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.     

LOGISTIC REGRESSION COMPARED TO NORMAL DISCRIMINATION FOR NON-NORMAL POPULATIONS'

An investigation is undertaken of the logistic regression procedure for estimating the posterior probability of an object belonging to one of two populations. The asymptotic bias and mean square error associated with the procedure are derived for univariate populations whose distributions satisfy the general Day-Kerridge model for which the logistic form is valid for the posterior probability. These properties are compared with those of the normal discrimination method based on the classical assumption of normal populations with common variances. The asymptotic relative efficiency of logistic regression is considered on the basis of asymptotic mean square error.     

Caliper pair-matching on a continuous variable in case-control studies

The use of matched pairs has been criticized as being less efficient than estimators based on random samples. This paper compares the mean square error of an analysis of covariance estimator based on random samples with two estimators based on caliper matched pairs. The first of these is a simple mean difference estimator and the second a regression estimator suggested by Rubin (1973b). Under conditions which commonly occur in epidemiologic case-control studies, both of the matched pair estimators can have smaller mean square errors than analysis o f covariance estimator. When there is a weak relationship between the matching and response variate, the mean difference estimator has a lower mean square error than the regression estimator.     

Using Adjacent-category Logits Procedure for Estimating Receiver Operating Characteristic Surface

This article presents a semiparametric method for estimating receiver operating characteristic surface under density ratio model. The construction of the proposed method is based on the adjacent-category logit model and the empirical likelihood approach. A bootstrap approach for the VUS estimator inference is presented. In a simulation study, the proposed estimator is compared with the existing parametric and nonparametric estimators in terms of bias, standard error, and mean square error. Finally, a real data example and some discussions on the proposed method are provided.     

Nonparametric probability density estimation using normalized b–splines

We present a new nonparametric density estimate based on normalized tensor B–Splines. We show under the expected conditions that the non- parametric density estimate converges in mean square error and integrated mean square error. Results of simulations are also presented.     

Modeling Conditional Covariances With Economic Information Instruments

We propose a new model for conditional covariances based on predetermined idiosyncratic shocks as well as macroeconomic and own information instruments. The specification ensures positive definiteness by construction, is unique within the class of linear functions for our covariance decomposition, and yields a simple yet rich model of covariances. We introduce a property, invariance to variate order, that assures estimation is not impacted by a simple reordering of the variates in the system. Simulation results using realized covariances show smaller mean absolute errors (MAE) and root mean square errors (RMSE) for every element of the covariance matrix relative to a comparably specified BEKK model with own information instruments. We also find a smaller mean absolute percentage error (MAPE) and root mean square percentage error (RMSPE) for the entire covariance matrix. Supplementary materials for practitioners as well as all Matlab code used in the article are available online.     

Preliminary Test Estimators Induced by Three Large Sample Tests for Stochastic Constraints in a Regression Model with Multivariate Student-t Error

In this article, we consider the preliminary test approach to the estimation of the regression parameter in a multiple regression model with multivariate Student-t distribution. The preliminary test estimators (PTE) based on the Wald (W), Likelihood Ratio (LR), and Lagrangian Multiplier (LM) tests are given under the suspicion of stochastic constraints occurring. The bias, mean square error matr ix (MSEM), and weighted mean square error (WMSE) of the proposed estimators are derived and compared. The conditions of superiority of the proposed estimators are obtained. Finally, we conclude that the optimum choice of the level of significance becomes the traditional choice by using the W test.     

Estimation of reliability for a multicomponent survival stress-strength model based on exponential distributions

Uniformly minimum variance unbiased estimator (UMVUE) of reliability in stress-strength model (known stress) is obtained for a multicomponent survival model based on exponential distributions for parallel system. The variance of this estimator is compared with Cramer-Rao lower bound (CRB) for the variance of unbiased estimator of reliability, and the mean square error (MSE) of maximum likelihood estimator of reliability in case of two component system.     

A note on mse coverage intervals in a partial spline model

A partial spline model is used to estimate an unknown function which is smooth except for some break points. Assuming the break points are known, a Generalized Cross-Validated smoothing spline estimation method is proposed. Some interval estimation methods for the magnitude of the discontinuities based on the mean square error are introduced and investigated.     

A new class of unbiased estimators of the variance of the systematic sample mean

We propose a class of estimators of the variance of the systematic sample mean, which is unbiased under the assumption that the population follows a superpopulation model that satisfies some mild conditions. The approach is based on the separate estimation of the portion of the variance due to the systematic component of the model and that due to the stochastic component. In particular, we deal with two estimators belonging to the proposed class that are based on moving averages and local polynomials to estimate the systematic component of the model. The latter estimators are unbiased under the assumption that the population follows a linear trend and the errors are homoscedastic and uncorrelated. Through a simulation study we show that these estimators generally outperform, in terms of bias and mean square error, the usual estimator based on the first differences also when the superpopulation model departs significantly from linearity and the errors are heteroscedastic.     

Local improvement of best linear unbiased estimation and admissibility under the weakly singular gauss-markov model

Under the weakly singular Gauss-Markov model, the class of linearly admissible estimators for the expectation of the observable random vector with respect to the mean square error criterion is considered. It is demonstrated that this class admits linearly admissible estimators for an arbitrary estimable parametric function, which locally improve the best linear estimator with respect to the mean square error matrix criterion.     

Mean square error and efficiency of the least squares estimator over interval constraints

We derive the mean square error of an interval constrained least squares estimator (INCLS) for a regression model. We then show that the INCLS estimator dominates, in mean square error, the unconstrained least squares estimator provided the regression residuais are normally distri'iiuted and Ynat Yrie imposed coii-

straint is satisfied or nearly satisfied.     

Comparisons of the alternative biased estimators for the distributed lag models

The finite distributed lag models include highly correlated variables as well as lagged and unlagged values of the same variables. Some problems are faced for this model when applying the ordinary least squares (OLS) method or econometric models such as Almon and Koyck models. The primary aim of this study is to compare performances of alternative estimators to the OLS estimator defined by combining the Almon estimator with some estimators using Almon (1965) data. A simulation study with different model parameters is performed and the estimators are compared according to the root mean square error (RMSE) and prediction mean square error (PMSE).     

Shrunken estimators via a pitman nearness criterion

Shrunken estimators have traditionally been developed and studied using mean square error (MSE). Recent research on Pitman nearness (PN), however, indicates that it is an interesting, “intrinsic”, alternative to the mean square error (MSE) criterion for investigating estimators. Thus, we develop a shrunken estimator for the mean of a multivariate normal distribution based on minimizing PN, instead of MSE, Further, since the shrinkage factor of this estimator depends on unknown parameters, we examine two approaches for determining this factor: (1) “plug-in” estimates, (2) a range of values for the factor based on an approximate cońfidence interval for the Pitman Nearness probability. A numerical example is given.     

On multivariate linear regression shrinkage and reduced-rank procedures

An alternate derivation of the canonical analysis shrinkage prediction procedure of Breiman and Friedman (1997. J. Roy. Statist. Soc. B 59, 3–54) is presented for the multivariate linear model. It is based on consideration of prediction mean square error matrix, and bias of the squared sample canonical correlations. A modified procedure involving partial canonical correlation analysis is also introduced and discussed.     

Performance of extrapolation based on Pitman's measure of closeness in spatial regression models with extended skew t innovations

Pitman's measure of closeness and mean square error of prediction are two well-known criteria for comparison between estimators and also between predictors. In a stationary first order multiplicative spatial autoregressive model, interpolation and extrapolation are compared based on these two criteria. A wide class of different innovation types are also studied containing Gaussian, exponential, asymmetric Laplace and extended skew t distributions.     

Selecting a test level for random effects

A method is presented for selecting an a-level to use when testing for group difference in a one-way classification random effects model. The a-level is chosen to make the power of the test equal to .5 when the parameters are such that between group mean square and total mean square are equally good minimum expected squared error estimators of the variance of y the estimator of the mean     

Some remarks on a ridge-type-estimator and good prior means

Pliskin (1987) compared modified ridge regression estimators based on prior information with respect to their mean square error matrices. A further characterization of good prior mean is given here, and the case of different ridge parameters is also considered.     

短面板约束下固定效应空间动态面板模型的估计与模拟

空间动态面板数据(SDPD)模型中被解释变量初值极易带有内生性,采用一般拟极大似然(QML)方法容易造成参数估计偏误,特别是当样本结构为n大T小的时候。鉴于此,本文在一般QML基础上,通过重塑误差项的方差-协方差矩阵,修正拟似然函数表达式,得到修正QML,进而估计短面板下含空间、时间、误差三类关联项的固定效应SDPD模型,基于数值模拟和实例应用检验一般QML与修正QML的估计效果。数值模拟结果表明：修正QML比一般QML更精确、更稳健,均方误差修正率随样本短面板结构的增大而增大。实例应用不仅重新评估环境规制与技术创新之间的空间效应,回归结果也再次证实从数值模拟中得出的结论。     

Asymptotic confidence intervals in ridge regression based on the Edgeworth expansion

Ridge Regression techniques have been found useful to reduce mean square errors of parameter estimates when multicollinearity is present. But the usefulness of the method rest not only upon its ability to produce good parameter estimates, with smaller mean squared error than Ordinary Least Squares, but also on having reasonable inferential procedures. The aim of this paper is to develop asymptotic confidence intervals for the model parameters based on Ridge Regression estimates and the Edgeworth expansion. Some simulation experiments are carried out to compare these confidence intervals with those obtained from the application of Ordinary Least Squares. Also, an example will be provided based on the well known data set of Hald.