Inference of time-varying regression models

We consider parameter estimation, hypothesis testing and variable selection for partially time-varying coefficient models. Our asymptotic theory has the useful feature that it can allow dependent, nonstationary error and covariate processes. With a two-stage method, the parametric component can be estimated with a $n^{1/2}$-convergence rate. A simulation-assisted hypothesis testing procedure is proposed for testing significance and parameter constancy. We further propose an information criterion that can consistently select the true set of significant predictors. Our method is applied to autoregressive models with time-varying coefficients. Simulation results and a real data application are provided.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1010 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Spline regression for zero-inflated models

We propose a regression model for count data when the classical generalized linear model approach is too rigid due to a high outcome of zero counts and a nonlinear influence of continuous covariates. Zero-Inflation is applied to take into account the presence of excess zeros with separate link functions for the zero and the nonzero component. Nonlinearity in covariates is captured by spline functions based on B-splines. Our algorithm relies on maximum-likelihood estimation and allows for adaptive box-constrained knots, thus improving the goodness of the spline fit and allowing for detection of sensitivity changepoints. A simulation study substantiates the numerical stability of the algorithm to infer such models. The AIC criterion is shown to serve well for model selection, in particular if nonlinearities are weak such that BIC tends to overly simplistic models. We fit the introduced models to real data of children's dental sanity, linking caries counts with the so-called Body-Mass-Index (BMI) and other socioeconomic factors. This reveals a puzzling nonmonotonic influence of BMI on caries counts which is yet to be explained by clinical experts

Efficient quantile regression for heteroscedastic models

Quantile regression (QR) provides estimates of a range of conditional quantiles. This stands in contrast to traditional regression techniques, which focus on a single conditional mean function. Lee et al. [Regularization of case-specific parameters for robustness and efficiency. Statist Sci. 2012;27(3):350–372] proposed efficient QR by rounding the sharp corner of the loss. The main modification generally involves an asymmetric ℓ₂ adjustment of the loss function around zero. We extend the idea of ℓ₂ adjusted QR to linear heterogeneous models. The ℓ₂ adjustment is constructed to diminish as sample size grows. Conditions to retain consistency properties are also provided

Measuring influence in dynamic regression models

This article presents a methodology to build measures of influence in regression models with time series data. We introduce statistics that measure the influence of each observation on the parameter estimates and on the forecasts. These statistics take into account the autocorrelation of the sample. The first statistic can be decomposed to measure the change in the univariate ARIMA parameters, the transfer function parameters and the interaction between both. For independent data they reduce to the D statistics considered by Cook in the standard regression modelo These statistics can be easily computed using standard time series software. Their performance is analyzed in an example in which they seem to be useful to identify important events, such as additive outliers and trend shifts, in time series data

Mixture regression for observational data, with application to functional regression models

In a regression analysis, suppose we suspect that there are several heterogeneous groups in the population that a sample represents. Mixture regression models have been applied to address such problems. By modeling the conditional distribution of the response given the covariate as a mixture, the sample can be clustered into groups and the individual regression models for the groups can be estimated simultaneously. This approach treats the covariate as deterministic so that the covariate carries no information as to which group the subject is likely to belong to. Although this assumption may be reasonable in experiments where the covariate is completely determined by the experimenter, in observational data the covariate may behave differently across the groups. Thus the model should also incorporate the heterogeneity of the covariate, which allows us to estimate the membership of the subject from the covariate. In this paper, we consider a mixture regression model where the joint distribution of the response and the covariate is modeled as a mixture. Given a new observation of the covariate, this approach allows us to compute the posterior probabilities that the subject belongs to each group. Using these posterior probabilities, the prediction of the response can adaptively use the covariate. We introduce an inference procedure for this approach and show its properties concerning estimation and prediction. The model is explored for the functional covariate as well as the multivariate covariate. We present a real-data example where our approach outperforms the traditional approach, using the well-analyzed Berkeley growth study data