Another generalization of the logarithmic series and the geometric distributions

A new generalization of the logarithmic series distribution is presented based on a generalized negative binomial distribution obtained from a generalized Poisson distribution compounded with the truncated gamma distribution. By length biasing this generalized log-series distribution, another generalized geometric distribution is uresented. For the generalized log-series distribution, maximum likelihood estimators are developed and an example is presented for illustration.     

On a multivariate log-gamma distribution and the use of the distribution in the Bayesian analysis

In this article, we propose a new generalized multivariate log-gamma distribution. We consider the usage of the proposed multivariate distribution as the prior distribution in the Bayesian analysis. The generalized multivariate log-gamma distribution allows for the inclusion of prior knowledge on correlations between model parameters when likelihood is not in the form of a normal distribution. Use of the proposed distribution in the Bayesian analysis of log-linear models is also discussed.     

On the bivariate weighted exponential distribution based on the generalized exponential distribution

In this paper, we introduce a bivariate weighted exponential distribution based on the generalized exponential distribution. This class of distributions generalizes the bivariate distribution with weighted exponential marginals (BWE). We derive different properties of this new distribution. It is a four-parameter distribution, and the maximum-likelihood estimator of unknown parameters cannot be obtained in explicit forms. One data set has been re-analyzed and it is observed that the proposed distribution provides better fit than the BWE distribution.     

Criticism of the predictive distribution

The predictive distribution is a mixture of the original distribution model and is used for predicting a future observation. Therein, the mixing distribution is the posterior distribution of the distribution parameters in the Bayesian inference. The mixture can also be computed for the frequentist inference because the Bayesian posterior distribution has the same meaning as a frequentist confidence interval. I present arguments against the concept of predictive distribution. Examples illustrate these. The most important argument is that the predictive distribution can depend on the parameterization. An improvement of the theory of the predictive distribution is recommended.     

Reducing bias of the maximum-likelihood estimation for the truncated Pareto distribution

The Pareto distribution is an important distribution in statistics, which has been widely used in finance, physics, hydrology, geology, astronomy, and so on. Even though the parameter estimation for the Pareto distribution has been well established in the literature, the estimation problem for the truncated Pareto distribution becomes complex. This article investigates the bias and mean-squared error of the maximum-likelihood estimation for the truncated Pareto distribution, and some useful results are obtained.     

An extension of the half-normal distribution

In this paper we introduce a new distribution, namely, the slashed half-normal distribution and it can be seen as an extension of the half-normal distribution. It is shown that the resulting distribution has more kurtosis than the ordinary half-normal distribution. Moments and some properties are derived for the new distribution. Moment estimators and maximum likelihood estimators can computed using numerical procedures. Results of two real data application are reported where model fitting is implemented by using maximum likelihood estimation. The applications illustrate the better performance of the new distribution.     

On estimation of the PDF and CDF of the Lindley distribution

This article addresses two methods of estimation of the probability density function (PDF) and cumulative distribution function (CDF) for the Lindley distribution. Following estimation methods are considered: uniformly minimum variance unbiased estimator (UMVUE) and maximum likelihood estimator (MLE). Since the Lindley distribution is more flexible than the exponential distribution, the same estimators have been found out for the exponential distribution and compared. Monte Carlo simulations and a real data analysis are performed to compare the performances of the proposed methods of estimation.     

On the Exact and Near-Exact Distributions of the Product of Generalized Gamma Random Variables and the Generalized Variance

In this article, the authors first obtain the exact distribution of the logarithm of the product of independent generalized Gamma r.v.’s (random variables) in the form of a Generalized Integer Gamma distribution of infinite depth, where all the rate and shape parameters are well identified. Then, by a routine transformation, simple and manageable expressions for the exact distribution of the product of independent generalized Gamma r.v.’s are derived. The method used also enables us to obtain quite easily very accurate, manageable and simple near-exact distributions in the form of Generalized Near-Integer Gamma distributions. Numerical studies are carried out to assess the precision of different approximations to the exact distribution and they show the high accuracy of the approximations provided by the near-exact distributions. As particular cases of the exact distributions obtained we have the distribution of the product of independent Gamma, Weibull, Frechet, Maxwell-Boltzman, Half-Normal, Rayleigh, and Exponential distributions, as well as the exact distribution of the generalized variance, the exact distribution of discriminants or Vandermonde determinants and the exact distribution of any linear combination of generalized Gumbel distributions, as well as yet the distribution of the product of any power of the absolute value of independent Normal r.v.’s.     

Modelling the diameter distribution of forest stands using the Burr distribution

SUMMARY For the past 20 years, forest modellers have largely used the Weibull distribution to summarize diameter distributions and to predict timber volume in a forest at clearfell time. The wider family of Burr distributions is investigated here as an alternative to the Weibull distribution. Using data from 20 permanent sample plots of Pinus radiata, we compare Burr and Weibull fits, showing that the Burr distribution enhances precision by roughly 13 . The effectiveness of the Burr distribution for prediction is found to be similar to that of the Weibull distribution.     

Approximation of the Distribution of the Location Parameter in the Growth Curve Model

Abstract.  In this paper an Edgeworth-type approximation of order O(n ⁻² ) to the density of the estimator of the location parameter in the growth curve model has been derived. The approximation is a mixture of a normal and a Kotz-type distribution, thus being an elliptical distribution. A condition for unimodality of the mixture was found and marginal distribution of a subvector of the mixture distribution was derived. Finally, a small example was given to demonstrate an application of the approximation.     

The two-sided power distribution for the treatment of the uncertainty in PERT

The aim of this paper is to include the Two-Sided Power (TSP) distribution in the PERT methodology making use of the advantages that this four-parameter distribution offers. In order to be completely determined, a distribution of this type needs, the same as the beta distribution, a new datum apart from the three usual values a (pessimistic), m (most likely) and b (optimistic). To solve this question, when using the beta distribution in the PERT context, we are looking for the maximum similarity with the normal and so it is required that the distribution has the same variance as the normal or its same kurtosis, giving rise to the constant variance and mesokurtic families, respectively. Nevertheless, while this approach can be only applied to the beta distribution for some values in the range of the standardized mode, in the case of the TSP distribution this methodology leads always to a solution. A detailed analysis comparing the beta and TSP distribution based on their PERT means and variances is presented indicating better results for the second. We are very grateful for the comments and suggestions of two anonymous referees.     

The exact confidence interval for the scale parameter and the mvue of the laplace distribution

The exact distribution of the sample median, and of the maximum likelihood estimator of the scale parameter of the Laplace distribution is derived. Tables of Teans, variances and the distribution functions of the corresponding dislributions are evaluacted. Exact ,solutions to the problem of confidence interval and hypothesrs testing for the scale paramrter are provided. The minimum variance unbiased estimator (MVUE) of the p.d.f. of the Laplace distribution when the location parameter is known is also given.     

An approximation of the inverse moments of the positive hypergeometric distribution

The negative moments of the positive hyper geometric distribution are often approximated by the inverse of the positive moments of this distribution. In this paper, a suitable approximation to the positive hypergeometric distribution is used to obtain the negative moments.     

THE non-null distribution of the likelihood ratio criterion for testing the hypothesis that the covariance matrix is diagonal

This article deals with the exact non-null distribution of the likelihood ratio criterion for testing the hypothesis that the covariance matrix in a multinormal distribution is diagonal. The exact non-null moments as well as the exact non-null distribution are derived. The distribution is also expressed in computable form with the help of inverse Mellin transform and calculus of residues. The results obtained in this article are useful in studying the power of testing several correlation coefficients simultaneously.     

An approximation to the distribution of the product of two dependent correlation coefficients

Many methodological studies depend on the product of two dependent correlation coefficients. However, the behavior of the distribution of the product of two dependent correlation coefficients is not well known. The distribution of sets of correlation coefficients has been well studied, but not the distribution of the product of two dependent correlation coefficients. The present study derives an approximation to the distribution of the product of two dependent correlation coefficients with a closed form, resulting in a Pearson Type I distribution. A simulation study is also conducted to assess the accuracy of the approximation.     

On the mvue for the distribution in the multivariate normal regression

In the multivariate normal regression setting, the estimability of a distribution is studied generalizing earlier results for the univariate case. The MVUE of an estimable distribution is obtained.     

Discordancy Tests Based on the Likelihood Ratio for the Bipolar Watson Distribution on the Hypersphere

As the Watson distribution is frequently used for modeling axial data, it is important to investigate the existence of possible outliers in samples from this distribution. Then, we develop for the bipolar Watson distribution defined on the hypersphere, some tests of discordancy of an outlier or several outliers en bloc based on the likelihood ratio, supposing an alternative model of contamination of slippage type. We evaluate the performance of these tests of discordancy of an outlier and we also compare some tests of discordancy of an outlier available for this distribution.     

A Powerful Method of Assessing the Fit of the Lognormal Distribution

When faced with the problem of goodness-of-fit to the Lognormal distribution, testing methods typically reduce to comparing the empirical distribution function of the corresponding logarithmic data to that of the normal distribution. In this article, we consider a family of test statistics which make use of the moment structure of the Lognormal law. In particular, a continuum of moment conditions is employed in the construction of a new statistic for this distribution. The proposed test is shown to be consistent against fixed alternatives, and a simulation study shows that it is more powerful than several classical procedures, including those utilizing the empirical distribution function. We conclude by applying the proposed method to some, not so typical, data sets.     

A generalization of the beta–binomial distribution

Summary.  The paper describes a distribution generated by the Gaussian hypergeometric function that may be seen as a generalization of the beta–binomial distribution. It is expressed as a generalized beta mixture of a binomial distribution. This new mixing distribution allows the existence of a mode and an antimode, which is very useful for fitting some data sets. Two examples illustrate the greater versatility of the new distribution compared with the beta–binomial distribution.     

On the choice of the prior distribution in hypergeometric sampling

Information in a statistical procedure arising from sources other than sampling is called prior information, and its incorporation into the procedure forms the basis of the Bayesian approach to statistics. Under hypergeometric sampling, methodology is developed which quantifies the amount of information provided by the sample data relative to that provided by the prior distribution and allows for a ranking of prior distributions with respect to conservativeness, where conservatism refers to restraint of extraneous information embedded in any prior distribution. The most conservative prior distribution from a specified class (each member of which carries the available prior information) is that prior distribution within the class over which the likelihood function has the greatest average domination. Four different families of prior distributions are developed by considering a Bayesian approach to the formation of lots. The most conservative prior distribution from each of the four families of prior distributions is determined and compared for the situation when no prior information is available. The results of the comparison advocate the use of the Polya (beta-binomial) prior distribution in hypergeometric sampling.