Guided Censored Regression

Parametrically guided non‐parametric regression is an appealing method that can reduce the bias of a non‐parametric regression function estimator without increasing the variance. In this paper, we adapt this method to the censored data case using an unbiased transformation of the data and a local linear fit. The asymptotic properties of the proposed estimator are established, and its performance is evaluated via finite sample simulations.     

Difference-based estimation and model identification for panel data semiparametric models with cross-section dependence

Abstract

In this article, we consider a panel data partially linear regression model with fixed effect and non parametric time trend function. The data can be dependent cross individuals through linear regressor and error components. Unlike the methods using non parametric smoothing technique, a difference-based method is proposed to estimate linear regression coefficients of the model to avoid bandwidth selection. Here the difference technique is employed to eliminate the non parametric function effect, not the fixed effects, on linear regressor coefficient estimation totally. Therefore, a more efficient estimator for parametric part is anticipated, which is shown to be true by the simulation results. For the non parametric component, the polynomial spline technique is implemented. The asymptotic properties of estimators for parametric and non parametric parts are presented. We also show how to select informative ones from a number of covariates in the linear part by using smoothly clipped absolute deviation-penalized estimators on a difference-based least-squares objective function, and the resulting estimators perform asymptotically as well as the oracle procedure in terms of selecting the correct model.     

On Efficient Estimators of the Proportion of True Null Hypotheses in a Multiple Testing Setup

We consider the problem of estimating the proportion θ of true null hypotheses in a multiple testing context. The setup is classically modelled through a semiparametric mixture with two components: a uniform distribution on interval [0,1] with prior probability θ and a non‐parametric density f . We discuss asymptotic efficiency results and establish that two different cases occur whether f vanishes on a non‐empty interval or not. In the first case, we exhibit estimators converging at a parametric rate, compute the optimal asymptotic variance and conjecture that no estimator is asymptotically efficient (i.e. attains the optimal asymptotic variance). In the second case, we prove that the quadratic risk of any estimator does not converge at a parametric rate. We illustrate those results on simulated data.     

Goodness‐of‐Fit Test for Monotone Functions

Abstract. In this article, we develop a test for the null hypothesis that a real‐valued function belongs to a given parametric set against the non‐parametric alternative that it is monotone, say decreasing. The method is described in a general model that covers the monotone density model, the monotone regression and the right‐censoring model with monotone hazard rate. The criterion for testing is an ‐distance between a Grenander‐type non‐parametric estimator and a parametric estimator computed under the null hypothesis. A normalized version of this distance is shown to have an asymptotic normal distribution under the null, whence a test can be developed. Moreover, a bootstrap procedure is shown to be consistent to calibrate the test.     

Non parametric mixture of strictly monotone regression models

In this article, we propose the non parametric mixture of strictly monotone regression models. For implementation, a two-step procedure is derived. We further establish the asymptotic normality of the resultant estimator and demonstrate its good performance through numerical examples.     

A Copula‐Based Non‐parametric Measure of Regression Dependence

Abstract. This article presents a framework for comparing bivariate distributions according to their degree of regression dependence. We introduce the general concept of a regression dependence order (RDO). In addition, we define a new non‐parametric measure of regression dependence and study its properties. Besides being monotone in the new RDOs, the measure takes on its extreme values precisely at independence and almost sure functional dependence, respectively. A consistent non‐parametric estimator of the new measure is constructed and its asymptotic properties are investigated. Finally, the finite sample properties of the estimate are studied by means of a small simulation study.     

Shape Constrained Non‐parametric Estimators of the Baseline Distribution in Cox Proportional Hazards Model

Abstract. We investigate non‐parametric estimation of a monotone baseline hazard and a decreasing baseline density within the Cox model. Two estimators of a non‐decreasing baseline hazard function are proposed. We derive the non‐parametric maximum likelihood estimator and consider a Grenander type estimator, defined as the left‐hand slope of the greatest convex minorant of the Breslow estimator. We demonstrate that the two estimators are strongly consistent and asymptotically equivalent and derive their common limit distribution at a fixed point. Both estimators of a non‐increasing baseline hazard and their asymptotic properties are obtained in a similar manner. Furthermore, we introduce a Grenander type estimator for a non‐increasing baseline density, defined as the left‐hand slope of the least concave majorant of an estimator of the baseline cumulative distribution function, derived from the Breslow estimator. We show that this estimator is strongly consistent and derive its asymptotic distribution at a fixed point.     

Asymptotic results for parametric estimation in inadequate two phases regression models

We study the asymptotic properties, consistency, asymptotic normality, of the least squares estimator in a non linear regression problem. The model uses a parametric class Л of functions, but we do not assume that the unknown function belongs to that class. Л is here a class of continuous functions with a discontinuity in the first derivative. The problem of making a choice between two classes of that type is also studied.     

Non‐parametric Regression Tests Using Dimension Reduction Techniques

Abstract. Testing for parametric structure is an important issue in non‐parametric regression analysis. A standard approach is to measure the distance between a parametric and a non‐parametric fit with a squared deviation measure. These tests inherit the curse of dimensionality from the non‐parametric estimator. This results in a loss of power in finite samples and against local alternatives. This article proposes to circumvent the curse of dimensionality by projecting the residuals under the null hypothesis onto the space of additive functions. To estimate this projection, the smooth backfitting estimator is used. The asymptotic behaviour of the test statistic is derived and the consistency of a wild bootstrap procedure is established. The finite sample properties are investigated in a simulation study.     

Testing Monotonicity of Regression Functions – An Empirical Process Approach

We propose several new tests for monotonicity of regression functions based on different empirical processes of residuals and pseudo‐residuals. The residuals are obtained from an unconstrained kernel regression estimator whereas the pseudo‐residuals are obtained from an increasing regression estimator. Here, in particular, we consider a recently developed simple kernel‐based estimator for increasing regression functions based on increasing rearrangements of unconstrained non‐parametric estimators. The test statistics are estimated distance measures between the regression function and its increasing rearrangement. We discuss the asymptotic distributions, consistency and small sample performances of the tests.     

Comparison of spline estimator at various levels of autocorrelation in smoothing spline non parametric regression for longitudinal data

The purpose of this research are: (1) to obtain spline function estimation in non parametric regression for longitudinal data with and without considering the autocorrelation between data of observation within subject, (2) to develop the algorithm that generates simulation data with certain autocorrelation level based on size of sample (N) and error variance (EV), and (3) to establish shape of spline estimator in non parametric regression for longitudinal data to simulation with various level of autocorrelation, as well as compare DM and TM approaches in predicting spline estimator in the data simulation with different of autocorrelation observational data on within subject. The results of the application are as follows: (a) implementation of smoothing spline with penalized weighted least square (PWLS) approach with or without consideration of autocorrelation in general (in all sizes and all error variances levels) provides significantly different spline estimator when the autocorrelation level >0.8; (b) based on size comparison, spline estimator in non parametric regression smoothing spline with PLS approach with (DM), or without (DM) consideration of autocorrelation showed significantly different result in level of autocorrelation > 0.8 (in overall size, moderate and large sample size), and > 0.7 (in small sample size); (c) based on level of variance, spline estimator in non parametric regression smoothing spline with PLS approach with (DM), or without (DM) consideration of autocorrelation showed significantly different result in level of autocorrelation > 0.8 (in overall level of variance, moderate and large variance), and > 0.7 (in small variance).     

Adaptive Warped Kernel Estimators

In this work, we develop a method of adaptive non‐parametric estimation, based on ‘warped’ kernels. The aim is to estimate a real‐valued function s from a sample of random couples (X,Y). We deal with transformed data (Φ(X),Y), with Φ a one‐to‐one function, to build a collection of kernel estimators. The data‐driven bandwidth selection is performed with a method inspired by Goldenshluger and Lepski (Ann. Statist., 39, 2011, 1608). The method permits to handle various problems such as additive and multiplicative regression, conditional density estimation, hazard rate estimation based on randomly right‐censored data, and cumulative distribution function estimation from current‐status data. The interest is threefold. First, the squared‐bias/variance trade‐off is automatically realized. Next, non‐asymptotic risk bounds are derived. Lastly, the estimator is easily computed, thanks to its simple expression: a short simulation study is presented.     

Guaranteed Local Maximum Likelihood Detection of a Change Point in Nonparametric Logistic Regression

We consider nonparametric logistic regression and propose a generalized likelihood test for detecting a threshold effect that indicates a relationship between some risk factor and a defined outcome above the threshold but none below it. One important field of application is occupational medicine and in particular, epidemiological studies. In epidemiological studies, segmented fully parametric logistic regression models are often threshold models, where it is assumed that the exposure has no influence on a response up to a possible unknown threshold, and has an effect beyond that threshold. Finding efficient methods for detection and estimation of a threshold is a very important task in these studies. This article proposes such methods in a context of nonparametric logistic regression. We use a local version of unknown likelihood functions and show that under rather common assumptions the asymptotic power of our test is one. We present a guaranteed non asymptotic upper bound for the significance level of the proposed test. If applying the test yields the acceptance of the conclusion that there was a change point (and hence a threshold limit value), we suggest using the local maximum likelihood estimator of the change point and consider the asymptotic properties of this estimator.     

A New Regression Model: Modal Linear Regression

The mode of a distribution provides an important summary of data and is often estimated on the basis of some non‐parametric kernel density estimator. This article develops a new data analysis tool called modal linear regression in order to explore high‐dimensional data. Modal linear regression models the conditional mode of a response Y given a set of predictors x as a linear function of x . Modal linear regression differs from standard linear regression in that standard linear regression models the conditional mean (as opposed to mode) of Y as a linear function of x . We propose an expectation–maximization algorithm in order to estimate the regression coefficients of modal linear regression. We also provide asymptotic properties for the proposed estimator without the symmetric assumption of the error density. Our empirical studies with simulated data and real data demonstrate that the proposed modal regression gives shorter predictive intervals than mean linear regression, median linear regression and MM‐estimators.     

Double-smoothed drift estimation of jump-diffusion model

In this paper, we consider the non parametric drift estimation for the jump-diffusion model. We employ the double-smoothed non parametric method, and establish the strong consistency and asymptotic normality for estimator.     

Focused information criterion and model averaging based on weighted composite quantile regression

We study the focused information criterion and frequentist model averaging and their application to post‐model‐selection inference for weighted composite quantile regression (WCQR) in the context of the additive partial linear models. With the non‐parametric functions approximated by polynomial splines, we show that, under certain conditions, the asymptotic distribution of the frequentist model averaging WCQR‐estimator of a focused parameter is a non‐linear mixture of normal distributions. This asymptotic distribution is used to construct confidence intervals that achieve the nominal coverage probability. With properly chosen weights, the focused information criterion based WCQR estimators are not only robust to outliers and non‐normal residuals but also can achieve efficiency close to the maximum likelihood estimator, without assuming the true error distribution. Simulation studies and a real data analysis are used to illustrate the effectiveness of the proposed procedure.     

ROC Curves in Non‐Parametric Location‐Scale Regression Models

Abstract. The receiver operating characteristic (ROC) curve is a tool of extensive use to analyse the discrimination capability of a diagnostic variable in medical studies. In certain situations, the presence of a covariate related to the diagnostic variable can increase the discriminating power of the ROC curve. In this article, we model the effect of the covariate over the diagnostic variable by means of non‐parametric location‐scale regression models. We propose a new non‐parametric estimator of the conditional ROC curve and study its asymptotic properties. We also present some simulations and an illustration to a data set concerning diagnosis of diabetes.     

Statistic inference for a single-index ARCH-M model

For a single-index autoregressive conditional heteroscedastic (ARCH-M) model, estimators of the parametric and non parametric components are proposed by the profile likelihood method. The research results had shown that all the estimators have consistency and the parametric estimators have asymptotic normality. We extend this line of research by deriving the asymptotic normality of the non parametric estimator. Based on the asymptotic properties, we propose Wald statistic and generalized likelihood ratio statistic to investigate the testing problems for ARCH effect and goodness of fit, respectively. A simulation study is conducted to evaluate the finite-sample performance of the proposed estimation methodology and testing procedure.     

Statistical inference under imputation for proportional hazard model with missing covariates

Missing covariate data are common in biomedical studies. In this article, by using the non parametric kernel regression technique, a new imputation approach is developed for the Cox-proportional hazard regression model with missing covariates. This method achieves the same efficiency as the fully augmented weighted estimators (Qi et al. 2005. Journal of the American Statistical Association, 100:1250) and has a simpler form. The asymptotic properties of the proposed estimator are derived and analyzed. The comparisons between the proposed imputation method and several other existing methods are conducted via a number of simulation studies and a mouse leukemia data.     

Asymptotic normality for a non parametric estimator of conditional quantile with left-truncated data

In this paper, we construct a non parametric estimator of conditional distribution function by the double-kernel local linear approach for left-truncated data, from which we derive the weighted double-kernel local linear estimator of conditional quantile. The asymptotic normality of the proposed estimators is also established. Finite-sample performance of the estimator is investigated via simulation.