Random sampling from the watson distribution

An envelope-rejection method is used to generate random variates from the Watson distribution. The method is compact and is competitive with, if not superior to, the existing sampling algorithms. For the girdle form of the Watson distribution, a faster algorithm is proposed. As a result, Johnson's sampling algorithm for the Bingham distribution is improved.     

A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution

Summary.  A useful discrete distribution (the Conway–Maxwell–Poisson distribution) is revived and its statistical and probabilistic properties are introduced and explored. This distribution is a two-parameter extension of the Poisson distribution that generalizes some well-known discrete distributions (Poisson, Bernoulli and geometric). It also leads to the generalization of distributions derived from these discrete distributions (i.e. the binomial and negative binomial distributions). We describe three methods for estimating the parameters of the Conway–Maxwell–Poisson distribution. The first is a fast simple weighted least squares method, which leads to estimates that are sufficiently accurate for practical purposes. The second method, using maximum likelihood, can be used to refine the initial estimates. This method requires iterations and is more computationally intensive. The third estimation method is Bayesian. Using the conjugate prior, the posterior density of the parameters of the Conway–Maxwell–Poisson distribution is easily computed. It is a flexible distribution that can account for overdispersion or underdispersion that is commonly encountered in count data. We also explore two sets of real world data demonstrating the flexibility and elegance of the Conway–Maxwell–Poisson distribution in fitting count data which do not seem to follow the Poisson distribution.     

An extension of the gamma distribution

ABSTRACT

The gamma distribution has been widely used in many research areas such as engineering and survival analysis. We present an extension of this distribution, called the Kummer beta gamma distribution, having greater flexibility to model scenarios involving skewed data. We derive analytical expressions for some mathematical quantities. The estimation of parameters is approached by the maximum likelihood method and Bayesian analysis. The likelihood ratio and formal goodness-of-fit tests are used to compare the presented distribution with some of its sub-models and non nested models. A real data set is used to illustrate the importance of the distribution.     

The generalized Poisson-binomial distribution and the computation of its distribution function

The Poisson-binomial distribution is useful in many applied problems in engineering, actuarial science and data mining. The Poisson-binomial distribution models the distribution of the sum of independent but non-identically distributed random indicators whose success probabilities vary. In this paper, we extend the Poisson-binomial distribution to a generalized Poisson-binomial (GPB) distribution. The GPB distribution corresponds to the case where the random indicators are replaced by two-point random variables, which can take two arbitrary values instead of 0 and 1 as in the case of random indicators. The GPB distribution has found applications in many areas such as voting theory, actuarial science, warranty prediction and probability theory. As the GPB distribution has not been studied in detail so far, we introduce this distribution first and then derive its theoretical properties. We develop an efficient algorithm for the computation of its distribution function, using the fast Fourier transform. We test the accuracy of the developed algorithm by comparing it with enumeration-based exact method and the results from the binomial distribution. We also study the computational time of the algorithm under various parameter settings. Finally, we discuss the factors affecting the computational efficiency of the proposed algorithm and illustrate the use of the software package.     

Optimal estimation for the Pareto distribution

This paper proposes an optimal estimation method for the shape parameter, probability density function and upper tail probability of the Pareto distribution. The new method is based on a weighted empirical distribution function. The exact efficiency functions of the estimators relative to the existing estimators are derived. The paper gives L ₁-optimal and L ₂-optimal weights for the new weighted estimator. Monte Carlo simulation results confirm the theoretical conclusions. Both theoretical and simulation results show that the new estimation method is more efficient relative to several existing methods in many situations.     

Slashed generalized exponential distribution

In this paper, we introduce an extension of the generalized exponential (GE) distribution, making it more robust against possible influential observations. The new model is defined as the quotient between a GE random variable and a beta-distributed random variable with one unknown parameter. The resulting distribution is a distribution with greater kurtosis than the GE distribution. Probability properties of the distribution such as moments and asymmetry and kurtosis are studied. Likewise, statistical properties are investigated using the method of moments and the maximum likelihood approach. Two real data analyses are reported illustrating better performance of the new model over the GE model.     

Modified slash distribution

In this paper we introduce a modified slash distribution obtained by modifying the usual slash distribution. This new distribution is based on the quotient of two independent random variables, whose distributions are the normal and the power of an exponential distribution of scale parameter equals to two, respectively. In this way, the result is a new distribution whose kurtosis values are greater when compared with that of the slash distribution. We study the density, some properties, moments, kurtosis and make inferences by the method of moments and maximum likelihood. We introduce a multivariate version of this new distribution. Moreover, we provide two illustrations with real data showing that the new distribution fits better the data than the ordinary slash distribution.     

Design-based confidence limits for a distribution function

A method for constructing confidence limits for a distribution function is proposed. This method is a simple modification of the common method based on a normal approximation to the distribution of the estimated distribution function. The methods differ in how the estimated standard errors are used. The coverage properties of the two methods are compared in a simulation study. Coverage probabilities for the proposed method are found to be much closer to the nominal levels, particularly in the tails of the population distribution.     

A generalized binomial distribution

A new generalization of the binomial distribution is introduced that allows dependence between trials, nonconstant probabilities of success from trial to trial, and which contains the usual binomial distribution as a special case. Along with the number of trials and an initial probability of ‘success’, an additional parameter that controls the degree of correlation between trials is introduced. The resulting class of distributions includes the binomial, unirnodal distributions, and bimodal distributions. Formulas for the moments, mean, and variance of this distribution are given along with a method for fitting the distribution to sample data.     

A simple exact method for testing hypotheses about the shape parameter of a log-normal distribution

The log-normal distribution is a useful lifetime distribution in many areas. The survival function of a log-normal distribution cannot be expressed in close forms. This makes it difficult to develop exact statistical methods for parameter estimation when censoring occurs. This article proposes a simple and exact method for conducting statistical tests about the shape parameter of a log-normal distribution. Necessary tables are provided based on Monte Carlo simulation. The method can be used for type II censored data. Comparing with existing exact methods, this method uses fewer tables and is easier for calculations.     

Moments of order statistics of the standard two-sided power distribution

The standard two-sided power distribution is a flexible distribution having uniform, power function and triangular as subdistributions, and it is a reasonable alternative to the Laplace distribution in some cases. In this work, computationally efficient expressions for moments of order statistics, expressions for L-moments, and asymptotic results for sample extrema are derived. Then a simulation study is performed for the location-scale estimation problem of a small data set by considering the maximum likelihood estimation method and the best linear unbiased estimation method based on the moments of order statistics.     

Maximum likelihood estimation of the logarithmic series distribution

This paper discusses the maximum likelihood estimation of the parameter of the logarithmic series distribution. The univariate case is treated in Part I, the multivariate case in Part II. A simple numerical estimation procedure is suggested using a fixed point approach. Convergence to the maximum likelihood estimator is shown. In Part III convergence rate is proven to be linear which is also demonstrated through example. In addition, comparisons with Newton’s method and the secant method in the univariate case, and with Newton’s method and the projected gradient method in the multivariate case are provided.     

Zero-modified power series distribution and its Hurdle distribution version

This paper presents a new family of distributions for count data, the so called zero-modified power series (ZMPS), which is an extension of the power series (PS) distribution family, whose support starts at zero. This extension consists in modifying the probability of observing zero of each PS distribution, enabling the new zero-modified distribution to appropriately accommodate data which have any amount of zero observations (for instance, zero-inflated or zero-deflated data). The Hurdle distribution version of the ZMPS distribution is presented. PS distributions included in the proposed ZMPS family are the Poisson, Generalized Poisson, Geometric, Binomial, Negative Binomial and Generalized Negative Binomial distributions. The paper also describes the properties and particularities of the new distribution family for count data. The distribution parameters are estimated via maximum likelihood method and the use of the new family is illustrated in three real data sets. We emphasize that the new distribution family can accommodate sets of count data without any previous knowledge on the characteristic of zero-inflation or zero-deflation present in the data.     

Bayesian estimation for the Pareto income distribution

The Bayes estimators of the Gini index, the mean income and the proportion of the population living below a prescribed income level are obtained in this paper on the basis of censored income data from a pareto income distribution. The said estimators are obtained under the assumptions of a two-parameter exponential prior distribution and the usual squared error loss function. This work is also extended to the case when the income data are grouped and the exact incomes for the individuals in the population are not available. The method for the assessment of the hyperparameters is also outlined. Finally, the results are generalized for the doubly truncated gamma prior distribution. Now deceased.     

Some computational aspects of the generalized von Mises distribution

This article deals with some important computational aspects of the generalized von Mises distribution in relation with parameter estimation, model selection and simulation. The generalized von Mises distribution provides a flexible model for circular data allowing for symmetry, asymmetry, unimodality and bimodality. For this model, we show the equivalence between the trigonometric method of moments and the maximum likelihood estimators, we give their asymptotic distribution, we provide bias-corrected estimators of the entropy, the Akaike information criterion and the measured entropy for model selection, and we implement the ratio-of-uniforms method of simulation.     

Simulation of the p-generalized Gaussian distribution

In this paper, we introduce the p-generalized polar methods for the simulation of the p-generalized Gaussian distribution. On the basis of geometric measure representations, the well-known Box–Muller method and the Marsaglia–Bray rejecting polar method for the simulation of the Gaussian distribution are generalized to simulate the p-generalized Gaussian distribution, which fits much more flexibly to data than the Gaussian distribution and has already been applied in various fields of modern sciences. To prove the correctness of the p-generalized polar methods, we give stochastic representations, and to demonstrate their adequacy, we perform a comparison of six simulation techniques w.r.t. the goodness of fit and the complexity. The competing methods include adapted general methods and another special method. Furthermore, we prove stochastic representations for all the adapted methods.     

Some properties of the dirichlet-multinomial distribution and its use in prior elicitation

Two results on the unimodality of the Dirichlet-multinomial distribution are proved, and a further result is alos proved on the identifiability of mixtures of multinomial distributions. These properties are used in developing a method for eliciting a Dirchlet prior distribution. The elicitation method is based on the mode, and region around the mode, of the Dirichlet-multinomial predictive distribution.     

The Gamma–Kumaraswamy distribution: An alternative to Gamma distribution

In this article, a new generalization of the Kumaraswamy distribution, namely the Gamma–Kumaraswamy distribution, is defined and studied. Various properties of the Gamma–Kumaraswamy are obtained. The structural analysis of the distribution in this article includes limiting behavior, mode, quantiles, moments, skewness, kurtosis, Shannon’s entropy, and order statistics. The method of maximum likelihood estimation is proposed for estimating the model parameters. For illustrative purposes, two real datasets are analyzed as application of the Gamma–Kumaraswamy distribution.     

Residual-based specification of the random-effects distribution for cluster data

We propose a method for specifying the distribution of random effects included in a model for cluster data. The class of models we consider includes mixed models and frailty models whose random effects and explanatory variables are constant within clusters. The method is based on cluster residuals obtained by assuming that the random effects are equal between clusters. We exhibit an asymptotic relationship between the cluster residuals and variations of the random effects as the number of observations increases and the variance of the random effects decreases. The asymptotic relationship is used to specify the random-effects distribution. The method is applied to a frailty model and a model used to describe the spread of plant diseases.     

Optimal B-robust estimators for the parameters of the Burr XII distribution

The Burr XII distribution offers a flexible alternative to the distributions that play important role for modelling data in reliability, risk and process capability. However, estimating the shape parameters of the Burr XII distribution is a challenging problem. The classical estimation methods such as maximum likelihood and least squares are often used to estimate the parameters of the Burr XII distribution, but these methods are very sensitive to the outliers in the data. Thus, a robust estimation method alternative to the classical methods is needed to find robust estimators that are less sensitive to the outliers in the data. The purpose of this paper is to use the optimal B-robust estimation method [Hampel FR, Ronchetti EM, Rousseeuw PJ, Stahel WA. Robust statistics: the approach based on influence functions. New York: Wiley; 1986] to obtain robust estimators for the shape parameters of the Burr XII distribution. The simulation results show that the optimal B-robust estimators generally outperform the classical estimators in terms of the bias and root mean square errors when there are outliers in data.