Estimating the parameters of Poisson-exponential models

This paper proposes two methods of estimation for the parameters in a Poisson-exponential model. The proposed methods combine the method of moments with a regression method based on the empirical moment generating function. One of the methods is an adaptation of the mixed-moments procedure of Koutrouvelis & Canavos (1999). The asymptotic distribution of the estimator obtained with this method is derived. Finite-sample comparisons are made with the maximum likelihood estimator and the method of moments. The paper concludes with an exploratory-type analysis of real data based on the empirical moment generating function.     

Parameter estimates and tests of fit for infinite mixther distrirutions

Al though mixtures form a rich class of probability models, they often present difficulties for statistical inference. Likelihood functions are sometimes unbounded at certain values of the parameters, and densities often have no closed form. These features complicate hoth maximum-likelihood estimation and tests of fit based on the empirical distribution function. New inferential methods using sample characteristic functions (Cfs) and moment generating functions (MGFs) seem well-suited to mixtures. since these transforms often take simple form/ This paper reports a simulation study of the properties of estimators and tests of fit based on CFs, MGFs, and sample moments when applied to three specific families of thick tailed mixture distributios.     

Inference based on adaptive grid selection of probability transforms

Probability transform-based inference, for example, characteristic function-based inference, is a good alternative to likelihood methods when the probability density function is unavailable or intractable. However, a set of grids needs to be determined to provide an effective estimator based on probability transforms. This paper is concerned with parametric inference based on adaptive selection of grids. By employing a closeness measure to evaluate the asymptotic variance of the transform-based estimator, we propose a statistical inference procedure, accompanied with adaptive grid selection. The selection algorithm aims for a small set of grids, and yet the resulting estimator can be highly efficient. Generally, the asymptotic variance is very close to that of the maximum likelihood estimator.     

Some simple estimators for the two-parameter gamma distribution

ABSTRACT

In this paper, we propose two new simple estimation methods for the two-parameter gamma distribution. The first one is a modified version of the method of moments, whereas the second one makes use of some key properties of the distribution. We then derive the asymptotic distributions of these estimators. Also, bias-reduction methods are suggested to reduce the bias of these estimators. The performance of the estimators are evaluated through a Monte Carlo simulation study. The probability coverages of confidence intervals are also discussed. Finally, two examples are used to illustrate the proposed methods.     

General mathematical properties,regression and applications of the log-gamma-generated family

The construction of some wider families of continuous distributions obtained recently has attracted applied statisticians due to the analytical facilities available for easy computation of special functions in programming software. We study some general mathematical properties of the log-gamma-generated (LGG) family defined by Amini, MirMostafaee, and Ahmadi (2014 Amini, M., S. M. T. K. MirMostafaee, and J. Ahmadi. 2014. Log-gamma-generated families of distributions. Statistics 48:913–32.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). It generalizes the gamma-generated class pioneered by Risti? and Balakrishnan (2012 Risti?, M. M., and N. Balakrishnan. 2012. The gamma exponentiated exponential distribution. Journal of Statistical Computation and Simulation 82:1191–206.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). We present some of its special models and derive explicit expressions for the ordinary and incomplete moments, generating and quantile functions, mean deviations, Bonferroni and Lorenz curves, Shannon entropy, Rényi entropy, reliability, and order statistics. Models in this family are compared with nested and non nested models. Further, we propose and study a new LGG family regression model. We demonstrate that the new regression model can be applied to censored data since it represents a parametric family of models and therefore can be used more effectively in the analysis of survival data. We prove that the proposed models can provide consistently better fits in some applications to real data sets.     

Book Reviews

Books reviewed:
R.R. Peck, L.D. Haugh and A. Goodman; Statistical Case Studies. A Collaboration between Academe and Industry
Christian Gouriéroux;ARCH Models and Financial Applications
E.L. Lehmann;Elements of Large-Sample Theory
R.B. Schinazi; Classical and Spatial Stochastic Processes     

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.     

A Dimensional CLT for Non-Central Wilks' Lambda in Multivariate Analysis

Abstract.  We consider the non-central distribution of the classical Wilks' lambda statistic for testing the general linear hypothesis in MANOVA. We prove that as the dimension of the observation vector goes to infinity, Wilks' lambda obeys a central limit theorem under simple growth conditions on the non-centrality matrix. In one case we also prove a stronger result: the saddlepoint cumulative distribution function (CDF) approximation for the standardized version of Wilks' lambda converges uniformly on compact sets to the standard normal CDF. These theoretical results go some way towards explaining why saddlepoint approximations to the distribution of Wilks' lambda retain excellent accuracy in high-dimensional cases.     

New inference procedures for generalized Poisson distributions

A common feature for compound Poisson and Katz distributions is that both families may be viewed as generalizations of the Poisson law. In this paper, we present a unified approach in testing the fit to any distribution belonging to either of these families. The test involves the probability generating function, and it is shown to be consistent under general alternatives. The asymptotic null distribution of the test statistic is obtained, and an effective bootstrap procedure is employed in order to investigate the performance of the proposed test with real and simulated data. Comparisons with classical methods based on the empirical distribution function are also included.     

Asymptotic expansions for the moments of serial correlation coefficients

This work is concerned with evaluating the moments of a number of serial correlation coefficients which arise in various ways and where the observations are from the first order autoregressive Gaussian process with known zero mean. The forms considered have biases whose main parts (of order 0(n^-1) , where n is the sample size) are substantially different. They are the intra-class correlation,the maximum likelihood estimators and an estimator whose main part of the bias is sere. The moments are obtained as asymptotic expansions in terms of the parameter of the process and to terms of order 0(n^-3). It is found that removing certain end terms in the denominator of a serial correlation has the effect of reducing the magnitude of the main part of its bias considerably and in one case completely eliminating it. This work extends the results of various authors,e.g.Kandall(1954), Marriott and pope(1954) and white (1961) in the special cases of the first order autogressive process.     

Classification of dichotomous and continuous variables with incomplete samples

This Article Considers the problem of classifiying an observation consisting of both binary and continuous variables based on two general incomplete training samples one from each of the two given populations. The location linear model adopted by krzanowski 1975 forms the basis of our inverstigation. For a given location, When the common dispersion matrix as Well as the corresponding cell probabilities for the underlying populations are known, exact distribution of the conditional maximum likelihood classification rule is derived. The overall error rate can be obtained and is based on linear cominations of independent non– Chi– Distributions. large sample result for the case where the cell probabilities are unknown is also available.     

On Second Order Admissibilities in Two-Parameter Logistic Regression Model

In two-parameter family of distribution, conditions for a modified maximum likelihood estimator to be second-order admissible are given. Applying these results to two-parameter logistic regression model, it is shown that the maximum likelihood estimator is always second-order inadmissible and the Rao-Blackwellized minimum logit chi-squared estimator is second-order admissible if and only if the number of the doses is greater than or equal to 6.     

On a modified version of intervened negative binomial distribution

In this paper, a generalized version of the intervened negative binomial distribution of Kumar and Sreeja [On intervened negative binomial distribution and some of its properties. Statistica. 2012;72:395–404] is considered and studied some of its properties. Certain methods of estimation of the parameters of the distribution are discussed and illustrated with the help of real life data sets. A test procedure is suggested for testing the intervention parameter and a simulation study is conducted for examining the performance of the estimators.     

The alpha–beta skew normal distribution: properties and applications

This paper introduces a new class of skew distributions by extending the alpha skew normal distribution proposed by Elal-Olivero [Elal-Olivero, D. Alpha-skew-normal distribution. Proyecciones. 2010;29:224–240]. Statistical properties of the new family are studied in details. In particular, explicit expressions for the moments and the shape parameters including the skewness and the kurtosis coefficients and the moment generating function are derived. The problem of estimating parameters on the basis of a random sample coming from the new class of distribution is considered. To examine the performance of the obtained estimators, a Monte Carlo simulation study is conducted. Flexibility and usefulness of the proposed family of distributions are illustrated by analysing three real data sets.     

Goodness of fit for the poisson distribution based on the probability generating function

For testing the fit of a discrete distribution, use of the probability generating function and its empirical counterpart has been suggested in Koeherlakota and Kocherlakota (1986). In the present paper, a particular functional of the corresponding empirical probability generating function process is proposed as a measure to test the discrepancy between the evidence and the hypothesis. The asymptotic behavior of the empirical probability generating function when a parameter is estimated is obtained, The study is exemplified for the Poisson case only but the procedure can be extended to other discrete distributions.     

A conditionally-unbiased estimator of population size based on plant-capture in continuous time

This paper proposes an estimator of the unknown size of a target population to which has been added a planted population of known size. The augmented population is observed for a fixed time and individuals are sighted according to independent Poisson processes. These processes may be time-inhomogeneous, but, within each population, the intensity function is the same for all individuals. When the two populations have the same intensity function, known results on factorial series distributions suggest that the proposed estimator is approximately unbiased and provide a useful estimator of standard deviation. Except for short sampling times, computational results confirm that the proposed population-size estimator is nearly unbiased, and indicate that it gives a better overall performance than existing estimators in the literature. The robustness of this performance is investigated in situations in which it cannot be assumed that the behaviour of the plants matches that of individuals from the target population.     

Generalized empirical likelihood inference in partial linear regression model for longitudinal data

In this paper, empirical likelihood inference for longitudinal data within the framework of partial linear regression models are investigated. The proposed procedures take into consideration the correlation within groups without involving direct estimation of nuisance parameters in the correlation matrix. The empirical likelihood method is used to estimate the regression coefficients and the baseline function, and to construct confidence intervals. A nonparametric version of Wilk's theorem for the limiting distribution of the empirical likelihood ratio is derived. Compared with methods based on normal approximations, the empirical likelihood does not require consistent estimators for the asymptotic variance and bias. The finite sample behaviour of the proposed method is evaluated with simulation and illustrated with an AIDS clinical trial data set.     

Parametric estimation of mixtures of two uniform distributions

In this paper, we consider a mixture of two uniform distributions and derive L-moment estimators of its parameters. Three possible ways of mixing two uniforms, namely with neither overlap nor gap, with overlap, and with gap, are studied. The performance of these L-moment estimators in terms of bias and efficiency is compared to that obtained by means of the conventional method of moments (MM), modified maximum likelihood (MML) method and the usual maximum likelihood (ML) method. These intensive simulations reveal that MML estimators are the best in most of the cases, and the L-moment estimators are less subject to bias in estimation for some mixtures and more efficient in most of the cases than the conventional MM estimators. The L-moment estimators are, in some cases, more efficient than the ML and MML estimators.     

Improved two-parameter estimators for the negative binomial and Poisson regression models

Negative binomial regression (NBR) and Poisson regression (PR) applications have become very popular in the analysis of count data in recent years. However, if there is a high degree of relationship between the independent variables, the problem of multicollinearity arises in these models. We introduce new two-parameter estimators (TPEs) for the NBR and the PR models by unifying the two-parameter estimator (TPE) of Özkale and Kaç?ranlar [The restricted and unrestricted two-parameter estimators. Commun Stat Theory Methods. 2007;36:2707–2725]. These new estimators are general estimators which include maximum likelihood (ML) estimator, ridge estimator (RE), Liu estimator (LE) and contraction estimator (CE) as special cases. Furthermore, biasing parameters of these estimators are given and a Monte Carlo simulation is done to evaluate the performance of these estimators using mean square error (MSE) criterion. The benefits of the new TPEs are also illustrated in an empirical application. The results show that the new proposed TPEs for the NBR and the PR models are better than the ML estimator, the RE and the LE.     

Testing the Fit of Gamma Distributions Using the Empirical Moment Generating Function

ABSTRACT

This article presents goodness-of-fit tests for two and three-parameter gamma distributions that are based on minimum quadratic forms of standardized logarithmic differences of values of the moment generating function and its empirical counterpart. The test statistics can be computed without reliance to special functions and have asymptotic chi-squared distributions. Monte Carlo simulations are used to compare the proposed test for the two-parameter gamma distribution with goodness-of-fit tests employing empirical distribution function or spacing statistics. Two data sets are used to illustrate the various tests.