Kernel estimators of the ROC Curve with censored data

Receiver operating characteristic (ROC) curves are often used to study the two sample problem in medical studies. However, most data in medical studies are censored. Usually a natural estimator is based on the Kaplan-Meier estimator. In this paper we propose a smoothed estimator based on kernel techniques for the ROC curve with censored data. The large sample properties of the smoothed estimator are established. Moreover, deficiency is considered in order to compare the proposed smoothed estimator of the ROC curve with the empirical one based on Kaplan-Meier estimator. It is shown that the smoothed estimator outperforms the direct empirical estimator based on the Kaplan-Meier estimator under the criterion of deficiency. A simulation study is also conducted and a real data is analyzed.     

AN ADAPTIVE VERSION OF GLIMM＇S SCHEME

This article describes a local error estimator for Glimm＇s scheme for hyperbolic systems of conservation laws and uses it to replace the usual random choice in Glimm＇s scheme by an optimal choice. As a by-product of the local error estimator, the procedure provides a global error estimator that is shown numerically to be a very accurate estimate of the error in L1 （R） for all times. Although there is partial mathematical evidence for the error estimator proposed, at this stage the error estimator must be considered ad- hoc. Nonetheless, the error estimator is simple to compute, relatively inexpensive, without adjustable parameters and at least as accurate as other existing error estimators. Numerical experiments in 1-D for Burgers＇ equation and for Euler＇s system are performed to measure the asymptotic accuracy of the resulting scheme and of the error estimator.     

Some properties for the estimators in linear mixed models

Linear mixed models (LMMs) have become an important statistical method for analyzing cluster or longitudinal data. In most cases, it is assumed that the distributions of the random effects and the errors are normal. This paper removes this restrictions and replace them by the moment conditions. We show that the least square estimators of fixed effects are consistent and asymptotically normal in general LMMs. A closed-form estimator of the covariance matrix for the random effect is constructed and its consistent is shown. Based on this, the consistent estimate for the error variance is also obtained. A simulation study and a real data analysis show that the procedure is effective.     

L1-Norm Estimation and Random Weighting Method in a Semiparametric Model

In this paper, the L_1-norm estimators and the random weighted statistic for a semiparametric regression model are constructed, the strong convergence rates of estimators are obtain under certain conditions, the strong efficiency of the random weighting method is shown. A simulation study is conducted to compare the L_1-norm estimator with the least square estimator in term of approximate accuracy, and simulation results are given for comparison between the random weighting method and normal approximation method.     

对称熵损失下的一类指数族刻度参数估计

In this paper we investigate the estimator for the rth power of the scale parameter in a class of exponential family under symmetric entropy loss L（θ, δ） = v（θ/δ ＋ δ/θ - 2）. An exact form of the minimum risk equivariant estimator under symmetric entropy loss is given, and the minimaxity of the minimum risk equivariant estimator is proved. The results with regard to admissibility and inadmissibility of a class of linear estimators of the form cT（X） ＋ d are given, where T（X） Gamma（v, θ）.     

Distribution estimation with smoothed auxiliary information

Distribution estimation is very important in order to make statistical inference for parameters or its functions based on this distribution.In this work we propose an estimator of the distribution of some variable with non-smooth auxiliary information,for example,a symmetric distribution of this variable.A smoothing technique is employed to handle the non-differentiable function.Hence,a distribution can be estimated based on smoothed auxiliary information.Asymptotic properties of the distribution estimator are derived and analyzed.The distribution estimators based on our method are found to be significantly efficient than the corresponding estimators without these auxiliary information.Some simulation studies are conducted to illustrate the finite sample performance of the proposed estimators.     

AN EFFICIENT SEQUENTIAL DESIGN FOR SENSITIVITY EXPERIMENTS

In sensitivity experiments, the response is binary and each experimental unit has a critical stimulus level that cannot be observed directly. It is often of interest to estimate extreme quantiles of the distribution of these critical stimulus levels over the tested products. For this purpose a new sequential scheme is proposed with some commonly used models. By using the bootstrap repeated-sampling principle, reasonable prior distributions based on a historic data set are specified. Then, a Bayesian strategy for the sequential procedure is provided and the estimator is given. Further, a high order approximation for such an estimator is explored and its consistency is proven. A simulation study shows that the proposed method gives superior performances over the existing methods.     

Moderate Deviations for M-estimators in Linear Models with φ-mixing Errors

In this paper, the moderate deviations for the M-estimators of regression parameter in a linear model are obtained when the errors form a strictly stationary φ-mixing sequence. The results are applied to study many different types of M-estimators such as Huber's estimator, Lp-regression estimator, least squares estimator and least absolute deviation estimator.     

A class of weighted estimators for additive hazards model in case-cohort studies

Case-cohort sampling is a commonly used and efficient method for studying large cohorts. In many situations, some covariates are easily measured on all cohort subjects, and surrogate measurements of the expensive covariates also may be observed. In this paper, to make full use of the covariate data collected outside the case-cohort sample, we propose＇a class of weighted estimators with general time-varying weights for the additive hazards model, and the estimators are shown to be consistent and asymptotically normal. We also identify the estimator within this class that maximizes efficiency, and simulation studies show that the efficiency gains of the proposed estimator over the existing ones can be substantial in practical situations. A real example is provided.     

Weighted Profile Least Squares Estimation for a Panel Data Varying-Coefficient Partially Linear Model

This paper is concerned with inference of panel data varying-coefficient partially linear models with a one-way error structure. The model is a natural extension of the well-known panel data linear model （due to Baltagi 1995） to the setting of semiparametric regressions. The authors propose a weighted profile least squares estimator （WPLSE） and a weighted local polynomial estimator （WLPE） for the parametric and nonparametric components, respectively. It is shown that the WPLSE is asymptotically more efficient than the usual profile least squares estimator （PLSE）, and that the WLPE is also asymptotically more efficient than the usual local polynomial estimator （LPE）. The latter is an interesting result. According to Ruckstuhl, Welsh and Carroll （2000） and Lin and Carroll （2000）, ignoring the correlation structure entirely and ＂pretending＂ that the data are really independent will result in more efficient estimators when estimating nonparametric regression with longitudinal or panel data. The result in this paper shows that this is not true when the design points of the nonparametric component have a closeness property within groups. The asymptotic properties of the proposed weighted estimators are derived. In addition, a block bootstrap test is proposed for the goodness of fit of models, which can accommodate the correlations within groups illustrate the finite sample performances of the Some simulation studies are conducted to proposed procedures.     

Asymptotic Normality of a Combined Regression Estimator

In this paper, we propose a combined regression estimator by using a parametric estimator and a nonparametric estimator of the regression function. The asymptotic distribution of this estimator is obtained for cases where the parametric regression model is correct, incorrect, and approximately correct. These distributional results imply that the combined estimator is superior to the kernel estimator in the sense that it can never do worse than the kernel estimator in terms of convergence rate and it has the same convergence rate as the parametric estimator in the case where the parametric model is correct. Unlike the parametric estimator, the combined estimator is robust to model misspecification. In addition, we also establish the asymptotic distribution of the estimator of the weight given to the parametric estimator in constructing the combined estimator. This can be used to construct consistent tests for the parametric regression model used to form the combined estimator.     

随机变量二次型的协方差在混合效应模型中的应用

本文提出方差分量ANOVA估计的一种改进方法, 证明了对于一般的方差分量模型, 只要方差分量的ANOVA估计存在就可以通过此方法给出其改进形式, 并且在均方误差意义下优于ANOVA估计. 特别地, 对于单向分类随机效应模型, Kelly和Mathew^[1]对ANOVA估计的改进就是我们提出的改进方法的特殊形式, 这也给出了此类改进估计在均方误差意义下优于ANOVA估计的另一种合理的解释. 同时, 本文又将此思想应用到对谱分解估计的改进上. 本文应用协方差的简单性质证明了对带有一个随机效应的方差分量模型, 当随机效应的协方差阵只有一个非零特征值时, 随机效应方差分量谱分解估计在均方误差意义下总是优于ANOVA估计. 本文最后将第三节的结论推广到广义谱分解估计下, 同时给出广义谱分解估计待定系数的一个合理的取值.     

两个半相依模型回归系数的改进估计

对于两个半相依回归系统的未知回归系数,本文首先借鉴文献中给出的两步协方差改进估计的方法给出两种两步协方差改进估计序列,并给出其与两步估计等价的条件和均方误差意义下的优良性; 其次,我们对文献中给出的一种两步估计作简单改进,使得改进后的估计在更大的参数空间内优于最小二乘估计. 再次,本文另辟蹊径, 构造了一种新的估计,同样地,此估计也具有更好的小样本性质.本文最后一节讨论了Pitman准则下两步估计的优良性.     

Asymptotic properties of conditional maximum likelihood estimator in a certain exponential model

The conditional maximum likelihood estimator is suggested as an alternative to the maximum likelihood estimator and is favorable for an estimator of a dispersion parameter in the normal distribution, the inverse-Gaussian distribution, and so on. However, it is not clear whether the conditional maximum likelihood estimator is asymptotically efficient in general. Consider the case where it is asymptotically efficient and its asymptotic covariance depends only on an objective parameter in an exponential model. This remand implies that the exponential model possesses a certain parallel foliation. In this situation, this paper investigates asymptotic properties of the conditional maximum likelihood estimator and compares the conditional maximum likelihood estimator with the maximum likelihood estimator. We see that the bias of the former is more robust than that of the latter and that two estimators are very close, especially in the sense of bias-corrected version. The mean Pythagorean relation is also discussed.     

AN ADAPTIVE VERSION OF GLIMM''S SCHEME

This article describes a local error estimator for Glimm's scheme for hyperbolic systems of conservation laws and uses it to replace the usual random choice in Glimm's scheme by an optimal choice.As a by-product of the local error estimator,the procedure provides a global error estimator that is shown numerically to be a very accurate estimate of the error in L1(R) for all times.Although there is partial mathematical evidence for the error estimator proposed,at this stage the error estimator must be considered adhoc.Nonetheless,the error estimator is simple to compute,relatively inexpensive,without adjustable parameters and at least as accurate as other existing error estimators.Numerical experiments in 1-D for Burgers' equation and for Euler's system are performed to measure the asymptotic accuracy of the resulting scheme and of the error estimator.     

分位点函数的光滑非参数估计的BAHADUR表示

文中对分位函数给出了具有更广泛应用的光滑分位估计,证明了该光滑分位估计的逐点和一致的Bahadur强表示定理;并由此结果推导了估计的重对数律,强逼近等深刻结果。     

Estimating the Innovation Distribution in Nonlinear Autoregressive Models

The usual estimator for the expectation of a function under the innovation distribution of a nonlinear autoregressive model is the empirical estimator based on estimated innovations. It can be improved by exploiting that the innovation distribution has mean zero. We show that the resulting estimator is efficient if the innovations are estimated with an efficient estimator for the autoregression parameter. Efficiency of this estimator is necessary except when the expectation of the function can be estimated adaptively. Analogous results hold for heteroscedastic models.     

Large and moderate deviations principles for kernel estimators of the multivariate regression

In this paper, we prove large deviations principle for the Nadaraya-Watson estimator and for the semi-recursive kernel estimator of the regression in the multidimensional case. Under suitable conditions, we show that the rate function is a good rate function. We thus generalize the results already obtained in the one-dimensional case for the Nadaraya-Watson estimator. Moreover, we give a moderate deviations principle for these two estimators. It turns out that the rate function obtained in the moderate deviations principle for the semi-recursive estimator is larger than the one obtained for the Nadaraya-Watson estimator.