Switching Order Statistics through Random Power Contractions

This paper investigates a new random contraction scheme which complements the length‐biasing and convolution contraction schemes considered in the literature. A random power contraction is used with order statistics, leading to new and elegant characterizations of the power distribution. In view of Rossberg's counter‐example of a non‐exponential law with exponentially distributed spacings of order statistics, possibly the most appealing consequence of the result is a characterization of the exponential distribution via an independent exponential shift of order statistics.     

Characterization of Lindley distribution by truncated moments

In the past few years, the Lindley distribution has gained popularity for modeling lifetime data as an alternative to the exponential distribution. This paper provides two new characterizations of the Lindley distribution. The first characterization is based on a relation between left truncated moments and failure rate function. The second characterization is based on a relation between right truncated moments and reversed failure rate function.     

The beta generalized exponential distribution

A new distribution called the beta generalized exponential distribution is proposed. It includes the beta exponential and generalized exponential (GE) distributions as special cases. We provide a comprehensive mathematical treatment of this distribution. The density function can be expressed as a mixture of generalized exponential densities. This is important to obtain some mathematical properties of the new distribution in terms of the corresponding properties of the GE distribution. We derive the moment generating function (mgf) and the moments, thus generalizing some results in the literature. Expressions for the density, mgf and moments of the order statistics are also obtained. We discuss estimation of the parameters by maximum likelihood and obtain the information matrix that is easily numerically determined. We observe in one application to a real skewed data set that this model is quite flexible and can be used effectively in analyzing positive data in place of the beta exponential and GE distributions.     

Waiting time paradox and size biased sampling

Feller's (1971) 'Waiting time for the bus paradox1 is explained by the fact that the long Interarrival times have a better chance of covering an arbitrary point in time than do short interarrival times, This gives rise to size biased sampling and the length of the Interarrival time that contains the arbitrary point has a moment distribution, This paper deals with the characterization of the exponential distribution based on this 'Waiting time paradox'. Similar characterizations of the binomial, the negative binomial, the Poisson and the geometric distributions are obtained.     

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.     

The Lack of Memory Property in the Density Form for the Bivariate Setup

The lack of memory property is a characterizing property of the exponential distribution in the continuous domain. In the bivariate setup different generalizations of the same are available in terms of survival function. We extend this lack of memory property in terms of bivariate probability density function and examine its characterization properties. In this process the density version of the lack of memory property can be interlinked with conditionally specified exponential distribution, bivariate reciprocal coordinate subtangent of the density curve and a few other derived measures.     

Generalized Linear Failure Rate Distribution

The exponential and Rayleigh are the two most commonly used distributions for analyzing lifetime data. These distributions have several desirable properties and nice physical interpretations. Unfortunately, the exponential distribution only has constant failure rate and the Rayleigh distribution has increasing failure rate. The linear failure rate distribution generalizes both these distributions which may have non increasing hazard function also. This article introduces a new distribution, which generalizes linear failure rate distribution. This distribution generalizes the well-known (1) exponential distribution, (2) linear failure rate distribution, (3) generalized exponential distribution, and (4) generalized Rayleigh distribution. The properties of this distribution are discussed in this article. The maximum likelihood estimates of the unknown parameters are obtained. A real data set is analyzed and it is observed that the present distribution can provide a better fit than some other very well-known distributions.     

Two-Sided Generalized Exponential Distribution

We derive a generalization of the exponential distribution by making log transformation of the standard two-sided power distribution. We show that this new generalization is in fact a mixture of a truncated exponential distribution and truncated generalized exponential distribution introduced by Gupta and Kundu [Generalized exponential distributions. Aust. N. Z. J. Stat. 41(1999):173–188]. The newly defined distribution is more flexible for modeling data than the ordinary exponential distribution. We study its properties, estimate the parameters, and demonstrate it on some well-known real data sets comparing other existing methods.     

Generalized Weighted Exponential Distribution

The new class of weighted exponential (WE) distributions obtained by Gupta and Kundu (2009) by implementing Azzalini's method to the exponential distribution. In this study, we generalize the WE distribution to a new class of generalized weighted exponential (GWE) distribution. Several statistical and reliability properties of this new class of distribution are obtained. Estimation and inference procedure for distribution parameters are investigated. Finally, we show that the proposed model can provide better fit than the recent class of weighted exponential by using two real data examples.     

Regression mean residual life of a parallel system of dependent components

In this paper, we consider a system consisting of two dependent components and we are interested in the average remaining life of the component that fails last when (i) the first failure occurs at time t and (ii) the first failure occurs after time t. For both the cases, expressions are derived in the case of general bivariate normal distribution and a class of bivariate exponential distribution including bivariate exponential distribution of Arnold and Strauss, absolutely continuous bivariate exponential distribution of Block and Basu, bivariate exponential distribution of Raftery, Freund's bivariate exponential distribution and Gumbel's bivariate exponential distribution.     

Modified Sarhan–Balakrishnan singular bivariate distribution

Recently Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527] introduced a new bivariate distribution using generalized exponential and exponential distributions. They discussed several interesting properties of this new distribution. Unfortunately, they did not discuss any estimation procedure of the unknown parameters. In this paper using the similar idea as of Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527], we have proposed a singular bivariate distribution, which has an extra shape parameter. It is observed that the marginal distributions of the proposed bivariate distribution are more flexible than the corresponding marginal distributions of the Marshall–Olkin bivariate exponential distribution, Sarhan–Balakrishnan's bivariate distribution or the bivariate generalized exponential distribution. Different properties of this new distribution have been discussed. We provide the maximum likelihood estimators of the unknown parameters using EM algorithm. We reported some simulation results and performed two data analysis for illustrative purposes. Finally we propose some generalizations of this bivariate model.     

Distribution of a linear function of correlated ordered variables

In this paper, we have considered the problem of finding the distribution of a linear combination of the minimum and the maximum for a general bivariate distribution. The general results are used to obtain the required distribution in the case of bivariate normal, bivariate exponential of Arnold and Strauss, absolutely continuous bivariate exponential distribution of Block and Basu, bivariate exponential distribution of Raftery, Freund's bivariate exponential distribution and Gumbel's bivariate exponential distribution. The distributions of the minimum and maximum are obtained as special cases.     

The McDonald Gompertz distribution: Properties and applications

This article introduces a five-parameter lifetime model called the McDonald Gompertz (McG) distribution to extend the Gompertz, generalized Gompertz, generalized exponential, beta Gompertz, and Kumaraswamy Gompertz distributions among several other models. The hazard function of new distribution can be increasing, decreasing, upside-down bathtub, and bathtub shaped. We obtain several properties of the McG distribution including moments, entropies, quantile, and generating functions. We provide the density function of the order statistics and their moments. The parameter estimation is based on the usual maximum likelihood approach. We also provide the observed information matrix and discuss inferences issues. The flexibility and usefulness of the new distribution are illustrated by means of application to two real datasets.     

Some characterizations of distributions by truncated moments

In this paper we extend GUPTA'S (1975) resLilt and show that the constant value of a truncated moment characterizes the exponential distribution. Similar results are prov¬ed in the discrete case and it is shown that two consecutive factorial moments are enough to determine the distribution. However, under mild conditions, the constancy of one fac¬torial moment is enough to guarantee that the distribution is geometric. Considering the truncation on the right, a general method of obtaining the distribution, whenever £(h(X) | Xj) is known, is fxmbitud in tnr continuous and in the discrete case. Several of the known characterization theorems toilow trivially trorn our results     

Designing an effective exponential cusum chart without the use of nomographs

The development of control charts for monitoring processes associated with very low rates of nonconformities is increasingly becoming more important as manufacturing processes become more capable. Since the rate of nonconformities can typically be modeled by a simple homogeneous Poisson process, the perspective of monitoring the interarrival times using the exponential distribution becomes an alternative. Gan (1994) developed a CUSUM-based approach for monitoring the exponential mean. In this paper, we propose an alternative CUSUM-based approach based on its ease of implementation. We also provide a study of the relative performance of the two approaches.     

A new lifetime model with decreasing failure rate

In this paper, we introduce a new lifetime distribution by compounding exponential and Poisson–Lindley distributions, named the exponential Poisson–Lindley (EPL) distribution. A practical situation where the EPL distribution is most appropriate for modelling lifetime data than exponential–geometric, exponential–Poisson and exponential–logarithmic distributions is presented. We obtain the density and failure rate of the EPL distribution and properties such as mean lifetime, moments, order statistics and Rényi entropy. Furthermore, estimation by maximum likelihood and inference for large samples are discussed. The paper is motivated by two applications to real data sets and we hope that this model will be able to attract wider applicability in survival and reliability.     

MAXIMUM LIKELIHOOD ESTIMATION FOR GENERALISED LOGISTIC DISTRIBUTIONS

ABSTRACT

Maximum likelihood estimation for the type I generalised logistic distributions is investigated. We show that the maximum likelihood estimation usually exists, except when the so-called embedded model problem occurs. A full set of embedded distributions is derived, including Gumbel distribution and a two-parameter reciprocal exponential distribution. Properties relating the embedded distributions are given. We also provide criteria to determine when the embedded distribution occurs. Examples are given for illustration.     

The beta exponentiated Weibull distribution

The Weibull distribution is one of the most important distributions in reliability. For the first time, we introduce the beta exponentiated Weibull distribution which extends recent models by Lee et al. [Beta-Weibull distribution: some properties and applications to censored data, J. Mod. Appl. Statist. Meth. 6 (2007), pp. 173–186] and Barreto-Souza et al. [The beta generalized exponential distribution, J. Statist. Comput. Simul. 80 (2010), pp. 159–172]. The new distribution is an important competitive model to the Weibull, exponentiated exponential, exponentiated Weibull, beta exponential and beta Weibull distributions since it contains all these models as special cases. We demonstrate that the density of the new distribution can be expressed as a linear combination of Weibull densities. We provide the moments and two closed-form expressions for the moment-generating function. Explicit expressions are derived for the mean deviations, Bonferroni and Lorenz curves, reliability and entropies. The density of the order statistics can also be expressed as a linear combination of Weibull densities. We obtain the moments of the order statistics. The expected information matrix is derived. We define a log-beta exponentiated Weibull regression model to analyse censored data. The estimation of the parameters is approached by the method of maximum likelihood. The usefulness of the new distribution to analyse positive data is illustrated in two real data sets.     

On characterizations of the logistic distribution

We modify and extend George and Mudholkar's [1981. A characterization of the logistic distribution by a sample median. Ann. Inst. Statist. Math. 33, 125–129] characterization result about the logistic distribution, which is in terms of the sample median and Laplace distribution. Moreover, we give some new characterization results in terms of the smallest order statistics and the exponential distribution.     

Tests of fit for exponentiality based on a characterization via the mean residual life function

We study two new omnibus goodness of fit tests for exponentiality, each based on a characterization of the exponential distribution via the mean residual life function. The limiting null distributions of the tests statistics are the same as the limiting null distributions of the Kolmogorov-Smirnov and Cramér-von Mises statistics proposed when testing the simple hypothesis that the distribution of the sample variables is uniform on the interval [0, 1]. Work supported by the Deutsche Forschungsgemeinschaft