Bivariate Lomax Distribution Research Articles

Bivariate Lomax Distribution Based on Clayton Copula: Estimation and Prediction

Lomax distribution has been studied by many statisticians due to its important role in reliability modeling and lifetime testing. The bivariate Lomax distribution is an important lifetime distribution in survival analysis. In this article, a bivariate Lomax distribution is constructed based on Clayton copula. The joint probability density function and the joint cumulative distribution function are derived in closed forms. Some properties of this bivariate distribution are discussed. The maximum likelihood and Bayes estimators for the unknown parameters are derived. Also, the maximum likelihood and Bayesian two-sample prediction for the future observations are obtained. The performance of the proposed bivariate distribution is examined using a simulation study. Finally, one real data set under the proposed distribution is considered to illustrate its flexibility and applicability for real-life applications.

Objective Bayesian inference for the reliability in a bivariate Lomax distribution

Objective Bayesian inference for the reliability in a bivariate Lomax distribution

Inference on P(X < Y) in Bivariate Lomax model based on progressive type II censoring.

This article considers the estimation of the stress-strength reliability parameter, θ = P(X < Y), when both the stress (X) and the strength (Y) are dependent random variables from a Bivariate Lomax distribution based on a progressive type II censored sample. The maximum likelihood, the method of moments and the Bayes estimators are all derived. Bayesian estimators are obtained for both symmetric and asymmetric loss functions, via squared error and Linex loss functions, respectively. Since there is no closed form for the Bayes estimators, Lindley’s approximation is utilized to derive the Bayes estimators under these loss functions. An extensive simulation study is conducted to gauge the performance of the proposed estimators based on three criteria, namely, relative bias, mean squared error, and Pitman nearness probability. A real data application is provided to illustrate the performance of our proposed estimators through bootstrap analysis.

Efficient Estimation of a Scale Parameter of Bivariate Lomax Distribution by Ranked Set Sampling

Ranked set sampling (RSS) is an efficient technique for estimating parameters and is applicable whenever ranking on a set of sampling units can be done easily by a judgment method or based on an auxiliary variable. In this paper, we assume [Formula: see text]to have bivariate Lomax distribution where a study variable [Formula: see text]is difficult and/or expensive to measure and is correlated with an auxiliary variable [Formula: see text] which is readily measurable. The auxiliary variable is used to rank the sampling units. In this article, we propose an estimator for the scale parameter of bivariate Lomax distribution using some of the modified RSS schemes. Efficiency comparison of the proposed estimators is performed numerically as well as graphically. A simulation study is also performed to demonstrate the performance of the proposed estimators. Finally, we implement the results to real-life datasets. AMS classification codes: 62D05, 62F07, 62G30

Posterior propriety of bivariate lomax distribution under objective priors

The Lomax or Pareto II, distribution has been quite widely used for reliability modeling and life testing, and applied to the sizes of computer files on servers, and even application in the biological sciences. Especially, the bivariate Lomax distribution is considered for a two components system which works under interdependency circumstances. We here develop the noninformative priors such as the probability matching priors and the reference priors for the parameters in the bivariate lomax population. It turns out that the reference prior satisfy a first order matching criterion only for some parameter groupings. We also check the conditions for the propriety of posterior distributions under the general prior class including the matching priors and the reference priors. It is a quite interesting that the posterior distributions under the reference priors for some parameter groupings are proper, but those for the other parameter groupings are improper. However all of the matching priors give the proper posteriors.

On Concomitants of Order Statistics from Farlie-Gumbel-Morgenstern Bivariate Lomax Distribution and its Application in Estimation

On Concomitants of Order Statistics from Farlie-Gumbel-Morgenstern Bivariate Lomax Distribution and its Application in Estimation

Estimation of scale parameter of a Bivariate Lomax distribution by Ranked Set Sampling

Concept of Ranked Set Sampling is applicable whenever ranking on a set of sampling units can be done easily by a judgment method or based on an auxiliary variable. In this work, we consider a study variable Y correlated with auxiliary variable X which is used to rank the sampling units. Further (X, Y) is assumed to have bivariate Lomax distribution. We obtain an unbiased estimator of scale parameter of study variable Y based on ranked set sample and censored ranked set sample. Efficiency comparison of these estimators with usual SRS estimator is also performed.

Parameter estimation for the bivariate Lomax distribution based on censored samples

In this paper, we estimate the parameters of the bivariate Lomax distribution of Marshall-Olkin bsed on right censored sample. The advantage of using the EM algorithm is that the observed data vector is viewed as being incomplete but regarded as an observable function of complete data. Then the EM algorithm exploits the reduced complexity of maximum likelihood estimation for complete data. We also derive the standard deviations of the estimates for this bivariate Lomax distribution. A simulation study is conducted to compare the estimated values with the true values.

Estimation of the bivariate generalized Lomax distribution parameters based on censored samples

Recently it is observed that the generalized exponential distribution can be used quite effectively to analyze lifetime data in one dimension. This paper extends Marshall and Olkin's bivariate exponential model to the Generalized Bivariate Lomax (BGL) Distribution. The cumulative distribution function, the probability density function and the Marginal distribution of the BGL distribution are reached. The maximum likelihood estimation procedure is derived for the estimation of the BGL parameters based on Censored Samples when all parameters are unknown and also obtain the observed Fisher information matrix. A special case of the distribution of the BGL distribution is reached in a closed form when one of the parameters is known. Simulation study was analyzed, and a numerical comparison is made between the proposed estimation procedure of the BGL distribution and Block-Basu estimation technique.

A NOTE ON THE BIVARIATE PARETO DISTRIBUTION

The Fisher information matrix plays a significant role i inference in connection with estimation and properties of variance of estimators. Using Bivariate Lomax distribution, we can define statistical model and drive the Fisher information matrix of Bivariate Lomax distribution. In this paper, we correct the wrong of the paper [7].

Structure of generalized bivariate Lomax distribution based on dependence

In this paper, we study the dependence structure of the generalized bivariate Lomax family. We also obtain some association measures and the local dependence function due to Kotz and Nadarajah [25]. In addition, we derive entropy, mutual information and quadratic mutual information measures for this family and discuss about them. Furthermore, we compare local dependence function and Pearson’s via a numerical study.

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.

Sums, products, and ratios for the bivariate lomax distribution

We derive the distributions of S=X+Y, D=X-Y, P=XY and W=X/(X+Y) and the corresponding moment properties when X and Y follow the bivariate lomax distribution. Some expressions involve the Gauss hypergeometric function. We also provide an extensive tabulation of the associated percentage points. This tabulation—obtained using intensive computing power—will be of use to practitioners of the lomax distribution.

Characterizations Using Local Dependence Function

In this article, we study the properties of the local dependence function, introduced by Holland and Wang [Holland, P. W., Wang, Y. J. (1987). Dependence function for continuous bivariate densities. Commun. Statist. Theory and Methods 16:863–876] in the context of reliability modeling. It is shown that one component of the vector failure with the local dependence function determines the joint bivariate density uniquely. Furthermore, we present characterizations for bivariate Lomax distribution, bivariate Dirichlet distribution, and bivariate normal distribution using local dependence function and regression functions.

Some General Characterizations of the Bivariate Gumbel Distribution and the Bivariate Lomax Distribution Based on Truncated Expectations

Recently attempts have been made to characterize probability distributions via truncated expectations in both univariate and multivariate cases. In this paper we will use a well known theorem of Lau and Rao (1982) to obtain some characterization results, based on the truncated expectations of a functionh, for the bivariate Gumbel distribution, a bivariate Lomax distribution, and a bivariate power distribution. The results of the paper subsume some earlier results appearing in the literature.

Bivariate Extension of Lomax and Finite Range Distributions through Characterization Approach

In the univariate setup the Lomax distribution is being widely used for stochastic modelling of decreasing failure rate life components. It also serves as a useful model in the study of labour turnover, queueing theory, and biological analysis. A bivariate extension of the Lomax distribution given in Lindley and Singpurwalla (1986) fails to cover the case of independence. Our present attempt is to obtain the unique determination of a bivariate Lomax distribution through characterization results. In this process we also obtain bivariate extensions of the exponential and a finite range distributions. The bivariate Lomax distribution thus obtained is a member of the Arnold (1990) flexible family of Pareto distributions and the bivariate exponential distribution derived here is identical with that of Gumbel (1960). Various properties of the proposed extensions are presented.

A Characterization of Lindley and Singpurwalla's Bivariate Lomax Distribution

Bivariate Lomax distribution due to Lindley and Singpurwalla has been characterized through the bivariate extension of a univariate characterizing property of the Lomax distribution.

A characterization of Gumbel's bivariate exponential and lindley and Singpurwalla's bivariate lomax distributions

A bivariate exponential distribution due to Gumbel and a bivariate Lomax distribution due to Lindley and Singpurwalla have been characterized through a bivariate extension of a univariate characterizing property of the exponential and the Lomax distribution; the characterization is based on mean residual lives and hazard rates.

A characterization of Gumbel's bivariate exponential and lindley and Singpurwalla's bivariate lomax distributions

A bivariate exponential distribution due to Gumbel and a bivariate Lomax distribution due to Lindley and Singpurwalla have been characterized through a bivariate extension of a univariate characterizing property of the exponential and the Lomax distribution; the characterization is based on mean residual lives and hazard rates.