general formulae for computing the first two moments of the TMVT distribution under the double truncation. We formulate the results as analytic matrix expressions, which can be directly computed in existing software. Results for the left and right truncation can be viewed as special cases. We then apply the slice sampling algorithm to generate random variates from the TMVT distribution by introducing auxiliary variables. This strategic approach can result in a series of full conditional densities that are of uniform distributions. Finally, several examples and practical applications are given to illustrate the effectiveness and importance of the proposed results.     

Simulation of right and left truncated gamma distributions by mixtures

We study the properties of truncated gamma distributions and we derive simulation algorithms which dominate the standard algorithms for these distributions. For the right truncated gamma distribution, an optimal accept–reject algorithm is based on the fact that its density can be expressed as an infinite mixture of beta distribution. For integer values of the parameters, the density of the left truncated distributions can be rewritten as a mixture which can be easily generated. We give an optimal accept–reject algorithm for the other values of the parameter. We compare the efficiency of our algorithm with the previous method and show the improvement in terms of minimum acceptance probability. The algorithm proposed here has an acceptance probability which is superior to e/4.     

A new lifetime distribution for a series-parallel system: properties,applications and estimations under progressive type-II censoring

A compound class of zero truncated Poisson and lifetime distributions is introduced. A specialization is paved to a new three-parameter distribution, called doubly Poisson-exponential distribution, which may represent the lifetime of units connected in a series-parallel system. The new distribution can be obtained by compounding two zero truncated Poisson distributions with an exponential distribution. Among its motivations is that its hazard rate function can take different shapes such as decreasing, increasing and upside-down bathtub depending on the values of its parameters. Several properties of the new distribution are discussed. Based on progressive type-II censoring, six estimation methods [maximum likelihood, moments, least squares, weighted least squares and Bayes (under linear-exponential and general entropy loss functions) estimations] are used to estimate the involved parameters. The performance of these methods is investigated through a simulation study. The Bayes estimates are obtained using Markov chain Monte Carlo algorithm. In addition, confidence intervals, symmetric credible intervals and highest posterior density credible intervals of the parameters are obtained. Finally, an application to a real data set is used to compare the new distribution with other five distributions.     

Modeling Bimodal Discrete Data Using Conway-Maxwell-Poisson Mixture Models

Bimodal truncated count distributions are frequently observed in aggregate survey data and in user ratings when respondents are mixed in their opinion. They also arise in censored count data, where the highest category might create an additional mode. Modeling bimodal behavior in discrete data is useful for various purposes, from comparing shapes of different samples (or survey questions) to predicting future ratings by new raters. The Poisson distribution is the most common distribution for fitting count data and can be modified to achieve mixtures of truncated Poisson distributions. However, it is suitable only for modeling equidispersed distributions and is limited in its ability to capture bimodality. The Conway–Maxwell–Poisson (CMP) distribution is a two-parameter generalization of the Poisson distribution that allows for over- and underdispersion. In this work, we propose a mixture of CMPs for capturing a wide range of truncated discrete data, which can exhibit unimodal and bimodal behavior. We present methods for estimating the parameters of a mixture of two CMP distributions using an EM approach. Our approach introduces a special two-step optimization within the M step to estimate multiple parameters. We examine computational and theoretical issues. The methods are illustrated for modeling ordered rating data as well as truncated count data, using simulated and real examples.     

A class of rectangle-screened multivariate normal distributions and its applications

A screening problem is tackled by proposing a parametric class of distributions designed to match the behavior of the partially observed screened data. This class is obtained from the nontruncated marginal of the rectangle-truncated multivariate normal distributions. Motivations for the screened distribution as well as some of the basic properties, such as its characteristic function, are presented. These allow us a detailed exploration of other important properties that include closure property in linear transformation, in marginal and conditional operations, and in a mixture operation as well as the first two moments and some sampling distributions. Various applications of these results to the statistical modelling and data analysis are also provided.     

Power binomial exponential distribution: Modeling,simulation and application

ABSTRACT

The binomial exponential 2 (BE2) distribution was proposed by Bakouch et al. as a distribution of a random sum of independent exponential random variables, when the sample size has a zero truncated binomial distribution. In this article, we introduce a generalization of BE2 distribution which offers a more flexible model for lifetime data than the BE2 distribution. The hazard rate function of the proposed distribution can be decreasing, increasing, decreasing–increasing–decreasing and unimodal, so it turns out to be quite flexible for analyzing non-negative real life data. Some statistical properties and parameters estimation of the distribution are investigated. Three different algorithms are proposed for generating random data from the new distribution. Two real data applications regarding the strength data and Proschan's air-conditioner data are used to show that the new distribution is better than the BE2 distribution and some other well-known distributions in modeling lifetime data.     

Confidence intervals for discrete distributions based on a single observation

Confidence intervals are developed for the mode of a discrete unimodal distribution in the case where only a single observation is available. These intervals are centered on either the observation, X, or a weighted average of X with a constant, b, chosen by the investigator. Intervals are derived for nonrestricted unimodal distributions, for unimodal distributions with a symmetry property, and for a family of two-sided truncated geometric distributions.     

Discrete Truncated Power‐Law Distributions

Discrete power‐law distributions have significant consequences for understanding many phenomena in practice, and have attracted much attention in recent decades. However, in many practical applications, there exists a natural upper bound for the probability tail. In this paper, we develop maximum likelihood estimates for truncated discrete power‐law distributions based on the upper order statistics, and large sample properties are mentioned as well. Monte Carlo simulation is carried out to examine the finite sample performance of the estimates. Applications in real cyber attack data and peak gamma‐ray intensity of solar flares are highlighted.     

A Sieve model for extreme values

From the class of extreme value distributions, we focus on the set of heavy-tailed distributions which produce low-frequency, high-cost events. The regular Pareto distribution is the basic model of choice, being the simplest heavy-tailed distribution. Real data suggest that modifications of the Pareto distribution may be a better fit; an alternative model is the truncated Pareto distribution (TPD). For further study, this paper proposed a TPD Sieve class of distributions. The properties and estimation on the Sieve class are also discussed. We fit the models to the largest Black Sea bass caught in Buzzard's Bay, MA, USA and the costliest Atlantic hurricanes from 1900 to 2005. Using measures of model adequacy, the TPD Sieve model is generally found to be the best-fitting model.     

The beta exponentiated Weibull distribution

The Weibull distribution is one of the most important distributions in reliability. For the first time, we introduce the beta exponentiated Weibull distribution which extends recent models by Lee et al. [Beta-Weibull distribution: some properties and applications to censored data, J. Mod. Appl. Statist. Meth. 6 (2007), pp. 173–186] and Barreto-Souza et al. [The beta generalized exponential distribution, J. Statist. Comput. Simul. 80 (2010), pp. 159–172]. The new distribution is an important competitive model to the Weibull, exponentiated exponential, exponentiated Weibull, beta exponential and beta Weibull distributions since it contains all these models as special cases. We demonstrate that the density of the new distribution can be expressed as a linear combination of Weibull densities. We provide the moments and two closed-form expressions for the moment-generating function. Explicit expressions are derived for the mean deviations, Bonferroni and Lorenz curves, reliability and entropies. The density of the order statistics can also be expressed as a linear combination of Weibull densities. We obtain the moments of the order statistics. The expected information matrix is derived. We define a log-beta exponentiated Weibull regression model to analyse censored data. The estimation of the parameters is approached by the method of maximum likelihood. The usefulness of the new distribution to analyse positive data is illustrated in two real data sets.     

BIVARIATE DISTRIBUTION WITH TRUNCATED POISSON MARGINAL DISTRIBUTIONS

ABSTRACT

A bivariate distribution, whose marginal distributions are truncated Poisson distributions, is developed as a product of truncated Poisson distributions and a multiplicative factor. The multiplicative factor takes into account the correlation, either positive or negative, between the two random variables. The distributional properties of this model are studied and the model is fitted to a real life bivariate data.     

An Overview of Asymptotic Properties of Estimators in Truncated Distributions

In this article, a method is proposed to get the limiting distributions and asymptotic properties of estimators based on the minimum and/or maximum of a given srs of a truncated distribution. Following a common outline, a review is carried out by considering different kinds of truncated distributions, some new results are also developed.     

On conditional correlations defined by a class of nonlinearly truncated distributions and their applications

This article derives and studies several types of conditional correlations. The correlations are obtained by a class of two-piece scale mixture skew-normal distributions. The class is obtained by applying a set of nonlinear constraints to the bivariate scale mixture of normal distributions. The correlations of the class are invariant with respect to the choice of the scale mixing function, however, they are dependent upon the type of the nonlinear truncation. Moreover, their respective upper and lower limits are no longer 1.00 and?1.00. They are useful for the truncated data analysis, the multivariate interdependence methods (such as the principal component analysis and the factor analysis), and the random truncation modelling. Some distributional properties and the Bayesian computation of the correlations are considered when developing necessary theories and providing illustrative examples, respectively. Two applications are also given to demonstrate the usefulness of the conditional correlations in a multivariate analysis.     

On Conditional and Partial Correlation

A graphical technique, similar in spirit to probability plotting, can be used to judge whether a Poisson model is appropriate for an observed frequency distribution. This “Poissonness plot” can equally be applied to truncated Poisson situations. It provides a type of robustness for detecting isolated discrepancies in otherwise well-behaved frequency distributions.