A new semiparametric Weibull cure rate model: fitting different behaviors within GAMLSS

ABSTRACT

We propose a new semiparametric Weibull cure rate model for fitting nonlinear effects of explanatory variables on the mean, scale and cure rate parameters. The regression model is based on the generalized additive models for location, scale and shape, for which any or all distribution parameters can be modeled as parametric linear and/or nonparametric smooth functions of explanatory variables. We present methods to select additive terms, model estimation and validation, where all computational codes are presented in a simple way such that any R user can fit the new model. Biases of the parameter estimates caused by models specified erroneously are investigated through Monte Carlo simulations. We illustrate the usefulness of the new model by means of two applications to real data. We provide computational codes to fit the new regression model in the R software.     

Robust inference for parsimonious model-based clustering

We introduce a robust clustering procedure for parsimonious model-based clustering. The classical mclust framework is robustified through impartial trimming and eigenvalue-ratio constraints (the tclust framework, which is robust but not affine invariant). An advantage of our resulting mtclust approach is that eigenvalue-ratio constraints are not needed for certain model formulations, leading to affine invariant robust parsimonious clustering. We illustrate the approach via simulations and a benchmark real data example. R code for the proposed method is available at https://github.com/afarcome/mtclust.     

Bayesian adaptive lasso with variational Bayes for variable selection in high-dimensional generalized linear mixed models

This article describes a full Bayesian treatment for simultaneous fixed-effect selection and parameter estimation in high-dimensional generalized linear mixed models. The approach consists of using a Bayesian adaptive Lasso penalty for signal-level adaptive shrinkage and a fast Variational Bayes scheme for estimating the posterior mode of the coefficients. The proposed approach offers several advantages over the existing methods, for example, the adaptive shrinkage parameters are automatically incorporated, no Laplace approximation step is required to integrate out the random effects. The performance of our approach is illustrated on several simulated and real data examples. The algorithm is implemented in the R package glmmvb and is made available online.     

The McDonald Gumbel model

ABSTRACT

We propose a new model called the McDonald Gumbel distribution, the major advantage of which is its ability to fit asymmetric real data that can not be properly adjusted by existing distributions. This model contains as special models the Gumbel, exponentiated Gumbel (Persson and Rydén, 2010 Persson, K., Rydén, J. (2010). Exponentiated Gumbel distribution for estimation of return levels of significant wave height. J. Environ. Statist. 1:1–12. [Google Scholar]), beta Gumbel (Nadarajah and Kotz, 2004 Nadarajah, S., Kotz, S. (2004). The beta Gumbel distribution. Mathemat. Prob. Eng.323–332.[Crossref], [Web of Science ®] , [Google Scholar]), Kumaraswamy Gumbel distributions, among others. We obtain the ordinary moments, quantile and generating functions and mean deviations. The method of maximum likelihood is used to fit the proposed distribution. The applicability of the new model is illustrated by means of two real data sets.     

A general algorithm for non-parametric maximum likelihood estimator of stochastically ordered survival functions from case 2 interval-censored data

In this paper, we study an algorithm to compute the non-parametric maximum likelihood estimator of stochastically ordered survival functions from case 2 interval-censored data. The algorithm, simply denoted by SQP (sequential quadratic programming), re-parameterizes the likelihood function to make the order constraints as a set of linear constraints, approximates the log-likelihood function as a quadratic function, and updates the estimate by solving a quadratic programming. We particularly consider two stochastic orderings, simple and uniform orderings, although the algorithm can also be applied to many other stochastic orderings. We illustrate the algorithm using the breast cancer data reported in Finkelstein and Wolfe (1985 Finkelstein, D. M., and R. A. Wolfe. 1985. A semiparametric model for regression analysis of interval-censored failure time data. Biometrics 41:933–45. [Google Scholar]).     

Log-epsilon-skew normal: A generalization of the log-normal distribution

Abstract

The log-normal distribution is widely used to model non-negative data in many areas of applied research. In this paper, we introduce and study a family of distributions with non-negative reals as support and termed the log-epsilon-skew normal (LESN) which includes the log-normal distributions as a special case. It is related to the epsilon-skew normal developed in Mudholkar and Hutson (2000 Mudholkar, G. S., and A. D. Hutson. 2000. The epsilon-skew-normal distribution for analyzing near-normal data. Journal of Statistical Planning and Inference 83 (2):291–309. doi:10.1016/S0378-3758(99)00096-8.[Crossref], [Web of Science ®] , [Google Scholar]) the way the log-normal is related to the normal distribution. We study its main properties, hazard function, moments, skewness and kurtosis coefficients, and discuss maximum likelihood estimation of model parameters. We summarize the results of a simulation study to examine the behavior of the maximum likelihood estimates, and we illustrate the maximum likelihood estimation of the LESN distribution parameters to two real world data sets.     

Survival analysis for a new compounded bivariate failure time distribution in shock and competing risk models via an EM algorithm

Abstract

In this paper, two bivariate models based on the proposed methods of Marshall and Olkin are introduced. In the first model, the new bivariate distribution is presented based on the proposed method of Marshall and Olkin (1967 Marshall, A. W., and I. Olkin. 1967. A multivariate exponential distribution. Journal of the American Statistical Association 62 (317):30–44. doi: 10.2307/2282907.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) which has natural interpretations, and it can be applied in fatal shock models or in competing risks models. In the second model, the proposed method of Marshall and Olkin (1997 Marshall, A. W., and I. Olkin. 1997. A new method of adding a parameter to a family of distributions with application to the exponential and weibull families. Biometrika 84 (3):641–52. doi: 10.1093/biomet/84.3.641.[Crossref], [Web of Science ®] , [Google Scholar]) is generalized to bivariate case and a new bivariate distribution is introduced. We call these new distributions as the bivariate Gompertz (BGP) distribution and bivariate Gompertz-geometric (BGPG) distribution, respectively. Moreover, the BGP model can be obtained as a special case of the BGPG model. Then, we present various properties of the new bivariate models. In this regard, the joint and conditional density functions, the joint cumulative distribution function can be obtained in compact forms. Also, the aging properties and the bivariate hazard gradient are discussed. This model has five unknown parameters and the maximum likelihood estimators cannot be obtained in explicit form. We propose to use the EM algorithm to compute the maximum likelihood estimators of the unknown parameters, and it is computationally quite tractable. Also, Monte Carlo simulations are performed to investigate the effectiveness of the proposed algorithm. Finally, we analyze three real data sets for illustrative purposes.     

Zero-Inflated NGINAR(1) process

In this paper, we develop a zero-inflated NGINAR(1) process as an alternative to the NGINAR(1) process (Risti?, Nasti?, and Bakouch 2009 Risti?, M. M., A. S. Nasti?, and H. S. Bakouch. 2009. A new geometric first-order integer-valued autoregressive (NGINAR(1)) process. Journal of Statistical Planning and Inference 139:2218–26.[Crossref], [Web of Science ®] , [Google Scholar]) when the number of zeros in the data is larger than the expected number of zeros by the geometric process. The proposed process has zero-inflated geometric marginals and contains the NGINAR(1) process as a particular case. In addition, various properties of the new process are derived such as conditional distribution and autocorrelation structure. Yule-Walker, probability based Yule-Walker, conditional least squares and conditional maximum likelihood estimators of the model parameters are derived. An extensive Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples. Forecasting performances of the model are discussed. Application to a real data set shows the flexibility and potentiality of the new model.     

A Log-Linear Regression Model for the Beta-Weibull Distribution

We introduce the log-beta Weibull regression model based on the beta Weibull distribution (Famoye et al., 2005 Famoye , F. , Lee , C. , Olumolade , O. ( 2005 ). The beta-Weibull distribution . Journal of Statistical Theory and Applications 4 : 121 – 136 . [Google Scholar]; Lee et al., 2007 Lee , C. , Famoye , F. , Olumolade , O. ( 2007 ). Beta-Weibull distribution: Some properties and applications to censored data . Journal of Modern Applied Statistical Methods 6 : 173 – 186 .[Crossref] , [Google Scholar]). We derive expansions for the moment generating function which do not depend on complicated functions. The new regression model represents a parametric family of models that includes as sub-models several widely known regression models that can be applied to censored survival data. We employ a frequentist analysis, a jackknife estimator, and a parametric bootstrap for the parameters of the proposed model. We derive the appropriate matrices for assessing local influences on the parameter estimates under different perturbation schemes and present some ways to assess global influences. Further, for different parameter settings, sample sizes, and censoring percentages, several simulations are performed. In addition, the empirical distribution of some modified residuals are displayed and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be extended to a modified deviance residual in the proposed regression model applied to censored data. We define martingale and deviance residuals to evaluate the model assumptions. The extended regression model is very useful for the analysis of real data and could give more realistic fits than other special regression models.     

Nonparametric test for stratum effects in the cox model with interval-censored data

The testing of the stratum effects in the Cox model is an important and commonly asked question in medical research as well as in many other fields. In this paper, we will discuss the problem where one observes interval-censored failure time data and generalize the procedure given in Sun and Yang (2000 Sun, J., and I. Yang. 2000. Nonparametric test for stratum effects in the cox model. Lifetime Data Analysis 6:321–30.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) for right-censored data. The asymptotic distribution of the new test statistic is established and the simulation study conducted for the evaluation of the finite sample properties of the method suggests that the generalized procedure seems to work well for practical situations. An application is provided.     

Testing economic “genetic pleiotropy” for Box-Cox linear model

Abstract

Genetic pleiotropy occurs when a single gene influences two or more seemingly unrelated phenotypic traits. It is significant to detect pleiotropy and understand its causes. However, most current statistical methods to discover pleiotropy mainly test the null hypothesis that none of the traits is associated with a variant, which departures from the null to test just one associated trait or k associated traits. Schaid et al. (2016 Schaid, D. J., X. Tong, B. Larrabee, R. B. Kennedy, G. A. Poland, and J. P. Sinnwell. 2016. Statistical methods for testing genetic pleiotropy. Genetics 204 (2):483–97. doi:10.1534/genetics.116.189308.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) first proposed a sequential testing framework to analyze pleiotropy based on a linear model and a multivariate normal distribution. In this paper, we analyze the Economic pleiotropy which occurs when an economic action or policy influences two or more economic phenomena. In this paper, we extend the linear model to Box-Cox transformation model and proposed a new decision method. It improves the efficiency of hypothesis test and controls the Type I error. We then apply the method using economic data to multivariate sectoral employments in response to governmental expenditures and provide a quantitative assessment and some insights of different impacts from economic policy.     

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.     

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.     

Using a GLM to Decompose the Symmetry Model in Square Contingency Tables with Ordered Categories

In this paper, we are employing the generalized linear model (GLM) in the form 𝓁_ij=Xλ to decompose the symmetry model into the class of models discussed in Tomizawa (1992 Tomizawa, S. 1992. Quasi-diagonals-parameter symmetry model for square contingency tables with ordered categories. Calcutta Statist. Assoc. Bull., 39: 53–61.  [Google Scholar]). In this formulation, the random component would be the observed counts f _ij with an underlying Poisson distribution. This approach utilizes the non-standard log-linear model and our focus in this paper therefore relates to models that are decompositions of the complete symmetry model. That is, models that are implied by the symmetry models. We develop factor and regression variables required for the implementation of these models in SAS PROC GENMOD and SPSS PROC GENLOG. We apply this methodology to analyse the three 4×4 contingency table, one of which is the Japanese Unaided distance vision data. Results obtained in this study are consistent with those from the numerous literature on the subject. We further extend our applications to the 6×6 Brazilian social mobility data. We found that both the quasi linear diagonal-parameters symmetry (QLDPS) and the quasi 2-ratios parameter symmetry (Q2RPS) models fit the Brazilian data very well. Parsimonious models being the QLDPS and the quasi-conditional symmetry (QCS) models. The SAS and SPSS programs for implementing the models discussed in this paper are presented in Appendices A, B and C.     

Bayesian process monitoring schemes for the two-parameter exponential distribution

In this paper a Bayesian procedure is applied to obtain control limits for the location and scale parameters, as well as for a one-sided upper tolerance limit in the case of the two-parameter exponential distribution. An advantage of the upper tolerance limit is that it monitors the location and scale parameter at the same time. By using Jeffreys’ non-informative prior, the predictive distributions of future maximum likelihood estimators of the location and scale parameters are derived analytically. The predictive distributions are used to determine the distribution of the “run-length” and expected “run-length”. A dataset given in Krishnamoorthy and Mathew (2009 Krishnamoorthy, K., and T. Mathew. 2009. Statistical Tolerance Regions: Theory, Applications and Computation. Wiley Series in Probability and Statistics.[Crossref] , [Google Scholar]) are used for illustrative purposes. The data are the mileages for some military personnel carriers that failed in service. The paper illustrates the flexibility and unique features of the Bayesian simulation method.     

A nonparametric specification test for the volatility functions of diffusion processes

This paper develops a new test for the parametric volatility function of a diffusion model based on nonparametric estimation techniques. The proposed test imposes no restriction on the functional form of the drift function and has an asymptotically standard normal distribution under the null hypothesis of correct specification. It is consistent against any fixed alternatives and has nontrivial asymptotic power against a class of local alternatives with proper rates. Monte Carlo simulations show that the test performs well in finite samples and generally has better power performance than the nonparametric test of Li (2007 Li, F. (2007). Testing the parametric specification of the diffusion function in a diffusion process. Econometric Theory 23(2):221–250.[Crossref], [Web of Science ®] , [Google Scholar]) and the stochastic process-based tests of Dette and Podolskij (2008 Dette, H., Podolskij, M. (2008). Testing the parametric form of the volatility in continuous time diffusion models–a stochastic process approach. Journal of Econometrics 143(1):56–73.[Crossref], [Web of Science ®] , [Google Scholar]). When applying the test to high frequency data of EUR/USD exchange rate, the empirical results show that the commonly used volatility functions fit more poorly when the data frequency becomes higher, and the general volatility functions fit relatively better than the constant volatility function.     

Test for high dimensional regression coefficients of partially linear models

Abstract

Partially linear models attract much attention to investigate the association between predictors and the response variable when the dependency on some predictors may be nonlinear. However, the hypothesis test for significance of predictors is still challenging, especially when the number of predictors is larger than sample size. In this paper, we reconsider the test procedure of Zhong and Chen (2011 Zhong, P., and S. Chen. 2011. Tests for high-dimensional regression coefficients with factorial designs. Journal of the American Statistical Association 106 (493):260–74. doi:10.1198/jasa.2011.tm10284.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) when regression models have nonlinear components, and propose a generalized U-statistic for testing the linear components of the high dimensional partially linear models. The asymptotic properties of test statistic are obtained under null and alternative hypotheses, where the effect of nonlinear components should be considered and thus is different from that in linear models. Through simulation studies, we demonstrate good finite-sample performance of the proposed test in comparison with the existing methods. The practical utility of our proposed method is illustrated by a real data example.     

Fitting competing risks data to bivariate Pareto models

This paper revisits two bivariate Pareto models for fitting competing risks data. The first model is the Frank copula model, and the second one is a bivariate Pareto model introduced by Sankaran and Nair (1993 Sankaran, P. G., and N. U. Nair. 1993. A bivariate Pareto model and its applications to reliability. Naval Research Logistics 40 (7):1013–1020. doi:10.1002/1520-6750(199312)40:7%3c1013::AID-NAV3220400711%3e3.0.CO;2-7.[Crossref], [Web of Science ®] , [Google Scholar]). We discuss the identifiability issues of these models and develop the maximum likelihood estimation procedures including their computational algorithms and model-diagnostic procedures. Simulations are conducted to examine the performance of the maximum likelihood estimation. Real data are analyzed for illustration.     

On a class exhibiting trend change in average failure rate

Abstract

Weak convergence and moment convergence issues are investigated for the New Better than Average Failure Rate (NBAFR) family (introduced by Loh (1984 Loh, W. Y. 1984. A new generalization of the class of NBU distributions. IEEE Transactions on Reliability R-33 :97–113[Crossref], [Web of Science ®] , [Google Scholar])). We explore the validity of these results in the context of a more general ageing class that we introduce. We prove some new properties of this class and derive its interrelationships with other non-monotonic ageing families. Reliability and moment bounds are obtained and an interesting characterization of exponentiality is proved. Special cases of our results lead to new theorems for the NBAFR class. Finally weak convergence and related issues are established for this class.     

New influence diagnostics in ridge regression

We occasionally find that a small subset of the data exerts a disproportionate influence on the fitted regression model. We would like to locate these influential points and assess their impact on the model. However, the existence of influential data is complicated by the presence of collinearity (see, e.g. [15 E. Walker and J. Birch, Influence measures in ridge regression, Technometrics 30 (1989), pp. 221–227. doi: 10.1080/00401706.1988.10488370[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]). In this article we develop a new influence statistic for one or a set of observations in linear regression dealing with collinearity. We show that this statistic has asymptotically normal distribution and is able to detect a subset of high ridge leverage outliers. Using this influence statistic we also show that when ridge regression is used to mitigate the effects of collinearity, the influence of some observations can be drastically modified. As an illustrative example, simulation studies and a real data set are analysed.