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ABSTRACT: Recently the proportional reversed hazard model has received a considerable amount of attention in the statistical literature.
The main aim of this paper is to introduce a bivariate proportional reversed hazard model and discuss its different properties.
In most of the cases the joint probability distribution function can be expressed in compact forms. The maximum likelihood
estimators cannot be expressed in explicit forms in most of the cases. EM algorithm has been proposed to compute the maximum
likelihood estimators of the unknown parameters. For illustrative purposes two data sets have been analyzed and the performances
are quite satisfactory.
KeywordsJoint probability density function–Conditional probability density function–Maximum likelihood estimators–EM algorithm
04/2012; 72(2):236-253.
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ABSTRACT: 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.
KeywordsBivariate exchangeable distribution–Dependence properties–Clayton copula–Hazard rate–Maximum likelihood estimators–Pseudo generator
AStA Advances in Statistical Analysis 04/2012; 95(2):169-185. · 0.44 Impact Factor
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ABSTRACT: Recently, Gupta and Gupta [Analyzing skewed data by power-normal model, Test 17 (2008), pp. 197–210] proposed the power-normal distribution for which normal distribution is a special case. The power-normal distribution is a skewed distribution, whose support is the whole real line. Our main aim of this paper is to consider bivariate power-normal distribution, whose marginals are power-normal distributions. We obtain the proposed bivariate power-normal distribution from Clayton copula, and by making a suitable transformation in both the marginals. Lindley–Singpurwalla distribution also can be used to obtain the same distribution. Different properties of this new distribution have been investigated in detail. Two different estimators are proposed. One data analysis has been performed for illustrative purposes. Finally, we propose some generalizations to multivariate case also along the same line and discuss some of its properties.
Statistics. 05/2011; iFirst(2011).
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ABSTRACT: Introducing a shape parameter to an exponential model is nothing new. There are many ways to introduce a shape parameter to an exponential distribution. The different methods may result in variety of weighted exponential (WE) distributions. In this article, we have introduced a shape parameter to an exponential model using the idea of Azzalini, which results in a new class of WE distributions. This new WE model has the probability density function (PDF) whose shape is very close to the shape of the PDFS of Weibull, gamma or generalized exponential distributions. Therefore, this model can be used as an alternative to any of these distributions. It is observed that this model can also be obtained as a hidden truncation model. Different properties of this new model have been discussed and compared with the corresponding properties of well-known distributions. Two data sets have been analysed for illustrative purposes and it is observed that in both the cases it fits better than Weibull, gamma or generalized exponential distributions.
Statistics. 12/2009; 43(6):621-634.
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ABSTRACT: Recently it has been observed that the generalized exponential distribution can be used quite effectively to analyze lifetime data in one dimension. The main aim of this paper is to define a bivariate generalized exponential distribution so that the marginals have generalized exponential distributions. It is observed that the joint probability density function, the joint cumulative distribution function and the joint survival distribution function can be expressed in compact forms. Several properties of this distribution have been discussed. We suggest to use the EM algorithm to compute the maximum likelihood estimators of the unknown parameters and also obtain the observed and expected Fisher information matrices. One data set has been re-analyzed and it is observed that the bivariate generalized exponential distribution provides a better fit than the bivariate exponential distribution.
Journal of Multivariate Analysis 01/2009; 100(4):581-593. · 0.88 Impact Factor
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ABSTRACT: Recently two-parameter generalized exponential distribution has been introduced by the authors. In this paper we consider the Bayes estimators of the unknown parameters under the assumptions of gamma priors on both the shape and scale parameters. The Bayes estimators cannot be obtained in explicit forms. Approximate Bayes estimators are computed using the idea of Lindley. We also propose Gibbs sampling procedure to generate samples from the posterior distributions and in turn computing the Bayes estimators. The approximate Bayes estimators obtained under the assumptions of non-informative priors, are compared with the maximum likelihood estimators using Monte Carlo simulations. One real data set has been analyzed for illustrative purposes.
Computational Statistics & Data Analysis 02/2008; · 1.03 Impact Factor
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ABSTRACT: In this paper we propose a very convenient way to generate gamma random variables using generalized exponential distribution, when the shape parameter lies between 0 and 1. The new method is compared with the most popular Ahrens and Dieter method and the method proposed by Best. Like Ahrens and Dieter and Best methods our method also uses the acceptance–rejection principle. But it is observed that our method has greater acceptance proportion than Ahrens and Dieter or Best methods.
Computational Statistics & Data Analysis 02/2007; · 1.03 Impact Factor
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IEEE Transactions on Reliability. 01/2006; 55:270-280.
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Metrika 02/2005; 61(3):291-308. · 0.67 Impact Factor
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ABSTRACT: The two-parameter generalized exponential distribution was recently introduced by Gupta and Kundu (Austral. New Zealand J. Statist. 40 (1999) 173). It is observed that the Generalized Exponential distribution can be used quite eeectively to analyze skewed data set as an alternative to the more popular log-normal distribution. In this paper, we use the ratio of the maximized likelihoods in choosing between the log-normal and generalized exponential distributions. We obtain asymptotic distributions of the logarithm of the ratio of the maximized likelihoods and use them to determine the required sample size to discriminate between the two distributions for a user speciÿed probability of correct selection and tolerance limit. c 2003 Published by Elsevier B.V.
Journal of Statistical Planning and Inference 01/2005; 127:213-227. · 0.72 Impact Factor
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ABSTRACT: Recently the two-parameter generalized exponential distribution was introduced by the authors. It is observed that a generalized exponential distribution has several properties which are quite similar to a gamma distribution. It is also observed that a generalized exponential distribution can be used quite effectively in many situations where a skewed distribution is needed. In this paper, we use the ratio of the maximized likelihoods in choosing between a generalized exponential distribution and a gamma distribution. We obtain asymptotic distributions of the logarithm of the ratio of the maximized likelihoods under null hypotheses and use them to determine the sample size needed to discriminate between two overlapping families of distributions for a user specified probability of correct selection and a tolerance limit.
Journal of Statistical Computation and Simulation 03/2004; 74:107-121. · 0.50 Impact Factor
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Communication in Statistics- Theory and Methods 01/2004; No. 12(pp. 3095–3102):3095-3102. · 0.27 Impact Factor
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ABSTRACT: Recently the two-parameter generalized exponential (GE) distribution was introduced by the authors. It is observed that a GE distribution can be considered for situations where a skewed distribution for a non-negative random variable is needed. The ratio of the maximized likelihoods (RML) is used in discriminating between Weibull and GE distributions. Asymptotic distributions of the logarithm of the RML under null hypotheses are obtained and they are used to determine the minimum sample size required in discriminating between two overlapping families of distributions for a user specified probability of correct selection and tolerance limit.
Computational Statistics & Data Analysis 02/2003; · 1.03 Impact Factor
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ABSTRACT: Recently a new distribution, named as generalized exponential distribution has been introduced and studied quite extensively by the authors. Generalized exponential distribution can be used as an alternative to gamma or Weibull distribution in many situations. In a companion paper, the authors considered the maximum likelihood estimation of the dierent parameters of a generalized exponential distribution and discussed some of the testing of hypothesis problems. In this paper we mainly consider ®ve other estimation procedures and compare their performances through numerical simulations.
Comput. Simul. 01/2000; 00.
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ABSTRACT: The three-parameter gamma and three-parameter Weibull distributions are commonly used for analysing any lifetime data or skewed data. Both distributions have several desirable properties, and nice physical interpretations. Because of the scale and shape parameters, both have quite a bit of flexibility for analysing different types of lifetime data. They have increasing as well as decreasing hazard rate depending on the shape parameter. Unfortunately both distributions also have certain drawbacks. This paper considers a three-parameter distribution which is a particular case of the exponentiated Weibull distribution originally proposed by Mudholkar, Srivastava & Freimer (1995) when the location parameter is not present. The study examines different properties of this model and observes that this family has some interesting features which are quite similar to those of the gamma family and the Weibull family, and certain distinct properties also. It appears this model can be used as an alternative to the gamma model or the Weibull model in many situations. One dataset is provided where the three-parameter generalized exponential distribution fits better than the three-parameter Weibull distribution or the three-parameter gamma distribution.
Australian & New Zealand Journal of Statistics 05/1999; 41(2):173 - 188. · 0.44 Impact Factor
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ABSTRACT: The three-parameter gamma and three-parameter Weibull distributions are commonly used for analysing any lifetime data or skewed data. Both distributions have several desirable properties, and nice physical interpretations. Because of the scale and shape parameters, both have quite a bit of flexibility for analysing different types of lifetime data. They have in-creasing as well as decreasing hazard rate depending on the shape parameter. Unfortunately both distributions also have certain drawbacks. This paper considers a three-parameter dis-tribution which is a particular case of the exponentiated Weibull distribution originally proposed by Mudholkar, Srivastava & Freimer (1995) when the location parameter is not present. The study examines different properties of this model and observes that this family has some interesting features which are quite similar to those of the gamma family and the Weibull family, and certain distinct properties also. It appears this model can be used as an alternative to the gamma model or the Weibull model in many situations. One dataset is provided where the three-parameter generalized exponential distribution fits better than the three-parameter Weibull distribution or the three-parameter gamma distribution.
01/1999; 41:173-188.
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ABSTRACT: We consider a particular two dimensional model, which has a wide applications in statistical signal processing and texture classifications. We prove the consistency of the least squares estimators of the model parameters and also obtain the asymptotic distribution of the least squares estimators. We observe the strong consistency of the least squares estimators when the errors are independent and identically distributed double array random variables. We show that the asymptotic distribution of the least squares estimators are multivariate normal. It is observed that the asymptotic dispersion matrix coincides with the Cramer-Rao lower bound. This paper generalizes some of the existing one dimensional results to the two dimensional case. Some numerical experiments are performed to see how the asymptotic results work for finite samples.
Metrika 11/1998; 48(2):83-97. · 0.67 Impact Factor
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ABSTRACT: Block and Basu bivariate exponential distribution is one of the most popular abso-lutely continuous bivariate distribution. Extensive work has been done on the Block and Basu bivariate exponential model over the past several decades. Interestingly it is observed that the Block and Basu bivariate exponential model can be extended to the Weibull model also. We call this new model as the Block and Basu bivariate Weibull model. We consider different properties of the Block and Basu bivariate Weibull model. The Block and Basu bivariate Weibull model has four unknown parameters and the maximum likelihood estimators cannot be obtained in closed form. To compute the the maximum likelihood estimators directly, one needs to solve a four dimensional op-timization problem. We propose to use the EM algorithm for computing the maximum likelihood estimators of the unknown parameters. The proposed EM algorithm can be carried out by solving one non-linear equation at each EM step. Our method can be used to compute the maximum likelihood estimators for the Block and Basu bivariate exponential model also. One data analysis has been preformed for illustrative purpose.
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ABSTRACT: In this paper we discuss different properties of the two generalizations of the logistic distributions, which can be used to model the data exhibiting a unimodal density having some skewness present. The first generalization is carried out using the basic idea of Azzalini [2] and we call it as the skew logistic distribution. It is observed that the density function of the skew logistic distribution is always unimodal and log-concave in nature. But the distribution function, failure rate function and different moments can not be obtained in explicit forms and therefore it becomes quite difficult to use it in practice. The second generalization we propose as a proportional reversed hazard family with the base line distribution as the logistic distribution. It is also known in the literature as the Type-I generalized logistic distribution. The density function of the proportional reversed hazard logistic distribution may be asymmetric but it is always unimodal and log-concave. The distribution function, hazard function are in compact forms and the different moments can be obtained in terms of the ψ function and its derivatives. We have proposed different estimators and performed one data analysis for illustrative purposes.
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ABSTRACT: This paper deals with the estimation of R = P [Y < X] when X and Y are two independent Weibull distributions with different scale parameters but having the same shape parameter. The maximum likelihood estimator and the approximate maximum likelihood estimator of R are proposed. We obtain the asymptotic distribution of the maximum likelihood estimator of R. Based on the asymptotic distribution, the confidence interval of R can be obtained. We also propose two bootstrap confidence intervals. We consider the Bayesian estimate of R and propose the corresponding credible interval for R. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a real data set has also been presented for illustrative purposes.
