An autoregressive process with geometric α-laplace marginals

Geometric Laplace and geometric α-Laplace distributions are studied and certain limit properties are derived. An autoregressive process with geometric Laplace stationary marginal distribution is introduced and its properties are studied. These results are generalized to geometric α-Laplace case also and applications are discussed.     

A bivariate generalized geometric distribution with applications

This paper proposes a bivariate version of the univariate discrete generalized geometric distribution considered by Gómez–Déniz (2010 Gómez–Déniz, E. (2010). Another generalization of the geometric distribution. Test 19:399–415.[Crossref], [Web of Science ®] , [Google Scholar]). The proposed bivariate distribution can have a positive or negative correlation coefficient which can be used for modeling bivariate-dependent count data. After discussing some of its properties, maximum likelihood estimation is discussed. Two illustrative examples are given for fitting and demonstrating the usefulness of the new bivariate distribution proposed here.     

Uniform-Geometric distribution

In this paper, a new discrete distribution called Uniform-Geometric distribution is proposed. Several distributional properties including survival function, moments, skewness, kurtosis, entropy and hazard rate function are discussed. Estimation of distribution parameter is studied by methods of moments, proportions and maximum likelihood. A simulation study is performed to compare the performance of the different estimates in terms of bias and mean square error. Two real data applications are also presented to see that new distribution is useful in modelling data.     

Truncation invariant dependence structures

In this paper, a special class of m-dimensional distribution functions which can be uniquely determined in terms of their 2-dimensional marginals is studied. The members of the class can be characterized as having truncation invariant dependence structure. The representation given in this paper provides a physical meaning to the multivariate Cook-Johnson distribution, and introduces a systematic way of generating higher dimensional distributions by using rich 2-dimensional distributions provided that the 2-dimensional marginals are compatible. A class of 3-dimensional multivariate normal distribution has been generated and bounds in terms of lower dimensional marginals are provided.     

Discrete Weibull geometric distribution and its properties

In this article, the discrete analog of Weibull geometric distribution is introduced. Discrete Weibull, discrete Rayleigh, and geometric distributions are submodels of this distribution. Some basic distributional properties, hazard function, random number generation, moments, and order statistics of this new discrete distribution are studied. Estimation of the parameters are done using maximum likelihood method. The applications of the distribution is established using two datasets.     

On the distribution of the estimated mean from nonstandard mixtures of distributions

The distribution of the estimated mean of the nonstandard mixture of distributions that has a discrete probability mass at zero and a gamma distribution for positive values is derived. Furthermore, for the studied nonstandard mixture of distributions, the distribution of the standardized statistic (estimator - true mean)/standard deviation of estimator is derived. The results are used to study the accuracy of the confidence interval for the mean based on a large sample approximation. Quantiles for the standardized statistic are also calculated.     

On zero-distorted generalized geometric distribution

ABSTRACT

We propose a new generalized geometric distribution which permits inflation/deflation of the zero count probability and study some of its properties. We also present an actuarial application of this distribution and fit it to three datasets used by other researchers. It is observed that the proposed distribution fits reasonably well to these data. Further, in a regression setup, the performance of this distribution is studied vis–a–vis other competing distributions used for explaining variability in a response variable.     

An approximation to the distribution of the product of two dependent correlation coefficients

Many methodological studies depend on the product of two dependent correlation coefficients. However, the behavior of the distribution of the product of two dependent correlation coefficients is not well known. The distribution of sets of correlation coefficients has been well studied, but not the distribution of the product of two dependent correlation coefficients. The present study derives an approximation to the distribution of the product of two dependent correlation coefficients with a closed form, resulting in a Pearson Type I distribution. A simulation study is also conducted to assess the accuracy of the approximation.     

A lifetime distribution motivated by parallel and series structures

A new three-parameter distribution with decreasing, increasing, and bathtub-shaped hazard rates obtained by compounding geometric, power series, and exponential distributions is introduced. It includes some well-known distributions as particular cases. Various mathematical properties of the new distribution as well as details of the maximum likelihood estimation and a sensitivity analysis for its parameters are presented. Finally, two real data applications are presented.     

Characterizations of Bivariate and Multivariate Life Distributions Based on Reciprocal Subtangent

The subtangent is the projection of the tangent upon the axis of abscissa. The usefulness of the reciprocal subtangent as a measure of the survival and density curves has earlier been reported in the literature for univariate distributions. This measure was generalized for bivariate and multivariate setups and related characterization problems were examined. The conditionally specified bivariate exponential distribution has been uniquely determined from the local constancy of the bivariate reciprocal subtangents. The case of global constancy and other related results have been studied.

Conditionally specified bivariate Lomax distribution and normal distribution were also studied. Further, the conditionally specified multivariate exponential distribution was uniquely determined from the local constancy of the multivariate reciprocal subtangents.     

Generalized Poisson–Lindley Distribution

An extended version of the compound Poisson distribution is obtained by compounding the Poisson distribution with the generalized Lindley distribution. Estimation of the parameters is discussed using the method of moments and maximum likelihood estimators. Examples are given of the fitting of this distribution to data, and the fit is compared with that obtained using other distributions.     

On a Bivariate Pólya-Aeppli Distribution

In this article, we use the bivariate Poisson distribution obtained by the trivariate reduction method and compound it with a geometric distribution to derive a bivariate Pólya-Aeppli distribution. We then discuss a number of properties of this distribution including the probability generating function, correlation structure, probability mass function, recursive relations, and conditional distributions. The generating function of the tail probabilities is also obtained. Moment estimation of the parameters is then discussed and illustrated with a numerical example.     

Survival functions for the frailty models based on the discrete compound Poisson process

Frailty models are often used to model heterogeneity in survival analysis. The distribution of the frailty is generally assumed to be continuous. In some circumstances, it is appropriate to consider discrete frailty distributions. Having zero frailty can be interpreted as being immune, and population heterogeneity may be analysed using discrete frailty models. In this paper, survival functions are derived for the frailty models based on the discrete compound Poisson process. Maximum likelihood estimation procedures for the parameters are studied. We examine the fit of the models to earthquake and the traffic accidents’ data sets from Turkey.     

The unit extended Weibull families of distributions and its applications

In this paper, two new general families of distributions supported on the unit interval are introduced. The proposed families include several known models as special cases and define at least twenty (each one) new special models. Since the list of well-being indicators may include several double bounded random variables, the applicability for modeling those is the major practical motivation for introducing the distributions on those families. We propose a parametrization of the new families in terms of the median and develop a shiny application to provide interactive density shape illustrations for some special cases. Various properties of the introduced families are studied. Some special models in the new families are discussed. In particular, the complementary unit Weibull distribution is studied in some detail. The method of maximum likelihood for estimating the model parameters is discussed. An extensive Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples. Applications to the literacy rate in Brazilian and Colombian municipalities illustrate the usefulness of the two new families for modeling well-being indicators.     

Gompertz-power series distributions

ABSTRACT

In this article, we introduce the Gompertz power series (GPS) class of distributions which is obtained by compounding Gompertz and power series distributions. This distribution contains several lifetime models such as Gompertz-geometric (GG), Gompertz-Poisson (GP), Gompertz-binomial (GB), and Gompertz-logarithmic (GL) distributions as special cases. Sub-models of the GPS distribution are studied in details. The hazard rate function of the GPS distribution can be increasing, decreasing, and bathtub-shaped. We obtain several properties of the GPS distribution such as its probability density function, and failure rate function, Shannon entropy, mean residual life function, quantiles, and moments. The maximum likelihood estimation procedure via a EM-algorithm is presented, and simulation studies are performed for evaluation of this estimation for complete data, and the MLE of parameters for censored data. At the end, a real example is given.     

Time series models associated with Mittag-Leffler type distributions and its properties

ABSTRACT

We have provided a fractional generalization of the Poisson renewal processes by replacing the first time derivative in the relaxation equation of the survival probability by a fractional derivative of order α(0 < α ? 1). A generalized Laplacian model associated with the Mittag-Leffler distribution is examined. We also discuss some properties of this new model and its relevance to time series. Distribution of gliding sums, regression behaviors, and sample path properties are studied. Finally we introduce the q-Mittag-Leffler process associated with the q-Mittag-Leffler distribution.     

A new generalized two-sided class of distributions with an emphasis on two-sided generalized normal distribution

The ordinary-G class of distributions is defined to have the cumulative distribution function (cdf) as the value of the cdf of the ordinary distribution F whose range is the unit interval at G, that is, F(G), and it generalizes the ordinary distribution. In this work, we consider the standard two-sided power distribution to define other classes like the beta-G and the Kumaraswamy-G classes. We extend the idea of two-sidedness to other ordinary distributions like normal. After studying the basic properties of the new class in general setting, we consider the two-sided generalized normal distribution with maximum likelihood estimation procedure.     

The role of weighted distributions in stochastic modeling

C. R. Rao pointed out that “The role of statistical methodology is to extract the relevant information from a given sample to answer specific questions about the parent population” and raised the question “What population does a sample represent”? Wrong specification can lead to invalid inference giving rise to a third kind of error. Rao introduced the concept of weighted distributions as a method of adjustment applicable to many situations.

In this paper, we study the relationship between the weighted distributions and the parent distributions in the context of reliability and life testing. These relationships depend on the nature of the weight function and give rise to interesting connections between the different ageing criteria of the two distributions. As special cases, the length biased distribution, the equilibrium distribution of the backward and forward recurrence times and the residual life distribution, which frequently arise in practice, are studied and their relationships with the original distribution are examined. Their survival functions, failure rates and mean residual life functions are compared and some characterization results are established.     

Statistical analysis of the Lomax–Logarithmic distribution

In this paper we introduce a three-parameter lifetime distribution following the Marshall and Olkin [New method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika. 1997;84(3):641–652] approach. The proposed distribution is a compound of the Lomax and Logarithmic distributions (LLD). We provide a comprehensive study of the mathematical properties of the LLD. In particular, the density function, the shape of the hazard rate function, a general expansion for moments, the density of the rth order statistics, and the mean and median deviations of the LLD are derived and studied in detail. The maximum likelihood estimators of the three unknown parameters of LLD are obtained. The asymptotic confidence intervals for the parameters are also obtained based on asymptotic variance–covariance matrix. Finally, a real data set is analysed to show the potential of the new proposed distribution.