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Identification of a subgroup of patients who may be sensitive to a specific treatment is an important problem in precision medicine. This article considers the case where the treatment effect is assessed by longitudinal measurements, such as quality of life scores assessed over the duration of a clinical trial, and the subset is determined by a continuous baseline covariate, such as age and expression level of a biomarker. Recently, a linear mixed threshold regression model has been proposed but it assumes the longitudinal measurements are normally distributed. In many applications, longitudinal measurements, such as quality of life data obtained from answers to questions on a Likert scale, may be restricted in a fixed interval because of the floor and ceiling effects and, therefore, may be skewed. In this article, a threshold longitudinal Tobit quantile regression model is proposed and a computational approach based on alternating direction method of multipliers algorithm is developed for the estimation of parameters in the model. In addition, a random weighting method is employed to estimate the variances of the parameter estimators. The proposed procedures are evaluated through simulation studies and applications to the data from clinical trials. [ABSTRACT FROM AUTHOR]

Many methods have been developed to study nonparametric function-on-function regression models. Nevertheless, there is a lack of model selection approach to the regression function as a functional function with functional covariate inputs. To study interaction effects among these functional covariates, in this article, we first construct a tensor product space of reproducing kernel Hilbert spaces and build an analysis of variance (ANOVA) decomposition of the tensor product space. We then use a model selection method with the L 1 criterion to estimate the functional function with functional covariate inputs and detect interaction effects among the functional covariates. The proposed method is evaluated using simulations and stroke rehabilitation data. [ABSTRACT FROM AUTHOR]

Regression models with a functional response and functional covariate have received significant attention recently. While various nonparametric and semiparametric models have been developed, there is an urgent need for model selection and diagnostic methods. In this article, we develop a unified framework for estimation and model selection in nonparametric function-on-function regression. We propose a general nonparametric functional regression model with the model space constructed through smoothing spline analysis of variance (SS ANOVA). The proposed model reduces to some of the existing models when selected components in the SS ANOVA decomposition are eliminated. We propose new estimation procedures under either L1 or L2 penalty and show that the combination of the SS ANOVA decomposition and L1 penalty provides powerful tools for model selection and diagnostics. We establish consistency and convergence rates for estimates of the regression function and each component in its decomposition under both the L1 and L2 penalties. Simulation studies and real examples show that the proposed methods perform well. Technical details and additional simulation results are available in online . [ABSTRACT FROM AUTHOR]

Relative error approaches are more of concern compared to absolute error ones such as the least square and least absolute deviation, when it needs scale invariant of output variable, for example with analyzing stock and survival data. A relative error estimation procedure based on the h-likelihood is developed to avoid heavy and intractable integration for a multiplicative regression model with random effect. Statistical properties of the parameters and random effect in the model are studied. Numerical studies including simulation and real examples show the proposed estimation procedure performs well. [ABSTRACT FROM AUTHOR]

The least product relative error (LPRE) estimator and test statistic to test linear hypotheses of regression parameters in the multiplicative regression model are studied when the number of covariate variables increases with the sample size. Some properties of the LPRE estimator and test statistic are obtained such as consistency, Bahadur presentation, and asymptotic distributions. Furthermore, we extend the LPRE to a more general relative error criterion and provide their statistical properties. Numerical studies including simulations and two real examples show that the proposed estimation performs well. [ABSTRACT FROM AUTHOR]

Gaussian process regression (GPR) model has been widely used to fit data when the regression function is unknown and its nice properties have been well established. In this article, we introduce an extended t -process regression (eTPR) model, a nonlinear model which allows a robust best linear unbiased predictor (BLUP). Owing to its succinct construction, it inherits many attractive properties from the GPR model, such as having closed forms of marginal and predictive distributions to give an explicit form for robust procedures, and easy to cope with large dimensional covariates with an efficient implementation. Properties of the robustness are studied. Simulation studies and real data applications show that the eTPR model gives a robust fit in the presence of outliers in both input and output spaces and has a good performance in prediction, compared with other existed methods. [ABSTRACT FROM AUTHOR]

As a result of novel data collection technologies, it is now common to encounter data in which the number of explanatory variables collected is large, while the number of variables that actually contribute to the model remains small. Thus, a method that can identify those variables with impact on the model without inferring other noneffective ones will make analysis much more efficient. Many methods are proposed to resolve the model selection problems under such circumstances, however, it is still unknown how large a sample size is sufficient to identify those 'effective' variables. In this paper, we apply sequential sampling method so that the effective variables can be identified efficiently, and the sampling is stopped as soon as the 'effective' variables are identified and their corresponding regression coefficients are estimated with satisfactory accuracy, which is new to sequential estimation. Both fixed and adaptive designs are considered. The asymptotic properties of estimates of the number of effective variables and their coefficients are established, and the proposed sequential estimation procedure is shown to be asymptotically optimal. Simulation studies are conducted to illustrate the performance of the proposed estimation method, and a diabetes data set is used as an example. [ABSTRACT FROM AUTHOR]

This paper considers an adaptive method based on relative error criteria to select grouped variables and estimate parameters simultaneously for the multiplicative regression model. The oracle properties of the proposed method are established. Simulation studies indicate that the proposed relative error approach has better performance to detect effective groups relative to several existing methods based on absolute error criteria, for example, the least squares and the least absolute deviation. Application is illustrated with the body fat and the birth weight datasets. [ABSTRACT FROM AUTHOR]

A least product relative error criterion is proposed for multiplicative regression models. It is invariant under scale transformation of the outcome and covariates. In addition, the objective function is smooth and convex, resulting in a simple and uniquely defined estimator of the regression parameter. It is shown that the estimator is asymptotically normal and that the simple plug-in variance estimation is valid. Simulation results confirm that the proposed method performs well. An application to body fat calculation is presented to illustrate the new method. [ABSTRACT FROM AUTHOR]