The generalized linear exponential distribution

This paper deals with a new generalization of the linear exponential distribution. This distribution is called the generalized linear exponential distribution (GLED). Some statistical properties such as moments, modes and quantiles are derived. The failure rate function and the mean residual lifetime are also discussed. The maximum likelihood estimators of the parameters are obtained using a simulation study. Real data are used to determine whether the GLED is better than other well-known distributions in modeling lifetime data or not.     

Exponentiated generalized linear exponential distribution

A new generalization of the linear exponential distribution is recently proposed by Mahmoud and Alam [1], called as the generalized linear exponential distribution. Another generalization of the linear exponential was introduced by Sarhan and Kundu  and , named as the generalized linear failure rate distribution. This paper proposes a more generalization of the linear exponential distribution which generalizes the two. We refer to this new generalization as the exponentiated generalized linear exponential distribution. The new distribution is important since it contains as special sub-models some widely well known distributions in addition to the above two models, such as the exponentiated Weibull distribution among many others. It also provides more flexibility to analyze complex real data sets. We study some statistical properties for the new distribution. We discuss maximum likelihood estimation of the distribution parameters. Three real data sets are analyzed using the new distribution, which show that the exponentiated generalized linear exponential distribution can be used quite effectively in analyzing real lifetime data.     

A new generalized linear exponential distribution and its applications

A new generalized linear exponential distribution （NCLED） is considered in this paper which can be deemed as a new and more flexible extension of linear exponential distribution. Some statistical properties for the NGLED such as the hazard rate function, moments, quantiles are given. The maximum likelihood estimations （MLE） of unknown parameters are also discussed. A simulation study and two real data analyzes are carried out to illustrate that the new distribution is more flexible and effective than other popular distributions in modeling lifetime data.     

Bayesian inference and life testing plans for generalized exponential distribution

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. This work was partially supported by a grant from the Department of Science and Technology, Government of India     

Bivariate Hermite interpolation on the exponential curve

Periodica Mathematica Hungarica - We study Hermite interpolation problems on the exponential curve $$y=e^x$$ in $$\mathbb {R}^2$$ . We construct some kind of regular Hermite interpolation schemes...     

Expectation identities of lower generalized order statistics from generalized exponential distribution and a characterization

In this paper, we study the explicit expressions and some recurrence relations for single and product moments of lower generalized order statistics from generalized exponential distribution. The results include as particular cases the above relations for moments of order statistics and lower records. Further, using a recurrence relation for single moments we obtain a characterization of generalized exponential distribution.     

Estimation under failure-censored step-stress life test for the generalized exponential distribution parameters

Accelerated life testing of materials is used to collect failure data quickly when the lifetime of a specimen under use condition is too long. This article considers estimates of the generalized exponential distribution parameters under step-stress partially accelerated life testing with Type-II censoring. The maximum likelihood approach is applied to derive point and asymptotic confidence interval estimations of the model parameters. The performance of the estimators is evaluated numerically for different parameter values and different sample sizes via their mean square error. Also, the average confidence intervals lengths and the associated coverage probabilities are obtained. A simulation study is conducted for illustration.     

Bivariate inverse gaussian distribution

Summary A bivariate inverse Gaussian (IG) density function is constructed. Relations of the bivariate IG distribution to the normal and χ² distributions are established. The corresponding bivariate random walk (RW) density function is obtained. The properties and behaviour of bivariate IG distribution are studied for large parametric values. Moment estimates of the five parameters are given and applications are pointed out. A generalization to the multivariate IG distribution is proposed.     

Bayesian estimation for generalized exponential distribution based on progressive type-I interval censoring

In this study, we consider the Bayesian estimation of unknown parameters and reliability function of the generalized exponential distribution based on progressive type-I interval censoring. The Bayesian estimates of parameters and reliability function cannot be obtained as explicit forms by applying squared error loss and Linex loss functions, respectively; thus, we present the Lindley’s approximation to discuss these estimations. Then, the Bayesian estimates are compared with the maximum likelihood estimates by using the Monte Carlo simulations.     

NOTES ON GENERALIZED EXPONENTIAL DICHOTOMIES

ＮＯＴＥＳＯＮＧＥＮＥＲＡＬＩＺＥＤＥＸＰＯＮＥＮＴＩＡＬＤＩＣＨＯＴＯＭＩＥＳ￥ＺＨＡＮＧＷＥＩＮＩＡＮ（张伟年）（ＣｅｎｔｅｒｆｏｆＭａｔｈｅｍａｔｉｃａｌＳｃｉｅｎｃｅｓＣＩＣＡ，ｔｈｅＣｈｉｎｅｓｅＡｃａｄｅｍｙｏｆＳｃｉｅｎｃｅｓ，Ｃｈｅｎ...     

A characterization of generalized exponential dichotomy

This paper studies some important properties of the notion generalized exponential dichotomy. A new notion called generalized bounded growth is introduced to describe the characterization of generalized exponential dichotomy. The relations between generalized bounded growth and generalized exponential dichotomy are established.     

The exponential integral distribution

The exponential integral distribution is introduced. The gamma distribution is a subclass of this distribution. The higher order exponential integrals are closely related to both the gamma and the beta function. The physical relevance of this new distribution is discussed.     

Strongly topological linearization with generalized exponential dichotomy

It has been proved that if x^′=A(t)x$x^{\prime }=A\left(t\right)x$ has a generalized exponential dichotomy and f(t,x)$f(t,x)$ satisfies certain conditions, then the nonlinear system x^′=A(t)x+f(t,x)$x^{\prime }=A\left(t\right)x+f(t,x)$ is topologically equivalent to its linear system x^′=A(t)x$x^{\prime }=A\left(t\right)x$. In this paper, we prove that if the condition |A(t)|≤M⋅a(t)$\left|A\left(t\right)\right|\le M\cdot a\left(t\right)$ is added, then x^′=A(t)x+f(t,x)$x^{\prime }=A\left(t\right)x+f(t,x)$ is strongly topologically equivalent to x^′=A(t)x$x^{\prime }=A\left(t\right)x$, where M$M$ is some positive number and a(t)$a\left(t\right)$ is the eigenfunction of the generalized exponential dichotomy, and therefore the corresponding solutions of x^′=A(t)x+f(t,x)$x^{\prime }=A\left(t\right)x+f(t,x)$ and x^′=A(t)x$x^{\prime }=A\left(t\right)x$ have the same stability.     

Tolerance interval for exponential distribution

Tolerance interval is a kind of interval that assures the probability of at least a given proportion of population falls into the interval attains to a fixed level. It is widely needed in various industrial practices and business activities, such as product design, reliability analysis, and quality inspection. However, comparing to its widely needs, the research on it is still quite limited. In this paper, we propose a numerical method to compute the tolerance interval for exponential distribution. As the simulation study illustrates, our method performs consistently well as the sample size varies. In particular, its good performance for small sample endows itself broadly potential usefulness in practice.     

A generalized two dimensional Emden-Fowler equation with exponential nonlinearity

In this paper we study the asymptotic behavior of two-dimensional equation in divergence form with exponential nonlinearity. We discuss also the construction of singular limits in symmetric case. Received: 27 October 1999 / Accepted: 17 July 2000 / Published online: 23 April 2001     

A characterization of the exponential distribution

Summary A necessary and sufficient condition is given under which the minimum ofX _k /a _k ,k=1, 2, ...,n has the same distribution asX ₁, whereX's are i.i.d. positive random variables anda's are given positive constants. The Institute of Statistical Mathematics