Empirical likelihood analysis of the Buckley-James estimator

The censored linear regression model, also referred to as the accelerated failure time (AFT) model when the logarithm of the survival time is used as the response variable, is widely seen as an alternative to the popular Cox model when the assumption of proportional hazards is questionable. Buckley and James [Linear regression with censored data, Biometrika 66 (1979) 429-436] extended the least squares estimator to the semiparametric censored linear regression model in which the error distribution is completely unspecified. The Buckley-James estimator performs well in many simulation studies and examples. The direct interpretation of the AFT model is also more attractive than the Cox model, as Cox has pointed out, in practical situations. However, the application of the Buckley-James estimation was limited in practice mainly due to its illusive variance. In this paper, we use the empirical likelihood method to derive a new test and confidence interval based on the Buckley-James estimator of the regression coefficient. A standard chi-square distribution is used to calculate the P-value and the confidence interval. The proposed empirical likelihood method does not involve variance estimation. It also shows much better small sample performance than some existing methods in our simulation studies.     

Completeness and unbiased estimation of mean vector in the multivariate group sequential case

We consider estimation after a group sequential test about a multivariate normal mean, such as a χ² test or a sequential version of the Bonferroni procedure. We derive the density function of the sufficient statistics and show that the sample mean remains to be the maximum likelihood estimator but is no longer unbiased. We propose an alternative Rao-Blackwell type unbiased estimator. We show that the family of distributions of the sufficient statistic is not complete, and there exist infinitely many unbiased estimators of the mean vector and none has uniformly minimum variance. However, when restricted to truncation-adaptable statistics, completeness holds and the Rao-Blackwell estimator has uniformly minimum variance.     

Uncertainty under a multivariate nested-error regression model with logarithmic transformation

This work aims to predict exponentials of mixed effects under a multivariate linear regression model with one random factor. Such quantities are of particular interest in prediction problems where the dependent variable is the logarithm of the variable that is the object of inference. Bias-corrected empirical predictors of the target quantities are defined. A second-order approximation for the mean crossed product error of two of these predictors is obtained, where the mean squared error is a particular case. An estimator of the mean crossed product error with second-order bias is proposed. Finally, results are illustrated through an application related to small area estimation.     

Lower bound of risk in linear unbiased estimation and its application

Summary Lower bound of risk in linear unbiased estimation and its connection with the existence of a uniformly minimum variance linear unbiased estimator is considered.     

Simultaneous estimation of parameters in exponential families

Summary The paper considers estimation of the natural parameter vector or the mean vector from independent distributions each belonging to the one-parameter discrete or absolutely continuous exponential family. The usual estimators (maximum likelihood, minimum variance unbiased or best invariant) are improved simultaneously under various weighted squared error losses. Research supported by the NSF Grant Number MCS-8202116.     

The restricted EM algorithm under inequality restrictions on the parameters

One of the most powerful algorithms for maximum likelihood estimation for many incomplete-data problems is the EM algorithm. The restricted EM algorithm for maximum likelihood estimation under linear restrictions on the parameters has been handled by Kim and Taylor (J. Amer. Statist. Assoc. 430 (1995) 708-716). This paper proposes an EM algorithm for maximum likelihood estimation under inequality restrictions A₀β?0, where β is the parameter vector in a linear model W=Xβ+ε and ε is an error variable distributed normally with mean zero and a known or unknown variance matrix Σ>0. Some convergence properties of the EM sequence are discussed. Furthermore, we consider the consistency of the restricted EM estimator and a related testing problem.     

Estimation of location and scale parameters for the Burr XII distribution using generalized order statistics

The minimum variance linear unbiased estimators (MVLUE), the best linear invariant estimators (BLIE) and the maximum likelihood estimators (MLE) based on n-selected generalized order statistics are presented for the parameters of the Burr XII distribution.     

New insights into best linear unbiased estimation and the optimality of least-squares

New results in matrix algebra applied to the fundamental bordered matrix of linear estimation theory pave the way towards obtaining additional and informative closed-form expressions for the best linear unbiased estimator (BLUE). The results prove significant in several respects. Indeed, more light is shed on the BLUE structure and on the working of the OLS estimator under nonsphericalness in (possibly) singular models.     

Unbiased estimation in the multivariate natural exponential family with simple quadratic variance function

We give expansions for the unbiased estimator of a parametric function of the mean vector in a multivariate natural exponential family with simple quadratic variance function. This expansion is given in terms of a system of multivariate orthogonal polynomials with respect to the density of the sample mean. We study some limit properties of the system of orthogonal polynomials. We show that these properties are useful to establish the limit distribution of unbiased estimators.     

Empirical likelihood for LAD estimators in infinite variance ARMA models

In this paper, we use an empirical likelihood method to construct confidence regions for the stationary ARMA(p,q) models with infinite variance. An empirical log-likelihood ratio is derived by the estimating equation of the self-weighted LAD estimator. It is proved that the proposed statistic has an asymptotic standard chi-squared distribution. Simulation studies show that in a small sample case, the performance of empirical likelihood method is better than that of normal approximation of the LAD estimator in terms of the coverage accuracy.     

Preliminary test estimation for spectra

For a general non-Gaussian stationary linear process, quasi-maximum likelihood estimation of a subset of the parameters of the spectral density is considered when the complementary subset is suspected to be superfluous. A preliminary test quasi-maximum likelihood estimator (q-MLE) of parameters is introduced and, in the light of its mean square error, is compared with the restricted and unrestricted q-MLE.     

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.     

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.     

Simultaneous estimation of Poisson means

In the simultaneous estimation of means from independent Poisson distributions, an estimator is developed which incorporates a prior mean and variance for each Poisson mean estimated. This estimator possesses substantially smaller risk than the usual estimator in a region of the parameter space and seems superior to other estimators proposed to estimate p Poisson means. It is indicated through two asymptotic results that, unlike the conjugate Bayes estimator, the risk of the estimator does not greatly exceed the risk of the usual estimator outside of the region of risk improvement.     

Improving on the maximum likelihood estimators of the means in Poisson decomposable graphical models

In this article we study the simultaneous estimation of the means in Poisson decomposable graphical models. We derive some classes of estimators which improve on the maximum likelihood estimator under the normalized squared losses. Our estimators are based on the argument in Chou [Simultaneous estimation in discrete multivariate exponential families, Ann. Statist. 19 (1991) 314-328.] and shrink the maximum likelihood estimator depending on the marginal frequencies of variables forming a complete subgraph of the conditional independence graph.     

测量误差数据下线性模型的几乎无偏岭估计

文章讨论带测量误差的线性模型中参数估计的问题.当带测量误差的线性模型存在复共线的时候,通过几乎无偏估计的思想,提出了几乎无偏岭估计,并对估计的性质进行分析.通过研究发现几乎无偏岭估计不但能克服复共线性,同时有比较小的均方误差.     

On improved estimation of normal precision matrix and discriminant coefficients

The problem of estimating the precision matrix of a multivariate normal distribution model is considered with respect to a quadratic loss function. A number of covariance estimators originally intended for a variety of loss functions are adapted so as to obtain alternative estimators of the precision matrix. It is shown that the alternative estimators have analytically smaller risks than the unbiased estimator of the precision matrix. Through numerical studies of risk values, it is shown that the new estimators have substantial reduction in risk. In addition, we consider the problem of the estimation of discriminant coefficients, which arises in linear discriminant analysis when Fisher's linear discriminant function is viewed as the posterior log-odds under the assumption that two classes differ in mean but have a common covariance matrix. The above method is also adapted for this problem in order to obtain improved estimators of the discriminant coefficients under the quadratic loss function. Furthermore, a numerical study is undertaken to compare the properties of a collection of alternatives to the “unbiased” estimator of the discriminant coefficients.     

Nonparametric estimation of an extreme-value copula in arbitrary dimensions

Inference on an extreme-value copula usually proceeds via its Pickands dependence function, which is a convex function on the unit simplex satisfying certain inequality constraints. In the setting of an i.i.d. random sample from a multivariate distribution with known margins and an unknown extreme-value copula, an extension of the Capéraà-Fougères-Genest estimator was introduced by D. Zhang, M. T. Wells and L. Peng [Nonparametric estimation of the dependence function for a multivariate extreme-value distribution, Journal of Multivariate Analysis 99 (4) (2008) 577-588]. The joint asymptotic distribution of the estimator as a random function on the simplex was not provided. Moreover, implementation of the estimator requires the choice of a number of weight functions on the simplex, the issue of their optimal selection being left unresolved.A new, simplified representation of the CFG-estimator combined with standard empirical process theory provides the means to uncover its asymptotic distribution in the space of continuous, real-valued functions on the simplex. Moreover, the ordinary least-squares estimator of the intercept in a certain linear regression model provides an adaptive version of the CFG-estimator whose asymptotic behavior is the same as if the variance-minimizing weight functions were used. As illustrated in a simulation study, the gain in efficiency can be quite sizable.     

Estimation of the mean vector of a multivariate normal distribution: subspace hypothesis

This paper considers the estimation of the mean vector θ of a p-variate normal distribution with unknown covariance matrix Σ when it is suspected that for a p×r known matrix B the hypothesis θ=Bη, η∈R^r may hold. We consider empirical Bayes estimators which includes (i) the unrestricted unbiased (UE) estimator, namely, the sample mean vector (ii) the restricted estimator (RE) which is obtained when the hypothesis θ=Bη holds (iii) the preliminary test estimator (PTE), (iv) the James-Stein estimator (JSE), and (v) the positive-rule Stein estimator (PRSE). The biases and the risks under the squared loss function are evaluated for all the five estimators and compared. The numerical computations show that PRSE is the best among all the five estimators even when the hypothesis θ=Bη is true.     

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.