On Some New α-Modified Binomial and Poisson Distributions and Their Applications

A new class of α-modified binomial distribution has been proposed, and its distributional properties like probability generating function (pgf), moments, and their interrelations have been studied. Two new α-modified Poisson distributions and Poisson distribution have been obtained as limiting distributions. Modified binomial and Poisson distributions introduced by Berg and Jaworski (1988 Berg , S. , Jaworski , J. ( 1988 ). Modified binomial and Poisson distributions with application in random mapping theory . J. Statist. Plann. Infer. 18 : 313 – 322 . [Google Scholar]) have been seen as particular cases. Mixture distributions of α-modified binomial distributions have been derived. A new distributions called α-modified binomial distributions of type j, their moment properties, limiting distributions as α-modified Poisson distribution of type j, their different convolution properties, pgf, parameter estimators have been studied. Two more new distributions namely Doubly α-modified binomial distributions of type (i, j) and α-modified weighted generalized Poisson distributions of type (j ? 1) have also been studied. Various α-modified binomial and Poisson distributions of Berg and Mutafchiev (1990 Berg , S. , Mutafchiev , L. ( 1990 ). Random mapping with an attracting center: Lagrangian distributions and a regression function . J. Appl. Probab. 27 : 622 – 636 . [Google Scholar]) and Berg and Nowicki (1991 Berg , S. , Nowicki , K. ( 1991 ). Statistical inference for a class of modified power series distributions with applications to random mapping theory . J. Statist. Plann. Infer. 28 : 247 – 261 . [Google Scholar]) have been seen as special cases. Application of some of these proposed distributions have been identified.     

On the Properties of the Likelihood Function of Spanos' Conditional t Heteroskedastic Model

This article investigates the properties of the likelihood function of Spanos’ conditional t heteroskedastic model (Spanos, 1994 Spanos , A. ( 1994 ). On modeling heteroskedasticity: The student's t and elliptical linear regression models . Econometric Theor. 10 : 286 – 315 .[Crossref], [Web of Science ®] , [Google Scholar]) On modeling heteroskedasticity: the student's t and elliptical linear regression models. It is shown that estimability of the degrees of freedom of t distribution and the block-diagonality of the information matrix of the joint likelihood function with respect to conditional mean parameters and remaining parameters hold for the model. The joint maximum likelihood estimator and its inference based on the t-statistic and χ²-statistic are examined in finite samples by simulation when the degrees of freedom is known and unknown.     

On the performance of the poisson and the negative binomial ridge predictors

Mansson and Shukur (2011 Mansson, K., Shukur, G. (2011). A Poisson ridge regression estimator. Economic Modelling 28:1475–1481. [Google Scholar]) investigated the performance of the Poisson ridge regression (PRR) estimator in terms of the mean square error (MSE) criterion. Similarly, Mansson (2012 Mansson, K. (2012). On ridge estimators for the negative binomial regression model. Economic Modelling 29:178–184. [Google Scholar]) investigated the performance of the Negative binomial ridge regression (NBRR) according to the MSE criterion. But there is no any analysis of the predictive performance of the PRR and NBRR estimators. Therefore, we define the PRR and the NBRR predictors to evaluate their predictive performances according to the prediction mean squared error under the target function. The Monte Carlo simulations and the real life numerical example are conducted to investigate the defined predictors' performance.     

Some Remarks on Testing Overdispersion in Zero-Inflated Poisson and Binomial Regression Models

This note extends the score test statistics for overdispersion in Poisson and binomial regression models (Dean, 1992 Dean , C. B. ( 1992 ). Testing for overdispersion in Poisson and binomial regression models . J. Amer. Statist. Assoc. 87 : 451 – 457 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) to the zero-inflated models. Some general results are obtained, and examples illustrate the application of the extended results.     

The exponential Poisson logarithmic distribution

Abstract

In this article, we proposed a new three parameter lifetime distribution motivated mainly by lifetime issues, which generalizes the Exponential Poisson distribution proposed by Cancho et al. (2011) Cancho, V.G., Louzada-Neto, F., Barriga, G.D. (2011). The poisson-exponential lifetime distribution. Computat. Statist. Data Anal. 55:677–686.[Crossref], [Web of Science ®] , [Google Scholar]. We derive various standard mathematical properties of the proposed model including a formal proof of its probability density function and hazard rate function. The inference via the maximum likelihood approach is discussed. The performance of the maximum likelihood estimators, the likelihood ratio test and its power are studied by simulation. Finally, the proposed model is fitted to two real data sets and it is compared with several models.     

An approximation to the convolution of gamma distributions

In general, the exact distribution of a convolution of independent gamma random variables is quite complicated and does not admit a closed form. Of all the distributions proposed, the gamma-series representation of Moschopoulos (1985 Moschopoulos, P. G. (1985). The distribution of the sum of independent gamma random variables. Annals of the Institute of Statistical Mathematics 37Part A:541–544. [Google Scholar]) is relatively simple to implement but for particular combinations of scale and/or shape parameters the computation of the weights of the series can result in complications with too much time consuming to allow a large-scale application. Recently, a compact random parameter representation of the convolution has been proposed by Vellaisamy and Upadhye (2009 Vellaisamy, P., Upadhye, N. S. (2009). On the sums of compound negative binomial and gamma random variables. Journal of Applied Probability 46:272–283.[Crossref], [Web of Science ®] , [Google Scholar]) and it allows to give an exact interpretation to the weights of the series. They describe an infinite discrete probability distribution. This result suggested to approximate Moschopoulos’s expression looking for an approximating theoretical discrete distribution for the weights of the series. More precisely, we propose a general negative binomial distribution. The result is an “excellent” approximation, fast and simple to implement for any parameter combination.     

The Alpha-power Tobit Model

The main object of this article is to propose an extension of the tobit model for which the error distribution follows the power-normal distribution (Gupta and Gupta, 2008 Gupta , D. , Gupta , R. C. ( 2008 ). Analyzing skewed data by power normal model . Test 17 : 197 – 210 .[Crossref], [Web of Science ®] , [Google Scholar]). Inference is dealt with by using the likelihood approach. Simulation studies and application to a real data set are used to demonstrate the usefulness of the extension.     

New Properties of the Kumaraswamy Distribution

The Kumaraswamy distribution is very similar to the Beta distribution but has the key advantage of a closed-form cumulative distribution function. This makes it much better suited than the Beta distribution for computation-intensive activities like simulation modeling and the estimation of models by simulation-based methods. However, in spite of the fact that the Kumaraswamy distribution was introduced in 1980, further theoretical research on the distribution was not developed until very recently (Garg, 2008 Garg , M. ( 2008 ). On distribution of order statistics from Kumaraswamy distribution . Kyunpook Mathemat. J. 48 : 411 – 417 .[Crossref] , [Google Scholar]; Jones, 2009 Jones , M. C. ( 2009 ). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages . Statist. Methodol. 6 ( 1 ): 70 – 81 .[Crossref] , [Google Scholar]; Mitnik, 2009 Mitnik , P. ( 2009 ). The Kumaraswamy distribution: a median-dispersion re-parameterization for regression modeling and simulation-based estimation. Available at SSRN: http://ssrn.com/abstract=1231587 . [Google Scholar]; Nadarajah, 2008 Nadarajah , S. ( 2008 ). On the distribution of Kumaraswamy . J. Hydrol. 348 : 568 – 569 .[Crossref], [Web of Science ®] , [Google Scholar]). This article contributes to this recent research and: (a) shows that Kumaraswamy variables exhibit closeness under exponentiation and under linear transformation; (b) derives an expression for the moments of the general form of the distribution; (c) specifies some of the distribution's limiting distributions; and (d) introduces an analytical expression for the mean absolute deviation around the median as a function of the parameters of the distribution, and establishes some bounds for this dispersion measure and for the variance.     

Finite-sample distributions of the Wald,likelihood ratio,and Lagrange multiplier test statistics in the classical linear model

In this note, it is shown that the finite-sample distributions of the Wald, likelihood ratio, and Lagrange multiplier statistics in the classical linear regression model are members of the generalized beta model introduced by McDonald and Xu (1995a McDonald, J.B., Xu, Y.J. (1995a). A generalization of the beta distribution with applications. J. Econom. 66:133–152.[Crossref], [Web of Science ®] , [Google Scholar]). This is useful for examining the properties of these test statistics. For example, this characterization makes it easy to find distribution, quantile, and density functions for each test statistic, makes it clear why Wald tests may overreject the null hypothesis using asymptotic critical values, and formalizes the fact that the Lagrange multiplier statistic follows a distribution with bounded support.     

An extended Birnbaum–Saunders distribution: Theory,estimation, and applications

Abstract

Birnbaum and Saunders (1969a Birnbaum, Z.W., Saunders, S.C. (1969a). A new family of life distributions. J. Appl. Probab. 6:319–327.[Crossref], [Web of Science ®] , [Google Scholar]) pioneered a lifetime model which is commonly used in reliability studies. Based on this distribution, a new model called the gamma Birnbaum–Saunders distribution is proposed for describing fatigue life data. Several properties of the new distribution including explicit expressions for the ordinary and incomplete moments, generating and quantile functions, mean deviations, density function of the order statistics, and their moments are derived. We discuss the method of maximum likelihood and a Bayesian approach to estimate the model parameters. The superiority of the new model is illustrated by means of three failure real data sets. We also propose a new extended regression model based on the logarithm of the new distribution. The last model can be very useful to the analysis of real data and provide more realistic fits than other special regression models.     

A new lifetime model with different types of failure rate

A new class of lifetime distributions, which can exhibit with upside-down bathtub-shaped, bathtub-shaped, decreasing, and increasing failure rates, is introduced. The new distribution is constructed by compounding generalized Weibull and logarithmic distributions, leading to improvement on the lifetime distribution considered in Dimitrakopoulou et al. (2007 Dimitrakopoulou, T., K. Adamidis, and S. Loukas. 2007. A lifetime distribution with an upside-down bathtub-shaped hazard function. IEEE Transactions on Reliability 56:308–11.[Crossref], [Web of Science ®] , [Google Scholar]) by having no restriction on the shape parameter and extending the result studied by Tahmasbi and Rezaei (2008 Tahmasbi, R., and S. Rezaei. 2008. A two-parameter lifetime distribution with decreasing failure rate. Computational Statistics and Data Analysis 52:3889–901.[Crossref], [Web of Science ®] , [Google Scholar]) in the general form. The proposed model includes the exponential–logarithmic and Weibull–logarithmic distributions as special cases. Various statistical properties of the proposed class are discussed. Furthermore, estimation via the maximum likelihood method and the Fisher information matrix are discussed. Applications to real data demonstrate that the new class of distributions is more flexible than other recently proposed classes.     

Stirling's Formula and Its Extensions: Heuristic Approaches

Walsh (1995 Walsh , D. P. ( 1995 ). Equating Poisson and normal probability functions to derive Stirling's formula . Amer. Statist. 49 : 270 – 271 .[Taylor & Francis Online] , [Google Scholar]) introduced a heuristic approach to motivate Stirling's formula by equating a Poisson probability to an analogous value from a normal density function. We explore similar heuristics to derive approximations for various binomial, negative binomial, and multinomial coefficients. Also, using heuristics markedly different from those of Walsh, we develop an approximation of (nk)! for positive integers n (large) and k. These heuristics are then used to validate Stirling's formula for Γ(nα) where α is a positive real number. To derive each of our approximations we use a different probability distribution, and hence each section may serve as pedagogical module.     

Empirical likelihood ratio-based goodness-of-fit test for the logistic distribution

The logistic distribution has been used to model growth curves in survival analysis and biological studies. In this article, we propose a goodness-of-fit test for the logistic distribution based on the empirical likelihood ratio. The test is constructed based on the methodology introduced by Vexler and Gurevich [17 A. Vexler and G. Gurevich, Empirical likelihood ratios applied to goodness-of-fit tests based on sample entropy, Comput. Stat. Data Anal. 54 (2010), pp. 531–545. doi: 10.1016/j.csda.2009.09.025[Crossref], [Web of Science ®] , [Google Scholar]]. In order to compute the test statistic, parameters of the distribution are estimated by the method of maximum likelihood. Power comparisons of the proposed test with some known competing tests are carried out via simulations. Finally, an illustrative example is presented and analyzed.     

An Improvement of Poisson Approximation for Sums of Dependent Bernoulli Random Variables

This article uses the Stein-Chen method to obtain new non uniform bounds on the error of the cumulative distribution function of sums of dependent Bernoulli random variables and the Poisson cumulative distribution function. The bounds obtained in the present study are sharper than those reported in Teerapabolarn and Neammanee (2006 Teerapabolarn, K., Neammanee, K. (2006). Poisson approximation for sums of dependent Bernoulli random variables. Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis. 22:87–99. [Google Scholar]). Examples are provided to illustrate applications of the obtained results.     

Negative variance components for non-negative hierarchical data with correlation,over-, and/or underdispersion

The concept of negative variance components in linear mixed-effects models, while confusing at first sight, has received considerable attention in the literature, for well over half a century, following the early work of Chernoff [7 H. Chernoff, On the distribution of the likelihood ratio, Ann. Math. Statist. 25 (1954), pp. 573–578.[Crossref] , [Google Scholar]] and Nelder [21 J.A. Nelder, The interpretation of negative components of variance, Biometrika 41 (1954), pp. 544–548.[Crossref], [Web of Science ®] , [Google Scholar]]. Broadly, negative variance components in linear mixed models are allowable if inferences are restricted to the implied marginal model. When a hierarchical view-point is adopted, in the sense that outcomes are specified conditionally upon random effects, the variance–covariance matrix of the random effects must be positive-definite (positive-semi-definite is also possible, but raises issues of degenerate distributions). Many contemporary software packages allow for this distinction. Less work has been done for generalized linear mixed models. Here, we study such models, with extension to allow for overdispersion, for non-negative outcomes (counts). Using a study of trichomes counts on tomato plants, it is illustrated how such negative variance components play a natural role in modeling both the correlation between repeated measures on the same experimental unit and over- or underdispersion.     

A new log-location regression model: estimation,influence diagnostics and residual analysis

We introduce a log-linear regression model based on the odd log-logistic generalized half-normal distribution [7 G.M. Cordeiro, M. Alizadeh, R.R. Pescim, and E.M.M. Ortega, The odd log-logistic generalized half-normal lifetime distribution: Properties and applications, Comm. Statist. Theory Methods (2015), accepted for publication. [Google Scholar]]. Some of its structural properties including explicit expressions for the density function, quantile and generating functions and ordinary moments are derived. We estimate the model parameters by the maximum likelihood method. For different parameter settings, proportion of censoring and sample size, some simulations are performed to investigate the behavior of the estimators. We derive the appropriate matrices for assessing local influence diagnostics on the parameter estimates under different perturbation schemes. We also define the martingale and modified deviance residuals to detect outliers and evaluate the model assumptions. In addition, we demonstrate that the extended regression model can be very useful in the analysis of real data and provide more realistic fits than other special regression models. The potentiality of the new regression model is illustrated by means of a real data set.     

The generalized exponential cure rate model with covariates

In this article, we consider a parametric survival model that is appropriate when the population of interest contains long-term survivors or immunes. The model referred to as the cure rate model was introduced by Boag 1 Boag, J. W. 1949. Maximum likelihood estimates of the proportion of patients cured by cancer therapy. J. R. Stat. Soc. Ser. B, 11: 15–53.  [Google Scholar] in terms of a mixture model that included a component representing the proportion of immunes and a distribution representing the life times of the susceptible population. We propose a cure rate model based on the generalized exponential distribution that incorporates the effects of risk factors or covariates on the probability of an individual being a long-time survivor. Maximum likelihood estimators of the model parameters are obtained using the the expectation-maximisation (EM) algorithm. A graphical method is also provided for assessing the goodness-of-fit of the model. We present an example to illustrate the fit of this model to data that examines the effects of different risk factors on relapse time for drug addicts.     

The Beta Exponential-Geometric Distribution

A new four-parameter distribution with decreasing, increasing, and upside-down bathtub failure rate called the beta exponential-geometric distribution is proposed. The new distribution, generated from the logit of a beta random variable, extends the exponential-geometric distribution of Adamidis and Loukas (1998 Adamidis , K. , Loukas , S. ( 1998 ). A lifetime distribution with decreasing failure rate . Statistics and Probability Letters 39 : 35 – 42 .[Crossref], [Web of Science ®] , [Google Scholar]) and some other distributions. A comprehensive mathematical treatment of this distribution is provided. Some expressions for the moment generating function, moments, order statistics, and Rényi entropy of the new distribution are derived. Estimation of the stress-strength parameter is also obtained. The model parameters are estimated by the maximum likelihood method and Fisher information matrix is discussed. Finally, an application to a real data set is illustrated.     

A new cure rate model based on the Yule–Simon distribution with application to a melanoma data set

In this paper, a new survival cure rate model is introduced considering the Yule–Simon distribution [12 H.A. Simon, On a class of skew distribution functions, Biometrika 42 (1955), pp. 425–440.[Crossref], [Web of Science ®] , [Google Scholar]] to model the number of concurrent causes. We study some properties of this distribution and the model arising when the distribution of the competing causes is the Weibull model. We call this distribution the Weibull–Yule–Simon distribution. Maximum likelihood estimation is conducted for model parameters. A small scale simulation study is conducted indicating satisfactory parameter recovery by the estimation approach. Results are applied to a real data set (melanoma) illustrating the fact that the model proposed can outperform traditional alternative models in terms of model fitting.     

A Unified Approach to Ordering Comparison of GPS Distributions with their Mixtures

Recently, several authors have been concerned with ordering comparison of known distributions of the family of generalized power series (GPS) distributions with their mixtures in various senses. In this article, we shall employ a unified approach and obtain similar results, more generally, for all members of the class of the GPS distributions. Some of the previous findings of Misra et al. (2003 Misra , N. , Singh , H. , Harner , E. J. ( 2003 ). Stochastic comparisons of Poisson and binomial random variables with their mixtures . Statist. Probab. Lett. 65 : 279 – 290 .[Crossref], [Web of Science ®] , [Google Scholar]), Alamatsaz and Abbasi (2008 Alamatsaz , M. H. , Abbasi , S. ( 2008 ). Ordering comparison of negative binomial random variables with their mixtures . Statist. Probab. Lett. 78 : 2234 – 2239 . [Google Scholar]), and Aghababaei Jazi and Alamatsaz (2010 Aghababaei Jazi , M. , Alamatsaz , M. H. ( 2010 ). Ordering comparison of logarithmic series random variables with their mixtures . Commun. Statist. Theor. Meth. 39 : 3252 – 3263 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) in this connection, then, follow as corollaries. Further, we have derived some more ordering comparison results.