A review of model selection procedures relevant to the weibull distribution

A number of goodness-of-fit and model selection procedures related to the Weibull distribution are reviewed. These procedures include probability plotting, correlation type goodness-of-fit tests, and chi-square goodness-of-fit tests. Also the Kolmogorow-Smirniv, Kuiper, and Cramer-Von Mises test statistics for completely specified hypothesis based on censored data are reviewed, and these test statistics based on complete samples for the unspecified parameters case are considered. Goodness-of-fit tests based on sample spacings, and a goodness-of-fit test for the Weibull process, is also discussed.

Model selection procedures for selecting between a Weibull and gamma model, a Weibull and lognormal model, and for selecting from among all three models are considered. Also tests of exponential versus Weibull and Weibull versus generalized gamma are mentioned.     

Estimating the parameters of a truncated weibull distribution

The problem of nonexistence of the maximum likelihood estimators (m.l.e.) with positive probability is investigated for the truncated Weibull distribution. Similar nonexistence of the m.l.e. is known for some other distributions such as truncated exponential, truncated normal, and one parameter truncated gamma. Modified likelihood estimators, which exist with probability one, are given and compared with the m.l.e.     

Comparisons of certain distribution functions

Certain common distributions, including the exponential, gamma, Gumbel. Weibull and normal are compared by means of partial orderings, extending results given in LISEK (1978).     

Using the singly truncated normal distribution to analyze satellite data

This paper proposes the singly truncated normal distribution as a model for estimating radiance measurements from satellite-borne infrared sensors. These measurements are made in order to estimate sea surface temperatures which can be related to radiances. Maximum likelihood estimation is used to provide estimates for the unknown parameters. In particular, a procedure is described for estimating clear radiances in the presence of clouds and the Kolmogorov-Smirnov statistic is used to test goodness-of-fit of the measurements to the singly truncated normal distribution. Tables of quantile values of the Kolmogorov-Smirnov statistic for several values of the truncation point are generated from Monie Carlo experiment Mnally a numerical emample using satetic data is presented to illustrate the application of the procedures.     

Smooth goodness of fit tests for the weibull distribution with singly censored data

The smooth goodness of fit tests are generalized to singly censored data and applied to the problem of testing Weibull (or extreme value) fit. Smooth tests, Pearson-type tests, and the spacings tests proposed by Mann, Schemer, and Fertig (1973) are compared on the basis of local asymptotic relative efficiency with respect to the asymptotic best test against generalized gamma alternatives, The smooth test of order one Is found to be most efficient for the generalized gamma alternatives.     

Analysis of accelerated life tests with weibull failure distribution via generalized linear models

Efficient industrial experiments for reliability analysis of manufactured goods may consist in subjecting the units to higher stress levels than those of the usual working conditions. This results in the so called "accelerated life tests" where, for each pre-fixed stress level, the experiment ends after the failure of a certain pre-fixed proportion of units or a certain test time is reached. The aim of this paper is to determine estimates of the mean lifetime of the units under usual working conditions from censored failure data obtained under stress conditions. This problem is approached through generalized linear modelling and related inferential techniques, considering a Weibull failure distribution and a log-linear stress-response relationship. The general framework considered has as particular cases, the Inverse Power Law model, the Eyring model, the Arrhenius model and the generalized Eyring model. In order to illustrate the proposed methodology, a numerical example is provided.     

Estimation for two-parameter weibull distribution and extreme-value distribution under multiply type-II censoring

Compared to Type-II censoring, multiply Type-II censoring is a more general, yet mathematically and numerically much more complicated censoring scheme. For multiply Type II censored data from a two-parameter Weibull distribution, we propose several estimators, including MLE, approximate MLE, and estimators corresponding to the BLUE and BLIE from estimating parameters in extreme-value distribution. An approximately unbiased estimator for the shape parameter is also proposed which has the smallest MSE. Numerical examples show that this estimator is the best in terms of bias and MSE. Numerical examples also show that the approximate MLE which admits a closed form is better for estimating the scale parameter.     

Modeling asset returns with alternative stable distributions

In the 1960's Benoit Mandelbrot and Eugene Fama argued strongly in favor of the stable Paretian distribution as a model for the unconditional distribution of asset returns. Although a substantial body of subsequent empirical studies supported this position, the stable Paretian model plays a minor role in current empirical work.

While in the economics and finance literature stable distributions are virtually exclusively associated with stable Paretian distributions, in this paper we adopt a more fundamental view and extend the concept of stability to a variety of probabilistic schemes. These schemes give rise to alternative stable distributions, which we compare empirically using S&P 500 stock return data. In this comparison the Weibull distribution, associated with both the nonrandom-minimum and geometric-random summation schemes dominates the other stable distributions considered-including the stable Paretian model.     

Modeling asset returns with alternative stable distributions *

In the 1960's Benoit Mandelbrot and Eugene Fama argued strongly in favor of the stable Paretian distribution as a model for the unconditional distribution of asset returns. Although a substantial body of subsequent empirical studies supported this position, the stable Paretian model plays a minor role in current empirical work.

While in the economics and finance literature stable distributions are virtually exclusively associated with stable Paretian distributions, in this paper we adopt a more fundamental view and extend the concept of stability to a variety of probabilistic schemes. These schemes give rise to alternative stable distributions, which we compare empirically using S&P 500 stock return data. In this comparison the Weibull distribution, associated with both the nonrandom-minimum and geometric-random summation schemes dominates the other stable distributions considered-including the stable Paretian model.     

Nonparametric testing of closeness between two unknown distribution functions

Based on the kernel integrated square difference and applying a central limit theorem for degenerate V-statistic proposed by Hall (1984), this paper proposes a consistent nonparametric test of closeness between two unknown density functions under quite mild conditions. We only require the unknown density functions to be bounded and continuous. Monte Carlo simulations show that the proposed tests perform well for moderate sample sizes.     

Conditional maxima and inferences for the truncated exponential distribution

This paper derives the conditional distribution of the maximum given the sample total for a random sample from the truncated exponential distribution. Based on that result, the paper develops tests or associated confidence intervals for the truncation parameter θ with another parameter θ assumed unknown.     

Estimation of weibull parameters for grouped data with competing risks

We consider the problem of estimating Weibull parameters for grouped data when competing risks are present. We propose two simple methods of estimation and derive their asymptotic properties. A Monte Carlo study was carried out to evaluate the performance of these two methods.     

Umvu estimator of the probability in the geometric distribution with unknown truncation parameter

The uniformly minimum variance unbiased estimator of the probability in the geometric distribution with unknown truncation parameter is constructed.     

Parameters estimation for weibull distributed lifetimes under progressive censoring with random removeals

This paper considers the estimation problem when lifetimes are Weibull distributed and are collected under a Type-II progressive censoring with random removals, where the number of units removed at each failure time follows a uniform discrete distribution. The expected time of this censoring plan is discussed and compared numerically to that under a Type II censoring without removal. Maximum likelihood estimator of the parameters and their asymptotic variances are derived.     

Some results on the truncated multivariate t distribution

The use of truncated distributions arises often in a wide variety of scientific problems. In the literature, there are a lot of sampling schemes and proposals developed for various specific truncated distributions. So far, however, the study of the truncated multivariate t (TMVT) distribution is rarely discussed. In this paper, we first present general formulae for computing the first two moments of the TMVT distribution under the double truncation. We formulate the results as analytic matrix expressions, which can be directly computed in existing software. Results for the left and right truncation can be viewed as special cases. We then apply the slice sampling algorithm to generate random variates from the TMVT distribution by introducing auxiliary variables. This strategic approach can result in a series of full conditional densities that are of uniform distributions. Finally, several examples and practical applications are given to illustrate the effectiveness and importance of the proposed results.     

Moment expressions and summary statistics for the complete and truncated weibull distribution

Traditionally, the moments of the Weibull distribution have been calculated using the standard Weibull (Johnson and Kotz, 1970) . This article will expand on that idea and cover the truncated cases for the standard Weibull distributions. Also, the same techniques used for the standard form will be used to derive the moment expressions for the three-parameter complete and truncated Weibull distributions. The summary statistics are then calculated from the moment expressions. Weibull moments involve the gamma and incomplete gamma functions.     

On the construction of bayesian prediction limits for the weibull distribution

This paper is concerned with a BAYESian construction of the prediction limits for the Weibull distribution as an example of extreme value distributions. Thus, considering Weibull and Uniform distributions for the parameters, the predictive functions, which may lead to approximative evaluation of the prediction limits, is determined by using simulation methods     

Some results on interval estimation for the two-parameter weibull or extreme-value distribution

Two statistics based on simple, closed form estimators are examined for use in interval estimation of reliability and of the location parameter of the extreme-value distribution. Properties of the estimators are studied by Monte Carlo simulation, and procedures for interval estimation and tests of hypotheses for the location parameter and reliability are provided.     

Exponential family distribution with unknown truncation parameter: two sample problem

The uniformly most powerful unbiased tests are formulated for two sample problem of a given continuous distribution belonging to the exponential family with unknown scale and truncation parameters. The two-parameter exponential and Paretc distributions are considered in examples.