The biases associated with maximum likelihood methods of estimation of the multivariate logistic risk function

The posterior probability of an object belonging to one of two populations can be estimated using multivariate logistic regression. The bias associated with this procedure is derived In the context of normal populations with different mean vectors and a common covariance matrix and is compared with the bias of the classical method based on this normality assumption, -It Is found that the bias of the more robust procedure of logistic regression is of a lower order than that of the normality based method.     

Bayes Estimation of Intraclass Correlation Coefficients Under Unequal Family Sizes

Bayesian inference for the intraclass correlation ρ is considered under unequal family sizes. We obtain the posterior distribution of ρ and then compare the performance of the Bayes estimator (posterior mean of ρ) with that of Srivastava's (1984) estimator through simulation. Simulation study shows that the Bayes estimator performs better than the Srivastava's estimator in terms of lower mean square error. We also obtain large sample posteriors of ρ based on the asymptotic posterior distribution and based on the Laplace approximation.     

Bias Adjustment Minimizing the Asymptotic Mean Square Error

A method of bias adjustment which minimizes the asymptotic mean square error is presented for an estimator typically given by maximum likelihood. Generally, this adjustment includes unknown population values. However, in some examples, the adjustment can be done without population values. In the case of a logit, a reasonable fixed value for the adjustment is found, which gives the asymptotic mean square error smaller than those of the asymptotically unbiased estimator and the maximum likelihood estimator. The weighted-score method, which yields directly the estimator with the minimized asymptotic mean square error, is also given.     

On a generalized stein estimator of regression coefficients

This paper studies a generalized Stein estimator of regression coefficients. The small disturbance approximations for the bias and mean square error matrix of the estimator are derived and a necessary and sufficient condition is obtained for the estimator to dominate the ordinary least squares estimator under the mean square error criterion.     

On the Estimation of the Probability Density by Trigonometric Series

We present in this article an estimator based on a new orthogonal trigonometric series. We give its statistical properties (bias, variance, mean square error, and mean integrated square error) and the asymptotic properties (convergence of variance, convergence of the mean square error, convergence of the mean integrated square error, uniform convergence in probability, and the rate of convergence of the mean integrated square error). The comparison by simulation on a test density between the estimator obtained from a new trigonometric series with Fejer estimator also based on orthogonal trigonometric series, shows that our estimator is more performant in the sense of the mean integrated square error.     

Absolute error criteria for bandwidth selection in density estimation from censored data

In Kernel density estimation, a criticism of bandwidth selection techniques which minimize squared error expressions is that they perform poorly when estimating tails of probability density functions. Techniques minimizing absolute error expressions are thought to result in more uniform performance and be potentially superior. An asympotic mean absolute error expression for nonparametric kernel density estimators from right-censored data is developed here. This expression is used to obtain local and global bandwidths that are optimal in the sense that they minimize asymptotic mean absolute error and integrated asymptotic mean absolute error, respectively. These estimators are illustrated fro eight data sets from known distributions. Computer simulation results are discussed, comparing the estimation methods with squared-error-based bandwidth selection for right-censored data.     

Asymptotic Behavior of the Hazard Rate Kernel Estimator Under Truncated and Censored Data

In this article, we give the asymptotic mean integrated squared error and the mean squared error for the kernel estimator of the hazard rate from truncated and censored data. Martingale techniques and combinatory calculus are used to obtain these results. A probability bound and the optimal bandwidth choice are also given.     

Accounting for Uncertainty in Heteroscedasticity in Nonlinear Regression

Toxicologists and pharmacologists often describe toxicity of a chemical using parameters of a nonlinear regression model. Thus estimation of parameters of a nonlinear regression model is an important problem. The estimates of the parameters and their uncertainty estimates depend upon the underlying error variance structure in the model. Typically, a priori the researcher would not know if the error variances are homoscedastic (i.e., constant across dose) or if they are heteroscedastic (i.e., the variance is a function of dose). Motivated by this concern, in this paper we introduce an estimation procedure based on preliminary test which selects an appropriate estimation procedure accounting for the underlying error variance structure. Since outliers and influential observations are common in toxicological data, the proposed methodology uses M-estimators. The asymptotic properties of the preliminary test estimator are investigated; in particular its asymptotic covariance matrix is derived. The performance of the proposed estimator is compared with several standard estimators using simulation studies. The proposed methodology is also illustrated using a data set obtained from the National Toxicology Program.     

Confidence Corridors for Multivariate Generalized Quantile Regression

We focus on the construction of confidence corridors for multivariate nonparametric generalized quantile regression functions. This construction is based on asymptotic results for the maximal deviation between a suitable nonparametric estimator and the true function of interest, which follow after a series of approximation steps including a Bahadur representation, a new strong approximation theorem, and exponential tail inequalities for Gaussian random fields. As a byproduct we also obtain multivariate confidence corridors for the regression function in the classical mean regression. To deal with the problem of slowly decreasing error in coverage probability of the asymptotic confidence corridors, which results in meager coverage for small sample sizes, a simple bootstrap procedure is designed based on the leading term of the Bahadur representation. The finite-sample properties of both procedures are investigated by means of a simulation study and it is demonstrated that the bootstrap procedure considerably outperforms the asymptotic bands in terms of coverage accuracy. Finally, the bootstrap confidence corridors are used to study the efficacy of the National Supported Work Demonstration, which is a randomized employment enhancement program launched in the 1970s. This article has supplementary materials online.     

An improved class of estimators for the population mean

Starting from the Rao (Commun Stat Theory Methods 20:3325–3340, 1991) regression estimator, we propose a class of estimators for the unknown mean of a survey variable when auxiliary information is available. The bias and the mean square error of the estimators belonging to the class are obtained and the expressions for the optimum parameters minimizing the asymptotic mean square error are given in closed form. A simple condition allowing us to improve the classical regression estimator is worked out. Finally, in order to compare the performance of some estimators with the regression one, a simulation study is carried out when some population parameters are supposed to be unknown.     

Bayesian and maximum likelihood estimations of the inverted exponentiated half logistic distribution under progressive Type II censoring

In this paper, the estimation of parameters, reliability and hazard functions of a inverted exponentiated half logistic distribution (IEHLD) from progressive Type II censored data has been considered. The Bayes estimates for progressive Type II censored IEHLD under asymmetric and symmetric loss functions such as squared error, general entropy and linex loss function are provided. The Bayes estimates for progressive Type II censored IEHLD parameters, reliability and hazard functions are also obtained under the balanced loss functions. However, the Bayes estimates cannot be obtained explicitly, Lindley approximation method and importance sampling procedure are considered to obtain the Bayes estimates. Furthermore, the asymptotic normality of the maximum likelihood estimates is used to obtain the approximate confidence intervals. The highest posterior density credible intervals of the parameters based on importance sampling procedure are computed. Simulations are performed to see the performance of the proposed estimates. For illustrative purposes, two data sets have been analyzed.     

Laplace approximations to means and variances with asymptotic modes

The moment-generating function method, which is proposed by Tierney et al. [1989a. Fully exponential Laplace approximations to expectations and variances of nonpositive functions. J. Amer. Statist. Assoc. 84, 710–716], is an asymptotic technique of approximating a posterior mean of a general function by approximating the moment-generating function (MGF), and then differentiating it. In this article, we give approximations to the posterior means and variances by combining the MGF method and the Laplace approximations with asymptotic modes. We prove that asymptotic errors of the approximate means and variances are of order n^-2$n^{-2}$ and of order n^-3$n^{-3}$, respectively. Our approximation is closely related to a standard-form approximation, and is given without evaluating the exact posterior mode and third derivatives of the log-likelihood function. The MGF method also improves numerical instability of the fully exponential Laplace approximation for a predictive mean in logistic regression.     

函数型变量倾斜分位回归模型及其应用

本文考虑函数型数据的结构特征,针对两类函数型变量分位回归模型（函数型因变量对标量自变量和函数型因变量对函数型自变量）,基于函数型倾斜分位曲线的定义构建新型函数型倾斜分位回归模型。对于第二类模型,本文分别考虑样条基函数对模型系数展开和函数型主成分基函数对函数型自变量展开,得到倾斜分位回归模型的基本形式。参数估计采用成分梯度Boosting算法最小化加权非对称损失函数,提高计算效率。在理论上证明了倾斜分位回归模型的系数估计量均服从渐近正态分布。模拟和实证研究结果显示,倾斜分位回归模型比已有的逐点分位回归模型具有更好的拟合效果。根据积分均方预测误差准则,本文提出的模型有一致较好的预测能力。     

A note on the central limit theorem for the bootstrap mean

Following Yang (1988), a simple, more self-contained derivation of the asymptotic normality of the bootstrap mean is presented, as well as other asymptotic results. The derivations are appropriate for beginning graduate students in statistics, relying only on fundamental notions of probability theory and analysis,     

A simulation study of properties of a regressive logistic model

A regressive logistic model for the analysis of data with dependent binary observations is constructed by successively conditioning on preceding observations. The properties of this model are investigated and compared to those of the ordinary logistic regression model in which the dependence is not considered, using computer simulation. Comparison criteria include the magnitude of the bias and the total mean square error (MSE) of the regression coefficient β and the significance level. The results suggest the regressive model significantly improves the estimation of the regression coefficient     

Bayesian estimation and prediction based on lognormal record values

In this paper we consider the problems of estimation and prediction when observed data from a lognormal distribution are based on lower record values and lower record values with inter-record times. We compute maximum likelihood estimates and asymptotic confidence intervals for model parameters. We also obtain Bayes estimates and the highest posterior density (HPD) intervals using noninformative and informative priors under square error and LINEX loss functions. Furthermore, for the problem of Bayesian prediction under one-sample and two-sample framework, we obtain predictive estimates and the associated predictive equal-tail and HPD intervals. Finally for illustration purpose a real data set is analyzed and simulation study is conducted to compare the methods of estimation and prediction.     

Asymptotic cumulants of the estimator of the canonical parameter in the exponential family

Asymptotic cumulants of the maximum likelihood estimator of the canonical parameter in the exponential family are obtained up to the fourth order with the added higher-order asymptotic variance. In the case of a scalar parameter, the corresponding results with and without studentization are given. These results are also obtained for the estimators by the weighted score, especially for those using the Jeffreys prior. The asymptotic cumulants are used for reducing bias and mean square error to improve a point estimator and for interval estimation to have higher-order accuracy. It is shown that the kurtosis to squared skewness ratio of the sufficient statistic plays a fundamental role.     

On Semiparametric Mode Regression Estimation

It has been found that, for a variety of probability distributions, there is a surprising linear relation between mode, mean, and median. In this article, the relation between mode, mean, and median regression functions is assumed to follow a simple parametric model. We propose a semiparametric conditional mode (mode regression) estimation for an unknown (unimodal) conditional distribution function in the context of regression model, so that any m-step-ahead mean and median forecasts can then be substituted into the resultant model to deliver m-step-ahead mode prediction. In the semiparametric model, Least Squared Estimator (LSEs) for the model parameters and the simultaneous estimation of the unknown mean and median regression functions by the local linear kernel method are combined to infer about the parametric and nonparametric components of the proposed model. The asymptotic normality of these estimators is derived, and the asymptotic distribution of the parameter estimates is also given and is shown to follow usual parametric rates in spite of the presence of the nonparametric component in the model. These results are applied to obtain a data-based test for the dependence of mode regression over mean and median regression under a regression model.     

Asymptotic relative efficiency of wald tests in measurement error models

In this paper, asymptotic relative efficiency (ARE) of Wald tests for the Tweedie class of models with log-linear mean, is considered when the aux¬iliary variable is measured with error. Wald test statistics based on the naive maximum likelihood estimator and on a consistent estimator which is obtained by using Nakarnura's (1990) corrected score function approach are defined. As shown analytically, the Wald statistics based on the naive and corrected score function estimators are asymptotically equivalents in terms of ARE. On the other hand, the asymptotic relative efficiency of the naive and corrected Wald statistic with respect to the Wald statistic based on the true covariate equals to the square of the correlation between the unobserved and the observed co-variate. A small scale numerical Monte Carlo study and an example illustrate the small sample size situation.     

Expansions of the coverage probabilities of prediction region based on a shrinkage estimator

An expansion formula for the coverage probability of prediction region based on a shrinkage estimator proposed by Joshi [Joshi, V. M. (1967). Inadmissibility of the usual confidence sets for the mean of a multivariate normal population. Ann. Math. Statist., 38, 1868–1875.] is obtained. Its error bound is evaluated in terms of a function of an unknown parameter. Applying this result, three types of asymptotic expansions are derived. These expansions show inadmissibility of the usual prediction region.