Generalized linear regression models incorporating original outcome distributions

ABSTRACT

In this article, we propose an approach for incorporating continuous and discrete original outcome distributions into the usual exponential family regression models. The new approach is an extension of the works of Suissa (1991 Suissa, S. (1991). Binary methods for continuous outcomes: A parametric alternative. J. Clin. Epidemiol. 44:241–248.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) and Suissa and Blais (1995 Suissa, S., Blais, L. (1995). Binary regression with continuous outcomes. Stat. Med. 14:247–255.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), which present methods to estimate the risk of an event defined in a sample subspace of an original continuous outcome variable. Simulation studies are presented in order to illustrate the performance of the developed methodology. Real data sets are analyzed by using the proposed models.     

A Strong Limit Theorem for Weighted Sums of Negatively Dependent Random Variables

In this article, we establish a new complete convergence theorem for weighted sums of negatively dependent random variables. As corollaries, many results on the almost sure convergence and complete convergence for weighted sums of negatively dependent random variables are obtained. In particular, the results of Jing and Liang (2008 Jing, B.Y., Liang, H.Y. (2008). Strong limit theorems for weighted sums of negatively associated random variables. J. Theor. Probab. 21:890–909.[Crossref], [Web of Science ®] , [Google Scholar]), Sung (2012 Sung, S.H. (2012). Complete convergence for weighted sums of negatively dependent random variables. Stat. Pap. 53:73–82.[Crossref], [Web of Science ®] , [Google Scholar]), and Wu (2010) can be obtained.     

A Simple Corrected Score for Logistic Regression with Errors-in-Covariates

We develop a simple corrected score for logistic regression with errors-in-covariates. The new method is an extension of the consistent functional methods proposed by Huang and Wang (2001) and is closely related to the corrected score method by Nakamura (1990 Nakamura, T. (1990). Corrected score function for errors-in-variables models: Methodology and application to generalized linear models. Biometrika. 77:127–137.[Crossref], [Web of Science ®] , [Google Scholar]) and Stefanski (1989 Stefanski, L.A. (1989). Unbiased estimation of a nonlinear function a normal mean with application to measurement error models. Commun. Stat. Ser. A - Theory Methods. 18:4335–4358.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The new method requires that the measurement error distribution is known, but does not require normality. The new method yields a consistent and asymptotically normal estimator under regularity conditions. We examine the finite-sample performance of the new estimator through simulation studies. Finally, we illustrate the new method by applying it to an AIDS study.     

On goodness-of-fit tests for Aalen's additive model based on left-truncated right-censored or doubly censored data

ABSTRACT

Gandy and Jensen (2005 Gandy, A., Jensen, U. (2005). On goodness-of-fit tests for Aalen's additive risk model. Scan. J. Stat. 32:425–445.[Crossref], [Web of Science ®] , [Google Scholar]) proposed goodness-of-fit tests for Aalen's additive risk model. In this article, we demonstrate that the approach of Gandy and Jensen (2005 Gandy, A., Jensen, U. (2005). On goodness-of-fit tests for Aalen's additive risk model. Scan. J. Stat. 32:425–445.[Crossref], [Web of Science ®] , [Google Scholar]) can be applied to left-truncated right-censored (LTRC) data and doubly censored data. A simulation study is conducted to investigate the performance of the proposed tests. The proposed tests are illustrated using heart transplant data.     

Restricted estimation and testing of hypothesis in linear measurement errors models

ABSTRACT

In this article, the linear models with measurement error both in the response and in the covariates are considered. Following Shalabh et al. (2007 Shalabh, Garg, G., Misra, N. (2007). Restricted regression estimation in measurement error models. Comput. Stat. Data Anal. 52:1149–1166.[Crossref], [Web of Science ®] , [Google Scholar], 2009 Shalabh, Garg, G., Misra, N. (2009). Use of prior information in the consistent estimation of regression coefficients in measurement error models. J. Multivariate Anal. 100:1498–1520.[Crossref], [Web of Science ®] , [Google Scholar]), we propose several restricted estimators for the regression coefficients. The consistency and asymptotic normality of the restricted estimators are established. Furthermore, we also discuss the superiority of the restricted estimators to unrestricted estimators under Pitman closeness criterion. We also develop several variance estimators and establish their asymptotic distributions. Wald-type statistics are constructed for testing the linear restrictions. Finally, Monte Carlo simulations are conducted to illustrate the finite-sample properties of the proposed estimators.     

Group Variable Selection with Oracle Property by Weight-Fused Adaptive Elastic Net Model for Strongly Correlated Data

This paper is the generalization of weight-fused elastic net (Fu and Xu, 2012 Fu, G., Xu, Q. (2012). Grouping variable selection by weight fused elastic net for multi-collinear data. Communications in Statistics-Simulation and Computation 41(2):205–221.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), which performs group variable selection by combining weight-fused LASSO(wfLasso) and elastic net (Zou and Hastie, 2005 Zou, H., Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 67(2):301–320.[Crossref], [Web of Science ®] , [Google Scholar]) penalties. In this study, the elastic net penalty is replaced by adaptive elastic net penalty (AdaEnet) (Zou and Zhang, 2009 Zou, H., Zhang, H. (2009). On the adaptive elastic-net with a diverging number of parameters. Annals of Statistics 37(4):1733–1751.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), and a new group variable selection algorithm with oracle property (Fan and Li, 2001 Fan, J., Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96(456):1348–1360.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]; Zou, 2006 Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the American Statistical Association 101(476):1418–1429.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) is obtained.     

Subjective Bayesian Analysis of the Elliptical Model

For the multivariate elliptical model subjective Bayesian estimators of the location vector and some functions of the characteristic matrix with the normal-inverse Wishart and the normal-Wishart as prior, respectively, are derived. Fang and Li (1999 Fang, K.T., Li, R.Z. (1999). Bayesian statistical inference on elliptical matrix distributions. J. Multivariate Anal. 70: 66–85.[Crossref], [Web of Science ®] , [Google Scholar]) considered the elliptical model for Bayesian analysis for an objective prior structure. In addition, the newly developed results are applied to the multivariate normal- and t-distribution. A performance study is done to evaluate the normal-gamma and normal-inverse gamma distributions as suitable priors. A practical application for the posterior distributions of the multivariate t-distribution is included by means of Gibbs sampling and a Metropolis-Hastings algorithm.     

Analysis of Order Statistics from Distributions with Regularly Varying Tails

In this article, necessary conditions for comparing order statistics from distributions with regularly varying tails are discussed in terms of various stochastic orders. A necessary and sufficient condition for stochastically comparing tail behaviors of order statistics is derived. The main results generalize and recover some results in Kleiber (2002 Kleiber, C. (2002). Variability ordering of heavy-tailed distributions with applications to order statistics. Statist. Probab. Lett. 58:381388.[Crossref], [Web of Science ®] , [Google Scholar], 2004 Kleiber, C. (2004). Lorenz ordering of order statistics from log-logistic and related distributions. J. Statist. Plann. Infer. 120:2004.[Crossref], [Web of Science ®] , [Google Scholar]). Extensions to coherent systems are mentioned as well.     

Aspects of Dependence in Generalized Farlie-Gumbel-Morgenstern Distributions

In this article, we obtained a dependence measure for generalized Farlie-Gumbel-Morgenstern (FGM) family in view of Kochar and Gupta (1987 Kochar , S. G. , Gupta , R. P. ( 1987 ). Competitors of Kendall-tau test for testing independence against PQD . Biometrika 74 ( 3 ): 664 – 669 .[Crossref], [Web of Science ®] , [Google Scholar]) and then compared this measure with Spearman's rho and Kendall's tau in FGM family. Moreover, we evaluated the empirical power of the class of distribution-free tests proposed by Kochar and Gupta (1987 Kochar , S. G. , Gupta , R. P. ( 1987 ). Competitors of Kendall-tau test for testing independence against PQD . Biometrika 74 ( 3 ): 664 – 669 .[Crossref], [Web of Science ®] , [Google Scholar], 1990 Kochar , S. G. , Gupta , R. P. ( 1990 ). Distribution-free tests based on sub-sample extrema for testing against positive dependence . Australian Journal of Statistics 32 : 45 – 51 .[Crossref] , [Google Scholar]) based on exact distribution of a U-statistics. This is derived via a simulation study for sample of sizes n = 6, 8, 10, 12, 16, and 20. Also, we compared our simulation results with those achieved by Amini et al. (2010 Amini , M. , Jabbari , H. , Mohtashami Borzadaran , G. R. , Azadbakhsh , M. ( 2010 ). Power comparison of independence test for the Farlie-Gumbel-Moregenstern family . Communications of the Korean Statistical Society 17 ( 4 ): 493 – 505 .[Crossref] , [Google Scholar]) and Güven and Kotz (2008 Güven , B. , Kotz , S. ( 2008 ). Test of independence for generalized Farlie-Gumbel-Morgenstern distributions . Journal of Computational and Applied Mathematics 212 : 102 – 111 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Inference in log-alpha-power and log-skew-normal multivariate models

ABSTRACT

Random vectors with positive components are common in many applied fields, for example, in meteorology, when daily precipitation is measured through a region Marchenko and Genton (2010 Marchenko, Y., Genton, M. (2010). Multivariate log-skew-elliptical distributions with applications to precipitation data. Environmetrics 21:318–340.[Crossref], [Web of Science ®] , [Google Scholar]). Frequently, the log-normal multivariate distribution is used for modeling this type of data. This modeling approach is not appropriate for data with high asymmetry or kurtosis. Consequently, more flexible multivariate distributions than the log-normal multivariate are required. As an alternative to this distribution, we propose the log-alpha-power multivariate and log-skew-normal multivariate models. The first model is an extension for positive data of the fractional order statistics model Durrans (1992 Durrans, S. (1992). Distributions of fractional order statistics in hydrology. Water Resour. Res. 28:1649–1655.[Crossref], [Web of Science ®] , [Google Scholar]). The second one is an extension of the log-skew-normal model studied by Mateu-Figueras and Pawlowsky-Glahn (2007 Mateu-Figueras, G., Pawlowsky-Glahn, V. (2007). The skew-normal distribution on the simplex. Commun. Stat.-Theory Methods 36:1787–1802.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). We study parameter estimation for these models by means of pseudo-likelihood and maximum likelihood methods. We illustrate the proposal analyzing a real dataset.     

On Augmented Franses Tests for Seasonal Unit Roots

This article extends the results reported in del Barrio Castro, Osborn and Taylor (2012 del Barrio Castro, T., Osborn, D.R., Taylor, A. M.R. (2012). On augmented HEGY tests for seasonal unit roots. Econometric Theor. 18:1121–1143.[Crossref], [Web of Science ®] , [Google Scholar]) to the approach followed by Franses (1991a Franses, P. H. (1991a). Model selection and seasonality in time series. Tibergen Institute Series, 18. [Google Scholar],b Franses, P.H. (1991b). Seasonality, non-stationarity and the forecasting of monthly time series. Int. J. Forecast. 7:199–208.[Crossref], [Web of Science ®] , [Google Scholar]) to test for seasonal unit roots, providing the asymptotic representation to the seasonal unit roots tests proposed by Franses for a general number of seasons S.     

Quantile-based Chernoff distance for truncated random variables

Several probability distributions such as power-Pareto distribution (see Gilchrist 2000 Gilchrist, W. 2000. Statistical modelling with quantile functions. Boca Raton, FL: Chapman and Hall/CRC.[Crossref] , [Google Scholar] and Hankin and Lee 2006 Hankin, R. K. S., and A. Lee. 2006. A new family of non-negative distributions. Australian and New Zealand Journal of Statistics 48:67–78.[Crossref], [Web of Science ®] , [Google Scholar]), various forms of lambda distributions (see Ramberg and Schmeiser 1974 Ramberg, J. S., and B. W. Schmeiser. 1974. An appropriate method for generating asymmetric random variables. Communications of the ACM 17:78–82.[Crossref], [Web of Science ®] , [Google Scholar] and Freimer et al. 1988 Freimer, M., S. Mudholkar, G. Kollia, and C. T. Lin. 1988. A study of the generalized lambda family. Communications in Statistics - Theory and Methods 17:3547–67.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Govindarajulu distribution (see Nair, Sankaran, and Vineshkumar 2012 Nair, U. N., P. G. Sankaran, and B. Vineshkumar. 2012. The Govindarajulu distribution: some properties and applications. Communications in Statistics—Theory and Methods 41:4391–406.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), etc., do not have manageable distribution functions, though they have tractable quantile functions. Hence, analytical study of the properties of Chernoff distance of two random variables associated with these distributions via traditional distribution function-based tool becomes difficult. To make this simple, in this paper, we introduce quantile-based Chernoff distance for (left or right) truncated random variables and study its various properties. Some useful bounds as well as characterization results are obtained.     

Autocovariance Function of the Fractionally Integrated Separable Spatial ARMA (FISSARMA) Models

Spatial modeling is important in many fields and there are various kinds of spatial models. One of such models is known as the fractionally integrated separable spatial ARMA (FISSARMA) model. In the area of time series analysis, Sowell (1992 Sowell, F. (1992). Maximum likelihood estimation of stationary univariate fractionally integrated time series models. J. Econ. 53:165–188.[Crossref], [Web of Science ®] , [Google Scholar]) has established the autocovariance function of the long-memory models using hypergeometric function. In this paper we will extend Sowell’s work for FISSARMA models.     

Alternative Approximations to Value-At-Risk: A Comparison

This article compares three value-at-risk (VaR) approximation methods suggested in the literature: Cornish and Fisher (1937 Cornish, E.A., Fisher, R.A. (1937). Moments and cumulants in the specification of distributions. Revue de l’Institut International de Statistique 5:307–320.[Crossref] , [Google Scholar]), Sillitto (1969 Sillitto, G.P. (1969). Derivation of approximants to the inverse distribution function of a continuous univariate population from the order statistics of a sample. Biometrika 56:641–650.[Crossref], [Web of Science ®] , [Google Scholar]), and Liu (2010 Liu, W.-H. (2010). Estimation and testing of portfolio value-at-risk based on L-comoment matrices. Journal of Futures Markets 30:897–908.[Crossref], [Web of Science ®] , [Google Scholar]). Simulation results are obtained for three families of distributions: student-t, skewed-normal, and skewed-t. We recommend the Sillitto approximation as the best method to evaluate the VaR when the financial return has an unknown, skewed, and heavy-tailed distribution.     

Bootstrap test of hypothesis for the multi-state models in survival analysis

Abstract

In the present paper we develop bootstrap tests of hypothesis, based on simulation, for the transition probability matrix arising in the context of a multi-state model. The bootstrap test statistic is based on the paper of Tattar and Vaman (2008 Tattar, P. N., Vaman, H. J. (2008). Testing transition probability matrix of a multi-state model with censored data. Lifetime Data Anal. 14(2):216–230.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), which develops a statistic for the testing problems concerning the transition probability matrix of the non homogeneous Markov process.     

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.     

On Fitting Transformation Model to Survey Data

In this article, we consider fitting a semiparametric linear model to survey data with censored observations. The specific goal of the paper is to extend the methods of Cheng et al. (1995 Cheng, S.C., Wei, L.J., Ying, Z. (1995). Analysis of transformation models with censored data. Biometrika 82(4):835–845.[Crossref], [Web of Science ®] , [Google Scholar]) and Chen et al. (2002 Chen, K., Jin, Z. Ying, Z. (2002). Semiparametric analysis of transformation models with censored data. Biometrika 89:659–668.[Crossref], [Web of Science ®] , [Google Scholar]) to the case when the sample has been drawn from a population using a complex sampling design. Similar to the approach of Lin (2000 Lin, D.Y. (2000). On fitting Cox’s proportional hazards models to survey data. Biometrika 87:37–47.[Crossref], [Web of Science ®] , [Google Scholar]), we regard the survey population as a random sample from an infinite universe and accounts for this randomness in the statistical inference. A simulation study is conducted to investigate the performance of the proposed estimators.     

Fubini theorem for non additive measures in the framework of g-expectation

Since the seminal paper of Ghirardato (1997 Ghirardato, P. 1997. On the independence for non-additive measures, with a Fubini theorem. Journal of Economic Theory 73:261–91.[Crossref], [Web of Science ®] , [Google Scholar]), it is known that Fubini theorem for non additive measures can be available only for functions as “slice-comonotonic” in the framework of product algebra. Later, inspired by Ghirardato (1997 Ghirardato, P. 1997. On the independence for non-additive measures, with a Fubini theorem. Journal of Economic Theory 73:261–91.[Crossref], [Web of Science ®] , [Google Scholar]), Chateauneuf and Lefort (2008 Chateauneuf, A., and J. P. Lefort. 2008. Some Fubini theorems on product σ-algebras for non-additive measures. International Journal of Approximate Reasoning 48:686–96.[Crossref], [Web of Science ®] , [Google Scholar]) obtained some Fubini theorems for non additive measures in the framework of product σ-algebra. In this article, we study Fubini theorem for non additive measures in the framework of g-expectation. We give some different assumptions that provide Fubini theorem in the framework of g-expectation.     

Some Results on Dynamic Generalized Survival Entropy

Recently, Zografos and Nadarajah (2005 Zografos, K., Nadarajah, S. (2005). Survival exponential entropies. IEEE Trans. Inform. Theor. 51:1239–1246.[Crossref], [Web of Science ®] , [Google Scholar]) proposed two measures of uncertainty based on the survival function, called the survival exponential entropy and the generalized survival exponential entropy. In this article, we explore properties of the generalized survival entropy and the dynamic version of it. We study conditions under which the generalized survival entropy of first order statistic can uniquely determines the parent distribution. The exponential, Pareto, and finite range distributions, which are commonly used in reliability, have been characterized using this generalized measure. Another measure of entropy is also introduced in analogy with cumulative entropy which has been proposed by Di Crescenzo and Longobardi (2009) and some properties of it are given.     

On the Additive Weibull Distribution

We derive explicit expressions for the moments, incomplete moments, quantile function and generating function of the additive Weibull model pioneered by Xie and Lai (1995 Xie, M., Lai, C.D. (1995). Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function. Reliab. Eng. Syst. Safety 52:87–93.[Crossref], [Web of Science ®] , [Google Scholar]), which is a quite flexible distribution for fitting lifetime data with bathtub-shaped failure rate function. In addition, we estimate the model parameters by maximum likelihood and determine the observed information matrix. The flexibility of the additive Weibull distribution is illustrated by means of one application to real data.