The Effect of Mis-Specification Between the Lognormal and Weibull Distributions on the Interval Estimation of a Quantile for Complete Data

The lognormal and Weibull distributions are the most popular distributions for modeling lifetime data. In practical applications, they usually fit the data at hand well. However, their predictions may lead to large differences. The main purpose of the present article is to investigate the impacts of mis-specification between the lognormal and Weibull distributions on the interval estimation of a pth quantile of the distributions for complete data. The coverage probabilities of the confidence intervals (CIs) with mis-specification are evaluated. The results indicate that for both the lognormal and the Weibull distributions, the coverage probabilities are significantly influenced by mis-specification, especially for a small or a large p on lower or upper tail of the distributions. In addition, based on the coverage probabilities with correct and mis-specification, a maxmin criterion is proposed to make a choice between these two distributions. The numerical results indicate that for p ≤ 0.05 and 0.6 ≤ p ≤ 0.8, Weibull distribution is suggested to evaluate CIs of a pth quantile of the distributions, while, for 0.2 ≤ p ≤ 0.5 and p = 0.99, lognormal distribution is suggested to evaluate CIs of a pth quantile of the distributions. Besides, for p = 0.9 and 0.95, lognormal distribution is suggested if the sample size is large enough, while, for p = 0.1, Weibull distribution is suggested if the sample size is large enough. Finally, a simulation study is conducted to evaluate the efficiency of the proposed method.     

Power-Law Adjusted Survival Models

A simple adjustment to parametric failure-time distributions, which allows for much greater flexibility in the shape of the hazard-rate function, is considered. Analytical expressions for the distributions of the power-law adjusted Weibull, gamma, log-gamma, generalized gamma, lognormal, and Pareto distributions are given. Most of these allow for bathtub-shaped and other multi-modal forms of the hazard rate. The new distributions are fitted to real failure-time data which exhibit a multi-modal hazard-rate function and the fits are compared.     

Resampling Plans for Frailty Models

Methods for selecting a distributional model for a random variable such as ambient air quality concentration are examined. Specific consideration is given to identification of a model from the exponential, lognormal, Weibull and gamma distributions. The performance of a likelihood ratio statistic and a Kolmogorov ratio statistic are examined by Monte Carlo simulation. On the basis of these results a procedure for increasing the probability of correct selection is proposed.     

Parameter measures of skewness

Several different measures of skewness are commonly used in place of γ₁, the third central moment divided by the cube of the standard deviation. The numerical values of these measures are compared in this paper for members of the gamma, lognormal or Weibull family of distributions and shown to vary considerably in most cases even when skewness and kurtosis are moderate.     

THE LOG-EIG DISTRIBUTION: A NEW PROBABILITY MODEL FOR LIFETIME DATA

ABSTRACT

In this paper, we propose a new probability model called the log-EIG distribution for lifetime data analysis. Some important properties of the proposed model and maximum likelihood estimation of its parameters are discussed. Its relationship with the exponential inverse Gaussian distribution is similar to that of the lognormal and the normal distributions. Through applications to well-known datasets, we show that the log-EIG distribution competes well, and in some instances even provides a better fit than the commonly used lifetime models such as the gamma, lognormal, Weibull and inverse Gaussian distributions. It can accommodate situations where an increasing failure rate model is required as well as those with a decreasing failure rate at larger times.     

A new generalization of the gamma distribution with application to negatively skewed survival data

We propose a three-parameter distribution referred to as the reflected- shifted-truncated gamma (RSTG) distribution to model negatively skewed data. Various properties of the proposed distribution are derived. The estimation of the model parameters is approached by maximum likelihood methods and the observed information matrix is derived. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation for both small and large samples. Using information theoretic criteria, we compare the RSTG distribution to the exponential, generalized F, generalized gamma, Gompertz, log-logistic, lognormal, Rayleigh, and Weibull distributions in three negatively skewed real datasets.     

A NEW APPROACH TO MAXIMUM LIKELIHOOD ESTIMATION OF THE THREE-PARAMETER GAMMA AND WEIBULL DISTRIBUTIONS

A new approach, is proposed for maximum likelihood (ML) estimation in continuous univariate distributions. The procedure is used primarily to complement the ML method which can fail in situations such as the gamma and Weibull distributions when the shape parameter is, at most, unity. The new approach provides consistent and efficient estimates for all possible values of the shape parameter. Its performance is examined via simulations. Two other, improved, general methods of ML are reported for comparative purposes. The methods are used to estimate the gamma and Weibull distributions using air pollution data from Melbourne. The new ML method is accurate when the shape parameter is less than unity and is also superior to the maximum product of spacings estimation method for the Weibull distribution.     

On characteristic functions of the wrapped lognormal and the wrapped Weibull distributions

The characteristic function plays a prominent role in determining the pdf of a circular model using the trigonometric moments. The characteristic functions of the wrapped lognormal and the wrapped Weibull distributions cannot be expressed in a closed form. Hence, numerical evaluation of the same is presented along with graphs. Also, certain population characteristics of the wrapped lognormal and the wrapped Weibull distributions are presented.     

Some Sampling Properties of a Model for Income Distribution

A recently proposed model for describing the distribution of income over a population, based on the Burr distribution, has been shown to fit better than the commonly used lognormal or gamma distributions. The current article extends that analysis by deriving the large-sample properties of the maximum likelihood estimates for this three-parameter model. Consequently, resulting confidence intervals for some measures of income inequality (including the Gini index) are used to further test the model's validity, as well as to examine apparent trends in inequality over time. Since these properties depend on the way the income data are grouped and censored, implications for choosing data-report intervals can be analyzed. Specifically, a choice between two common methods of reporting the data is shown to have an important impact on Gini index estimates.     

Generalized exponential distributions

The three-parameter gamma and three-parameter Weibull distributions are commonly used for analysing any lifetime data or skewed data. Both distributions have several desirable properties, and nice physical interpretations. Because of the scale and shape parameters, both have quite a bit of flexibility for analysing different types of lifetime data. They have increasing as well as decreasing hazard rate depending on the shape parameter. Unfortunately both distributions also have certain drawbacks. This paper considers a three-parameter distribution which is a particular case of the exponentiated Weibull distribution originally proposed by Mudholkar, Srivastava & Freimer (1995) when the location parameter is not present. The study examines different properties of this model and observes that this family has some interesting features which are quite similar to those of the gamma family and the Weibull family, and certain distinct properties also. It appears this model can be used as an alternative to the gamma model or the Weibull model in many situations. One dataset is provided where the three-parameter generalized exponential distribution fits better than the three-parameter Weibull distribution or the three-parameter gamma distribution.     

M31. Least squares solutions over interval restrictions

In this article, the three-parameter I.G. distribution is standardized with zero mean and unit variance. The third standard moment α₃ is employed as the shape parameter. Tables of the cumulative probability function are given as a function of the standardized variate z, and of the shape parameter, α₃. Various comparisons are made with the lognormal, Weibull, and gamma distributions.     

An empirical power study of multi-sample tests for exponentiality

Results are given of an empirical power study of three statistical procedures for testing for exponentiality of several independent samples. The test procedures are the Tiku (1974) test, a multi-sample Durbin (1975) test, and a multi-sample Shapiro–Wilk (1972) test. The alternative distributions considered in the study were selected from the gamma, Weibull, Lomax, lognormal, inverse Gaussian, and Burr families of positively skewed distributions. The general behavior of the conditional mean exceedance function is used to classify each alternative distribution. It is shown that Tiku's test generally exhibits overall greater power than either of the other two test procedures. For certain alternative distributions, Shapiro–Wilk's test is superior when the sample sizes are small.     

On finite mixtures of geometric and negative binomial distributions

Finite mixtures of distributions have been getting increasing use in the applied literature. In the continuous case, linear combinations of exponentials and gammas have been shown to be well suited for modeling purposes. In the discrete case, the focus has primarily been on continuous mixing, usually of Poisson distributions and typically using gammas to describe the random parameter, But many of these applications are forced, especially when a continuous mixing distribution is used. Instead, it is often prefe-rable to try finite mixtures of geometries or negative binomials, since these are the fundamental building blocks of all discrete random variables. To date, a major stumbling block to their use has been the lack of easy routines for estimating the parameters of such models. This problem has now been alleviated by the adaptation to the discrete case of numerical procedures recently developed for exponential, Weibull, and gamma mixtures. The new methods have been applied to four previously studied data sets, and significant improvements reported in goodness-of-fit, with resultant implications for each affected study.     

The beta generalized gamma distribution

For the first time, a new five-parameter distribution, called the beta generalized gamma distribution, is introduced and studied. It contains at least 25 special sub-models such as the beta gamma, beta Weibull, beta exponential, generalized gamma (GG), Weibull and gamma distributions and thus could be a better model for analysing positive skewed data. The new density function can be expressed as a linear combination of GG densities. We derive explicit expressions for moments, generating function and other statistical measures. The elements of the expected information matrix are provided. The usefulness of the new model is illustrated by means of a real data set.     

A study of moments and likelihood estimators of the odd Weibull distribution

The odd Weibull distribution is a three-parameter generalization of the Weibull and the inverse Weibull distributions having rich density and hazard shapes for modeling lifetime data. This paper explored the odd Weibull parameter regions having finite moments and examined the relation to some well-known distributions based on skewness and kurtosis functions. The existence of maximum likelihood estimators have shown with complete data for any sample size. The proof for the uniqueness of these estimators is given only when the absolute value of the second shape parameter is between zero and one. Furthermore, elements of the Fisher information matrix are obtained based on complete data using a single integral representation which have shown to exist for any parameter values. The performance of the odd Weibull distribution over various density and hazard shapes is compared with generalized gamma distribution using two different test statistics. Finally, analysis of two data sets has been performed for illustrative purposes.     

Characterizations of distributions through coincidences of semiparametric families

Semiparametric families are families that have both a real parameter and a parameter that is itself a distribution. A number of semiparametric families suitable for lifetime data are introduced: scale, power, frailty (proportional hazards), age, moment, Laplace transform, and convolution parameter families. The coincidence of two families provides a characterization of the underlying distribution. Characterizations of the Weibull, gamma, lognormal, and Gompertz distributions are obtained.     

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.     

A new family of generalized distributions

Kumaraswamy [Generalized probability density-function for double-bounded random-processes, J. Hydrol. 462 (1980), pp. 79–88] introduced a distribution for double-bounded random processes with hydrological applications. For the first time, based on this distribution, we describe a new family of generalized distributions (denoted with the prefix ‘Kw’) to extend the normal, Weibull, gamma, Gumbel, inverse Gaussian distributions, among several well-known distributions. Some special distributions in the new family such as the Kw-normal, Kw-Weibull, Kw-gamma, Kw-Gumbel and Kw-inverse Gaussian distribution are discussed. We express the ordinary moments of any Kw generalized distribution as linear functions of probability weighted moments (PWMs) of the parent distribution. We also obtain the ordinary moments of order statistics as functions of PWMs of the baseline distribution. We use the method of maximum likelihood to fit the distributions in the new class and illustrate the potentiality of the new model with an application to real data.     

Hierarchical likelihood approach for the Weibull frailty model

In this article, the proportional hazard model with Weibull frailty, which is outside the range of the exponential family, is used for analysing the right-censored longitudinal survival data. Complex multidimensional integrals are avoided by using hierarchical likelihood to estimate the regression parameters and to predict the realizations of random effects. The adjusted profile hierarchical likelihood is adopted to estimate the parameters in frailty distribution, during which the first- and second-order methods are used. The simulation studies indicate that the regression-parameter estimates in the Weibull frailty model are accurate, which is similar to the gamma frailty and lognormal frailty models. Two published data sets are used for illustration.     

A class of regression models for parallel and series systems with a random number of components

We extend the Weibull power series (WPS) class of distributions to the new class of extended Weibull power series (EWPS) class of distributions. The EWPS distributions are related to series and parallel systems with a random number of components, whereas the WPS distributions [Morais AL, Barreto-Souza W. A compound class of Weibull and power series distributions. Computational Statistics and Data Analysis. 2011;55:1410–1425] are related to series systems only. Unlike the WPS distributions, for which the Weibull is a limiting special case, the Weibull law is a particular case of the EWPS distributions. We prove that the distributions in this class are identifiable under a simple assumption. We also prove stochastic and hazard rate order results and highlight that the shapes of the EWPS distributions are markedly more flexible than the shapes of the WPS distributions. We define a regression model for the EWPS response random variable to model a scale parameter and its quantiles. We present the maximum likelihood estimator and prove its consistency and asymptotic normal distribution. Although series and parallel systems motivated the construction of this class, the EWPS distributions are suitable for modelling a wide range of positive data sets. To illustrate potential uses of this model, we apply it to a real data set on the tensile strength of coconut fibres and present a simple device for diagnostic purposes.