• Communications for Statistical Applications and Methods
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• v.21 no.2
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• pp.147-160
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• 2014

In this paper, we introduce the exponentiated Weibull-geometric (EWG) distribution which generalizes two-parameter exponentiated Weibull (EW) distribution introduced by Mudholkar et al. (1995). This proposed distribution is obtained by compounding the exponentiated Weibull with geometric distribution. We derive its cumulative distribution function (CDF), hazard function and the density of the order statistics and calculate expressions for its moments and the moments of the order statistics. The hazard function of the EWG distribution can be decreasing, increasing or bathtub-shaped among others. Also, we give expressions for the Renyi and Shannon entropies. The maximum likelihood estimation is obtained by using EM-algorithm (Dempster et al., 1977; McLachlan and Krishnan, 1997). We can obtain the Bayesian estimation by using Gibbs sampler with Metropolis-Hastings algorithm. Also, we give application with real data set to show the flexibility of the EWG distribution. Finally, summary and discussion are mentioned.

• International Journal of Reliability and Applications
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• v.9 no.1
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• pp.31-52
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• 2008

In this paper generalized version of the geometric distribution is introduced. This distribution can be considered as a two-parameter generalization of the discrete geometric distribution. The main statistical and reliability properties of this distribution are discussed. Two methods of estimation, namely maximum likelihood method and the method of moments are used to estimate the parameters of this distribution. Simulation is utilized to calculate these estimates and to study some of their properties. Also, asymptotic confidence limits are established for the maximum likelihood estimates. Finally, the appropriateness of this new distribution for a set of real data, compared with the geometric distribution, is shown by using the likelihood ratio test and the Kolmogorove-Smirnove test.

• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.859-870
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• 2008

In the present paper, we investigate the geometric structures of the Dirichlet manifold composed of the Dirichlet distribution. We show that the Dirichlet distribution is an exponential family distribution. We consider its dual structures and give its geometric metrics, and obtain the geometric structures of the lower dimension cases of the Dirichlet manifold. In particularly, the Beta distribution is a 2-dimensional Dirich-let distribution. Also, we construct an affine immersion of the Dirichlet manifold. At last, we give the e-flat hierarchical structures and the orthogonal foliations of the Dirichlet manifold. All these work will enrich the theoretical work of the Dirichlet distribution and will be great help for its further applications.

• Journal of the Korean Data and Information Science Society
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• v.8 no.2
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• pp.135-141
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• 1997

In this paper, it is investigated the properties of the transformed geometric Poisson process when the intensity function of the process is a distribution of the continuous random variable. If the intensity function of the transformed geometric Poisson process is a Pareto distribution then the transformed geometric Poisson process is a strongly P-process.

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1419-1443
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• 2021

The asymmetric Laplace distribution arises as a limiting distribution of geometric summations of independent and identically distributed random variables with finite second moments. The main purpose of this paper is to study the weak limit theorems for geometric summations of independent (not necessarily identically distributed) random variables together with convergence rates to asymmetric Laplace distributions. Using Trotter-operator method, the orders of approximations of the distributions of geometric summations by the asymmetric Laplace distributions are established in term of the "large-𝒪" and "small-o" approximation estimates. The obtained results are extensions of some known ones.

• Journal of applied mathematics & informatics
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• v.42 no.3
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• pp.709-723
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• 2024

We study the distributions of waiting times in variations of the geometric distribution of order k. Variation imposes length on the runs of successes and failures. We study two types of waiting time random variables. First, we consider the waiting time for a run of k consecutive successes the first time no sequence of consecutive k failures occurs prior, denoted by T^(k). Next, we consider the waiting time for a run of k consecutive failures the first time no sequence of k consecutive successes occurred prior, denoted by J^(k). In addition, we study the distribution of the weighted average. The exact formulae of the probability mass function, mean, and variance of distributions are also obtained.

• Journal of the Korean Data and Information Science Society
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• v.9 no.2
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• pp.305-310
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• 1998

In this paper, we introduce the conditions that the P-process has the intensity function which it is a standard form of gamma distribution. And we show that the transformed geometric Poisson process which the intensity function is a standard form of gamma distribution is a alternative sign P-process

• Communications for Statistical Applications and Methods
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• v.24 no.4
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• pp.397-419
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• 2017

A four-parameter extended fatigue lifetime model called the odd Birnbaum-Saunders geometric distribution is proposed. This model extends the odd Birnbaum-Saunders and Birnbaum-Saunders distributions. We derive some properties of the new distribution that include expressions for the ordinary moments and generating and quantile functions. The method of maximum likelihood and a Bayesian approach are adopted to estimate the model parameters; in addition, various simulations are performed for different parameter settings and sample sizes. We propose two new models with a cure rate called the odd Birnbaum-Saunders mixture and odd Birnbaum-Saunders geometric models by assuming that the number of competing causes for the event of interest has a geometric distribution. The applicability of the new models are illustrated by means of ethylene data and melanoma data with cure fraction.

• Korean Journal of Remote Sensing
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• v.19 no.3
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• pp.247-253
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• 2003

The mainly used technique to correct satellite images with geometric distortion is to develop a mathematical relationship between pixels on the image and corresponding points on the ground. Polynomial models with various transformations have been designed for defining the relationship between two coordinate systems. GCP based geometric correction has peformed overall plane to plane mapping. In the overall plane mapping, overall structure of a scene is considered, but local variation is discarded. The Region with highly variant height is rectified with distortion on overall plane mapping. To consider locally variable region in satellite image, TIN-based rectification on a satellite image is proposed in this paper. This paper describes the relationship between GCP distribution and rectification model through experimental result and analysis about each rectification model. We can choose a geometric correction model as the structural characteristic of a satellite image and the acquired GCP distribution.

• Structural Engineering and Mechanics
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• v.88 no.5
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• pp.405-417
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• 2023

This paper reveals theoretical research to the nonlinear dynamic response and initial geometric imperfections sensitivity of the spinning graphene platelets reinforced metal foams (GPLRMF) cylindrical shell under different boundary conditions in thermal environment. For the theoretical research, with the framework of von-Karman geometric nonlinearity, the GPLRMF cylindrical shell model which involves Coriolis acceleration and centrifugal acceleration caused by spinning motion is assumed to undergo large deformations. The coupled governing equations of motion are deduced using Euler-Lagrange principle and then solved by a combination of Galerkin's technique and modified Lindstedt Poincare (MLP) model. Furthermore, the impacts of a set of parameters including spinning velocity, initial geometric imperfections, temperature variation, weight fraction of GPLs, GPLs distribution pattern, porosity distribution pattern, porosity coefficient and external excitation amplitude on the nonlinear harmonic resonances of the spinning GPLRMF cylindrical shells are presented.