Shrinkage estimator in normal mean vector estimation based on conditional maximum likelihood estimators

Estimation of normal mean vector has broad applications such as small area estimation, estimation of nonparametric functions and estimation of wavelet coefficients. In this paper, we propose a new shrinkage estimator based on conditional maximum likelihood estimator incorporating with Stein’s risk unbiased estimator (SURE) when data have the normality. We present some theoretical work and provide numerical studies to compare with some existing methods.     

Statistical inference for quantiles in the frequency domain

For second-order stationary processes, the spectral distribution function is uniquely determined by the autocovariance function of the process. We define the quantiles of the spectral distribution function in frequency domain. The estimation of quantiles for second-order stationary processes is considered by minimizing the so-called check function. The quantile estimator is shown to be asymptotically normal. We also consider a hypothesis testing for quantiles in frequency domain and propose a test statistic associated with our quantile estimator, which asymptotically converges to standard normal under the null hypothesis. The finite sample performance of the quantile estimator is shown in our numerical studies.     

Nonparametric Estimation of the Finite-Time Survival Probability with Zero Initial Capital in the Classical Risk Model

In a classical risk model with zero initial capital and unknown claim-size distribution, we consider the statistical problem of estimating uniformly in t the (unknown) finite-time survival probability ?? ₀(t) at time t, given a sample of claim sizes. We construct an empirical estimator of the function ?? ₀(·) based on the sample of claim sizes, and using a functional approach we establish asymptotic statistical properties of our estimator with respect to supremum norm. We also consider numerical evaluation of finite-time survival probabilities and their empirical counterparts using the fast Fourier transform algorithm, and we carry out small-scale simulation studies of the behaviour of our estimator.     

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or the variance parameter of the normal distribution with a normal-inverse-gamma prior, we analytically calculate the Bayes posterior estimator with respect to a conjugate normal-inverse-gamma prior distribution under Stein's loss function. This estimator minimizes the Posterior Expected Stein's Loss (PESL). We also analytically calculate the Bayes posterior estimator and the PESL under the squared error loss function. The numerical simulations exemplify our theoretical studies that the PESLs do not depend on the sample, and that the Bayes posterior estimator and the PESL under the squared error loss function are unanimously larger than those under Stein's loss function. Finally, we calculate the Bayes posterior estimators and the PESLs of the monthly simple returns of the SSE Composite Index.     

Semiparametric estimation in regression with missing covariates using single-index models

We investigate semiparametric estimation of regression coefficients through generalized estimating equations with single-index models when some covariates are missing at random. Existing popular semiparametric estimators may run into difficulties when some selection probabilities are small or the dimension of the covariates is not low. We propose a new simple parameter estimator using a kernel-assisted estimator for the augmentation by a single-index model without using the inverse of selection probabilities. We show that under certain conditions the proposed estimator is as efficient as the existing methods based on standard kernel smoothing, which are often practically infeasible in the case of multiple covariates. A simulation study and a real data example are presented to illustrate the proposed method. The numerical results show that the proposed estimator avoids some numerical issues caused by estimated small selection probabilities that are needed in other estimators.

     

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.     

Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization

This paper studies the matrix completion problem under arbitrary sampling schemes. We propose a new estimator incorporating both max-norm and nuclear-norm regularization, based on which we can conduct efficient low-rank matrix recovery using a random subset of entries observed with additive noise under general non-uniform and unknown sampling distributions. This method significantly relaxes the uniform sampling assumption imposed for the widely used nuclear-norm penalized approach, and makes low-rank matrix recovery feasible in more practical settings. Theoretically, we prove that the proposed estimator achieves fast rates of convergence under different settings. Computationally, we propose an alternating direction method of multipliers algorithm to efficiently compute the estimator, which bridges a gap between theory and practice of machine learning methods with max-norm regularization. Further, we provide thorough numerical studies to evaluate the proposed method using both simulated and real datasets.     

Modified restricted Liu estimator in logistic regression model

?iray et al. (Commun Stat Simul Comput, 2014) proposed a restricted Liu estimator in logistic regression model with linear restrictions. However, this estimator did not satisfy the linear restrictions. In this paper, we introduce a modified restricted Liu estimator in logistic regression model with linear restrictions. Our results show that the new estimator satisfies the linear restrictions. We also discuss the properties of the new estimator under the matrix mean squared error criterion. Finally, a Monte Carlo study and a numerical example are given to show the performances of the new estimator.     

Estimating parameters in diffusion processes using an approximate maximum likelihood approach

We present an approximate Maximum Likelihood estimator for univariate Itô stochastic differential equations driven by Brownian motion, based on numerical calculation of the likelihood function. The transition probability density of a stochastic differential equation is given by the Kolmogorov forward equation, known as the Fokker-Planck equation. This partial differential equation can only be solved analytically for a limited number of models, which is the reason for applying numerical methods based on higher order finite differences.The approximate likelihood converges to the true likelihood, both theoretically and in our simulations, implying that the estimator has many nice properties. The estimator is evaluated on simulated data from the Cox-Ingersoll-Ross model and a non-linear extension of the Chan-Karolyi-Longstaff-Sanders model. The estimates are similar to the Maximum Likelihood estimates when these can be calculated and converge to the true Maximum Likelihood estimates as the accuracy of the numerical scheme is increased. The estimator is also compared to two benchmarks; a simulation-based estimator and a Crank-Nicholson scheme applied to the Fokker-Planck equation, and the proposed estimator is still competitive.     

Efficiency of a Liu-type estimator in semiparametric regression models

In this paper we consider the semiparametric regression model, y=Xβ+f+ε. Recently, Hu [11] proposed ridge regression estimator in a semiparametric regression model. We introduce a Liu-type (combined ridge-Stein) estimator (LTE) in a semiparametric regression model. Firstly, Liu-type estimators of both β and f are attained without a restrained design matrix. Secondly, the LTE estimator of β is compared with the two-step estimator in terms of the mean square error. We describe the almost unbiased Liu-type estimator in semiparametric regression models. The almost unbiased Liu-type estimator is compared with the Liu-type estimator in terms of the mean squared error matrix. A numerical example is provided to show the performance of the estimators.     

On-line Tracking of a Smooth Regression Function

We construct an on-line estimator with equidistant design for tracking a smooth function from Stone–Ibragimov–Khasminskii’s class. This estimator has the optimal convergence rate of risk to zero in sample size. The procedure for setting coefficients of the estimator is controlled by a single parameter and has a simple numerical solution. The off-line version of this estimator allows to eliminate a boundary layer. Simulation results are given. This work is partially supported by a fellowship from the Yitzhak and Chaya Weinstein Research Institute for Signal Processing at Tel Aviv University.     

Statistical inference for right-censored data with nonignorable missing censoring indicators

We consider the statistical inference for right-censored data when censoring indicators are missing but nonignorable, and propose an adjusted imputation product-limit estimator. The proposed estimator is shown to be consistent and converges to a Gaussian process. Furthermore, we develop an empirical processbased testing method to check the MAR (missing at random) mechanism, and establish asymptotic properties for the proposed test statistic. To determine the critical value of the test, a consistent model-based bootstrap method is suggested. We conduct simulation studies to evaluate the numerical performance of the proposed method and compare it with existing methods. We also analyze a real data set from a breast cancer study for an illustration.     

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.     

A posteriori error estimator based on gradient recovery by averaging for discontinuous Galerkin methods

We consider some (anisotropic and piecewise constant) diffusion problems in domains of R², approximated by a discontinuous Galerkin method with polynomials of any fixed degree. We propose an a posteriori error estimator based on gradient recovery by averaging. It is shown that this estimator gives rise to an upper bound where the constant is one up to some additional terms that guarantee reliability. The lower bound is also established. Moreover these additional terms are negligible when the recovered gradient is superconvergent. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests.     

Reliable a posteriori error estimation for state-constrained optimal control

We derive a reliable a posteriori error estimator for a state-constrained elliptic optimal control problem taking into account both regularisation and discretisation. The estimator is applicable to finite element discretisations of the problem with both discretised and non-discretised control. The performance of our estimator is illustrated by several numerical examples for which we also introduce an adaptation strategy for the regularisation parameter.     

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本文研究随机约束下线性回归模型中, 回归系数的加权混合估计与最小二乘估计的相对效率, 并且给出了相对效率的上下界限. 最后我们给出了一个例子来验证我们的理论结果.     

An a posteriori error estimator for the Lame equation based on equilibrated fluxes

Katharina Witowski ^{We derive a new a posteriori error estimator for the Lamésystem based on H(div)-conforming elements and equilibratedfluxes. It is shown that the estimator gives rise to an upperbound where the constant is one up to higher-order terms. Thelower bound is also established using Argyris elements. Thereliability and efficiency of the proposed estimator are confirmedby some numerical tests.}     

Asymptotic normality and efficiency of the maximum likelihood estimator for the parameter of a ballistic random walk in a random environment

We consider a one-dimensional ballistic random walk evolving in a parametric independent and identically distributed random environment. We study the asymptotic properties of the maximum likelihood estimator of the parameter based on a single observation of the path till the time it reaches a distant site. We prove asymptotic normality for this consistent estimator as the distant site tends to infinity and establish that it achieves the Cramér-Rao bound. We also explore in a simulation setting the numerical behavior of asymptotic confidence regions for the parameter value.     

Equilibrated error estimators for discontinuous Galerkin methods

We consider some diffusion problems in domains of ?^d, d = 2 or 3 approximated by a discontinuous Galerkin method with polynomials of any degree. We propose a new a posteriori error estimator based on H(div)‐conforming elements. It is shown that this estimator gives rise to an upper bound where the constant is one up to higher order terms. The lower bound is also established with a constant depending on the aspect ratio of the mesh, the dependence with respect to the coefficients being also traced. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Trimmed minimax estimator of a covariance matrix

Summary In the problem of estimating the covariance matrix of a multivariate normal population, James and Stein (Proc. Fourth Berkeley Symp. Math. Statist. Prob.,1, 361–380, Univ. of California Press) obtained a minimax estimator under a scale invariant loss. In this paper we propose an orthogonally invariant trimmed estimator by solving certain differential inequality involving the eigenvalues of the sample covariance matrix. The estimator obtained, truncates the extreme eigenvalues first and then shrinks the larger and expands the smaller sample eigenvalues. Adaptive version of the trimmed estimator is also discussed. Finally some numerical studies are performed using Monte Carlo simulation method and it is observed that the trimmed estimate shows a substantial improvement over the minimax estimator. The second author's research was supported by NSF Grant Number MCS 82-12968.