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Debasis Kundu - One of the best experts on this subject based on the ideXlab platform.
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a new method for generating Distributions with an application to Exponential Distribution
Communications in Statistics-theory and Methods, 2017Co-Authors: Abbas Mahdavi, Debasis KunduAbstract:ABSTRACTA new method has been proposed to introduce an extra parameter to a family of Distributions for more flexibility. A special case has been considered in detail, namely one-parameter Exponential Distribution. Various properties of the proposed Distribution, including explicit expressions for the moments, quantiles, mode, moment-generating function, mean residual lifetime, stochastic orders, order statistics, and expression of the entropies, are derived. The maximum likelihood estimators of unknown parameters cannot be obtained in explicit forms, and they have to be obtained by solving non linear equations only. Further, we consider an extension of the two-parameter Exponential Distribution also, mainly for data analysis purposes. Two datasets have been analyzed to show how the proposed models work in practice.
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bivariate discrete generalized Exponential Distribution
Statistics, 2017Co-Authors: Vahid Nekoukhou, Debasis KunduAbstract:ABSTRACTIn this paper, we develop a bivariate discrete generalized Exponential Distribution, whose marginals are discrete generalized Exponential Distribution as proposed by Nekoukhou, Alamatsaz and Bidram [Discrete generalized Exponential Distribution of a second type. Statistics. 2013;47:876–887]. It is observed that the proposed bivariate Distribution is a very flexible Distribution and the bivariate geometric Distribution can be obtained as a special case of this Distribution. The proposed Distribution can be seen as a natural discrete analogue of the bivariate generalized Exponential Distribution proposed by Kundu and Gupta [Bivariate generalized Exponential Distribution. J Multivariate Anal. 2009;100:581–593]. We study different properties of this Distribution and explore its dependence structures. We propose a new EM algorithm to compute the maximum-likelihood estimators of the unknown parameters which can be implemented very efficiently, and discuss some inferential issues also. The analysis of on...
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bivariate discrete generalized Exponential Distribution
arXiv: Methodology, 2017Co-Authors: Vahid Nekoukhou, Debasis KunduAbstract:In this paper we develop a bivariate discrete generalized Exponential Distribution, whose marginals are discrete generalized Exponential Distribution as proposed by Nekoukhou, Alamatsaz and Bidram ("Discrete generalized Exponential Distribution of a second type", Statistics, 47, 876 - 887, 2013). It is observed that the proposed bivariate Distribution is a very flexible Distribution and the bivariate geometric Distribution can be obtained as a special case of this Distribution. The proposed Distribution can be seen as a natural discrete analogue of the bivariate generalized Exponential Distribution proposed by Kundu and Gupta ("Bivariate generalized Exponential Distribution", Journal of Multivariate Analysis, 100, 581 - 593, 2009). We study different properties of this Distribution and explore its dependence structures. We propose a new EM algorithm to compute the maximum likelihood estimators of the unknown parameters which can be implemented very efficiently, and discuss some inferential issues also. The analysis of one data set has been performed to show the effectiveness of the proposed model. Finally we propose some open problems and conclude the paper.
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Marshall-Olkin generalized Exponential Distribution
METRON, 2015Co-Authors: Miroslav M. Ristić, Debasis KunduAbstract:Marshall and Olkin (Biometrika 641–652, 1997 ) introduced a new way of incorporating a parameter to expand a family of Distributions. In this paper we adopt the Marshall-Olkin approach to introduce an extra shape parameter to the two-parameter generalized Exponential Distribution. It is observed that the new three-parameter Distribution is very flexible. The probability density functions can be either a decreasing or an unimodal function. The hazard function of the proposed model, can have all the four major shapes, namely increasing, decreasing, bathtub or inverted bathtub types. Different properties of the proposed Distribution have been established. The new family of Distributions is analytically quite tractable, and it can be used quite effectively, to analyze censored data also. Maximum likelihood method is used to compute the estimators of the unknown parameters. Two data sets have been analyzed, and the results are quite satisfactory.
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absolute continuous bivariate generalized Exponential Distribution
AStA Advances in Statistical Analysis, 2011Co-Authors: Debasis Kundu, Rameshwar D GuptaAbstract:Generalized Exponential Distribution has been used quite effectively to model positively skewed lifetime data as an alternative to the well known Weibull or gamma Distributions. In this paper we introduce an absolute continuous bivariate generalized Exponential Distribution by using a simple transformation from a well known bivariate exchangeable Distribution. The marginal Distributions of the proposed bivariate generalized Exponential Distributions are generalized Exponential Distributions. The joint probability density function and the joint cumulative Distribution function can be expressed in closed forms. It is observed that the proposed bivariate Distribution can be obtained using Clayton copula with generalized Exponential Distribution as marginals. We derive different properties of this new Distribution. It is a five-parameter Distribution, and the maximum likelihood estimators of the unknown parameters cannot be obtained in closed forms. We propose some alternative estimators, which can be obtained quite easily, and they can be used as initial guesses to compute the maximum likelihood estimates. One data set has been analyzed for illustrative purposes. Finally we propose some generalization of the proposed model.
Saralees Nadarajah - One of the best experts on this subject based on the ideXlab platform.
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extended Exponential Distribution based on order statistics
Communications in Statistics-theory and Methods, 2017Co-Authors: Devendra Kumar, Sanku Dey, Saralees NadarajahAbstract:AbstractThe extended Exponential Distribution due to Nadarajah and Haghighi (2011) is an alternative to and always provides better fits than the gamma, Weibull and the exponentiated Exponential Distributions whenever the data contains zero values. We establish recurrence relations for the single and product moments of order statistics from the extended Exponential Distribution. These recurrence relations enable computation of the means, variances and covariances of all order statistics for all sample sizes in a simple and efficient manner. By using these relations, we tabulate the means, variances and covariances of order statistics and derive best linear unbiased estimates of the extended Exponential Distribution. Finally, a data application is provided.
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an extension of the Exponential Distribution
Statistics, 2011Co-Authors: Saralees Nadarajah, Firoozeh HaghighiAbstract:A generalization of the Exponential Distribution is presented. The generalization always has its mode at zero and yet allows for increasing, decreasing and constant hazard rates. It can be used as an alternative to the gamma, Weibull and exponentiated Exponential Distributions. A comprehensive account of the mathematical properties of the generalization is presented. A real data example is discussed to illustrate its applicability.
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The exponentiated Exponential Distribution: a survey
AStA Advances in Statistical Analysis, 2011Co-Authors: Saralees NadarajahAbstract:The exponentiated Exponential Distribution, a most attractive generalization of the Exponential Distribution, introduced by Gupta and Kundu (Aust. N. Z. J. Stat. 41:173–188, 1999) has received widespread attention. It appears, however, that many mathematical properties of this Distribution have not been known or have not been known in simpler/general forms. In this paper, we provide a comprehensive survey of the mathematical properties. We derive expressions for the moment generating function, characteristic function, cumulant generating function, the nth moment, the first four moments, variance, skewness, kurtosis, the nth conditional moment, the first four cumulants, mean deviation about the mean, mean deviation about the median, Bonferroni curve, Lorenz curve, Bonferroni concentration index, Gini concentration index, Renyi entropy, Shannon entropy, cumulative residual entropy, Song’s measure, moments of order statistics, L moments, asymptotic Distribution of the extreme order statistics, reliability, Distribution of the sum of exponentiated Exponential random variables, Distribution of the product of exponentiated Exponential random variables and the Distribution of the ratio of exponentiated Exponential random variables. We also discuss estimation by the method of maximum likelihood, including the case of censoring, and provide simpler expressions for the Fisher information matrix than those given by Gupta and Kundu. It is expected that this paper could serve as a source of reference for the exponentiated Exponential Distribution and encourage further research.
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the beta Exponential Distribution
Reliability Engineering & System Safety, 2006Co-Authors: Saralees Nadarajah, Samuel KotzAbstract:Abstract The Exponential Distribution is perhaps the most widely applied statistical Distribution for problems in reliability. In this note, we introduce a generalization—referred to as the beta Exponential Distribution—generated from the logit of a beta random variable. We provide a comprehensive treatment of the mathematical properties of the beta Exponential Distribution. We derive expressions for the moment generating function, characteristic function, the first four moments, variance, skewness, kurtosis, mean deviation about the mean, mean deviation about the median, Renyi entropy, Shannon entropy, the Distribution of sums and ratios, and the asymptotic Distribution of the extreme order statistics. We also discuss simulation issues, estimation by the methods of moments and maximum likelihood and provide an expression for the Fisher information matrix. We hope that this generalization will attract wider applicability in reliability.
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friday and patil s bivariate Exponential Distribution with application to drought data
Water Resources Management, 2006Co-Authors: Saralees Nadarajah, Arjun K. GuptaAbstract:Motivated by hydrological applications, we derive the exact Distributions of R = X + Y, P = XY and W = X/(X+Y) and the corresponding moment properties when X and Y follow Friday and Patil's bivariate Exponential Distribution. An application of the results is provided to drought data from Nebraska.
Biswabrata Pradhan - One of the best experts on this subject based on the ideXlab platform.
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generalized inverted Exponential Distribution under hybrid censoring
Statistical Methodology, 2014Co-Authors: Sanku Dey, Biswabrata PradhanAbstract:Abstract The hybrid censoring scheme is a mixture of Type-I and Type-II censoring schemes. Based on hybrid censored samples, we first derive the maximum likelihood estimators of the unknown parameters and the expected Fisher’s information matrix of the generalized inverted Exponential Distribution (GIED). Monte Carlo simulations are performed to study the performance of the maximum likelihood estimators. Next we consider Bayes estimation under the squared error loss function. These Bayes estimates are evaluated by applying Lindley’s approximation method, the importance sampling procedure and Metropolis–Hastings algorithm. The importance sampling technique is used to compute the highest posterior density credible intervals. Two data sets are analyzed for illustrative purposes. Finally, we discuss a method of obtaining the optimum hybrid censoring scheme.
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on progressively censored generalized Exponential Distribution
Test, 2009Co-Authors: Biswabrata Pradhan, Debasis KunduAbstract:In this paper, we consider the statistical inference of the unknown parameters of the generalized Exponential Distribution in presence of progressive censoring. We obtain maximum likelihood estimators of the unknown parameters using EM algorithm. We also compute the expected Fisher information matrix using the missing value principle. We then use these values to determine the optimal progressive censoring plans. Different optimality criteria are considered, and selected optimal progressive censoring plans are presented. One example has been provided for illustrative purposes.
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bayesian inference and life testing plans for generalized Exponential Distribution
Science China-mathematics, 2009Co-Authors: Debasis Kundu, Biswabrata PradhanAbstract:Recently generalized Exponential Distribution has received considerable attentions. In this paper, we deal with the Bayesian inference of the unknown parameters of the progressively censored generalized Exponential Distribution. It is assumed that the scale and the shape parameters have independent gamma priors. The Bayes estimates of the unknown parameters cannot be obtained in the closed form. Lindley’s approximation and importance sampling technique have been suggested to compute the approximate Bayes estimates. Markov Chain Monte Carlo method has been used to compute the approximate Bayes estimates and also to construct the highest posterior density credible intervals. We also provide different criteria to compare two different sampling schemes and hence to find the optimal sampling schemes. It is observed that finding the optimum censoring procedure is a computationally expensive process. And we have recommended to use the sub-optimal censoring procedure, which can be obtained very easily. Monte Carlo simulations are performed to compare the performances of the different methods and one data analysis has been performed for illustrative purposes.
Muhammad Aslam - One of the best experts on this subject based on the ideXlab platform.
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design of sampling plan for Exponential Distribution under neutrosophic statistical interval method
IEEE Access, 2018Co-Authors: Muhammad AslamAbstract:The sampling plan using the classical statistics under the Exponential Distribution can be applied only when there are certainty and clearness and in observations and parameters. But, in practice, it is not necessary that under some circumstances all the observations/parameters are determined. So, we cannot analyze them using the classical statistics which provides results in the determined values. The neutrosophic statistics which is the generation of classical statistics can be applied to the analysis when parameters/observations are incomplete, indeterminate, and vague imprecise. In this paper, we will design originally a sampling plan for the Exponential Distribution under the neutrosophic interval statistical method. The neutrosophic plan parameters of the proposed plan will be determined through the neutrosophic non-linear problem. The tables for various values of risk are presented for the use in the industrial. An example from the automobiles manufacturing industry is given to explain for the Exponential Distribution under the neutrosophic interval statistical method.
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acceptance sampling plans based on truncated life tests for weighted Exponential Distribution
Communications in Statistics - Simulation and Computation, 2017Co-Authors: Muhammad AslamAbstract:ABSTRACTGupta and Kundu proposed a new class of weighted Exponential Distributions using the idea of Azzalini. In this article, we develop an acceptance sampling plan for the weighted Exponential Distribution under a truncated life test. For various acceptance numbers, consumer’s confidence levels and values of the ratio of the experimental time to the specified mean lifetime, the minimum sample size necessary to ensure a certain mean lifetime are obtained. The operating characteristic function values and the associated producer’s risks are also presented. A numerical example is provided to illustrate the acceptance sampling plan.
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a new sampling plan under the Exponential Distribution
Communications in Statistics-theory and Methods, 2017Co-Authors: Muhammad Aslam, Muhammad AzamAbstract:ABSTRACTThis article considers a sampling plan when the quality characteristic follows the Exponential Distribution. We provide the exact approach and propose an approximated approach. In the proposed approximation, a new statistic combined with a Weibull transformation is used for the normal approximation. The plan parameters are obtained through the two-point approach at the acceptable quality level (AQL) and the limiting quality level (LQL). The tables for plan parameters are reported according to various values of the AQL and the LQL when the producer's and the consumer's risks are given. A real example is given to illustrate the proposed approximation approach.
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designing of a control chart using belief statistic for Exponential Distribution
Communications in Statistics - Simulation and Computation, 2016Co-Authors: Muhammad Aslam, Nasrullah KhanAbstract:ABSTRACTA new control chart is proposed by using the belief statistic for the Exponential Distribution. The structure of the proposed control chart is given to measure the average run length for the shifted process. The comparison of the proposed chart is given with the existing charts in terms of the average run lengths, which shows the outperformance of the proposed chart. The performance of the proposed control chart is also discussed with the help of simulated data.
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bayesian analysis of the two component mixture of inverted Exponential Distribution under quadratic loss function
International Journal of Physical Sciences, 2012Co-Authors: Muhammad Younas Majeed, Muhammad AslamAbstract:The inverted Exponential Distribution is studied as a prospective life Distribution. A two component mixture of inverted Exponential Distribution is considered in this paper. The Bayes estimators and Bayes posterior risk for the unknown parameters, and mixing weight of the mixture model are derived under quadratic loss function. For comparative study of these Bayes estimates uniform, improper and informative priors are considered. The Bayes and maximum likelihood estimators and Bayes posterior risks are viewed as a function of the test termination time. As a special case, the limiting expressions for these estimates are derived under the condition of infinite test termination time. Finally, a mixture data is simulated and numerical study is given to illustrate the results. Key words: Inverted Exponential Distribution, mixture models, Bayes estimates, quadratic loss function, fixed test termination time.
Sanku Dey - One of the best experts on this subject based on the ideXlab platform.
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extended Exponential Distribution based on order statistics
Communications in Statistics-theory and Methods, 2017Co-Authors: Devendra Kumar, Sanku Dey, Saralees NadarajahAbstract:AbstractThe extended Exponential Distribution due to Nadarajah and Haghighi (2011) is an alternative to and always provides better fits than the gamma, Weibull and the exponentiated Exponential Distributions whenever the data contains zero values. We establish recurrence relations for the single and product moments of order statistics from the extended Exponential Distribution. These recurrence relations enable computation of the means, variances and covariances of all order statistics for all sample sizes in a simple and efficient manner. By using these relations, we tabulate the means, variances and covariances of order statistics and derive best linear unbiased estimates of the extended Exponential Distribution. Finally, a data application is provided.
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weighted Exponential Distribution properties and different methods of estimation
Journal of Statistical Computation and Simulation, 2015Co-Authors: Sanku Dey, Sajid Ali, Chanseok ParkAbstract:In this article, we investigate various properties and methods of estimation of the Weighted Exponential Distribution. Although, our main focus is on estimation (from both frequentist and Bayesian point of view) yet, the stochastic ordering, the Bonferroni and the Lorenz curves, various entropies and order statistics are derived first time for the said Distribution. Different types of loss functions are considered for Bayesian estimation. Furthermore, the Bayes estimators and their respective posterior risks are computed and compared using Gibbs sampling. The different reliability characteristics including hazard function, stress and strength analysis, and mean residual life function are also derived. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and two real data sets have been analysed for illustrative purposes.
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generalized inverted Exponential Distribution under hybrid censoring
Statistical Methodology, 2014Co-Authors: Sanku Dey, Biswabrata PradhanAbstract:Abstract The hybrid censoring scheme is a mixture of Type-I and Type-II censoring schemes. Based on hybrid censored samples, we first derive the maximum likelihood estimators of the unknown parameters and the expected Fisher’s information matrix of the generalized inverted Exponential Distribution (GIED). Monte Carlo simulations are performed to study the performance of the maximum likelihood estimators. Next we consider Bayes estimation under the squared error loss function. These Bayes estimates are evaluated by applying Lindley’s approximation method, the importance sampling procedure and Metropolis–Hastings algorithm. The importance sampling technique is used to compute the highest posterior density credible intervals. Two data sets are analyzed for illustrative purposes. Finally, we discuss a method of obtaining the optimum hybrid censoring scheme.
