Nonparametric estimation of multivariate density with direct and auxiliary data and application

We consider the problem of multivariate density estimation, using samples from the distribution of interest as well as auxiliary samples from a related distribution. We assume that the data from the target distribution and the related distribution may occur individually as well as in pairs. Using nonparametric maximum likelihood estimator of the joint distribution, we derive a kernel density estimator of the marginal density. We show theoretically, in a simple special case, that the implied estimator of the marginal density has smaller integrated mean squared error than that of a similar estimator obtained by ignoring dependence of the paired observations. We establish consistency of the marginal density estimator under suitable conditions. We demonstrate small sample superiority of the proposed estimator over the estimator that ignores dependence of the samples, through a simulation study with dependent and non-normal populations. The application of the density estimator in nonparametric classification is also discussed. It is shown that the misclassification probability of the resulting classifier is asymptotically equivalent to that of the Bayes classifier. We also include a data analytic illustration.     

A robust alternative to the ratio estimator under non-normality

In sampling theory, the traditional ratio estimator is the most common estimator of the population mean when the correlation between study and auxiliary variables is positively high. We introduce a new ratio-type estimator based on the order statistics of a simple random sample. We show that this new estimator is considerably more efficient than the traditional ratio estimator under non-normality, and remarkably robust to data anomalies such as presence of outliers in data sets.     

Doubly robust semiparametric estimation for the missing censoring indicator model

We present a semiparametric analysis of an augmented inverse probability of non-missingness weighted (AIPW) estimator of a survival function for the missing censoring indicator model. Although the estimator is asymptotically less efficient than a Dikta semiparametric estimator, its advantage is the insulation that it offers against inconsistency due to misspecification. We present theoretical and numerical comparisons of the asymptotic variances when there is no misspecification. In addition, we derive the asymptotic variance of the AIPW estimator when there is partial misspecification. We also present a numerical robustness study that confirms the superiority of the AIPW estimator when there is misspecification.     

The multivariate least-trimmed squares estimator

In this paper we introduce the least-trimmed squares estimator for multivariate regression. We give three equivalent formulations of the estimator and obtain its breakdown point. A fast algorithm for its computation is proposed. We prove Fisher-consistency at the multivariate regression model with elliptically symmetric error distribution and derive the influence function. Simulations investigate the finite-sample efficiency and robustness of the estimator. To increase the efficiency of the estimator, we also consider a one-step reweighted estimator.     

Almost sure convergence of the Bartlett estimator

Summary We study the almost sure convergence of the Bartlett estimator for the asymptotic variance of the sample mean of a stationary weekly dependent process. We also study the a.\ s.\ behavior of this estimator in the case of long-range dependent observations. In the weakly dependent case, we establish conditions under which the estimator is strongly consistent. We also show that, after appropriate normalization, the estimator converges a.s. in the long-range dependent case as well. In both cases, our conditions involve fourth order cumulants and assumptions on the rate of growth of the truncation parameter appearing in the definition of the Bartlett estimator.     

A New Estimator for a Tail Index

We investigate properties of a new estimator for a tail index introduced by Davydov and co-workers. The main advantage of this estimator is the simplicity of the statistic used for the estimator. We provide results of simulation by comparing plots of our's and Hill's estimators.     

Complexity penalized support estimation

We consider the estimation of the support of a probability density function with iid observations. The estimator to be considered is a minimizer of a complexity penalized excess mass criterion. We present a fast algorithm for the construction of the estimator. The estimator is able to estimate supports which consists of disconnected regions. We will prove that the estimator achieves minimax rates of convergence up to a logarithmic factor simultaneously over a scale of Hölder smoothness classes for the boundary of the support. The proof assumes a sharp boundary for the support.     

Estimating spot volatility in the presence of infinite variation jumps

We propose a kernel estimator for the spot volatility of a semi-martingale at a given time point by using high frequency data, where the underlying process accommodates a jump part of infinite variation. The estimator is based on the representation of the characteristic function of Lévy processes. The consistency of the proposed estimator is established under some mild assumptions. By assuming that the jump part of the underlying process behaves like a symmetric stable Lévy process around 0, we establish the asymptotic normality of the proposed estimator. In particular, with a specific kernel function, the estimator is variance efficient. We conduct Monte Carlo simulation studies to assess our theoretical results and compare our estimator with existing ones.     

Asymptotics of Bayesian median loss estimation

We establish the consistency, asymptotic normality, and efficiency for estimators derived by minimizing the median of a loss function in a Bayesian context. We contrast this procedure with the behavior of two Frequentist procedures, the least median of squares (LMS) and the least trimmed squares (LTS) estimators, in regression problems. The LMS estimator is the Frequentist version of our estimator, and the LTS estimator approaches a median-based estimator as the trimming approaches 50% on each side. We argue that the Bayesian median-based method is a good tradeoff between the two Frequentist estimators.     

Correction in Bayesian nonparametric estimation in a series system or a competing-risk model

We revisit the Bayesian nonparametric estimator of the component reliability of a series system under a competing risk scenario. Unlike the estimator of Salinas-Torres et al. (2002), our estimator of the component survival function is the corresponding Bayes estimator under quadratic loss.     

嵌套病例对照研究中的两步估计方法（英）

本文讨论嵌套病例对照研究中相对危险率的估计问题，引入了相对危险率的两步估计，并在一般嵌套病例对照抽样的假设下讨论了相对危险率的两步估计的相合性问题，最后给出了几个例子。     

回归系数的广义岭型主相关估计及其优良性

In this paper, we propose a new biased estimator of the regression parameters, the generalized ridge and principal correlation estimator. We present its some properties and prove that it is superior to LSE （least squares estimator）, principal correlation estimator, ridge and principal correlation estimator under MSE （mean squares error） and PMC （Pitman closeness） criterion, respectively.     

The multiple imputations based Kaplan–Meier estimator

We derive the asymptotic distribution of the multiple imputations-based Kaplan–Meier estimator from right censored data with missing censoring indicators. We perform theoretical and numerical comparison studies with a competing semiparametric survival function estimator. We also carry out numerical studies to assess the performance of the proposed estimator when there is model misspecification.     

嵌套病例对照研究中的两步估计方法(英文)

本文讨论嵌套病例对照研究中相对危险率的估计问题,引入了相对危险率的两步估计,并在一般嵌套病例对照抽样的假设下讨论了相对危险率的两步估计的相合性问题．最后给出了几个例子．     

On Kernel-Based Intensity Estimation of Spatial Point Patterns on Linear Networks

We propose an extension of Diggle’s nonparametric edge-corrected kernel-based intensity estimator to the case of events coming from an inhomogenous point pattern on a linear network. We analyze its statistical properties, showing that it is an unbiased estimator of the first-order intensity; we also provide an expression for the variance, and comment on the appropriate bandwidth selection. Our estimator is compared with the current existing equal-split discontinuous kernel density estimator in terms of the mean integrated squared error (MISE). We then use our estimator on two real datasets. We first revisit street crimes in an area of Chicago, obtaining similar results to previously published ones based on a parametric intensity function. Then, we study network-based spatial events consisting of calls to the Police department reporting anti-social behavior in the city of Castellon (Spain).     

Pre-averaged kernel estimators for the drift function of a diffusion process in the presence of microstructure noise

We consider estimation of the drift function of a stationary diffusion process when we observe high-frequency data with microstructure noise over a long time interval. We propose to estimate the drift function at a point by a Nadaraya–Watson estimator that uses observations that have been pre-averaged to reduce the noise. We give conditions under which our estimator is consistent and asympotically normal. Its rate and asymptotic bias and variance are the same as those without microstructure noise. To use our method in data analysis, we propose a data-based cross-validation method to determine the bandwidth in the Nadaraya–Watson estimator. Via simulation, we study several methods of bandwidth choices, and compare our estimator to several existing estimators. In terms of mean squared error, our new estimator outperforms existing estimators.     

On Bayes and unbiased estimators of loss

We consider estimation of loss for generalized Bayes or pseudo-Bayes estimators of a multivariate normal mean vector, θ. In 3 and higher dimensions, the MLEX is UMVUE and minimax but is inadmissible. It is dominated by the James-Stein estimator and by many others. Johnstone (1988, On inadmissibility of some unbiased estimates of loss,Statistical Decision Theory and Related Topics, IV (eds. S. S. Gupta and J. O. Berger), Vol. 1, 361–379, Springer, New York) considered the estimation of loss for the usual estimatorX and the James-Stein estimator. He found improvements over the Stein unbiased estimator of risk. In this paper, for a generalized Bayes point estimator of θ, we compare generalized Bayes estimators to unbiased estimators of loss. We find, somewhat surprisingly, that the unbiased estimator often dominates the corresponding generalized Bayes estimator of loss for priors which give minimax estimators in the original point estimation problem. In particular, we give a class of priors for which the generalized Bayes estimator of θ is admissible and minimax but for which the unbiased estimator of loss dominates the generalized Bayes estimator of loss. We also give a general inadmissibility result for a generalized Bayes estimator of loss. Research supported by NSF Grant DMS-97-04524.     

聚集数据下回归系数混合估计的相对效率

讨论在聚集数据情形下,具有附加信息的线性回归模型的参数估计,提出了回归系数的聚集混合估计,研究了该估计相对于Peter—Karsten估计和相对于最小二乘估计的相对效率,得到了相对效率的上、下界.     

A family of estimators for multivariate kurtosis in a nonnormal linear regression model

In this paper, we propose a new estimator for a kurtosis in a multivariate nonnormal linear regression model. Usually, an estimator is constructed from an arithmetic mean of the second power of the squared sample Mahalanobis distances between observations and their estimated values. The estimator gives an underestimation and has a large bias, even if the sample size is not small. We replace this squared distance with a transformed squared norm of the Studentized residual using a monotonic increasing function. Our proposed estimator is defined by an arithmetic mean of the second power of these squared transformed squared norms with a correction term and a tuning parameter. The correction term adjusts our estimator to an unbiased estimator under normality, and the tuning parameter controls the sizes of the squared norms of the residuals. The family of our estimators includes estimators based on ordinary least squares and predicted residuals. We verify that the bias of our new estimator is smaller than usual by constructing numerical experiments.     

The a Posteriori Error Estimator of SDG Method for Variable Coefficients Time-Harmonic Maxwell's Equations

In this paper, we study the a posteriori error estimator of SDG method for variable coefficients time-harmonic Maxwell's equations. We propose two a posteriori error estimators, one is the recovery-type estimator, and the other is the residual-type estimator. We first propose the curl-recovery method for the staggered discontinuous Galerkin method (SDGM), and based on the super-convergence result of the postprocessed solution, an asymptotically exact error estimator is constructed. The residual-type a posteriori error estimator is also proposed, and it's reliability and effectiveness are proved for variable coefficients time-harmonic Maxwell's equations. The efficiency and robustness of the proposed estimators is demonstrated by the numerical experiments.