Restricted regression estimation in measurement error models

The problem of consistent estimation of the regression coefficients when some prior information about the regression coefficients is available is considered. Such prior information is expressed in the form of exact linear restrictions. The knowledge of covariance matrix of measurement errors that is associated with explanatory variables is used to construct the consistent estimators. Some consistent estimators are suggested which satisfy the exact linear restrictions also. Their asymptotic properties are derived and analytically analyzed under a multivariate ultrastructural model with not necessarily normally distributed measurement errors. The finite sample properties of the estimators are studied through a Monte-Carlo simulation experiment.     

Intertemporal asset allocation when the underlying factors are unobservable

The aim of this paper is to develop an optimal long-term bond investment strategy which can be applied to real market situations. This paper employs Merton’s intertemporal framework to accommodate the features of a stochastic interest rate and the time-varying dynamics of bond returns. The long-term investors encounter a partial information problem where they can only observe the market bond prices but not the driving factors of the variability of the interest rate and the bond return dynamics. With the assumption of Gaussian factor dynamics, we are able to develop an analytical solution for the optimal long-term investment strategies under the case of full information. To apply the best theoretical investment strategy to the real market we need to be aware of the existence of measurement errors representing the gap between theoretical and empirical models. We estimate the model based on data for the German securities market and then the estimation results are employed to develop long-term bond investment strategies. Because of the presence of measurement errors, we provide a simulation study to examine the performance of the best theoretical investment strategy. We find that the measurement errors have a great impact on the optimality of the investment strategies and that under certain circumstance the best theoretical investment strategies may not perform so well in a real market situation. In the simulation study, we also investigate the role of information about the variability of the stochastic interest rate and the bond return dynamics. Our results show that this information can indeed be used to advantage in making sensible long-term investment decisions.     

Estimating the width of a uniform distribution when data are measured with additive normal errors with known variance

The problem of estimating the width of the symmetric uniform distribution on the line when data are measured with normal additive error is considered. The main purpose is to discuss the efficiency of the maximum likelihood estimator and the moment method estimator. It is shown that the model is regular and that the maximum likelihood estimator is more efficient than the moment method estimator. A sufficient condition is also given for the existence of both estimators.     

Conditional mean estimation under asymmetric and heteroscedastic error by linear combination of quantile regressions

In this paper we propose a new estimator for regression problems in the form of the linear combination of quantile regressions. The proposed estimator is helpful for the conditional mean estimation when the error distribution is asymmetric and heteroscedastic.It is shown that the proposed estimator has the consistency under heteroscedastic regression model: Y=μ(X)+σ(X)·e, where X is a vector of covariates, Y is a scalar response, e is a zero mean random variable independent of X and σ(X) is a positive value function. When the error term e is asymmetric, we show that the proposed estimator yields better conditional mean estimation performance than the other estimators. Numerical experiments both in synthetic and real data are shown to illustrate the usefulness of the proposed estimator.     

Measurement error in statistical models of shape

Active shape models (ASMs) are popular and sophisticated methods of extracting features in (especially medical) images. Here we analyse the error in placing ASM points on the boundary of the feature. By using replications, a corrected covariance matrix is presented that should reduce the effects of placement error. We show analytically and via simulations that the cumulative variability for a given number of eigenvalues retained in principal components analysis (PCA) ought to be reduced by increasing levels of point-placement error. Results for predicted errors are in excellent agreement with the set-up parameters of two simulated shapes and with anecdotal evidence from the trained experts for real data taken from the OSTEODENT project. We derive an equation for the reliability of placing the points and we find values of 0.79 and 0.85 (where 0 = bad and 1 = good) for the two clinical experts for the OSTEODENT data. These analyses help us to understand the sources and effects of measurement error in shape models.     

On the performance of the flexible maximum entropy distributions within partially adaptive estimation

The partially adaptive estimation based on the assumed error distribution has emerged as a popular approach for estimating a regression model with non-normal errors. In this approach, if the assumed distribution is flexible enough to accommodate the shape of the true underlying error distribution, the efficiency of the partially adaptive estimator is expected to be close to the efficiency of the maximum likelihood estimator based on knowledge of the true error distribution. In this context, the maximum entropy distributions have attracted interest since such distributions have a very flexible functional form and nest most of the statistical distributions. Therefore, several flexible MaxEnt distributions under certain moment constraints are determined to use within the partially adaptive estimation procedure and their performances are evaluated relative to well-known estimators. The simulation results indicate that the determined partially adaptive estimators perform well for non-normal error distributions. In particular, some can be useful in dealing with small sample sizes. In addition, various linear regression applications with non-normal errors are provided.     

On the estimation of probability of error

This paper considers the problem of estimation of classification error in pattern recognition. A theorem is presented to obtain the changes in the eigenvalues and eigenvectors of matrices of the form S₂⁻¹ S₁, when there are changes of first order of smallness in the real symmetric matrices S_i, I = 1, 2. Based on this theory a computational algorithm is developed for the estimation of classification error of Fisher classifier, using leaving groups out method.     

Local influence assessment in heteroscedastic measurement error models

Functional heteroscedastic measurement error models are investigated aiming to assess the effects of perturbations of data on some inferential procedures. This goal is accomplished by resorting to methods of local influence. The techniques provide to the practitioner a valuable tool that enables to identify potential influential elements and to quantify the effects of perturbations in these elements on results of interest. An illustrative example with a real data set is also reported.     

On the martingale approximation of the estimation error of ARMA parameters

The aim of this paper is to prove a theorem which is instrumental in verifying Rissanen's tail condition for the estimation error of the parameters of a Gaussian ARMA process. We get an improved error bound for the martingale approximation of the estimation error for a wide class of ARMA processes.     

On the error behavior in linear minimum variance estimation problems

For linear systems the error covariance matrix for the unbiased, minimum variance estimate of the state does not depend upon any specific realization of the measurement sequence. Thus it can be examined to determine the expected behavior of the error in the estimate before actually using the filter in practice. In this paper, the general linear system that contains both plant and measurement noise is shown to exhibit a decomposition property that permits the derivation of upper and lower bounds upon the error covariance matrix. This decomposition allows systems containing either plant or measurement noise, but not both, to be considered separately. Some general characteristics of these simpler systems are discussed and conditions for the positive definiteness and vanishing of the error covariance matrix are established. It is seen that the presence of plant noise, in general, prevents the error from vanishing. Alternatively, the condition ofq-stage observability is seen to be sufficient to insure that the error covariance matrix asymptotically approaches the zero matrix for systems with noise-free plants. These results are used to establish very specific lower bounds. Through the application of the duality principle, they can be applied directly to the analysis of the linear regulator problem.     

直线时栅传感器全误差模型与误差修正方法研究

为了进一步提高直线时栅位移传感器的测量精度,在建立该传感器周期误差、阿贝误差、热膨胀误差的全误差模型基础上,提出了一种组合校准的方法,利用傅里叶谐波分析和材料线膨胀原理对直线时栅的各种误差进行修正,修正后的精度可达±0.5μm/m。实验证明:该方法解决了直线测量中误差难以分离的问题,同时解决了计算机连续自动采样问题,提高了标定效率,使该方法广泛地应用于生产实践成为可能。     

Theoretical and practical aspects of the quadratic error in the local linear estimation of the conditional density for functional data

The problem of the nonparametric local linear estimation of the conditional density of a scalar response variable given a random variable taking values in a semi-metric space is considered. Some theoretical and practical asymptotic properties of this estimator are established. The usefulness of the estimator is highlighted through the exact expression involved in the leading terms of the quadratic error, and by conducting a computational investigation to show the superiority of this estimation method for the conditional density and then for the conditional mode. Moreover, in order to verify the pertinence of the technique, from a practical point of view, it is applied to a real dataset.     

On estimation of a heteroscedastic measurement error model under heavy-tailed distributions

It is common in epidemiology and other fields that the analyzing data is collected with error-prone observations and the variances of the measurement errors change across observations. Heteroscedastic measurement error (HME) models have been developed for such data. This paper extends the structural HME model to situations in which the observations jointly follow scale mixtures of normal (SMN) distribution. We develop the EM algorithm to compute the maximum likelihood estimates for the model with and without equation error respectively, and derive closed forms of asymptotic variances. We also conduct simulations to verify the effective of the EM estimates and confirm their robust behaviors based on heavy-tailed SMN distributions. A practical application is reported for the data from the WHO MONICA Project on cardiovascular disease.     

Identification of polynomial input/output recursive models with simulation error minimisation methods

Polynomial input/output (I/O) recursive models are widely used in nonlinear model identification for their flexibility and representation capabilities. Several identification algorithms are available in the literature, which deal with both model selection and parameter estimation. Previous works have shown the limitations of the classical prediction error minimisation approach in this context, especially (but not only) when the disturbance contribution is unknown, and suggested the use of a simulation error minimisation (SEM) approach for a better selection of the I/O model. This article goes a step further by integrating the model selection procedure with a simulation-oriented parameter estimation algorithm. Notwithstanding the algorithmic and computational complexity of the proposed method, it is shown that it can sometimes achieve great performance improvements with respect to previously proposed approaches. Two different parameter estimation algorithms are suggested, namely a direct SEM optimisation algorithm, and an approximate method based on multi-step prediction iteration, which displays several convenient properties from the computational point of view. Several simulation examples are shown to demonstrate the effectiveness of the suggested SEM approaches.     

Bias-compensation-based least-squares estimation with a forgetting factor for output error models with white noise

In this paper, the bias-compensation-based recursive least-squares (LS) estimation algorithm with a forgetting factor is proposed for output error models. First, for the unknown white noise, the so-called weighted average variance is introduced. With this weighted average variance, a bias-compensation term is first formulated to achieve the bias-eliminated estimates of the system parameters. Then, the weighted average variance is estimated. Finally, the final estimation algorithm is obtained by combining the estimation of the weighted average variance and the recursive LS estimation algorithm with a forgetting factor. The effectiveness of the proposed identification algorithm is verified by a numerical example.     

Asymptotic error rates of the W and Z statistics when the training observations are dependent

In this paper consideration is given to the properties of the classification statistics W and Z, which were developed for use in discrimination problems with independent training observations. The relative behaviour of these two statistics when the training observations are dependent is investigated. For training observations following a stationary autoregressive process of order p, the asymptotic expansion of the expected error rates associated with W and Z are derived up to and including terms of the second order with respect to the reciprocals of the sample sizes. It is shown that neither Z nor W is absolutely superior to the other. Numerical results are given to show that their relative performance is dependent on the extent of correlation among the training observations and the size of the separation between the two populations, as measured by the Mahalanobis distance.     

GSA-based maximum likelihood estimation for threshold vector error correction model

The log-likelihood function of threshold vector error correction models is neither differentiable, nor smooth with respect to some parameters. Therefore, it is very difficult to implement maximum likelihood estimation (MLE) of the model. A new estimation method, which is based on a hybrid algorithm and MLE, is proposed to resolve this problem. The hybrid algorithm, referred to as genetic-simulated annealing, not only inherits aspects of genetic-algorithms (GAs), but also avoids premature convergence by incorporating elements of simulated annealing (SA). Simulation experiments demonstrate that the proposed method allows to estimate the parameters of larger cointegrating systems. Additionally, numerical results show that the hybrid algorithm does a better job than either SA or GA alone.     

Parametric (modified least squares) and non-parametric (Theil-Sen) linear regressions for predicting biophysical parameters in the presence of measurement errors

Remote sensing often involves the estimation of in situ quantities from remote measurements. Linear regression, where there are no non-linear combinations of regressors, is a common approach to this prediction problem in the remote sensing community. A review of recent remote sensing articles using univariate linear regression indicates that in the majority of cases, ordinary least squares (OLS) linear regression has been applied, with approximately half the articles using the in situ observations as regressors and the other half using the inverse regression with remote measurements as regressors. OLS implicitly assume an underlying normal structural data model to arrive at unbiased estimates of the response. OLS regression can be a biased predictor in the presence of measurement errors when the regression problem is based on a functional rather than structural data model. Parametric (Modified Least Squares) and non-parametric (Theil-Sen) consistent predictors are given for linear regression in the presence of measurement errors together with analytical approximations of their prediction confidence intervals. Three case studies involving estimation of leaf area index from nadir reflectance estimates are used to compare these unbiased estimators with OLS linear regression. A comparison to Geometric Mean regression, a standardized version of Reduced Major Axis regression, is also performed. The Theil-Sen approach is suggested as a potential replacement of OLS for linear regression in remote sensing applications. It offers simplicity in computation, analytical estimates of confidence intervals, robustness to outliers, testable assumptions regarding residuals and requires limited a priori information regarding measurement errors.     

On the role of error productions in syntactic error correction

Error productions are presented as a means of augmenting syntactic error correctors. Such productions are able to simply and efficiently handle a wide variety of difficult error situations. Furthermore, they can be employed without changing (or even knowing) the structure of the error correction algorithm being used. Examples of the use of error productions with the programming language Pascal are presented. Implementation and test results are included.