Estimation of association parameters in copula models for bivariate left-truncated and right-censored data

We investigate the problem of estimating the association between two related survival variables when they follow a copula model and bivariate left-truncated and right-censored data are available. By expressing truncation probability as the functional of marginal survival functions, we propose a two-stage estimation procedure for estimating the parameters of Archimedean copulas. The asymptotic properties of the proposed estimators are established. Simulation studies are conducted to investigate the finite sample properties of the proposed estimators. The proposed method is applied to a bivariate RNA data.     

Partial covariate adjusted regression

Covariate adjusted regression (CAR) is a recently proposed adjustment method for regression analysis where both the response and predictors are not directly observed [?entürk, D., Müller, H.G., 2005. Covariate adjusted regression. Biometrika 92, 75–89]. The available data have been distorted by unknown functions of an observable confounding covariate. CAR provides consistent estimators for the coefficients of the regression between the variables of interest, adjusted for the confounder. We develop a broader class of partial covariate adjusted regression (PCAR) models to accommodate both distorted and undistorted (adjusted/unadjusted) predictors. The PCAR model allows for unadjusted predictors, such as age, gender and demographic variables, which are common in the analysis of biomedical and epidemiological data. The available estimation and inference procedures for CAR are shown to be invalid for the proposed PCAR model. We propose new estimators and develop new inference tools for the more general PCAR setting. In particular, we establish the asymptotic normality of the proposed estimators and propose consistent estimators of their asymptotic variances. Finite sample properties of the proposed estimators are investigated using simulation studies and the method is also illustrated with a Pima Indians diabetes data set.     

相依函数型数据的局部回归估计的渐近正态性

函数型数据研究近年来为越来越多的学者所重视,其在天文,医药,经济现象,生态环境及工业制造等诸多方面均有重要应用.非参数统计是统计研究的一个重要方面,其中核函数估计和局部多项式方法是这一类研究中重要常用方法.函数型数据的非参数方法中以核函数估计方法较为常见,且其收敛速度与极限分布无论在独立情形还是相依情形都有理论结果.而局部多项式的研究在函数型数据背景下较为少见,原因在于将局部多项式方法推广到函数型数据背景一直是一个难题. Marin, Ferraty, Vieu [Journal of Nonparametric Statistics, 22 (5) (2010), pp.617-632] 提出了非参函数型模型的局部回归估计. 这种估计可以看作是局部多项式估计在函数型数据背景下的一个推广.这种方法提出后,许多学者进一步研究了这种方法,考察了这种方法的收敛速度和极限分布,并将这种方法应用到不同的模型中以适应实际需求.但是,前人的研究都要求数据具有独立同分布的性质.然而许多实际数据并不符合这一假设.本文研究了在相依函数型数据情形下局部回归估计的渐近正态性.由于估计方法有差异,核函数估计的研究方法无法直接推广到局部回归估计,而相依性结构也给研究带来了一些挑战,我们采用Bernstein分块方法将相依性问题转化为渐近独立的问题,从而得到了估计的渐近正态性.此外我们还采用数据模拟的方法进一步验证了渐近正态的结果.     

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.     

Some Asymptotic Results of Kernel Density Estimators Under Random Left-Truncation and Dependent Data

Problems with truncated data arise frequently in survival analyses and reliability applications. The estimation of the density function of the lifetimes is often of interest. In this article, the estimation of density function by the kernel method is considered, when truncated data are showing some kind of dependence. We apply the strong Gaussian approximation technique to study the strong uniform consistency for kernel estimators of the density function under a truncated dependent model. We also apply the strong approximation results to study the integrated square error properties of the kernel density estimators under the truncated dependent scheme.     

Shrinkage and Penalty Estimators of a Poisson Regression Model

In this paper we propose Stein‐type shrinkage estimators for the parameter vector of a Poisson regression model when it is suspected that some of the parameters may be restricted to a subspace. We develop the properties of these estimators using the notion of asymptotic distributional risk. The shrinkage estimators are shown to have higher efficiency than the classical estimators for a wide class of models. Furthermore, we consider three different penalty estimators: the LASSO, adaptive LASSO, and SCAD estimators and compare their relative performance with that of the shrinkage estimators. Monte Carlo simulation studies reveal that the shrinkage strategy compares favorably to the use of penalty estimators, in terms of relative mean squared error, when the number of inactive predictors in the model is moderate to large. The shrinkage and penalty strategies are applied to two real data sets to illustrate the usefulness of the procedures in practice.     

Nonparametric analysis of recurrent gap time data

Abstract

Recurrent event data are frequently encountered in longitudinal studies. In many applications, the times between successive recurrent events (gap times) are often of interest and lead to problems that have received much attention recently. In this article, using the approach of inverse probability-of-censoring weights (IPCW), we propose nonparametric estimators for the estimation of the bivariate distribution and survival functions for gap times of recurrent event data. We also consider the estimation of Kendall’s tau for two gap times by expressing it as an integral functional of the bivariate survival function. The asymptotic properties of the proposed estimators are established. Simulation studies are conducted to investigate their finite sample performance.     

Asymptotic Properties and Variance Estimators of the M-quantile Regression Coefficients Estimators

M-quantile regression is defined as a “quantile-like” generalization of robust regression based on influence functions. This article outlines asymptotic properties for the M-quantile regression coefficients estimators in the case of i.i.d. data with stochastic regressors, paying attention to adjustments due to the first-step scale estimation. A variance estimator of the M-quantile regression coefficients based on the sandwich approach is proposed. Empirical results show that this estimator appears to perform well under different simulated scenarios. The sandwich estimator is applied in the small area estimation context for the estimation of the mean squared error of an estimator for the small area means. The results obtained improve previous findings, especially in the case of heteroskedastic data.     

On the Local Polynomial Estimators of the Counting Process Intensity Function and its Derivatives

Abstract. We consider the properties of the local polynomial estimators of a counting process intensity function and its derivatives. By expressing the local polynomial estimators in a kernel smoothing form via effective kernels, we show that the bias and variance of the estimators at boundary points are of the same magnitude as at interior points and therefore the local polynomial estimators in the context of intensity estimation also enjoy the automatic boundary correction property as they do in other contexts such as regression. The asymptotically optimal bandwidths and optimal kernel functions are obtained through the asymptotic expressions of the mean square error of the estimators. For practical purpose, we suggest an effective and easy‐to‐calculate data‐driven bandwidth selector. Simulation studies are carried out to assess the performance of the local polynomial estimators and the proposed bandwidth selector. The estimators and the bandwidth selector are applied to estimate the rate of aftershocks of the Sichuan earthquake and the rate of the Personal Emergency Link calls in Hong Kong.     

Presmoothed Estimation with Left-Truncated and Right-Censored Data

We propose a new method to estimate the cumulative hazard function and the corresponding distribution function of survival times under randomly left-truncated and right-censored observations (LTRC). The new estimators are based on presmoothing ideas, the estimation of the conditional expectation m of the censoring indicator. An almost sure representation for both estimators is established, from which a strong consistency rate and asymptotic normality are derived. It is shown that the presmoothed modification leads to a gain in terms of asymptotic mean squared error. This efficiency with respect to the classical estimators is also shown in a simulation study. Finally, an application to a real data set is provided.     

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.     

Mean Estimation Under Successive Sampling with Calibration Estimators

The estimation of the finite population mean in successive occasions is investigated with calibration estimators in this article. We propose several estimators based on calibration techniques with arbitrary sampling design in each of the occasions. Asymptotic variance formulaes are derived for the proposed estimators. The properties of these estimators are studied via a simulation study and using natural populations.     

Jackknife empirical likelihood for the error variance in linear models

Variance estimation is a fundamental yet important problem in statistical modelling. In this paper, we propose jackknife empirical likelihood (JEL) methods for the error variance in a linear regression model. We prove that the JEL ratio converges to the standard chi-squared distribution. The asymptotic chi-squared properties for the adjusted JEL and extended JEL estimators are also established. Extensive simulation studies to compare the new JEL methods with the standard method in terms of coverage probability and interval length are conducted, and the simulation results show that our proposed JEL methods perform better than the standard method. We also illustrate the proposed methods using two real data sets.     

Understanding Estimators of Treatment Effects in Regression Discontinuity Designs

In this paper, we propose two new estimators of treatment effects in regression discontinuity designs. These estimators can aid understanding of the existing estimators such as the local polynomial estimator and the partially linear estimator. The first estimator is the partially polynomial estimator which extends the partially linear estimator by further incorporating derivative differences of the conditional mean of the outcome on the two sides of the discontinuity point. This estimator is related to the local polynomial estimator by a relocalization effect. Unlike the partially linear estimator, this estimator can achieve the optimal rate of convergence even under broader regularity conditions. The second estimator is an instrumental variable estimator in the fuzzy design. This estimator will reduce to the local polynomial estimator if higher order endogeneities are neglected. We study the asymptotic properties of these two estimators and conduct simulation studies to confirm the theoretical analysis.     

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.     

Non-parametric estimation of copula based mutual information

Abstract

Mutual information is a measure for investigating the dependence between two random variables. The copula based estimation of mutual information reduces the complexity because it is depend only on the copula density. We propose two estimators and discuss the asymptotic properties. To compare the performance of the estimators a simulation study is carried out. The methods are illustrated using real data sets.     

Regularized robust estimation in binary regression models

In this paper, we investigate robust parameter estimation and variable selection for binary regression models with grouped data. We investigate estimation procedures based on the minimum-distance approach. In particular, we employ minimum Hellinger and minimum symmetric chi-squared distances criteria and propose regularized minimum-distance estimators. These estimators appear to possess a certain degree of automatic robustness against model misspecification and/or for potential outliers. We show that the proposed non-penalized and penalized minimum-distance estimators are efficient under the model and simultaneously have excellent robustness properties. We study their asymptotic properties such as consistency, asymptotic normality and oracle properties. Using Monte Carlo studies, we examine the small-sample and robustness properties of the proposed estimators and compare them with traditional likelihood estimators. We also study two real-data applications to illustrate our methods. The numerical studies indicate the satisfactory finite-sample performance of our procedures.     

Robust Estimators for the Parameters of Multivariate Lognormal Distribution

Abstract

In this paper, we introduce a class of location and scale estimators for the p-variate lognormal distribution. These estimators are obtained by applying a log transform to the data, computing robust Fisher consistent estimators for the obtained Gaussian data and transforming those estimators for the lognormal using the relationship between the parameters of both distributions. We prove some of the properties of these estimators, such as Fisher consistency, robustness and asymptotic normality.     

Nonparametric Quantile Estimation with Correlated Failure Time Data

In biomedical studies, correlated failure time data arise often. Although point and confidence interval estimation for quantiles with independent censored failure time data have been extensively studied, estimation for quantiles with correlated failure time data has not been developed. In this article, we propose a nonparametric estimation method for quantiles with correlated failure time data. We derive the asymptotic properties of the quantile estimator and propose confidence interval estimators based on the bootstrap and kernel smoothing methods. Simulation studies are carried out to investigate the finite sample properties of the proposed estimators. Finally, we illustrate the proposed method with a data set from a study of patients with otitis media.     

Notes on interval estimation of the proportion ratio under a three-treatment three-period crossover trial

When comparing two experimental treatments with a placebo, we focus our attention on interval estimation of the proportion ratio (PR) of patient responses under a three-period crossover design. We propose a random effects exponential multiplicative risk model and derive asymptotic interval estimators in closed form for the PR between treatments and placebo. Using Monte Carlo simulations, we compare the performance of these interval estimators in a variety of situations. We use the data comparing two different doses of an analgesic with placebo for the relief of primary dysmenorrhea to illustrate the use of these interval estimators and the difference in estimates of the PR and odds ratio (OR) when the underlying relief rates are not small.