Kahadawala Cooray

A Generalization of the Half-Normal Distribution with Applications to Lifetime Data

A two-parameter family of lifetime distribution which is derived from a model for static fatigue is presented. This derivation follows from considerations of the relationship between static fatigue crack extension and the failure time of a certain specimen. The cumulative distribution function (cdf) of this new family is quite similar to the cdf of the half-normal distribution, and therefore this density is referred to as the generalized half-normal distribution (GHN). Furthermore, this GHN family is a special case of the three-parameter generalized gamma distribution. Even though the GHN distribution is a two-parameter distribution, the hazard rate function can form variety of shapes such as monotonically increasing, monotonically decreasing, and bathtub shapes. Some properties of this family are given, and examples are cited to compare with other commonly used failure time distributions such as Weibull, gamma, lognormal, and Birnbaum-Saunders.

Modeling actuarial data with a composite lognormal-Pareto model

The actuarial and insurance industries frequently use the lognormal and the Pareto distributions to model their payments data. These types of payment data are typically very highly positively skewed. Pareto model with a longer and thicker upper tail is used to model the larger loss data, while the larger data with lower frequencies as well as smaller data with higher frequencies are usually modeled by the lognormal distribution. Even though the lognormal model covers larger data with lower frequencies, it fades away to zero more quickly than the Pareto model. Furthermore, the Pareto model does not provide a reasonable parametric fit for smaller data due to its monotonic decreasing shape of the density. Therefore, taking into account the tail behavior of both small and large losses, we were motivated to look for a new avenue to remedy the situation. Here we introduce a two-parameter smooth continuous composite lognormal-Pareto model that is a two-parameter lognormal density up to an unknown threshold value and a two-parameter Pareto density for the remainder. The resulting two-parameter smooth density is similar in shape to the lognormal density, yet its upper tail is larger than the lognormal density and the tail behavior is quite similar to the Pareto density. Parameter estimation techniques and properties of this new composite lognormal-Pareto model are discussed and we compare its performance with the other commonly used models. A simulated example and a well-known fire insurance data set are analyzed to show the importance and applicability of this newly proposed composite lognormal-Pareto model.

Generalization of the Weibull distribution: the odd Weibull family

A three-parameter generalization of the Weibull distribution is presented to deal with general situations in modeling survival process with various shapes in the hazard function. This generalized Weibull distribution will be referred to as the odd Weibull family, as it is derived by considering the distributions of the odds of the Weibull and inverse Weibull families. As a result, the odd Weibull family is not only useful for testing goodness-of-fit of the Weibull and inverse Weibull as submodels, but it is also convenient for modeling and fitting different data sets, especially in the presence of censoring. The model parameters for uncensored data are estimated in two different ways because of the fact that the inverse transformation of the odd Weibull family does not change its density function. Adequacy of the model for the given uncensored data is illustrated by using the plot of scaled fitted total time on test (TTT) transforms. Furthermore, simulation studies are conducted to measure the discrepancy between empirical and fitted TTT transforms by using a previously proposed test statistic. Three different examples are, respectively, providedbasedondatafromsurvival, reliabilityandenvironmentalsciencestoillustrateincreasing, bathtub and unimodal failure rates.

Generalized Gumbel distribution

A generalization of the Gumbel distribution is presented to deal with general situations in modeling univariate data with broad range of skewness in the density function. This generalization is derived by considering a logarithmic transformation of an odd Weibull random variable. As a result, the generalized Gumbel distribution is not only useful for testing goodness-of-fit of Gumbel and reverse-Gumbel distributions as submodels, but it is also convenient for modeling and fitting a wide variety of data sets that are not possible to be modeled by well-known distributions. Skewness and kurtosis shapes of the generalized Gumbel distribution are illustrated by constructing the Galton’s skewness and Moor’s kurtosis plane. Parameters are estimated by using maximum likelihood method in two different ways due to the fact that the reverse transformation of the proposed distribution does not change its density function. In order to illustrate the flexibility of this generalization, wave and surge height data set is analyzed, and the fitness is compared with Gumbel and generalized extreme value distributions.

The Weibull–Pareto Composite Family with Applications to the Analysis of Unimodal Failure Rate Data

The Weibull distribution is composited with Pareto model to obtain a flexible, reliable long-tailed parametric distribution for modeling unimodal failure rate data. The hazard function of the composite family accommodates decreasing and unimodal failure rates, which are separated by the boundary line of the space of shape parameter, gamma, when it equals to a known constant. The least square and maximum likelihood parameter estimation techniques are discussed. The advantages of using the proposed family are demonstrated and compared by illustrating well-known examples: guinea pigs survival time data, head and neck cancer data, and nasopharynx cancer survival data.

The Folded Logistic Distribution

ABSTRACT Physical measurements like dimensions, including time, and angles in scientific experiments are frequently recorded without their algebraic sign. The directions of those physical quantities measured with respect to a frame of reference in most practical applications are considered to be unimportant and are ignored. As a consequence, the underlying distribution of measurements is replaced by a distribution of absolute measurements. When the underlying distribution is logistic, the resulting distribution is called the “folded logistic distribution”. Here, the properties of the folded logistic distribution will be presented and the techniques for estimating parameters will be given. The advantages of using this folded logistic distribution over the folded normal distribution will be discussed and some examples will be cited.

A log-linear regression model for the odd Weibull distribution with censored data

We introduce the log-odd Weibull regression model based on the odd Weibull distribution (Cooray, 2006). We derive some mathematical properties of the log-transformed distribution. The new regression model represents a parametric family of models that includes as sub-models some widely known regression models that can be applied to censored survival data. We employ a frequentist analysis and a parametric bootstrap for the parameters of the proposed model. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to assess global influence. Further, for different parameter settings, sample sizes and censoring percentages, some simulations are performed. In addition, the empirical distribution of some modified residuals are given and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be extended to a modified deviance residual in the proposed regression model applied to censored data. We define martingale and deviance residuals to check the model assumptions. The extended regression model is very useful for the analysis of real data.

Analyzing Grouped, Censored, and Truncated Data Using the Odd Weibull Family

In recent advances of Weibull generalization, the odd Weibull family has been shown to be useful not only for lifetime data modeling, but also discriminating between Weibull and inverse Weibull distributions. This three-parameter distribution accommodates seven different hazard shapes and a wide variety of shapes of the density function including bimodality. In addition, the odd Weibull parameters can be estimated in two different ways since the inverse transformation of the family does not change its density function. In this article, we adapted this two-way estimation method for analyzing grouped, censored, and truncated data that frequently encountered in survival analysis.

Bayesian estimators of the lognormal–Pareto composite distribution

In this paper, Bayesian methods with both Jeffreys and conjugate priors for estimating parameters of the lognormal–Pareto composite (LPC) distribution are considered. With Jeffreys prior, the posterior distributions for parameters of interest are derived and their properties are described. The conjugate priors are proposed and the conditional posterior distributions are provided. In addition, simulation studies are performed to obtain the upper percentage points of Kolmogorov–Smirnov and Anderson–Darling test statistics. Furthermore, these statistics are used to compare Bayesian and likelihood estimators. In order to clarify and advance the validity of Bayesian and likelihood estimators of the LPC distribution, well-known Danish fire insurance data-set is reanalyzed.

Analyzing lifetime data with long-tailed skewed distribution: the logistic-sinh family

A new two-parameter family of distribution is presented. It is derived to model the highly negatively skewed data with extreme observations. The new family of distribution is referred to as the logistic-sinh distribution, as it is derived from the logistic distribution by appropriately replacing an exponential term with a hyperbolic sine term. The resulting family provides not only negatively skewed densities with thick tails but also variety of monotonic density shapes. The space of shape parameter, lambda greater than zero is divided by boundary line of lambda equals one, into two regions over which the hazard function is, respectively, increasing and bathtub shaped. The maximum likelihood parameter estimation techniques are discussed by providing approximate coverage probabilities for uncensored samples. The advantages of using the new family are demonstrated and compared by illustrating well known examples.