Positive and negative non-separability for space–time covariance models

Separable spatio-temporal covariance models, defined as the product of purely spatial and purely temporal covariance functions, are often used in practice, but frequently they only represent a convenient assumption. On the other hand, non-separable models are receiving a lot of attention, since they are more flexible to handle empirical covariances showed up in applications. Different forms of non-separability for space–time covariance functions have been recently defined in the literature. In this paper, the notion of positive and negative non-separability is further formalized in order to distinguish between pointwise and uniform non-separability. Various well-known non-separable space–time stationary covariance models are analyzed and classified by using the new definition of non-separability. In particular, wide classes of non-separable spatio-temporal covariance functions, able to capture positive and negative non-separability, are proposed and some examples of these classes are given. General results concerning the non-separability of spatial–temporal covariance functions obtained by a linear combination of spatial–temporal covariance functions and some stability properties are also presented. These results can be helpful to generate as well as to select appropriate covariance models for describing space–time data.     

A generalised likelihood framework for partially observed capture–recapture–recovery models

We provide a closed form likelihood expression for multi-state capture–recapture–recovery data when the state of an individual may be only partially observed. The corresponding sufficient statistics are presented in addition to a matrix formulation which facilitates an efficient calculation of the likelihood. This likelihood framework provides a consistent and unified framework with many standard models applied to capture–recapture–recovery data as special cases.     

A new space–time multivariate approach for environmental data analysis

Air quality control usually requires a monitoring system of multiple indicators measured at various points in space and time. Hence, the use of space–time multivariate techniques are of fundamental importance in this context, where decisions and actions regarding environmental protection should be supported by studies based on either inter-variables relations and spatial–temporal correlations. This paper describes how canonical correlation analysis can be combined with space–time geostatistical methods for analysing two spatial–temporal correlated aspects, such as air pollution concentrations and meteorological conditions. Hourly averages of three pollutants (nitric oxide, nitrogen dioxide and ozone) and three atmospheric indicators (temperature, humidity and wind speed) taken for two critical months (February and August) at several monitoring stations are considered and space–time variograms for the variables are estimated. Simultaneous relationships between such sample space–time variograms are determined through canonical correlation analysis. The most correlated canonical variates are used for describing synthetically the underlying space–time behaviour of the components of the two sets.     

A generalized Q–Q plot for longitudinal data

Most biomedical research is carried out using longitudinal studies. The method of generalized estimating equations (GEEs) introduced by Liang and Zeger [Longitudinal data analysis using generalized linear models, Biometrika 73 (1986), pp. 13–22] and Zeger and Liang [Longitudinal data analysis for discrete and continuous outcomes, Biometrics 42 (1986), pp. 121–130] has become a standard method for analyzing non-normal longitudinal data. Since then, a large variety of GEEs have been proposed. However, the model diagnostic problem has not been explored intensively. Oh et al. [Modeldiagnostic plots for repeated measures data using the generalized estimating equations approach, Comput. Statist. Data Anal. 53 (2008), pp. 222–232] proposed residual plots based on the quantile–quantile (Q–Q) plots of the χ²-distribution for repeated-measures data using the GEE methodology. They considered the Pearson, Anscombe and deviance residuals. In this work, we propose to extend this graphical diagnostic using a generalized residual. A simulation study is presented as well as two examples illustrating the proposed generalized Q–Q plots.     

A new local estimation method for single index models for longitudinal data

Single index models are natural extensions of linear models and overcome the so-called curse of dimensionality. They are very useful for longitudinal data analysis. In this paper, we develop a new efficient estimation procedure for single index models with longitudinal data, based on Cholesky decomposition and local linear smoothing method. Asymptotic normality for the proposed estimators of both the parametric and nonparametric parts will be established. Monte Carlo simulation studies show excellent finite sample performance. Furthermore, we illustrate our methods with a real data example.     

Fitting semiparametric accelerated failure time models for nested case–control data

A nested case–control (NCC) study is an efficient cohort-sampling design in which a subset of controls are sampled from the risk set at each event time. Since covariate measurements are taken only for the sampled subjects, time and efforts of conducting a full scale cohort study can be saved. In this paper, we consider fitting a semiparametric accelerated failure time model to failure time data from a NCC study. We propose to employ an efficient induced smoothing procedure for rank-based estimating method for regression parameters estimation. For variance estimation, we propose to use an efficient resampling method that utilizes the robust sandwich form. We extend our proposed methods to a generalized NCC study that allows a sampling of cases. Finite sample properties of the proposed estimators are investigated via an extensive stimulation study. An application to a tumor study illustrates the utility of the proposed method in routine data analysis.     

A generalized scheffé method of multiple comparisons

Standard resulrs on the extrema of quotients of quadratic forms are extended to the non-negative definite case. The maximum and the set over which it is achieved are characterized explicitly both in terms of generalized inverse matrices and generalized eigenvalues. These results become the basis of Scheffe type multiple comparisons in the usual way. To demonstrate their application to statistics with singular covariance matrices, the method is detailed for Mantel-Haenszel, Breslow, and Cox statistics. An example is presented illustrating a situation where the proposed Scheffe type comparisons may be better than the pairwise method.     

A coordinate majorization descent algorithm for ℓ1 penalized learning

The glmnet package by Friedman et al. [Regularization paths for generalized linear models via coordinate descent, J. Statist. Softw. 33 (2010), pp. 1–22] is an extremely fast implementation of the standard coordinate descent algorithm for solving ?₁ penalized learning problems. In this paper, we consider a family of coordinate majorization descent algorithms for solving the ?₁ penalized learning problems by replacing each coordinate descent step with a coordinate-wise majorization descent operation. Numerical experiments show that this simple modification can lead to substantial improvement in speed when the predictors have moderate or high correlations.     

Mixture regression models for the gap time distributions and illness–death processes

Lifetime Data Analysis - The aim of this study is to provide an analysis of gap event times under the illness–death model, where some subjects experience “illness” before...     

Robust BCa–JaB method as a diagnostic tool for linear regression models

The Jackknife-after-bootstrap (JaB) technique originally developed by Efron [8 B. Efron, Jackknife-after-bootstrap standard errors and influence functions, J. R. Stat. Soc. 54 (1992), pp. 83–127. [Google Scholar]] has been proposed as an approach to improve the detection of influential observations in linear regression models by Martin and Roberts [12 M.A. Martin and S. Roberts, Jackknife-after-bootstrap regression influence diagnostics, J. Nonparametr. Stat. 22 (2010), pp. 257–269. doi: 10.1080/10485250903287906[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] and Beyaztas and Alin [2 U. Beyaztas and A. Alin, Jackknife-after-bootstrap method for detection of influential observations in linear regression model, Comm. Statist. Simulation Comput. 42 (2013), pp. 1256–1267. doi: 10.1080/03610918.2012.661908[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. The method is based on the use of percentile-method confidence intervals to provide improved cut-off values for several single case-deletion influence measures. In order to improve JaB, we propose using robust versions of Efron [7 B. Efron, Better bootstrap confidence intervals, J. Amer. Statist. Assoc. 82 (1987), pp. 171–185. doi: 10.1080/01621459.1987.10478410[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]’s bias-corrected and accelerated (BCa) bootstrap confidence intervals. In this study, the performances of robust BCa–JaB and conventional JaB methods are compared in the cases of DFFITS, Welsch's distance and modified Cook's distance influence diagnostics. Comparisons are based on both real data examples and through a simulation study. Our results reveal that under a variety of scenarios, our proposed method provides more accurate and reliable results, and it is more robust to masking effects.     

Joint mean–covariance model in generalized partially linear varying coefficient models for longitudinal data

In longitudinal data analysis, efficient estimation of regression coefficients requires a correct specification of certain covariance structure, and efficient estimation of covariance matrix requires a correct specification of mean regression model. In this article, we propose a general semiparametric model for the mean and the covariance simultaneously using the modified Cholesky decomposition. A regression spline-based approach within the framework of generalized estimating equations is proposed to estimate the parameters in the mean and the covariance. Under regularity conditions, asymptotic properties of the resulting estimators are established. Extensive simulation is conducted to investigate the performance of the proposed estimator and in the end a real data set is analysed using the proposed approach.     

Recent advances to model anisotropic space–time data

Building new and flexible classes of nonseparable spatio-temporal covariances and variograms has resulted a key point of research in the last years. The goal of this paper is to present an up-to-date overview of recent spatio-temporal covariance models taking into account the problem of spatial anisotropy. The resulting structures are proved to have certain interesting mathematical properties, together with a considerable applicability. In particular, we focus on the problem of modelling anisotropy through isotropy within components. We present the Bernstein class, and a generalisation of Gneiting’s approach (2002a) to obtain new classes of space–time covariance functions which are spatially anisotropic. We also discuss some methods for building covariance functions that attain negative values. We finally present several differentiation and integration operators acting on particular space–time covariance classes.      

Optimal generalized case–cohort sampling design under the additive hazard model

Generalized case–cohort designs have been proved to be a cost-effective way to enhance effectiveness in large epidemiological cohort. In generalized case–cohort design, we first select a subcohort from the underlying cohort by simple random sampling, and then sample a subset of the failures in the remaining subjects. In this article, we propose the inference procedure for the unknown regression parameters in the additive hazards model and develop an optimal sample size allocations to achieve maximum power at a given budget in generalized case–cohort design. The finite sample performance of the proposed method is evaluated through simulation studies. The proposed method is applied to a real data set from the National Wilm's Tumor Study Group.     

EM-based maximum likelihood parameter estimation for multivariate generalized hyperbolic distributions with fixed λ

Generalized Hyperbolic distribution (Barndorff-Nielsen 1977) is a variance-mean mixture of a normal distribution with the Generalized Inverse Gaussian distribution. Recently subclasses of these distributions (e.g., the hyperbolic distribution and the Normal Inverse Gaussian distribution) have been applied to construct stochastic processes in turbulence and particularly in finance, where multidimensional problems are of special interest. Parameter estimation for these distributions based on an i.i.d. sample is a difficult task even for a specified one-dimensional subclass (subclass being uniquely defined by ) and relies on numerical methods. For the hyperbolic subclass ( = 1), computer program hyp (Blæsild and Sørensen 1992) estimates parameters via ML when the dimensionality is less than or equal to three. To the best of the author's knowledge, no successful attempts have been made to fit any given subclass when the dimensionality is greater than three. This article proposes a simple EM-based (Dempster, Laird and Rubin 1977) ML estimation procedure to estimate parameters of the distribution when the subclass is known regardless of the dimensionality. Our method relies on the ability to numerically evaluate modified Bessel functions of the third kind and their logarithms, which is made possible by currently available software. The method is applied to fit the five dimensional Normal Inverse Gaussian distribution to a series of returns on foreign exchange rates.     

Statistical inference for the accelerated failure time model under two-stage generalized case–cohort design

Abstract

In this article, we propose a two-stage generalized case–cohort design and develop an efficient inference procedure for the data collected with this design. In the first-stage, we observe the failure time, censoring indicator and covariates which are easy or cheap to measure, and in the second-stage, select a subcohort by simple random sampling and a subset of failures in remaining subjects from the first-stage subjects to observe their exposures which are different or expensive to measure. We derive estimators for regression parameters in the accelerated failure time model under the two-stage generalized case–cohort design through the estimated augmented estimating equation and the kernel function method. The resulting estimators are shown to be consistent and asymptotically normal. The finite sample performance of the proposed method is evaluated through the simulation studies. The proposed method is applied to a real data set from the National Wilm’s Tumor Study Group.     

A two-stage estimation in the Clayton–Oakes model with marginal linear transformation models for multivariate failure time data

This paper considers the analysis of multivariate survival data where the marginal distributions are specified by semiparametric transformation models, a general class including the Cox model and the proportional odds model as special cases. First, consideration is given to the situation where the joint distribution of all failure times within the same cluster is specified by the Clayton–Oakes model (Clayton, Biometrika 65:141–151, l978; Oakes, J R Stat Soc B 44:412–422, 1982). A two-stage estimation procedure is adopted by first estimating the marginal parameters under the independence working assumption, and then the association parameter is estimated from the maximization of the full likelihood function with the estimators of the marginal parameters plugged in. The asymptotic properties of all estimators in the semiparametric model are derived. For the second situation, the third and higher order dependency structures are left unspecified, and interest focuses on the pairwise correlation between any two failure times. Thus, the pairwise association estimate can be obtained in the second stage by maximizing the pairwise likelihood function. Large sample properties for the pairwise association are also derived. Simulation studies show that the proposed approach is appropriate for practical use. To illustrate, a subset of the data from the Diabetic Retinopathy Study is used.