A discrete analog of Gumbel distribution: properties,parameter estimation and applications

A discrete version of the Gumbel distribution (Type-I Extreme Value distribution) has been derived by using the general approach of discretization of a continuous distribution. Important distributional and reliability properties have been explored. It has been shown that depending on the choice of parameters the proposed distribution can be positively or negatively skewed; possess long-tail(s). Log-concavity of the distribution and consequent results have been established. Estimation of parameters by method of maximum likelihood, method of moments, and method of proportions has been discussed. A method of checking model adequacy and regression type estimation based on empirical survival function has also been examined. A simulation study has been carried out to compare and check the efficacy of the three methods of estimations. The distribution has been applied to model three real count data sets from diverse application area namely, survival times in number of days, maximum annual floods data from Brazil and goal differences in English premier league, and the results show the relevance of the proposed distribution.     

A new discrete distribution: properties and applications in medical care

This paper proposes a simple and flexible count data regression model which is able to incorporate overdispersion (the variance is greater than the mean) and which can be considered a competitor to the Poisson model. As is well known, this classical model imposes the restriction that the conditional mean of each count variable must equal the conditional variance. Nevertheless, for the common case of well-dispersed counts the Poisson regression may not be appropriate, while the count regression model proposed here is potentially useful. We consider an application to model counts of medical care utilization by the elderly in the USA using a well-known data set from the National Medical Expenditure Survey (1987), where the dependent variable is the number of stays after hospital admission, and where 10 explanatory variables are analysed.     

The discrete Lindley distribution: properties and applications

Modelling count data is one of the most important issues in statistical research. In this paper, a new probability mass function is introduced by discretizing the continuous failure model of the Lindley distribution. The model obtained is over-dispersed and competitive with the Poisson distribution to fit automobile claim frequency data. After revising some of its properties a compound discrete Lindley distribution is obtained in closed form. This model is suitable to be applied in the collective risk model when both number of claims and size of a single claim are implemented into the model. The new compound distribution fades away to zero much more slowly than the classical compound Poisson distribution, being therefore suitable for modelling extreme data.     

A Discrete Distribution Arising as a Solution of a Linear Difference Equation: Extension of the Non Central Negative Binomial Distribution

This article considers a discrete distribution that arises as the dominant solution of a linear difference equation. Basic properties and various chance mechanisms that lead to this distribution are given. In particular, its formulation as a weighted distribution and a mixed Poisson process are proposed. Parameter estimation by (a) using a combination of observed frequencies and moments and (b) maximum likelihood are examined. An example of goodness of fit is considered.     

The quantile-based flattened logistic distribution: Some properties and applications

In this paper, the quantile-based flattened logistic distribution has been studied. Some classical and quantile-based properties of the distribution have been obtained. Closed form expressions of L-moments, L-moment ratios and expectation of order statistics of the distribution have been obtained. A quantile-based analysis concerning the method of matching L-moments estimation is employed to estimate the parameters of the proposed model. We further derive the asymptotic variance–covariance matrix of the matching L-Moments estimators of the proposed model. Finally, we apply the proposed model to simulated as well as two real life datasets and compare the fit with the logistic distribution.     

The robustness of the likelihood ratio test,the nonparametric sum rank test,and f-ratio tests when the populations are from the negative binomial family

In this study, the robustness of power and significance level of several statistical testing methods was evaluated under the assumption that the test populations were from Poisson, negative binomial, or geometric distributions. The F-ratio test, with or without appropriate transformations, was shown to be both safe and robust for all distributions examined.     

The McDonald Gompertz distribution: Properties and applications

This article introduces a five-parameter lifetime model called the McDonald Gompertz (McG) distribution to extend the Gompertz, generalized Gompertz, generalized exponential, beta Gompertz, and Kumaraswamy Gompertz distributions among several other models. The hazard function of new distribution can be increasing, decreasing, upside-down bathtub, and bathtub shaped. We obtain several properties of the McG distribution including moments, entropies, quantile, and generating functions. We provide the density function of the order statistics and their moments. The parameter estimation is based on the usual maximum likelihood approach. We also provide the observed information matrix and discuss inferences issues. The flexibility and usefulness of the new distribution are illustrated by means of application to two real datasets.     

Symbolic computation for approximating distributions of some families of one and two-sample nonparametric test statistics

A conditional saddlepoint approximation was provided by Gatto and Jammalamadaka (1999) for computing the distribution function of many test statistics based on dependent quantities like multinomial frequencies, spacing frequencies, etc. The considerable complexity of the formulas involved can be bypassed by symbolic computation. This article illustrates the effectiveness of symbolic computation to evaluate the saddlepoint approximation for the likelihood ratio, the exponential score, and the Wald-Wolfowitz test statistics. The case of composite hypotheses is also discussed.     

The exponentiated reduced Kies distribution: Properties and applications

Here we consider an exponentiated version of the reduced Kies distribution and discuss some of its properties. The parameters of the distribution are estimated by the method of maximum likelihood and illustrated with the help of certain real-life data sets. Asymptotic behavior of the maximum likelihood estimators of the parameters of the distribution is also studied by using certain simulated data sets.     

The Burr XII System of densities: properties,regression model and applications

We introduce a new class of distributions called the Burr XII system of densities with two extra positive parameters. We provide a comprehensive treatment of some of its mathematical properties. We estimate the model parameters by maximum likelihood. We assess the performance of the maximum likelihood estimators in terms of biases and mean squared errors by means of a simulation study. We also introduce a new family of regression models based on this system of densities. The usefulness of the proposed models is illustrated by means of three real data sets.