Seonjin Kim

Nonparametric functional central limit theorem for time series regression with application to self-normalized confidence interval

This paper is concerned with the inference of nonparametric mean function in a time series context. The commonly used kernel smoothing estimate is asymptotically normal and the traditional inference procedure then consistently estimates the asymptotic variance function and relies upon normal approximation. Consistent estimation of the asymptotic variance function involves another level of nonparametric smoothing. In practice, the choice of the extra bandwidth parameter can be difficult, the inference results can be sensitive to bandwidth selection and the normal approximation can be quite unsatisfactory in small samples leading to poor coverage. To alleviate the problem, we propose to extend the recently developed self-normalized approach, which is a bandwidth free inference procedure developed for parametric inference, to construct point-wise confidence interval for nonparametric mean function. To justify asymptotic validity of the self-normalized approach, we establish a functional central limit theorem for recursive nonparametric mean regression function estimates under primitive conditions and show that the limiting process is a Gaussian process with non-stationary and dependent increments. The superior finite sample performance of the new approach is demonstrated through simulation studies.

Multiple quantile regression analysis of longitudinal data

The objective of this paper is two-fold: to propose efficient estimation of multiple quantile regression analysis of longitudinal data and to develop a new test for the homogeneity of independent variable effects across multiple quantiles. Estimating multiple regression quantile coefficients simultaneously entails accommodating both association among the multiple quantiles and association among the repeated measurements of the response within subjects. We formulate simultaneous estimating equations using basis matrix expansion which accounts for the above-mentioned associations. The empirical likelihood method is adopted to estimate multiple regression quantile coefficients. Theoretical results show that the proposed simultaneous estimation is asymptotically more efficient than separate estimation of individual regression quantiles or ignoring the within-subject dependency. The proposed method also offers an empirical likelihood ratio test examining the homogeneity of the independent variable effects across the multiple quantiles. The Wilks theorem holds for the test statistic, and thus the test is easy to implement. Simulation studies and real data example of a multi-center AIDS cohort study are included to illustrate the proposed estimation and testing methods, and empirically examine their properties.

A nonparametric hypothesis test for heteroscedasticity

ABSTRACT In this paper, a hypothesis test for heteroscedasticity is proposed in a nonparametric regression model. The test statistic, which uses the residuals from a nonparametric fit of the mean function, is based on an adaptation of the well-known Levenes test. Using the recent theory for analysis of variance when the number of factor levels goes to infinity, the asymptotic distribution of the test statistic is established under the null hypothesis of homocedasticity and under local alternatives. Simulations suggest that the proposed test performs well in several situations, especially when the variance is a nonlinear function of the predictor.

Specification test for Markov models with measurement errors

Most existing works on specification testing assume that we have direct observations from the model of interest. We study specification testing for Markov models based on contaminated observations. The evolving model dynamics of the unobservable Markov chain is implicitly coded into the conditional distribution of the observed process. To test whether the underlying Markov chain follows a parametric model, we propose measuring the deviation between nonparametric and parametric estimates of conditional regression functions of the observed process. Specifically, we construct a nonparametric simultaneous confidence band for conditional regression functions and check whether the parametric estimate is contained within the band.

Efficient estimation for time-varying coefficient longitudinal models

ABSTRACT For estimation of time-varying coefficient longitudinal models, the widely used local least-squares (LS) or covariance-weighted local LS smoothing uses information from the local sample average. Motivated by the fact that a combination of multiple quantiles provides a more complete picture of the distribution, we investigate quantile regression-based methods to improve efficiency by optimally combining information across quantiles. Under the working independence scenario, the asymptotic variance of the proposed estimator approaches the Cramér–Rao lower bound. In the presence of dependence among within-subject measurements, we adopt a prewhitening technique to transform regression errors into independent innovations and show that the prewhitened optimally weighted quantile average estimator asymptotically achieves the Cramér–Rao bound for the independent innovations. Fully data-driven bandwidth selection and optimal weights estimation are implemented through a two-step procedure. Monte Carlo studies show that the proposed method delivers more robust and superior overall performance than that of the existing methods.

Model specification test in a semiparametric regression model for longitudinal data

We propose a model specification test for whether or not a postulated parametric model (null hypothesis) fits longitudinal data as well as a semiparametric model (alternative hypothesis) does. In the semiparametric model, we suppose that a baseline function of time is modeled nonparametrically, while the longitudinal covariate effect is assumed to be a parametric linear model. The existing kernel regression based likelihood ratio tests suffer from computing the likelihood function in the alternative hypothesis, because a specific parametric alternative is not desired. To circumvent this difficulty, we calibrate the semiparametric model to a regression model containing only the parametric parameters, and investigate the quadratic inference function in the calibrated model. The proposed approach yields an asymptotically unbiased parametric regression estimator without undersmoothing the baseline function. This provides us a simple and powerful test statistic that asymptotically follows a central chi-squared distribution with fixed degrees of freedom under the null hypothesis. Simulation studies show that the proposed test is able to identify the true parametric regression model consistently. We have also applied this test to real data and confirmed that the baseline function can be captured by a conjectured parametric form sufficiently well.