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.     

Estimation for parameters of interest in random effects growth curve models

In this paper, we consider the general growth curve model with multivariate random effects covariance structure and provide a new simple estimator for the parameters of interest. This estimator is not only convenient for testing the hypothesis on the corresponding parameters, but also has higher efficiency than the least-square estimator and the improved two-stage estimator obtained by Rao under certain conditions. Moreover, we obtain the necessary and sufficient condition for the new estimator to be identical to the best linear unbiased estimator. Examples of its application are given.     

A bivariate histogram density estimator: Consistency and asymptotic normality

This paper presents a nonparametric histogram density estimator based on the spacings of order statistics. This estimator generalizes to the bivariate case the univariate histogram estimator proposed by Van Ryzin (1973). The first of the two theorems in this paper gives conditions under which the estimator is pointwise strongly consistent. The second theorem provides conditions for the asymptotic normality of the estimator for points at which the density function possesses continuous partial derivatives of second order.     

Adaptive Fourier tester for statistical estimation

Based on Takenaka–Malmquist (TM) system, a new nonparametric estimator for probability density function is proposed. The TM estimation method is completely different from the existent density estimation methods in that the estimator depends on an approximate system with poles in a complex plane. Compared with the classic Fourier estimator, the TM estimator will offer more flexibility and adaptivity for real data due to the poles and nonlinearity of the phase of TM system. We compare the TM estimator with kernel, wavelet, and spline estimators by simulations. It shows that the introduced TM estimator is a more promising method than the existing and commonly used methods. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

广义自回归模型参数的收缩估计

运用经典方法结合参数的先验信息得到了广义一阶自回归模型中自相关系数的收缩估计的闭式表达式,它是通常极大似然估计与先验均值的加权平均,在适当的先验信息下优于原来的估计.     

Estimation of the entropy of a multivariate normal distribution

Motivated by problems in molecular biosciences wherein the evaluation of entropy of a molecular system is important for understanding its thermodynamic properties, we consider the efficient estimation of entropy of a multivariate normal distribution having unknown mean vector and covariance matrix. Based on a random sample, we discuss the problem of estimating the entropy under the quadratic loss function. The best affine equivariant estimator is obtained and, interestingly, it also turns out to be an unbiased estimator and a generalized Bayes estimator. It is established that the best affine equivariant estimator is admissible in the class of estimators that depend on the determinant of the sample covariance matrix alone. The risk improvements of the best affine equivariant estimator over the maximum likelihood estimator (an estimator commonly used in molecular sciences) are obtained numerically and are found to be substantial in higher dimensions, which is commonly the case for atomic coordinates in macromolecules such as proteins. We further establish that even the best affine equivariant estimator is inadmissible and obtain Stein-type and Brewster–Zidek-type estimators dominating it. The Brewster–Zidek-type estimator is shown to be generalized Bayes.     

Nonparametric estimators of the survival function with twice censored data

Patilea and Rolin (Ann Stat 34(2):925–938, 2006) proposed a product-limit estimator of the survival function for twice censored data. In this article, based on a modified self-consistent (MSC) approach, we propose an alternative estimator, the MSC estimator. The asymptotic properties of the MSC estimator are derived. A simulation study is conducted to compare the performance between the two estimators. Simulation results indicate that the MSC estimator outperforms the product-limit estimator and its advantage over the product-limit estimator can be very significant when right censoring is heavy.     

Nonparametric inference in multiplicative intensity model by discrete time observation

This paper deals with nonparametric inference problems in the multiplicative intensity model for counting processes. We propose a Nelson–Aalen type estimator based on discrete observation. The functional asymptotic normality of the estimator is proved. The limit process is the same as that in the continuous observation case, thus the proposed estimator based on discrete observation has the same properties as the Nelson–Aalen estimator based on continuous observation. For example, the asymptotic efficiency of proposed estimator is valid based on less information than the continuous observation case. A Kaplan–Meier type estimator is also discussed. Nonparametric goodness of fit test is considered, and an asymptotically distribution free test is proposed.     

Local linear estimator for stochastic differential equations driven by α-stable Lévy motions

We study the local linear estimator for the drift coefficient of stochastic differential equations driven by α-stable Lévy motions observed at discrete instants. Under regular conditions, we derive the weak consistency and central limit theorem of the estimator. Compared with Nadaraya-Watson estimator, the local linear estimator has a bias reduction whether the kernel function is symmetric or not under different schemes. A simulation study demonstrates that the local linear estimator performs better than Nadaraya-Watson estimator, especially on the boundary.     

利用样本分位数的Logistic总体分布参数的近似最佳线性无偏估计

Harter H_L.,Balakrishnan N.等先后讨论了Logistic总体分布参数的极大似然估计,近似极大似然估计;其后Ogawa J.,Lloyd E.H.,Kulldorff G.,Gupta S.S,及chan L.K. 等又先后讨论了Logistlic分布参数的最佳线性无偏估计及估计的相对效率等问题.令人遗憾的是:在大样本情形下,上述估计均难以求得.为缓解这一困难,本文讨论利用样本分位数的Logistic总体的近似最佳线性无偏估计,给出估计量的大样本性质,以及样本分位数不超过10情形下,估计量有渐近最大相对估计效率时样本分位数的选取方案等.     

Statistical Inference for the Parameter of Rayleigh Distribution Based on Progressively Typ e-I Interval Censored Sample

In this paper, the estimation of parameters based on a progressively type-I interval censored sample from a Rayleigh distribution is studied. Different methods of estimation are discussed. They include...     

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.     

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.     

Normal mixture quasi maximum likelihood estimation for non-stationary TGARCH(1,1) models

Although quasi maximum likelihood estimator based on Gaussian density (G-QMLE) is widely used to estimate GARCH-type models, it does not perform successfully when error distribution is either skewed or leptokurtic. This paper proposes normal mixture quasi-maximum likelihood estimator (NM-QMLE) for non-stationary TGARCH(1,1) models. We show that, under mild regular conditions, there is no consistent estimator for the intercept, and the proposed estimator for any other parameter is consistent.     

Duality Based A Posteriori Error Estimator for the Dirichlet Problem

The present work is devoted to the a posteriori error estimation for 2nd order elliptic problems with Dirichlet boundary conditions. Using the duality technique we derive a reliable and efficient a posteriori error estimator that measures the error in the energy norm. All the derivations are done on continuous level, and the estimator can be used in assessing the error of any approximate solution which belongs to the Sobolev space H¹, independently of the discretization method chosen. In particular, we make no use of the Galerkin orthogonality, which enables us to implement the estimator for measuring the error of the fictitious domain/penalty finite element method. The estimator is easily computable, and the only constant present in the estimator is the one from Friedrichs' inequality; the constant depends solely on the domain geometry, and the estimator is quite non‐sensitive to the error in the constant evaluation. Finally, we show how accurately the estimator captures the local error distribution, thus, creating a base for a justified adaptivity of an approximation.     

A Shrinkage Estimator for Combination of Bioassays

A shrinkage estimator and a maximum likelihood estimator are proposed in this paper for combination of bioassays. The shrinkage estimator is obtained in closed form which incorporates prior information just on the common log relative potency after the homogeneity test for combination of bioassays is accepted. It is a practical improvement over other estimators which require iterative procedure to obtain the estimator for the relative potency. A real data is also used to show the superiorities for the newly-proposed procedures.     

Estimating Functions for Nonlinear Time Series Models

This paper discusses the problem of estimation for two classes of nonlinear models, namely random coefficient autoregressive (RCA) and autoregressive conditional heteroskedasticity (ARCH) models. For the RCA model, first assuming that the nuisance parameters are known we construct an estimator for parameters of interest based on Godambe's asymptotically optimal estimating function. Then, using the conditional least squares (CLS) estimator given by Tjøstheim (1986, Stochastic Process. Appl., 21, 251–273) and classical moment estimators for the nuisance parameters, we propose an estimated version of this estimator. These results are extended to the case of vector parameter. Next, we turn to discuss the problem of estimating the ARCH model with unknown parameter vector. We construct an estimator for parameters of interest based on Godambe's optimal estimator allowing that a part of the estimator depends on unknown parameters. Then, substituting the CLS estimators for the unknown parameters, the estimated version is proposed. Comparisons between the CLS and estimated optimal estimator of the RCA model and between the CLS and estimated version of the ARCH model are given via simulation studies.     

An efficient estimator for the expectation of a bounded function under the residual distribution of an autoregressive process

Consider a stationary first-order autoregressive process, with i.i.d. residuals following an unknown mean zero distribution. The customary estimator for the expectation of a bounded function under the residual distribution is the empirical estimator based on the estimated residuals. We show that this estimator is not efficient, and construct a simple efficient estimator. It is adaptive with respect to the autoregression parameter.