Class Of Exponential Distributions Research Articles

Optimal confidence regions for the parameters of a general exponential class under Type-II progressive censoring

Under Type-II progressively censored data, joint confidence regions are proposed for the parameters of a general class of exponential distributions. The constrained optimization problem based on such censoring data can be adopted to obtain confidence regions for the unknown parameters of this general class with minimized size and a predetermined confidence level. The area of confidence sets are minimized by solving simultaneous non-linear equations. Two real data sets representing the duration of remission of leukemia patients and water level exceedances by River Nidd at Hunsingore located in New York, are analyzed by fitting appropriate well-known models. Further, numerical simulation study is performed to explain our procedures and findings here.

Generalized Bayesian inference study based on type-Ⅱ censored data from the class of exponential models

<p>Generalized Bayesian (GB) is a Bayesian approach based on the learning rate parameter (LRP) ($ 0 < \eta < 1 $) as a fraction of the power of the likelihood function. In this paper, we consider the GB method to perform inference studies for a class of exponential distributions. Generalized Bayesian estimators (GBE) and generalized empirical Bayesian estimators (GEBE) for the parameters of the considered distributions are obtained based on the censored type Ⅱ samples. In addition, generalized Bayesian prediction (GBP) and generalized empirical Bayesian prediction (GEBP) are considered using a one-sample prediction scheme. Monte Carlo simulations and illustrative example are performed for one parameter models to compare the performance of the GBE and GEBE estimation results and the GBP and GEBP prediction results for different values of the LRP.</p>

On inference in a class of exponential distribution under imperfect maintenance

This paper deals with statistical inference for lifetime data in presence of imperfect maintenance. For the maintenance model, the Sheu and Griffith model is considered. The lifetime distribution belongs to exponential distribution class. The maximum likelihood estimation procedure of the model parameters is discussed, and confidence intervals are provided using the asymptotic likelihood theory and bootstrap approach. Based on conjugate and discrete priors, Bayesian estimators of the model parameters are developed under symmetric and asymmetric loss functions. The proposed methodologies are applied to simulated data and sensitivity analysis to different parameters and data characteristics is carried out. The effect of model misspecification is also assessed within this class of distributions through a Monte Carlo simulation study. Finally, two datasets are analyzed for demonstrative aims.

On the odd inverse exponential class of distributions: Properties, applications and cure fraction regression

In this paper, the generalized odd inverse exponential family of distributions is developed and studied. Statistical properties such as moments, moment generating function, incomplete moment, characteristic function and order statistics of the new family were derived. Two special distributions; generalized odd inverse exponential Weibull and generalized odd inverse exponential Lomax distributions were proposed. Again, regression models with cure fractions were developed using the special distributions. Four applications of the new family using cancer datasets suggest that the special distributions provide better fit to the datasets than the other fitted models.

Type II General Exponential Class of Distributions

In this paper, a new class of continuous distributions with two extra positive parameters is introduced and is called the Type II General Exponential (TIIGE) distribution. Some special models are presented. Asymptotics, explicit expressions for the ordinary and incomplete moments, moment residual life, reversed residual life, quantile and generating functions and stress-strengh reliability function are derived. Characterizations of this family are obtained based on truncated moments, hazard function, conditional expectation of certain functions of the random variable are obtained. The performance of the maximum likelihood estimators in terms of biases, mean squared errors and confidence interval length is examined by means of a simulation study. Two real data sets are used to illustrate the application of the proposed class.

Randomly stopped minima and maxima with exponential-type distributions

Let {ξ1, ξ2,...} be a sequence of independent real-valued and possibly nonidentically distributed random variables. Suppose that η is a nonnegative, nondegenerate at 0 and integer-valued random variable, which is independent of {ξ1, ξ2,...}. In this paper, we consider conditions for {ξ1, ξ2,...} and η under which the distributions of the randomly stopped maxima and minima as well as randomly stopped maxima of sums and randomly stopped minima of sums belong to the class of exponential distributions.

Type I General Exponential Class of distributions

We introduce a new family of continuous distributions and study the mathematical properties of the new family. Some useful characterizations based on the ratio of two truncated moments and hazard function are also presented. We estimate the model parameters by the maximum likelihood method and assess its performance based on biases and mean squared errors in a simulation study framework.

Randomly stopped sums with exponential-type distributions

Assume that {ξ1, ξ2, …} are independent and possibly nonidentically distributed random variables. Suppose that η is a nonnegative, nondegenerate at zero and integer-valued random variable, which is independent of {ξ1, ξ2, …}. In this paper, we consider conditions for η and {ξ1, ξ2, …} under which the distribution of the random sum {ξ1 + ξ2 + … + ξη} belongs to the class of exponential distributions.

Analysis of progressively type-I interval censored competing risks data for a class of an exponential distribution

ABSTRACTIn this paper, we consider some problems of point estimation and point prediction when the competing risks data from a class of exponential distribution are progressive type-I interval censored. The maximum likelihood estimation and mid-point approximation method are proposed for the estimations of parameters. Also several point predictors of censored units such as the maximum likelihood predictor, the best unbiased predictor and the conditional median predictor are obtained. The methods discussed here are applied when the lifetime distributions of the latent failure times are independent and Weibull-distributed. Finally a simulation study is given by using Monte-Carlo simulations to compare the performances of the different methods and one data analysis has been presented for illustrative purposes.

Reliability analysis of a class of exponential distribution under record values

In this paper the estimation of the parameters as well as survival and hazard functions for a class of an exponential family are presented by using Bayesian and non-Bayesian approaches under record values. Maximum likelihood and interval estimation are derived for the model parameters, and Bayes estimators of reliability performances are obtained under symmetric and asymmetric loss functions, when the parameters have discrete and continuous priors, respectively. Finally, two numerical examples are presented to illustrate the proposed results. An algorithm is introduced to generate record data, then the numerical example is given by using Monte Carlo simulation and different estimates results are compared.

Using the maximum entropy method to estimate probability density functions for oceanographic and acoustic parameters

The maximum entropy (MaxEnt) method offers a powerful means of estimating the probability density functions (pdf’s) of measured data. Traditionally, pdf’s have been estimated by fitting data to particular distributions (e.g., Gaussian, log-normal, Rayleigh, chi-squared, etc.). This approach can lead to misleading results, especially if the data are highly skewed or kurtotic. First published in 1957 by Jaynes [E. T. Jaynes, Phys. Rev., 106, 620–630 (1957)], MaxEnt has gained wide use in recent years. The method uses Lagrangian multipliers to maximize the entropy, a measure of uncertainty, subject to specified constraints. The result is a pdf that Jaynes describes as being the ‘‘least biased estimate possible on the given information; i.e., it is maximally noncommittal with regard to missing information.’’ In other words, all possibilities under the given constraints are considered equally. This method is particularly useful in determining prior pdf’s based on limited measurements. MaxEnt pdf’s, which belong in the exponential class of distributions, can also be used to derive optimum processors in the maximum likelihood sense. This talk will discuss the theory and implementation of the MaxEnt method and emphasize its utility for modeling acoustic propagation in uncertain environments. [Work supported by The Undersea Signal Processing Program, Office of Naval Research.]

Testing Exponentiality Against Laplace Order Dominance-

We present a statistical procedure to test that a life distribution belongs to the class of exponential distributions against that it belongs to a class of alternatives based on the Laplace transform. The test has been shown to be consistent and the asymptotic distribution of the test statistic has been obtained. The performance of the test against various classes of alternatives has been studied by means of Monte Carlo simulation. An interesting characterization theorem for exponentials, which motivates our test procedure, has been proved.

Halphen Distribution System. II: Parameter and Quantile Estimation

In this paper, the second in a series of two, procedures for the estimation of parameters and quantiles of the Halphen type A, B, and IB distributions are presented. Because the Halphen distributions are members of the exponential class of distributions, parameters can be estimated from sufficient statistics, and maximum likelihood estimators should possess certain optimality characteristics. In some cases, the maximum likelihood system of equations does not allow a solution, and the limiting forms of the Halphen distributions, the gamma and inverse gamma, should alternatively be considered. The asymptotic variance of parameter estimators may be obtained by inverting the Fisher information matrix. The asymptotic variance of quantile estimators is obtained by classical first-order approximations. Practical experience with fitting the Halphen distributions is reported.

Solution of statistical problems for a class of exponential distributions of random variables

For a class of exponential-type distributions with the power index ranging from 0.25 to 8, a simple formula for an approximate evaluation of the distribution function is presented. The formula is then used for solving certain statistical problems.

Effect of Convective Boundary Layer on the Current Efficiency of a Membrane Bearing Nonuniformly Distributed Fixed Charges

The current efficiency η of a planar membrane bearing nonuniformly distributed fixed charges is analyzed theoretically. In particular, the effect of the convective boundary layer near the outer boundary of the membrane is discussed. Two types of fixed charge distribution are considered: a class of linear distribution and a class of exponential distribution. We show that, if the flow velocity in the membrane phase is small, increasing the inhomogeneity of fixed charge distribution is capable of raising η. However, if it is appreciable, the reverse is true. If a boundary is present, η decreases with the increase in current density; the reverse is true if it is absent. The former is due to the fact that the degree of concentration polarization in the boundary layer increases with current density. Although the existence of a boundary layer has the effect of reducing η, the convective flow of fluid perpendicular to membrane has a positive effect on η. If a boundary layer is absent and the fixed charges are di...

Approximation of distribution functions of random variables belonging to the class of exponential distributions

Simple formulas for an approximate evaluation of distribution functions belonging to the class of exponential distributions with the exponent ranging between 1 and 7.5 are presented. Errors of approximations for integer-valued and some fractional exponents are provided.

Some characterizations for the exponential class of distributions

A general characterization theorem, based on conditional expectation, is proved for the exponential class of distributions. The theorem is then applied to numerous discrete and continuous probability distributions of the exponential class providing specific characterizations for each one of them.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The generalized exponential time-frequency distribution

Time-frequency distributions (TFD) are joint time and frequency signal representations that, among other properties, maintain the true support of a signal's energy in both time and frequency. In addition to their mathematical elegance, TFDs can provide simultaneous resolution in time and frequency that exceeds that of the common spectrogram. In general, however, TFDs, also exhibit certain peculiarities that arise, in part, from the bilinear structure of the fundamental TFD form (L. Cohen's class). Perhaps most notable is the presence of spectral cross-term artifacts, a kind of spectral chaff that tends to impede visual understanding and interpretation of TFDs as instantaneous power spectrums. Several researchers have proposed and demonstrated a variety of TFDs, which through Cohen's (1989) form are defined through a kernel /spl phi/(/spl theta/,/spl tau/). Particularly notable among these is Choi and Williams' (1989) exponential distribution in which /spl phi/(/spl theta/,/spl tau/)=exp(-/spl theta//sup 2//spl tau//sup 2///spl sigma/). Of all distributions investigated, the exponential distribution is relatively immune to spectral cross-term generation and yet maintains high simultaneous time-frequency resolution. In a generalization of Choi and Williams' work, the author introduces the broader class of exponential distributions defined by the kernel exp(-|/spl theta/|/sup p/|/spl tau/|/sup q///spl sigma/) and investigate its properties. In particular, he shows that this generalized exponential distribution can exceed the time-frequency resolution performance of the exponential distribution and get also remain relatively free from spectral cross-term distortion.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A bayesian wls approach to generalized linear models

We use a Bayesian approach to fitting a linear regression model to transformations of the natural parameter for the exponential class of distributions. The usual Bayesian approach is to assume that a linear model exactly describes the relationship among the natural parameters. We assume only that a linear model is approximately in force. We approximate the theta-links by using a linear model obtained by minimizing the posterior expectation of a loss function.While some posterior results can be obtained analytically considerable generality follows from an exact Monte Carlo method for obtaining random samples of parameter values or functions of parameter values from their respective posterior distributions. The approach that is presented is justified for small samples, requires only one-dimensional numerical integrations, and allows for the use of regression matrices with less than full column rank. Two numerical examples are provided.

The Equivalence of Maximum Likelihood and Weighted Least Squares Estimates in the Exponential Family

Abstract In this article it is shown that weighted least squares estimators for the coefficients of a multiple regression model are the same as the maximum likelihood estimators when the dependent observations are from a member of the regular exponential class of distributions.