Functional data: local linear estimation of the conditional density and its application

In this paper, we introduce a new nonparametric estimation procedure of the conditional density of a scalar response variable given a random variable taking values in a semi-metric space. Under some general conditions, we establish both the pointwise and the uniform almost-complete consistencies with convergence rates of the conditional density estimator related to this estimation procedure. Moreover, we give some particular cases of our results which can also be considered as novel in the finite-dimensional setting. Notice also that the results of this paper are used to derive some asymptotic properties of the local linear estimator of the conditional mode.     

Asymptotic normality of a conditional hazard function estimate in the single index for quasi-associated data

Abstract

The main goal of this paper is to study the estimation of the conditional hazard function of a scalar response variable Y given a hilbertian random variable X in functional single-index model. We construct an estimator of this nonparametric function and we study its asymptotic properties, under quasi-associated structure. Precisely, we establish the asymptotic normality of the constructed estimator. We carried out simulation experiments to examine the behavior of this asymptotic property over finite sample data.     

FDA: strong consistency of the kNN local linear estimation of the functional conditional density and mode

In this paper we present a new estimator of the conditional density and mode when the co-variables are of functional kind. This estimator is a combination of both, the k-Nearest Neighbours procedure and the functional local linear estimation. Then, for each statistical parameter (conditional density or mode), results concerning the strong consistency and rate of convergence of the estimators are presented. Finally, their performances, for finite sample sizes, are illustrated by using simulated 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.     

Estimation of conditional mode with truncated,censored, and dependent data

In this paper, we studied the uniform convergence with rates for the kernel estimator of the conditional mode function for a left truncated and right censored model. It is assumed that the lifetime observations with multivariate covariates form a stationary α-mixing sequence. Also, the asymptotic normality of the estimator is established.     

Averaging estimation for conditional covariance models

Estimating conditional covariance matrices is important in statistics and finance. In this paper, we propose an averaging estimator for the conditional covariance, which combines the estimates of marginal conditional covariance matrices by Model Averaging MArginal Regression of Li, Linton, and Lu. This estimator avoids the “curse of dimensionality” problem that the local constant estimator of Yin et al. suffered from. We establish the asymptotic properties of the averaging weights and that of the proposed conditional covariance estimator. The finite sample performances are augmented by simulation. An application to portfolio allocation illustrates the practical superiority of the averaging estimator.     

Product-limit Estimators and Cox Regression with Missing Censoring Information

The Kaplan–Meier estimator of a survival function requires that the censoring indicator is always observed. A method of survival function estimation is developed when the censoring indicators are missing completely at random (MCAR). The resulting estimator is a smooth functional of the Nelson–Aalen estimators of certain cumulative transition intensities. The asymptotic properties of this estimator are derived. A simulation study shows that the proposed estimator has greater efficiency than competing MCAR-based estimators. The approach is extended to the Cox model setting for the estimation of a conditional survival function given a covariate.     

Quantile Regression for Location‐Scale Time Series Models with Conditional Heteroscedasticity

This paper considers quantile regression for a wide class of time series models including autoregressive and moving average (ARMA) models with asymmetric generalized autoregressive conditional heteroscedasticity errors. The classical mean‐variance models are reinterpreted as conditional location‐scale models so that the quantile regression method can be naturally geared into the considered models. The consistency and asymptotic normality of the quantile regression estimator is established in location‐scale time series models under mild conditions. In the application of this result to ARMA‐generalized autoregressive conditional heteroscedasticity models, more primitive conditions are deduced to obtain the asymptotic properties. For illustration, a simulation study and a real data analysis are provided.     

Bernstein conditional density estimation with application to conditional distribution and regression functions

In this paper we propose a smooth nonparametric estimation for the conditional probability density function based on a Bernstein polynomial representation. Our estimator can be written as a finite mixture of beta densities with data-driven weights. Using the Bernstein estimator of the conditional density function, we derive new estimators for the distribution function and conditional mean. We establish the asymptotic properties of the proposed estimators, by proving their asymptotic normality and by providing their asymptotic bias and variance. Simulation results suggest that the proposed estimators can outperform the Nadaraya–Watson estimator and, in some specific setups, the local linear kernel estimators. Finally, we use our estimators for modeling the income in Italy, conditional on year from 1951 to 1998, and have another look at the well known Old Faithful Geyser data.     

Analysis of transformation models with right-truncated data

In many applications, statistical data are frequently observed subject to a retrospective sampling criterion resulting in right-truncated data. In this article, a general class of semiparametric transformation models that include proportional hazards model and proportional odds model as special cases is studied for the analysis of right-truncated data. We proposed two estimators for regression coefficients. The first estimator is based on martingale estimating equations. The second estimator is based on the conditional likelihood function given the truncation times. The asymptotic properties of both estimators are derived. The finite sample performance is examined through a simulation study.     

Improved estimation for Poisson INAR(1) models

We consider the first-order Poisson autoregressive model proposed by McKenzie [Some simple models for discrete variate time series. Water Resour Bull. 1985;21:645–650] and Al-Osh and Alzaid [First-order integer valued autoregressive (INAR(1)) process. J Time Ser Anal. 1987;8:261–275], which may be suitable in situations where the time series data are non-negative and integer valued. We derive the second-order bias of the squared difference estimator [Weiß. Process capability analysis for serially dependent processes of Poisson counts. J Stat Comput Simul. 2012;82:383–404] for one of the parameters and show that this bias can be used to define a bias-reduced estimator. The behaviour of a modified conditional least-squares estimator is also studied. Furthermore, we access the asymptotic properties of the estimators here discussed. We present numerical evidence, based upon Monte Carlo simulation studies, showing that the here proposed bias-adjusted estimator outperforms the other estimators in small samples. We also present an application to a real data set.     

NONPARAMETRIC ESTIMATION OF CONDITIONAL CUMULATIVE HAZARDS FOR MISSING POPULATION MARKS

A new function for the competing risks model, the conditional cumulative hazard function, is introduced, from which the conditional distribution of failure times of individuals failing due to cause  j  can be studied. The standard Nelson–Aalen estimator is not appropriate in this setting, as population membership (mark) information may be missing for some individuals owing to random right-censoring. We propose the use of imputed population marks for the censored individuals through fractional risk sets. Some asymptotic properties, including uniform strong consistency, are established. We study the practical performance of this estimator through simulation studies and apply it to a real data set for illustration.     

Weak convergence for the conditional distribution function in a Koziol–Green model under dependent censoring

In this paper we consider the conditional Koziol–Green model of Braekers and Veraverbeke [2008. A conditional Koziol–Green model under dependent censoring. Statist. Probab. Lett., accepted for publication] in which they generalized the Koziol–Green model of Veraverbeke and Cadarso Suárez [2000. Estimation of the conditional distribution in a conditional Koziol–Green model. Test 9, 97–122] by assuming that the association between a censoring time and a time until an event is described by a known Archimedean copula function. They got in this way, an informative censoring model with two different types of informative censoring. Braekers and Veraverbeke [2008. A conditional Koziol–Green model under dependent censoring. Statist. Probab. Lett., accepted for publication] derived in this model a non-parametric Koziol–Green estimator for the conditional distribution function of the time until an event, for which they showed the uniform consistency and the asymptotic normality. In this paper we extend their results and prove the weak convergence of the process associated to this estimator. Furthermore we show that the conditional Koziol–Green estimator is asymptotically more efficient in this model than the general copula-graphic estimator of Braekers and Veraverbeke [2005. A copula-graphic estimator for the conditional survival function under dependent censoring. Canad. J. Statist. 33, 429–447]. As last result, we construct an asymptotic confidence band for the conditional Koziol–Green estimator. Through a simulation study, we investigate the small sample properties of this asymptotic confidence band. Afterwards we apply this estimator and its confidence band on a practical data set.     

Asymptotic properties of the kernel mode estimator under twice censorship model

In this article, we study the asymptotic properties of the kernel estimator of the mode and density function when the data are twice censored. More specifically, we first establish a strong uniform consistency over a compact set with a rate of the kernel density estimator and then we give the consistency with rate and asymptotic normality for the kernel mode estimator. An application to confidence bands is given.     

Non-parametric Quantile Regression with Censored Data

Abstract.  Censored regression models have received a great deal of attention in both the theoretical and applied statistics literature. Here, we consider a model in which the response variable is censored but not the covariates. We propose a new estimator of the conditional quantiles based on the local linear method, and give an algorithm for its numerical implementation. We study its asymptotic properties and evaluate its performance on simulated data sets.     

Estimation of regression parameters in generalized linear models for cluster correlated data with measurement error

Liang and Zeger (1986) introduced a class of estimating equations that gives consistent estimates of regression parameters and of their asymptotic variances in the class of generalized linear models for cluster correlated data. When the independent variables or covariates in such models are subject to measurement errors, the parameter estimates obtained from these estimating equations are no longer consistent. To correct for the effect of measurement errors, an estimator with smaller asymptotic bias is constructed along the lines of Stefanski (1985), assuming that the measurement error variance is either known or estimable. The asymptotic distribution of the bias-corrected estimator and a consistent estimator of its asymptotic variance are also given. The special case of a binary logistic regression model is studied in detail. For this case, methods based on conditional scores and quasilikelihood are also extended to cluster correlated data. Results of a small simulation study on the performance of the proposed estimators and associated tests of hypotheses are reported.     

Berry–Esseen type bound of conditional mode estimation under truncation and strong mixing assumptions

ABSTRACT

In order to investigate the convergence rate of the asymptotic normality for the estimator of the conditional mode function for the left-truncation model, we derive a Berry–Esseen type bound of the estimator when the lifetime observations with multivariate covariates form a stationary α-mixing sequence. The finite sample performance of the estimator of the conditional mode function is explored through simulations.     

An estimated‐score approach for dealing with missing covariate data in matched case–control studies

Matched case–control designs are commonly used in epidemiological studies for estimating the effect of exposure variables on the risk of a disease by controlling the effect of confounding variables. Due to retrospective nature of the study, information on a covariate could be missing for some subjects. A straightforward application of the conditional logistic likelihood for analyzing matched case–control data with the partially missing covariate may yield inefficient estimators of the parameters. A robust method has been proposed to handle this problem using an estimated conditional score approach when the missingness mechanism does not depend on the disease status. Within the conditional logistic likelihood framework, an empirical procedure is used to estimate the odds of the disease for the subjects with missing covariate values. The asymptotic distribution and the asymptotic variance of the estimator when the matching variables and the completely observed covariates are categorical. The finite sample performance of the proposed estimator is assessed through a simulation study. Finally, the proposed method has been applied to analyze two matched case–control studies. The Canadian Journal of Statistics 38: 680–697; 2010 © 2010 Statistical Society of Canada     

A nonparametric inverse probability weighted estimation for functional data with missing response data at random

This paper considers the nonparametric inverse probability weighted estimation for functional data with missing response data at random. Under mild conditions, the asymptotic properties of the proposed estimation method are established. Based on the resampling method, the estimation of the asymptotic variance of the proposed estimator is obtained. Finally, the finite sample properties of the proposed estimation method are investigated via Monte Carlo simulation studies. A real data analysis is given to illustrate the use of the proposed method.     

Consistency of information criteria for model selection with missing data

ABSTRACT

In this paper, we investigate the consistency of the Expectation Maximization (EM) algorithm-based information criteria for model selection with missing data. The criteria correspond to a penalization of the conditional expectation of the complete data log-likelihood given the observed data and with respect to the missing data conditional density. We present asymptotic properties related to maximum likelihood estimation in the presence of incomplete data and we provide sufficient conditions for the consistency of model selection by minimizing the information criteria. Their finite sample performance is illustrated through simulation and real data studies.