Cox regression of clustered event times with covariates missing not at random

Motivated by a recent tuberculosis (TB) study, this paper is concerned with covariates missing not at random (MNAR) and models the potential intracluster correlation by a frailty. We consider the regression analysis of right‐censored event times from clustered subjects under a Cox proportional hazards frailty model and present the semiparametric maximum likelihood estimator (SPMLE) of the model parameters. An easy‐to‐implement pseudo‐SPMLE is then proposed to accommodate more realistic situations using readily available supplementary information on the missing covariates. Algorithms are provided to compute the estimators and their consistent variance estimators. We demonstrate that both the SPMLE and the pseudo‐SPMLE are consistent and asymptotically normal by the arguments based on the theory of modern empirical processes. The proposed approach is examined numerically via simulation and illustrated with an analysis of the motivating TB study data.     

A semiparametric approach for modelling multivariate nonlinear time series

In this article, a semiparametric time‐varying nonlinear vector autoregressive (NVAR) model is proposed to model nonlinear vector time series data. We consider a combination of parametric and nonparametric estimation approaches to estimate the NVAR function for both independent and dependent errors. We use the multivariate Taylor series expansion of the link function up to the second order which has a parametric framework as a representation of the nonlinear vector regression function. After the unknown parameters are estimated by the maximum likelihood estimation procedure, the obtained NVAR function is adjusted by a nonparametric diagonal matrix, where the proposed adjusted matrix is estimated by the nonparametric kernel estimator. The asymptotic consistency properties of the proposed estimators are established. Simulation studies are conducted to evaluate the performance of the proposed semiparametric method. A real data example on short‐run interest rates and long‐run interest rates of United States Treasury securities is analyzed to demonstrate the application of the proposed approach. The Canadian Journal of Statistics 47: 668–687; 2019 © 2019 Statistical Society of Canada     

Testing homogeneity in semiparametric mixture case–control models

Parametric and semiparametric mixture models have been widely used in applications from many areas, and it is often of interest to test the homogeneity in these models. However, hypothesis testing is non standard due to the fact that several regularity conditions do not hold under the null hypothesis. We consider a semiparametric mixture case–control model, in the sense that the density ratio of two distributions is assumed to be of an exponential form, while the baseline density is unspecified. This model was first considered by Qin and Liang (2011 Qin, J., Liang, K.Y. (2011). Hypothesis testing in a mixture case–control model. Biometrics 67(1):182–198.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar], biometrics), and they proposed a modified score statistic for testing homogeneity. In this article, we consider alternative testing procedures based on supremum statistics, which could improve power against certain types of alternatives. We demonstrate the connection and comparison among the proposed and existing approaches. In addition, we provide a unified theoretical justification of the supremum test and other existing test statistics from an empirical likelihood perspective. The finite-sample performance of the supremum test statistics was evaluated in simulation studies.     

Estimation and Inference Procedures for Semiparametric Distribution Models with Varying Linear‐Index

More flexible semiparametric linear‐index regression models are proposed to describe the conditional distribution. Such a model formulation captures varying effects of covariates over the support of a response variable in distribution, offers an alternative perspective on dimension reduction and covers a lot of widely used parametric and semiparameteric regression models. A feasible pseudo likelihood approach, accompanied with a simple and easily implemented algorithm, is further developed for the mixed case with both varying and invariant coefficients. By justifying some theoretical properties on Banach spaces, the uniform consistency and asymptotic Gaussian process of the proposed estimator are also established in this article. In addition, under the monotonicity of distribution in linear‐index, we develop an alternative approach based on maximizing a varying accuracy measure. By virtue of the asymptotic recursion relation for the estimators, some of the achievements in this direction include showing the convergence of the iterative computation procedure and establishing the large sample properties of the resulting estimator. It is noticeable that our theoretical framework is very helpful in constructing confidence bands for the parameters of interest and tests for the hypotheses of various qualitative structures in distribution. Generally, the developed estimation and inference procedures perform quite satisfactorily in the conducted simulations and are demonstrated to be useful in reanalysing data from the Boston house price study and the World Values Survey.     

Statistical properties of detrended fluctuation analysis

The main goal of this work is to consider the detrended fluctuation analysis (DFA), proposed by Peng et al. [Mosaic organization of DNA nucleotides, Phys. Rev. E. 49(5) (1994), 1685–1689]. This is a well-known method for analysing the long-range dependence in non-stationary time series. Here we describe the DFA method and we prove its consistency and its exact distribution, based on the usual i.i.d. assumption, as an estimator for the fractional parameter d. In the literature it is well established that the nucleotide sequences present long-range dependence property. In this work, we analyse the long dependence property in view of the autoregressive moving average fractionally integrated ARFIMA(p, d, q) processes through the analysis of four nucleotide sequences. For estimating the fractional parameter d we consider the semiparametric regression method based on the periodogram function, in both classical and robust versions; the semiparametric R/S(n) method, proposed by Hurst [Long term storage in reservoirs, Trans. Am. Soc. Civil Eng. 116 (1986), 770–779] and the maximum likelihood method (see [R. Fox and M.S. Taqqu, Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series, Ann. Statist. 14 (1986), 517–532]), by considering the approximation suggested by Whittle [Hypothesis Testing in Time Series Analysis (1953), Hafner, New York].     

Several nonparametric and semiparametric approaches to linear mixed model regression

Mixed models are powerful tools for the analysis of clustered data and many extensions of the classical linear mixed model with normally distributed response have been established. As with all parametric (P) models, correctness of the assumed model is critical for the validity of the ensuing inference. An incorrectly specified P means model may be improved by using a local, or nonparametric (NP), model. Two local models are proposed by a pointwise weighting of the marginal and conditional variance–covariance matrices. However, NP models tend to fit to irregularities in the data and may provide fits with high variance. Model robust regression techniques estimate mean response as a convex combination of a P and a NP model fit to the data. It is a semiparametric method by which incomplete or incorrectly specified P models can be improved by adding an appropriate amount of the NP fit. We compare the approximate integrated mean square error of the P, NP, and mixed model robust methods via a simulation study and apply these methods to two real data sets: the monthly wind speed data from countries in Ireland and the engine speed data.