On the information content of partial observations

The modified zero order approach to estimating coefficients in the face of missing observations treats them as parameters to be estimated simultaneously with the missing observations. The paper then investigates (in the context of Han's generalized regression model)(i) when parameter estimators don't vary between using the partial data points and using only the complete ones (the informationless result), and (ii) large sample properties of the modified zero order estimator. It's found the sequential cut property is crucial to the informationless result for coefficient estimators; consistency of the modified zero order estimator depends on the percentage of observations with missing elements for large sample sizes or the sequential cut property.     

A comparison study of nonparametric imputation methods

Consider estimation of a population mean of a response variable when the observations are missing at random with respect to the covariate. Two common approaches to imputing the missing values are the nonparametric regression weighting method and the Horvitz-Thompson (HT) inverse weighting approach. The regression approach includes the kernel regression imputation and the nearest neighbor imputation. The HT approach, employing inverse kernel-estimated weights, includes the basic estimator, the ratio estimator and the estimator using inverse kernel-weighted residuals. Asymptotic normality of the nearest neighbor imputation estimators is derived and compared to kernel regression imputation estimator under standard regularity conditions of the regression function and the missing pattern function. A comprehensive simulation study shows that the basic HT estimator is most sensitive to discontinuity in the missing data patterns, and the nearest neighbors estimators can be insensitive to missing data patterns unbalanced with respect to the distribution of the covariate. Empirical studies show that the nearest neighbor imputation method is most effective among these imputation methods for estimating a finite population mean and for classifying the species of the iris flower data.     

Adaptive nonparametric density estimation with missing observations

It is well known that if some observations in a sample from the probability density are not available, then in general the density cannot be estimated. A possible remedy is to use an auxiliary variable that explains the missing mechanism. For this setting a data-driven estimator is proposed that mimics performance of an oracle that knows all observations from the sample. It is also proved that the estimator adapts to unknown smoothness of the density and its mean integrated squared error converges with a minimax rate. A numerical study, together with the analysis of a real data, shows that the estimator is feasible for small samples.     

A Class of Weighted Estimating Equations for Semiparametric Transformation Models with Missing Covariates

In survival analysis, covariate measurements often contain missing observations; ignoring this feature can lead to invalid inference. We propose a class of weighted estimating equations for right‐censored data with missing covariates under semiparametric transformation models. Time‐specific and subject‐specific weights are accommodated in the formulation of the weighted estimating equations. We establish unified results for estimating missingness probabilities that cover both parametric and non‐parametric modelling schemes. To improve estimation efficiency, the weighted estimating equations are augmented by a new set of unbiased estimating equations. The resultant estimator has the so‐called ‘double robustness’ property and is optimal within a class of consistent estimators.     

数据随机缺失下分位数回归模型的诱导光滑估计法

在数据随机缺失的分位数回归模型中,运用诱导光滑思想构造光滑的估计方程,得到了回归参数的诱导光滑估计及渐近协方差估计。接着证明了诱导光滑估计的渐近正态性质,并给出诱导光滑估计及其渐近协方差估计的算法。模拟研究表明新方法在有限样本中表现出色。     

A robust regression model for a first-order autoregressive time series with unequal spacing: application to water monitoring

Summary.  Time series arise often in environmental monitoring settings, which typically involve measuring processes repeatedly over time. In many such applications, observations are irregularly spaced and, additionally, are not distributed normally. An example is water monitoring data collected in Boston Harbor by the Massachusetts Water Resources Authority. We describe a simple robust approach for estimating regression parameters and a first-order autocorrelation parameter in a time series where the observations are irregularly spaced. Estimates are obtained from an estimating equation that is constructed as a linear combination of estimated innovation errors, suitably made robust by symmetric and possibly bounded functions. Under an assumption of data missing completely at random and mild regularity conditions, the proposed estimating equation yields consistent and asymptotically normal estimates. Simulations suggest that our estimator performs well in moderate sample sizes. We demonstrate our method on Secchi depth data collected from Boston Harbor.     

A note on robustness of D-optimal block designs for two-colour microarray experiments

Two-colour microarray experiments form an important tool in gene expression analysis. Due to the high risk of missing observations in microarray experiments, it is fundamental to concentrate not only on optimal designs but also on designs which are robust against missing observations. As an extension of Latif et al. (2009), we define the optimal breakdown number for a collection of designs to describe the robustness, and we calculate the breakdown number for various D-optimal block designs. We show that, for certain values of the numbers of treatments and arrays, the designs which are D-optimal have the highest breakdown number. Our calculations use methods from graph theory.     

A recursive approach for estimating missing observations in an univariate time series

A linear recursive technique that does not use the Kalman filter approach is proposed to estimate missing observations in an univariate time series. It is assumed that the series follows an invertible ARIMA model. The procedure is based on the restricted forecasting approach, and the recursive linear estimators are optimal in terms of minimum mean-square error.     

A minimax approach to missing values in linear regression

We consider the problem of estimating the parameter vector in the linear model when observations on the independent variables are partially missing or incorrect. A new estimator is developed which systematically combines prior restrictions on the exogenous variables with the incomplete data. We compare this method with the alternative strategy of deleting missing values.     

A Profile Conditional Likelihood Approach for the Semiparametric Transformation Regression Model with Missing Covariates

We propose a profile conditional likelihood approach to handle missing covariates in the general semiparametric transformation regression model. The method estimates the marginal survival function by the Kaplan-Meier estimator, and then estimates the parameters of the survival model and the covariate distribution from a conditional likelihood, substituting the Kaplan-Meier estimator for the marginal survival function in the conditional likelihood. This method is simpler than full maximum likelihood approaches, and yields consistent and asymptotically normally distributed estimator of the regression parameter when censoring is independent of the covariates. The estimator demonstrates very high relative efficiency in simulations. When compared with complete-case analysis, the proposed estimator can be more efficient when the missing data are missing completely at random and can correct bias when the missing data are missing at random. The potential application of the proposed method to the generalized probit model with missing continuous covariates is also outlined.     

Estimating the correlation coefficient in the presence of correlated observations from a bivariate normal population

Estimation of the correlation coefficient between two variates (p) in the presence of correlated observations from a bivar iate normal population is considered The estimated maximum likelihood estimator (EMLE), an estimate based on the maximum likelihood estimator (MLE), is proposed and studied for the estimation of p For the large sample case , approximate expressions foi the variance and the bias of the Pearson estimate of the correlation coefficient are derived. These expressions suggests that the Pearson’s estimator possesses high mean square error (MSE) in estimating ρ in comparison to the MLE The MSE is particularly high when the observations within clusters aie highly correlated. The Pearson’s estimate, the MLE, and the EMLE aie evaluated in a simulation study This study shows that the proposed EMLE pefoims bettei than the Pearson’s correlation coefficient except when the number of clusters is small.     

A wavelet lifting approach to long-memory estimation

Reliable estimation of long-range dependence parameters is vital in time series. For example, in environmental and climate science such estimation is often key to understanding climate dynamics, variability and often prediction. The challenge of data collection in such disciplines means that, in practice, the sampling pattern is either irregular or blighted by missing observations. Unfortunately, virtually all existing Hurst parameter estimation methods assume regularly sampled time series and require modification to cope with irregularity or missing data. However, such interventions come at the price of inducing higher estimator bias and variation, often worryingly ignored. This article proposes a new Hurst exponent estimation method which naturally copes with data sampling irregularity. The new method is based on a multiscale lifting transform exploiting its ability to produce wavelet-like coefficients on irregular data and, simultaneously, to effect a necessary powerful decorrelation of those coefficients. Simulations show that our method is accurate and effective, performing well against competitors even in regular data settings. Armed with this evidence our method sheds new light on long-memory intensity results in environmental and climate science applications, sometimes suggesting that different scientific conclusions may need to be drawn.     

Spectral estimation for locally stationary time series with missing observations

Time series arising in practice often have an inherently irregular sampling structure or missing values, that can arise for example due to a faulty measuring device or complex time-dependent nature. Spectral decomposition of time series is a traditionally useful tool for data variability analysis. However, existing methods for spectral estimation often assume a regularly-sampled time series, or require modifications to cope with irregular or ‘gappy’ data. Additionally, many techniques also assume that the time series are stationary, which in the majority of cases is demonstrably not appropriate. This article addresses the topic of spectral estimation of a non-stationary time series sampled with missing data. The time series is modelled as a locally stationary wavelet process in the sense introduced by Nason et al. (J. R. Stat. Soc. B 62(2):271–292, 2000) and its realization is assumed to feature missing observations. Our work proposes an estimator (the periodogram) for the process wavelet spectrum, which copes with the missing data whilst relaxing the strong assumption of stationarity. At the centre of our construction are second generation wavelets built by means of the lifting scheme (Sweldens, Wavelet Applications in Signal and Image Processing III, Proc. SPIE, vol. 2569, pp. 68–79, 1995), designed to cope with irregular data. We investigate the theoretical properties of our proposed periodogram, and show that it can be smoothed to produce a bias-corrected spectral estimate by adopting a penalized least squares criterion. We demonstrate our method with real data and simulated examples.     

Estimation in the Cox cure model with covariates missing not at random,with application to disease screening/prediction

In an attempt to provide a statistical tool for disease screening and prediction, we propose a semiparametric approach to analysis of the Cox proportional hazards cure model in situations where the observations on the event time are subject to right censoring and some covariates are missing not at random. To facilitate the methodological development, we begin with semiparametric maximum likelihood estimation (SPMLE) assuming that the (conditional) distribution of the missing covariates is known. A variant of the EM algorithm is used to compute the estimator. We then adapt the SPMLE to a more practical situation where the distribution is unknown and there is a consistent estimator based on available information. We establish the consistency and weak convergence of the resulting pseudo-SPMLE, and identify a suitable variance estimator. The application of our inference procedure to disease screening and prediction is illustrated via empirical studies. The proposed approach is used to analyze the tuberculosis screening study data that motivated this research. Its finite-sample performance is examined by simulation.     

Estimation and Inference Based on Neumann Series Approximation to Locally Efficient Score in Missing Data Problems

Abstract.  Theory on semi-parametric efficient estimation in missing data problems has been systematically developed by Robins and his coauthors. Except in relatively simple problems, semi-parametric efficient scores cannot be expressed in closed forms. Instead, the efficient scores are often expressed as solutions to integral equations. Neumann series was proposed in the form of successive approximation to the efficient scores in those situations. Statistical properties of the estimator based on the Neumann series approximation are difficult to obtain and as a result, have not been clearly studied. In this paper, we reformulate the successive approximation in a simple iterative form and study the statistical properties of the estimator based on the reformulation. We show that a doubly robust locally efficient estimator can be obtained following the algorithm in robustifying the likelihood score. The results can be applied to, among others, parametric regression, marginal regression and Cox regression when data are subject to missing values and the data are missing at random. A simulation study is conducted to evaluate the performance of the approach and a real data example is analysed to demonstrate the use of the approach.     

Estimation of single index model with missing response at random

Under missing at random, we estimate the unknown link function and the direction parameter in a single index model. The central limit theory and the convergence rate of the law of the iterated logarithm for the estimator of the direction parameter are derived, the optimal convergence rate for the estimator of the link function is obtained and simulation results support the theoretical results of the paper.     

Model selection of generalized partially linear models with missing covariates

In this paper, a generalized partially linear model (GPLM) with missing covariates is studied and a Monte Carlo EM (MCEM) algorithm with penalized-spline (P-spline) technique is developed to estimate the regression coefficients and nonparametric function, respectively. As classical model selection procedures such as Akaike's information criterion become invalid for our considered models with incomplete data, some new model selection criterions for GPLMs with missing covariates are proposed under two different missingness mechanism, say, missing at random (MAR) and missing not at random (MNAR). The most attractive point of our method is that it is rather general and can be extended to various situations with missing observations based on EM algorithm, especially when no missing data involved, our new model selection criterions are reduced to classical AIC. Therefore, we can not only compare models with missing observations under MAR/MNAR settings, but also can compare missing data models with complete-data models simultaneously. Theoretical properties of the proposed estimator, including consistency of the model selection criterions are investigated. A simulation study and a real example are used to illustrate the proposed methodology.     

Optimal and Robust Designs for Estimating the Concentration Curve and the AUC

The problem of interest is to estimate the concentration curve and the area under the curve (AUC) by estimating the parameters of a linear regression model with an autocorrelated error process. We introduce a simple linear unbiased estimator of the concentration curve and the AUC. We show that this estimator constructed from a sampling design generated by an appropriate density is asymptotically optimal in the sense that it has exactly the same asymptotic performance as the best linear unbiased estimator. Moreover, we prove that the optimal design is robust with respect to a minimax criterion. When repeated observations are available, this estimator is consistent and has an asymptotic normal distribution. Finally, a simulated annealing algorithm is applied to a pharmacokinetic model with correlated errors.     

Nonparametric Trend Estimation for Periodic Autoregressive Time Series

A periodically stationary time series has seasonal variances. A local linear trend estimation is proposed to accommodate unequal variances. A comparison of this proposed estimator with the estimator commonly used for a stationary time series is provided. The optimal bandwidth selection for this new trend estimator is discussed.     

缺失数据下的逆概率多重加权分位回归估计及其应用

数据缺失问题普遍存在于应用研究中。在随机缺失机制假定下,本文从模型推断角度出发,针对线性缺失分位回归模型,提出一种新的有效估计方法——逆概率多重加权（IPMW）估计。该方法是在逆概率加权（IPW）估计的基础上,结合倾向得分匹配及模型平均思想,经过多次估计,加权确定最终参数估计结果。该方法适用于响应变量是独立同分布或独立非同分布的情形,并适用于绝大多数缺失场景。经过理论推导及模拟研究发现,IPMW估计量在继承IPW估计量的优势上具有更稳健的性质。最后,将该方法应用于含有缺失数据的微观调查数据中,研究了经济较发达的准一线城市中等收入群体消费水平的影响因素,对比两种估计方法的估计结果及置信带,发现逆概率多重加权估计量的标准偏差更小,估计结果更稳健。