An optimal property of the “(r,k ) class estimators”

The purpose of this work is to display optimal conditions, in terms of the Mean Square Error criterion of the (r,k) class estimators. This will be done with respect to the ordinary ridge repression, principal components and ordinary least squares estimate.     

Optimal generalized logistic estimator

In this paper, we propose a new efficient estimator namely Optimal Generalized Logistic Estimator (OGLE) for estimating the parameter in a logistic regression model when there exists multicollinearity among explanatory variables. Asymptotic properties of the proposed estimator are also derived. The performance of the proposed estimator over the other existing estimators in respect of Scalar Mean Square Error criterion is examined by conducting a Monte Carlo simulation.     

Kernel estimation of regression function gradient

Abstract

This paper is focused on kernel estimation of the gradient of a multivariate regression function. Despite the importance of this topic, the progress in this area is rather slow. Our aim is to construct a gradient estimator using the idea of local linear estimator for a regression function. The quality of this estimator is expressed in terms of the Mean Integrated Square Error. We focus on a choice of bandwidth matrix. Further, we present some data-driven methods for its choice and develop a new approach. The performance of presented methods is illustrated using a simulation study and real data example.     

Study of partial least squares and ridge regression methods

This article considers both Partial Least Squares (PLS) and Ridge Regression (RR) methods to combat multicollinearity problem. A simulation study has been conducted to compare their performances with respect to Ordinary Least Squares (OLS). With varying degrees of multicollinearity, it is found that both, PLS and RR, estimators produce significant reductions in the Mean Square Error (MSE) and Prediction Mean Square Error (PMSE) over OLS. However, from the simulation study it is evident that the RR performs better when the error variance is large and the PLS estimator achieves its best results when the model includes more variables. However, the advantage of the ridge regression method over PLS is that it can provide the 95% confidence interval for the regression coefficients while PLS cannot.     

时间趋势平稳性检验的带宽选择及其影响

本文将基于最小化两类错误的概率选择带宽的思想引入KPSS检验。文章通过蒙特卡罗仿真实验,对基于最小化均方误差和最小化两类错误概率的KPSS检验的有限样本性质进行了比较;并对中国季度实际GDP对数序列的平稳性进行了检验。结果表明,基于最小化两类错误概率的检验比基于最小化均方误差的检验有更大的水平扭曲,但前者也有更大的检验势;并且在误差项自回归系数小于等于0.5的大多数情形下,基于最小化两类错误概率的检验有更好的有限样本性质;中国季度实际GDP对数序列是带漂移的单位根过程。     

Maximum likelihood estimation of variance components-a Monte Carlo study

The estimation of the parameter of a mixed model analysis of variance by maximum likelihood methods is discussed. The functional iteration method is studied and found to have good comptuational properties. The estimates are studied via Monte Carlo techniques and their small sample properties are observed; it is found that the MLE's may be biased but that they have good Mean Square Error properties.     

Best minimum bias linear estimators in gauss-markoff model

In this note we present a criterion for linear estimation which is similar to MV-MB-LE of Rao (1978) in Gauss-Markoff model (Y, XB, α²G). We call this criterion MMS-MB-LE (Minimum Mean Square Error-Minimum Bias-Linear Estimation)> Representations of solutions to such estimators similar to those of Rao (1978) are provided.     

A note on the estimation of variance of sample mean Using the knowledge of coefficient of variation in normal population

A class of estimators for the variance of sample mean is defined and its properties are studied in case of normal population. It is identified that the usual unbiased estimator, Singh, Pandey and Hirano (1973) -type estimator and Lee (1931) estimator are particular members of the proposed class of estimators. It is found that the minimum Mean Squared Error (MSE) of the proposed class of estimators is less than that of other estimators.     

Kalman filtering with censored measurements

This paper concerns Kalman filtering when the measurements of the process are censored. The censored measurements are addressed by the Tobit model of Type I and are one-dimensional with two censoring limits, while the (hidden) state vectors are multidimensional. For this model, Bayesian estimates for the state vectors are provided through a recursive algorithm of Kalman filtering type. Experiments are presented to illustrate the effectiveness and applicability of the algorithm. The experiments show that the proposed method outperforms other filtering methodologies in minimizing the computational cost as well as the overall Root Mean Square Error (RMSE) for synthetic and real data sets.     

Some conditions for optimality of the almost unbiased estimators of regression coefficients

The purpose of this paper is to examine the asymptotic properties of the operational almost unbiased estimator of regression coefficients which includes almost unbiased ordinary ridge estimator a s a special case. The small distrubance approximations for the bias and mean square error matrix of the estimator are derived. As a consequence, it is proved that, under certain conditions, the estimator is more efficient than a general class of estimators given by Vinod and Ullah (1981). Also it is shown that, if the ordinary ridge estimator (ORE) dominates the ordinary least squares estimator then the almost unbiased ordinary ridge estimator does not dominate ORE under the mean square error criterion.     

Minimum variance unbiased estimation of stress–strength reliability under bivariate normal and its comparisons

In many industrial and natural phenomena, we need the probability that a component is smaller than the other component. Under a stress–strength model, this is reliability of an item. Under independent setup, there are different approaches for the estimation of such reliability. Here, estimation is considered under the dependent case. Under bi-variate setup uniformly minimum variance unbiased estimator is obtained. Also comparison with available estimator based on Maximum Likelihood Estimate (MLE) is done through Mean Square Error (MSE) and bias. Also these are compared by computing L₁ distance between their distribution functions. From this idea and numerical computations, UMVUE appears to be good.     

Estimation of the best linear unbiased predictor for the mean with unequal sample sizes

When the samples selected from k normal populations are of unequal sizes, we consider the empirical best linear unbiased predictor, EBLUP, for the mean of each population. For fixed values of the means of these populations, conditions for the Mean Square Error (MSE) of the EBLUP to be smaller than the variance of the sample mean and, at the same time, for its absolute bias to be smaller than a specified fraction of the square root of its MSE are obtained. Preference of the EBLUP over the sample mean is examined for the estimation of the averages of the daily hospital expenses of the Standard Metropolitan Statistical Areas (SMSAs) of twenty states in the US.     

Matrix density estimation

The accuracy of orthogonal series types of density estimators can be conveniently measured in terms of their Mean Integrated Squared Error, or MISE. Further reduction In MISE is achieved by introducing certain weighting factors into the estimators. In this paper we consider optimal weighting matrices, and the result is a new class of density estimators, the collection of matrix density estimators.     

Ridge estimation in logistic regression

The variance of the Maximum Likelihood Estimator (MLE) of the slope parameter in a logistic regression model becomes large as the degree of collinearity among the explanatory variables increases. In a Monte Carlo study, we observed that a ridge type estimator is at least as good as, and often much better than, the MLE in terms of Total and Prediction Mean Squared Error criteria. Using a set of medical data it is illustrated that the ridge trace of the estimator considered here is a useful diagnostic tool in logistic regression analysis.     

Use of prior information for estimating the variance components

The MINQUE and its modifications are considered for estimating the variances of the balanced one-way random effects model. The effects of the a priori values on the estimators of the variances are examined in detail. The Mean Square Errors of the estimators are compared for variations in the prior values of the unknown variances.     

NONNEGATIVE ESTIMATORS FOR THE ONE-WAY RANDOM EFFECTS MODEL

The relative merits of ten estimators for the variance component of the balanced and unbalanced one-way random effects models are compared. Six of the estimators are nonnegative, two of which are obtained by modifying the Minimum Variance Quadratic Unbiased Estimator (MIVQUE) and the Weighted Least Square Estimator (WLS), and two more from the positive parts of these estimators. The Minimum Norm Quadratic Estimator (MINQE), which is nonnegative, is adjusted for reducing its bias. The nonnegative Minimum Mean Square Error Estimator (MIMSQE), the Analysis of Variance (ANOVA) and Unweighted Sums of Squares (USS) estimator are also included.     

基于变点理论的POT模型阈值确定方法——对操作风险经济资本的度量

对操作风险所要求的经济资本的度量以及配置能极大提高金融机构的风险控制能力。采用PCIT模型对操作风险度量时，阈值的选取是关键所在，它决定了拟合操作风险损失分布的近似程度。通过变点理论来定位Hill估计曲线开始进入稳定状态的位置，以精确地估计出阈值的大小。同时，为确保误差更小，结果更稳定，用平方误差积分法来估计POT模型的参数。结果表明，所改进的方法能为经济资本的度量提供有效的方法支持。     

A comparison of biased regression estimators using a pitman nearness criterion

Biased regression estimators have traditionally benn studied using the Mean Square Error (MSE) criterion. Usually these comparisons have been based on the sum of the MSE's of each of the individual parameters, i.e., a scaler valued measure that is the trace of the MSE matrix. However, since this summed MSE does not consider the covariance structure of the estimators, we propose the use of a Pitman Measure of Closeness (PMC) criterion (Keating and Gupta, 1984; Keating and Mason, 1985). In this paper we consider two versions of PMC. One of these compares the estimates and the other compares the resultant predicted values for 12 different regression estimators. These estimators represent three classes of estimators, namely, ridge, shrunken, and principal component estimators. The comparisons of these estimators using the PMC criteria are contrasted with the usual MSE criteria as well as the prediction mean square error. Included in the estimators is a relatively new estimator termed the generalized principal component estimator proposed by Jolliffe. This estimator has previously received little attention in the literature.     

ROBUST RIDGE REGRESSION BASED ON AN M-ESTIMATOR

Consider the linear regression model y =β₀1 +Xβ+ in the usual notation. It is argued that the class of ordinary ridge estimators obtained by shrinking the least squares estimator by the matrix (X¹X + kI)^-1X'X is sensitive to outliers in the ^variable. To overcome this problem, we propose a new class of ridge-type M-estimators, obtained by shrinking an M-estimator (instead of the least squares estimator) by the same matrix. Since the optimal value of the ridge parameter k is unknown, we suggest a procedure for choosing it adaptively. In a reasonably large scale simulation study with a particular M-estimator, we found that if the conditions are such that the M-estimator is more efficient than the least squares estimator then the corresponding ridge-type M-estimator proposed here is better, in terms of a Mean Squared Error criteria, than the ordinary ridge estimator with k chosen suitably. An example illustrates that the estimators proposed here are less sensitive to outliers in the y-variable than ordinary ridge estimators.     

Binary Dynamic Logit for Correlated Ordinal: estimation,application and simulation

We evaluate the estimation performance of the Binary Dynamic Logit model for correlated ordinal variables (BDLCO model), and compare it to GEE and Ordinal Logistic Regression performance in terms of bias and Mean Absolute Percentage Error (MAPE) via Monte Carlo simulation. Our results indicate that when the proportional-odds assumption does not hold, the proposed BDLCO method is superior to existing models in estimating correlated ordinal data. Moreover, this method is flexible in terms of modeling dependence and allows unequal slopes for each category, and can be used to estimate an apple bloom data set where the proportional-odds assumption is violated. We also provide a function in R to implement BDLCO.