Improved Estimation in Measurement Error Models Through Stein Rule Procedure

AbstractThis paper examines the role of Stein estimation in a linear ultrastructural form of the measurement errors model. It is demonstrated that the application of Stein rule estimation to the matrix of true values of regressors leads to the overcoming of the inconsistency of the least squares procedure and yields consistent estimators of regression coefficients. A further application may improve the efficiency properties of the estimators of regression coefficients. It is observed that the proposed family of estimators under some constraint on the characterizing scalar dominates the conventional consistent estimator with respect to the criterion of asymptotic risk under a specific quadratic loss function. Then the problem of prediction of the values of the study variable within the sample is considered, and it is found that the predictors based on the proposed family of estimators are always more efficient than the predictors based on the conventional estimator according to asymptotic predictive mean squared error criterion, although both are biased

Measurement error models for time series

Estimation for multivariate linear measurement error models with serially correlated observations is addressed;The asymptotic properties of some standard linear errors-in-variables regression parameter estimators are developed under an ultrastructural model in which the random components of the model follow a linear process. Under the same assumptions, the asymptotic properties of weighted method-of-moments estimators are derived. The large-sample results rest on the asymptotic properties of the sum of a linear function and a quadratic function of a sequence of serially correlated random vectors;Maximum likelihood estimation for the normal structural and functional models is addressed. For each model, first- and second-derivative matrices of the log-likelihood functions are given and Newton-Raphson maximum likelihood estimation procedures are considered. For the structural model, the assumption that the random components follow a multivariate autoregressive moving average process is used to develop autoregressive moving average and state-space models for the observation sequence. The state-space representation of the structural model leads to innovation sequences and associated derivative sequences that provide the basis for a Newton-Raphson procedure for the estimation of regression parameters and autocovariance parameters of the structural model. A modified state-space approach leads to a similar procedure for the estimation for the functional model. An extension of the state-space approach to maximum likelihood estimation for a structural model with combined time series and cross-sectional data is given

The limiting behavior of residuals from measurement error regressions

Multivariate measurement error regression models with normal errors are investigated and residuals, analogous to those of ordinary least squares, are defined. The limiting behavior of test statistics based on the residuals is determined;The residuals, properly standardized, are represented as a linear combination of two independent random vectors. This representation is used to show that the empirical process based on the standardized residuals converge to a unique Gaussian process, where the limit process is that of a normal sample standardized with estimated mean and variance. It is shown that many goodness-of-fit tests for normality based on the standardized residuals have the same limiting distribution as that of tests based on a sample of iid normal random vectors;Tests for outliers, for autocorrelation, and for homogeneity of variance are investigated. A test for autocorrelation is constructed by regressing the residuals on their lagged values and testing for zero coefficients. A test for homogeneity of variance is constructed by regressing the squared residuals on estimated values of the true independent variables and testing for zero coefficients. It is shown that the regression t-statistics and F-statistics for the autocorrelation test and for the homogeneity test converge to N(0, 1) and chi-square random variables, respectively;Monte Carlo studies are conducted to examine the adequacy of the asymptotic approximations in small samples. The large sample approximations are judged adequate even for models with only twenty degrees of freedom

Edgeworth expansions for errors-in-variables models

AbstractEdgeworth expansions for sums of independent but not identically distributed multivariate random vectors are established. The results are applied to get valid Edgeworth expansions for estimates of regression parameters in linear errors-in-variable models. The expansions for studentized versions are also developed. Further, Edgeworth expansions for the corresponding bootstrapped statistics are obtained. Using these expansions, the bootstrap distribution is shown to approximate the sampling distribution of the studentized estimators, better than the classical normal approximation

A computational approach for identifying the chemical factors involved in the glycosaminoglycans-mediated acceleration of amyloid fibril formation

BACKGROUND: Amyloid fibril formation is the hallmark of many human diseases, including Alzheimer's disease, type II diabetes and amyloidosis. Amyloid fibrils deposit in the extracellular space and generally co-localize with the glycosaminoglycans (GAGs) of the basement membrane. GAGs have been shown to accelerate the formation of amyloid fibrils in vitro for a number of protein systems. The high number of data accumulated so far has created the grounds for the construction of a database on the effects of a number of GAGs on different proteins. METHODOLOGY/PRINCIPAL FINDINGS: In this study, we have constructed such a database and have used a computational approach that uses a combination of single parameter and multivariate analyses to identify the main chemical factors that determine the GAG-induced acceleration of amyloid formation. We show that the GAG accelerating effect is mainly governed by three parameters that account for three-fourths of the observed experimental variability: the GAG sulfation state, the solute molarity, and the ratio of protein and GAG molar concentrations. We then combined these three parameters into a single equation that predicts, with reasonable accuracy, the acceleration provided by a given GAG in a given condition. CONCLUSIONS/SIGNIFICANCE: In addition to shedding light on the chemical determinants of the protein∶GAG interaction and to providing a novel mathematical predictive tool, our findings highlight the possibility that GAGs may not have such an accelerating effect on protein aggregation under the conditions existing in the basement membrane, given the values of salt molarity and protein∶GAG molar ratio existing under such conditions

Estimated generalized least squares estimation for the heterogeneous measurement error model

The measurement error model of interest is (UNFORMATTED TABLE OR EQUATION FOLLOWS)\eqalign y[subscript]t &= [beta][subscript]0 + x[subscript]t[beta][subscript]1 + q[subscript]t, Y[subscript]t &= y[subscript]t + w[subscript]t, X[subscript]t &= x[subscript]t + u[subscript]t, (TABLE/EQUATION ENDS)where Z[subscript]t = ( Y[subscript]t, X[subscript]t) is the observed p-dimensional vector, z[subscript]t = (y[subscript]t, x[subscript]t) is the true unknown random vector, q[subscript]t is the equation error, and the measurement errors a[subscript]t = (w[subscript]t, u[subscript]t) are distributed with mean zero and known variance [sigma][subscript]aatt. An estimated generalized least squares estimator of the mean and variance of z[subscript]t, denoted by [mu] and [sigma][subscript]zz respectively, is shown to have a limiting normal distribution under mild regularity conditions. An estimator of [beta][superscript]\u27 = ([beta][subscript]0, [beta][subscript]sp1\u27) based upon the proposed estimator of [mu] and [sigma][subscript]zz is constructed and shown to have a limiting normal distribution. The variances of the limiting distribution are less than or equal to the corresponding variances for other estimators that have been suggested for the heterogeneous error model. The estimated generalized least squares estimator also displayed smaller mean square error than other estimators in a Monte Carlo study. A program to implement the proposed estimators is developed. The iterated estimated generalized least squares estimators of the measurement error model are investigated. The limit of a modified iteration procedure is shown to be the maximum likelihood estimator for the normal distribution;Estimated generalized least squares estimation is considered for the general linear model, Y = X[beta] + u, where the variance of u is denoted by V[subscript]uu and the elements of X[superscript]\u27 V[subscript]spuu-1 X may increase at different rates. Sufficient conditions are given for the estimated generalized least squares estimator to be consistent and asymptotically equivalent to the generalized least squares estimator constructed with known V[subscript]uu. Consistent estimators of the normalizing matrix are developed, and the asymptotic distribution of a linear combination of the elements of the estimator is considered. The model where V[subscript]uu is a function of a fixed, finite number of parameters and the use of ordinary least squares residuals to estimate the parameters of V[subscript]uu are examined. Applications of the results to the trend model with first order autoregressive errors and to the measurement error model are given

On the Estimation of the Linear Relation when the Error Variances are known

The present article considers the problem of consistent estimation in measurement error models. A linear relation with not necessarily normally distributed measurement errors is considered. Three possible estimators which are constructed as different combinations of the estimators arising from direct and inverse regression are considered. The efficiency properties of these three estimators are derived and analyzed. The effect of non-normally distributed measurement errors is analyzed. A Monte-Carlo experiment is conducted to study the performance of these estimators in finite samples and the effect of a non-normal distribution of the measurement errors