A characterization of the geometric distribution

We show that for a simple random sample from a discrete distribution on the positive integers, the regression ofX _(2∶n) onX _(1∶n) is linear with unit slope if and only if the distribution is geometric.     

Some characterizations of discrete distributions based on weak records

Let X ₁, X ₂,... be iid random variables (rv's) with the support on nonnegative integers and let (W _n , n≥0) denote the corresponding sequence of weak record values. We obtain new characterization of geometric and some other discrete distributions based on different forms of partial independence of rv's W _n and W _n+r —W _n for some fixed n≥0 and r≥1. We also prove that rv's W ₀ and W _n+1 —W _n have identical distribution if and only if (iff) the underlying distribution is geometric.     

Stability for the Arnold-Ghosh characterization of the geometric distribution

Stability of the characterization of the geometric law by equidistribution of the spacing of two i.i.d. r.v.'s and one of them is studied.     

On simulation of weak records

In the present paper, we discuss algorithms of generation of weak records. These generation algorithms are based on two different methods. In the case, when the inverse function for the underlying distribution function can be obtained explicitly, the corresponding generation algorithms are built on the inverse-transform method. In the case, when the inverse function cannot be obtained explicitly, the algorithms are based on the rejection method. Generation algorithms of our paper are supplied with illustrative examples.     

A note on characterizations of the geometric distribution

Well-known characterizations of the geometric distribution via the independence of some contrast and the minimum in a sample of i.i.d. random variables are illustrated and supplemented.     

Weighted geometric distribution with new characterizations of geometric distribution

In this article, we derive a new generalized geometric distribution through a weight function, which can also be viewed as a discrete analog of weighted exponential distribution introduced by Gupta and Kundu (2009 Gupta, R. D., and D. Kundu. 2009. A new class of weighted exponential distributions. Statistics 43:621–34.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). We derive some distributional properties like moments, generating functions, hazard function, and infinite divisibility followed by different estimation methods to estimate the parameters. New characterizations of the geometric distribution are presented using the proposed generalized geometric distribution. The superiority of the proposed distribution to other competing models is demonstrated with the help of two real count datasets.     

Bias corrected MLEs for the Weibull distribution based on records

The maximum likelihood estimators of the Weibull distribution based on upper records are biased. Exact expressions are derived for constructing bias corrected MLEs. The performance of the bias corrected MLEs is compared with the MLEs by simulations and real data sets.     

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.     

On the problem of the unique characterization of discrete distributions by single regression of weak records

We consider the problem of the unique identification of discrete probability distributions by the single regression function of non-adjacent discrete weak record values. We present a new approach to this problem and we show that the uniqueness of the characterization is equivalent to the uniqueness of solution to a corresponding difference equation or an appropriate system of difference equations. This result is applied to obtain known as well as new characterizations of discrete distributions.     

A bivariate distribution with Lomax and geometric margins

We develop a stochastic model describing the joint distribution of $(X,N)$, where $N$ has a geometric distribution while $X$ is the sum of $N$ dependent, heavy-tail Pareto components. Models of this form arise in many applications, ranging from hydro-climatology to finance and insurance. We present fundamental properties of this vector, including marginal and conditional distributions, moments, representations, and parameter estimation. We also include an example from finance, illustrating modeling potential of this new bivariate distribution.     

Multivariate geometric skew-normal distribution

Azzalini [A class of distributions which include the normal. Scand J Stat. 1985;12:171–178] introduced a skew-normal distribution of which normal distribution is a special case. Recently, Kundu [Geometric skew normal distribution. Sankhya Ser B. 2014;76:167–189] introduced a geometric skew-normal distribution and showed that it has certain advantages over Azzalini's skew-normal distribution. In this paper we discuss about the multivariate geometric skew-normal (MGSN) distribution. It can be used as an alternative to Azzalini's skew-normal distribution. We discuss different properties of the proposed distribution. It is observed that the joint probability density function of the MGSN distribution can take a variety of shapes. Several characterization results have been established. Generation from an MGSN distribution is quite simple, hence the simulation experiments can be performed quite easily. The maximum likelihood estimators of the unknown parameters can be obtained quite conveniently using the expectation–maximization (EM) algorithm. We perform some simulation experiments and it is observed that the performances of the proposed EM algorithm are quite satisfactory. Furthermore, the analyses of two data sets have been performed, and it is observed that the proposed methods and the model work very well.     

The beta-Weibull geometric distribution

We propose a new distribution, the so-called beta-Weibull geometric distribution, whose failure rate function can be decreasing, increasing or an upside-down bathtub. This distribution contains special sub-models the exponential geometric [K. Adamidis and S. Loukas, A lifetime distribution with decreasing failure rate, Statist. Probab. Lett. 39 (1998), pp. 35–42], beta exponential [S. Nadarajah and S. Kotz, The exponentiated type distributions, Acta Appl. Math. 92 (2006), pp. 97–111; The beta exponential distribution, Reliab. Eng. Syst. Saf. 91 (2006), pp. 689–697], Weibull geometric [W. Barreto-Souza, A.L. de Morais, and G.M. Cordeiro, The Weibull-geometric distribution, J. Stat. Comput. Simul. 81 (2011), pp. 645–657], generalized exponential geometric [R.B. Silva, W. Barreto-Souza, and G.M. Cordeiro, A new distribution with decreasing, increasing and upside-down bathtub failure rate, Comput. Statist. Data Anal. 54 (2010), pp. 935–944; G.O. Silva, E.M.M. Ortega, and G.M. Cordeiro, The beta modified Weibull distribution, Lifetime Data Anal. 16 (2010), pp. 409–430] and beta Weibull [S. Nadarajah, G.M. Cordeiro, and E.M.M. Ortega, General results for the Kumaraswamy-G distribution, J. Stat. Comput. Simul. (2011). DOI: 10.1080/00949655.2011.562504] distributions, among others. The density function can be expressed as a mixture of Weibull density functions. We derive expansions for the moments, generating function, mean deviations and Rénvy entropy. The parameters of the proposed model are estimated by maximum likelihood. The model fitting using envelops was conducted. The proposed distribution gives a good fit to the ozone level data in New York.     

Selecting the best geometric distribution based on type-I censored data: a Bayesian approach

This paper considers the statistical reliability on discrete failure data and the selection of the best geometric distribution having the smallest failure probability from among several competitors. Using the Bayesian approach a Bayes selection rule based on type-I censored data is derived and its associated monotonicity is also obtained. An early selection rule which allows us to make a selection possible earlier than the censoring time of the life testing experiment is proposed. This early selection rule can be shown to be equivalent to the Bayes selection rule. An illustrative example is given to demonstrate the use and the performance of the early selection rule.     

A mixed bivariate distribution with exponential and geometric marginals

We study the joint distribution of X and N, where N has a geometric distribution and X is the sum of N i.i.d. exponential variables, independent of N. We present basic properties of this class of mixed bivariate distributions, and discuss their possible applications. Our results include marginal and conditional distributions, joint integral transforms, infinite divisibility, and stability with respect to geometric summation. We also discuss maximum likelihood estimation connected with this distribution. An example from finance, where N represents the number of consecutive positive daily log-returns of currency exchange rates, illustrates the modeling potential of these laws.