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This note discusses the radical technical inflation in error variance and the related standard error of test scores from both conceptual and empirical viewpoints. This technical inflation arises as a direct consequence of the technical underestimation of item-score correlation by the product–moment coefficient of correlation (PMC), which is embedded in the traditional estimators of reliability such as coefficients alpha, theta, omega, or rho (maximal reliability). Specifically, in educational settings where compilations usually include both easy and difficult items, the estimate by PMC may substantially deviate from the true association between an item and the score. Consequently, the use of traditional estimators of reliability leads to technically inflated estimates of standard errors, as the error variance related to these traditional measurement models is significantly inflated, resulting in deflated reliability estimates. In educational testing, employing deflation-corrected standard errors, calculated using deflation-corrected reliability estimators, would provide a more accurate measure of the test score's true precision. [ABSTRACT FROM AUTHOR]

Multicollinearity, a critical issue in regression analysis that can severely compromise the stability and accuracy of parameter estimates, arises when two or more variables exhibit correlation with each other. This paper solves this problem by introducing six new, improved two-parameter ridge estimators (ITPRE): NATPR1, NATPR2, NATPR3, NATPR4, NATPR5, and NATPR6. These ITPRE are designed to remove multicollinearity and improve the accuracy of estimates. A comprehensive Monte Carlo simulation analysis using the mean squared error (MSE) criterion demonstrates that all proposed estimators effectively mitigate the effects of multicollinearity. Among these, the NATPR2 estimator consistently achieves the lowest estimated MSE, outperforming existing ridge estimators in the literature. Application of these estimators to a real-world dataset further validates their effectiveness in addressing multicollinearity, underscoring their robustness and practical relevance in improving the reliability of regression models. [ABSTRACT FROM AUTHOR]

This note discusses the radical technical inflation in error variance and the related standard error of test scores from both conceptual and empirical viewpoints. This technical inflation arises as a direct consequence of the technical underestimation of item-score correlation by the product–moment coefficient of correlation (PMC), which is embedded in the traditional estimators of reliability such as coefficients alpha, theta, omega, or rho (maximal reliability). Specifically, in educational settings where compilations usually include both easy and difficult items, the estimate by PMC may substantially deviate from the true association between an item and the score. Consequently, the use of traditional estimators of reliability leads to technically inflated estimates of standard errors, as the error variance related to these traditional measurement models is significantly inflated, resulting in deflated reliability estimates. In educational testing, employing deflation-corrected standard errors, calculated using deflation-corrected reliability estimators, would provide a more accurate measure of the test score’s true precision.

Context. The article is devoted to overcoming the contradictions between the assumptions adopted in known methods of closedloop control system identification and the design and conditions of its operation. The article presents a new method of identifying the transfer functions matrix of a two-level closed-loop control system element, which functions under the conditions of multidimensional stationary centered random influences. Objective. The purpose of the study, the results of which are presented in this paper, is to extend the indirect identification method to the case of estimating one of the two-level closed-loop control system elements, dynamics model based on passive experiment data. Method. To solve the optimal identification problem, a variational method for minimizing the quality functional on the class of fractional-rational transfer function matrices was used. Results. As a result of the research, the identification problem formulation was formalized, the rules for obtaining experimental information about the input and output signals were determined, the rules for identifying the transfer functions matrix of a two-level closed-loop control system element, which minimizes the sum of the variances of identification errors in the frequency domain, and the verification of these rules was carried out. Conclusions. Justified rules allow to correctly determine transfer functions matrices of the closed-loop systems selected element when fulfilling the defined list of conditions. The closed-loop systems control paths signals analysis proves the possibility of the effect of changing these signals statistical means, even under conditions of only centered stationary input influences actions on the system. Based on this, the further development of research can be aimed at overcoming such effects. [ABSTRACT FROM AUTHOR]

A method of correlation-interferometric direction finding has been improved, which effectively solves the problem of radio direction finding of radio emission sources under conditions of exposure to one or two masking interference. The problem was solved using the selection of an unmasked fragment of the spatial spectrum of the signal and the reconstruction of the missing samples of its signal group. As a result of the synthesis of the proposed method, estimates of signal samples were obtained as exact solutions to the proposed energy balance equations. The resulting solutions provide a significant increase in the signal-to-interference ratio and, accordingly, direction-finding accuracy without increasing the number of reception channels of the antenna array. As a result of the simulation, the dependences of the standard deviation of the bearing estimate on the signal-to-noise ratio in the presence of interference were built. Under the influence of one or two masking interferences and a signal-to-interference ratio of 0 dB, the use of the known direction-finding method without interference selection produces an anomalously large direction-finding error of more than 0.42 degrees, which is practically independent of the signal-to-noise ratio. The direction-finding method with selection of spectral signal samples masked by interference reduces the direction-finding error to 0.22 degrees when exposed to one interference and to 0.3 degrees when exposed to two interferences. This is due to the presence of power losses of the usable signal during the selection of its samples masked by interference. The proposed method of direction finding with reconstruction of signal samples provides a significant gain in accuracy by 3–30 times compared to the method of selection of masked samples in the range of changes in the signal-to-noise ratio (–20.5) dB. The direction-finding error of the proposed method decreases with increasing signal/noise according to a hyperbolic dependence. It is advisable to use the proposed direction-finding method when masking no more than two samples of the signal group [ABSTRACT FROM AUTHOR]

Estimation of error variance in the case of genomic selection is a necessary step to measure the accuracy of the genomic selection model. For genomic selection, whole-genome high-density marker data is used where the number of markers is always larger than the sample size. This makes it difficult to estimate the error variance because the ordinary least square estimation technique cannot be used in the case of datasets where the number of parameters is greater than the number of individuals (i.e., p > n). In this article, two existing methods, viz. Refitted Cross Validation (RCV) and kfold-RCV, were suggested for such cases. Moreover, by considering the limitations of the above methods, two new methods, viz. Bootstrap-RCV and Ensemble method, have been proposed. Furthermore, an R package "varEst" has been developed, which contains four different functions to implement these error variance estimation methods in the case of Least Absolute Shrinkage and Selection Operator (LASSO), Least Squares Regression (LSR) and Sparse Additive Models (SpAM). The performances of the algorithms have been evaluated using simulated and real datasets. [ABSTRACT FROM AUTHOR]

In this paper, we apply the profile least-square method and inverse probability weighted method to define estimation of the error variance in partially linear varying-coefficient model when the covariates are missing at random. At the same time, we construct a jackknife estimator and jackknife empirical likelihood (JEL) statistic of the error variance, respectively. It is proved that the proposed estimators are asymptotical normality and the JEL statistic admits a limiting standard chi-square distribution. A simulation study is conducted to compare the JEL method with the normal approximation approach in terms of coverage probabilities and average interval lengths, and a comparison of the proposed estimators is done based on sample means, biases and mean square errors under different settings. Subsequently, a real data set is analyzed for illustration of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, robust Bayesian estimators of error variance in a normal linear model with uncertain hierarchical prior information are investigated. The posterior regret gamma minimax estimator, the least sensitive estimator and conditional gamma minimax estimator for error variance are obtained under two different classes of priors, respectively. A simulation study is used to compare the performance of the proposed estimators. A real data example is also given to illustrate the results. [ABSTRACT FROM AUTHOR]

Optimal RWIS network can be defined as an RWIS configuration where the total number of stations (RWIS density) are determined based on a well-established guideline and the locations are allocated systematically assuming that it will provide the maximum monitoring coverage of the network. This paper examines and quantifies the benefit of an optimized RWIS network and how these benefits impact traffic safety. The methodological framework presented herein builds upon our previous efforts in RWIS location-allocation, where the kriging variance is used as a performance indicator for monitoring coverage. In this study, the network coverage index (NCI) parameter is proposed to gauge RWIS network performance and quantitatively evaluate its impact on traffic safety. The findings of this study reveal a strong dependency between the NCI and the RWIS network configuration. In terms of traffic safety, the relationship between NCI and safety effectiveness can be expressed as a polynomial function, where the two are proportional to one another. In the state of Iowa, an RWIS network with 80% monitoring coverage (NCI = 0.8) can reduce additional 40 collisions per site annually compared to a network without RWIS stations. Based on the findings obtained in this study, road agencies and RWIS planners can now be assisted with conceptualizing the capabilities of an optimized RWIS network, which will help them increase monitoring coverage, and in the process, gain a quantitative understanding on its potential impact on traffic safety. [ABSTRACT FROM AUTHOR]

Variance estimation is fundamental in the statistical inference. Due to the nonlinearity of the variance estimator, Lin et al. [Jackknife empirical likelihood for the error variance in linear models. J Nonparametr Stat. 2017;29:151–166.] proposed the jackknife empirical likelihood method for the error variance in a linear regression model. However, people may have some prior information about the error variance. In this article, we propose the Bayesian jackknife empirical likelihood (BJEL) for the error variance in a linear regression model. The validity of the proposed method is verified, and the asymptotic normal properties for the BJEL are also established. A simulation study shows that the new approach for the small sample performs better than its frequentist counterpart. Two real data sets are also used to illustrate the proposed methods. [ABSTRACT FROM AUTHOR]

If a test is administered to n-persons, the test score can be viewed as a point in n – dimensional person space where each person is considered in an axis and the variables are represents as points or vectors. The orthogonality of the axes is better indication since individuals are assumed to be independent. Such presentation helps to make useful inferences about the subjects and various parameters of the test, item statistics along with geometrical interpretation of such computations. For a test consisting of one-right-rest wrong type items, the paper gives theoretical formulation of mental testing through person space, primarily in terms of length of observed score vector and angle separating the vector with the vector representing the maximum possible score in the test. In addition, method is described to obtain (i) test reliability, as per the theoretical definition, via a single administration of the test, which is extended to find reliability of a battery of tests (ii) difficulty, discriminating values of items and test along with their relationships. Empirical verification of the proposed methods are undertaken. [ABSTRACT FROM AUTHOR]

Measurement errors and missing data are often arise in practice. Under this circumstance, we focus on using jackknife empirical likelihood (JEL) and adjust jackknife empirical likelihood (AJEL) methods to construct confidence intervals for the error variance in linear models. Based on residuals of the models, the biased-corrected inverse probability weighted estimator of the error variance is introduced. Furthermore, we propose the jackknife estimator, jackknife and adjust jackknife empirical log-likelihood ratios of the error variance and establish their asymptotic distributions. Simulation studies in terms of coverage probability and average length of confidence intervals are conducted to evaluate the proposed method. A real data set is used to illustrate the proposed JEL and AJEL methods. [ABSTRACT FROM AUTHOR]

First lactation production records of 7,825 Mehsana buffaloes, sired by 200 sires maintained at Dudhsagar Research and Development Association, Dudhsagar Dairy, Mehsana over a period of 25 years (1989-2013), were utilised for estimation of breeding value of the first lactation milk yield using BLUP - Univariate sire model and animal model using WOMBAT software. The study aimed to find out the optimum number of daughters that may be sufficient for more accurate and unbiased estimate of breeding value of Mehsana buffalo bulls. The effectiveness of different models run with different daughters per sire for sire evaluation was judged on the basis of within-sire variance. The maximum and minimum estimated breeding values were 1924.04 l (BLUP-AM, 5 daughters/sire) and 1855.61 l (BLUP-SM, 30 daughters/sire), respectively. However, error variance estimated was highest when five daughters/sire were evaluated and lowest, when 30 daughters/sire were evaluated. The present study revealed that the sire evaluation with sires having minimum 5, 10 and 15 daughters are equally efficient. However, if we do evaluation with sires having daughters more than 20, 25 or 30, additional accuracy of approximately 2.5%, 3% and 5%, respectively may be obtained.

We propose a novel localization method using angle-of-arrival (AoA) measurements with two-step error variance-weighted least squares (TELS). The first step is to estimate a terminal location provisionally using least squares. The second step is to estimate the terminal location using weighted least squares, with the weights for each anchor and each evaluation-function term, calculated from the error variance based on the first step. The proposed method does not require previous information on the environment while achieving high performance. The simulation results indicate that a root mean square error (RMSE) of the proposed method is superior to that of the existing hybrid received signal strength (RSS)/AoA localization methods. When 11 anchors are deployed inside a cube with edge length 15 m, and the standard deviations of measurements are small, the RMSE of the proposed method reaches about 0.34 m. It is nearly equal to that of Cramer-Rao lower bound (CRLB) on AoA.

This study aimed to detect the contribution of multiple sources of the variance in the reliability of a test in mathematics by using the Generalizability Theory. The magnitude of the error variance that is explained from the facets (Tasks, Judges, and Occasions) in the total variance, and detect the conditions through which they can be better in levels of test reliability. The study sample consisted of (243) students from the fifth grade who were applied on a mathematics test that consisted of (12) Complex tasks in domain of numbers and operations on them, they were distributed on (4) formulas equally (application, inference, selection, and opinion), The performance of the persons was evaluated by (3) judges. The test was applied to them twice with a time two-week interval. The researchers used the completely crossed design (Person×Task×Judge×Occasion), and used (EduG) software to analyze the data. The results indicated that the largest sources of error variance on Generalizability coefficients were interaction (person-task), including after the Occasion contributed to improving Generalizability coefficients. The results have found that increasing the number of tasks and the number of occasions increases the Generalizability coefficients, and increasing the number of judges does not substantially increase the Generalizability coefficients. [ABSTRACT FROM AUTHOR]

Estimation of error variance in the case of genomic selection is a necessary step to measure the accuracy of the genomic selection model. For genomic selection, whole-genome high-density marker data is used where the number of markers is always larger than the sample size. This makes it difficult to estimate the error variance because the ordinary least square estimation technique cannot be used in the case of datasets where the number of parameters is greater than the number of individuals (i.e., p > n). In this article, two existing methods, viz. Refitted Cross Validation (RCV) and kfold-RCV, were suggested for such cases. Moreover, by considering the limitations of the above methods, two new methods, viz. Bootstrap-RCV and Ensemble method, have been proposed. Furthermore, an R package “varEst” has been developed, which contains four different functions to implement these error variance estimation methods in the case of Least Absolute Shrinkage and Selection Operator (LASSO), Least Squares Regression (LSR) and Sparse Additive Models (SpAM). The performances of the algorithms have been evaluated using simulated and real datasets.

Distribution state estimation (DSE) is an essential part of an active distribution network with high level of distributed energy resources. The challenges of accurate DSE with limited measurement data is a well-known problem. In practice, the operation and usability of DSE depend on not only the estimation accuracy but also the ability to predict error variance. This paper investigates the application of error covariance in DSE by using the augmented complex Kalman filter (ACKF). The Kalman filter method inherently provides state error covariance prediction. It can be utilized to accurately infer the error covariance of other parameters and provide a method to determine optimal measurement locations based on the sensitivity of error covariance to measurement noise covariance. This paper also proposes a generalized formulation of ACKF to allow scalar measurements to be incorporated into the complex-valued estimator. The proposed method is simulated by using modified IEEE 34-bus and IEEE 123-bus test feeders, and randomly generates the load data of complex-valued Wiener process. The ACKF method is compared with an equivalent formulation using the traditional weighted least squares (WLS) method and iterated extended Kalman filter (IEKF) method, which shows improved accuracy and computation performance.

We agree with Arocha that the common and exclusive focus on aggregate results of psychological studies creates problems. While a paradigm shift toward idiographic approaches or control theory might help, we argue that traditional approaches can accomplish more if measures of variability are taken seriously. We discuss three kinds of studies: within-person treatment studies, questionnaire-based studies, and replication studies. For each of these, we suggest ways to improve psychological meaningfulness by investigating variability surrounding aggregate results, without ending up in an either–or choice between aggregate results and separate, individual results. [ABSTRACT FROM AUTHOR]

Summary: Error variance estimation plays a key role in the analysis of homogeneous non‐parametric regression models. For a random design model, most methods in the literature for error variance estimation assume the independence between the predictor variable X and the error ε. In this work, we derive the optimal semi‐parametric efficiency bound for the error variance σ2=var(ϵ) without such an independence assumption. A residual‐based efficient estimator for σ2 is proposed and its asymptotic normality is established. An extensive simulation study is conducted, which shows that our proposed estimator works very favourably against competitors. A simple real‐data example is also presented. Derived the optimal semi‐parametric efficiency bound and efficient estimator for the error variance without the independence assumption [ABSTRACT FROM AUTHOR]

The performances of preliminary test estimators for error variance based on W, LR and LM tests in a normal linear model are considered in this paper. Firstly, the risks of the proposed estimators are derived and compared by theoretical analysis and numerical calculation, respectively. The results show that their risks are related to the equality constraint error and the critical value of test. Moreover, the minimum value of the risks can be achieved when the critical value of test equals to one. Secondly, the superiority conditions of the proposed estimators are discussed. Finally, the results are illustrated by a simulation example. [ABSTRACT FROM AUTHOR]

Sampling for chemical analysis is nearly always a multi-stage process where all stages contribute to the total uncertainty of the results reported. A sample, when extracted from the target, forgets its past, consequently, mistakes made in earlier stages of this process cannot be corrected at later stages, no matter how carefully and precisely they are executed. Primary sampling is the most important, and generally, its variance by far exceeds the variance of the laboratory measurement. That does not mean that, at the later stages when laboratory and analytical aliquots (or test portions) are prepared, the principles of the theory of sampling (TOS) can be neglected. Modern analytical instruments are designed to handle small samples (from milligrams to a few grams). In this case, if the sample is a mixture containing a low fraction of analyte containing particles, the constitution heterogeneity of the material may be so large that it ruins the whole analysis. Heterogeneity calculations and evaluation of the fundamental sampling error variances of the sample preparation steps are essential in developing fit-for-purpose analytical procedures. At the final steps of the sample preparation the new subsample is a significant part of parentsub-sample and account of that effect has to be taken in estimating the sampling variances. TOS offers tools for handling also these situations. Application of heterogeneity calculations are elucidated with two illustrative examples. In the first example the constitution heterogeneity is estimated for low concentration additives in chicken feed and in the second example the sample preparation is optimised for calibrating an IR instrument for estimating mineral impurities in a wollastonite concentrate. Heterogeneity assessments are important also when particle mixtures are processed and the efficiency of mixing is estimated. [ABSTRACT FROM AUTHOR]

The purpose of the present study was to apply the methodology developed by Raykov on modeling item-specific variance for the measurement of internal consistency reliability with longitudinal data. Participants were a randomly selected sample of 500 individuals who took on a professional qualifications test in Saudi Arabia over four different occasions. Data were analyzed by use of confirmatory factor analysis, and item error variance was corrected for item specificity. The estimation of reliability involved composite index omega. Results indicated that the initially low and unacceptable levels of internal consistency reliability approached acceptable levels after accounting for item-specific variance. Findings were verified by testing whether the difference estimates of internal consistency reliability deviated from a zero-mean distribution using 10,000 replicated samples assuming a known (symmetric) or unknown (asymmetric) population distribution of the difference reliability coefficients. Percentage improvement reliability estimates indices were also estimated along with their 95% confidence intervals. Two appendices provide annotated Mplus syntax files for future use. [ABSTRACT FROM AUTHOR]

When an assessment undergoes changes to the administration or instrument, bridge studies are typically used to try to ensure comparability of scores before and after the change. Among the most common and powerful is the common population linking design, with the use of a linear transformation to link scores to the metric of the original assessment. In the common population linking design, randomly equivalent samples receive the new and previous administration or instrument. However, conventional procedures to estimate error variances are not appropriate for scores linked in a bridge study, because the procedures neglect variance due to linking. A convenient approach is to estimate a variance component associated with the linking to add to the conventionally estimated error variance. Equations for the variance components in this approach are derived, and the approximations inherently made in this approach are shown and discussed. Exact error variances of linked scores, accounting for both conventional sources of variance (e.g., sampling) and linking variance together, are derived and discussed. The consequences of how linking changes how certain errors are related is considered mathematically. Specifically, the impacts of linking on the error variance for the comparison of two linked estimates (e.g., comparing the mean score of boys to the mean score of girls, after linking), for the comparison of scores across the two samples (e.g., comparing the mean score of boys in the new administration or instrument to the mean score of boys in the old administration or instrument), and for aggregating scores across the two samples (e.g., the mean score of boys across both administrations or instruments) are derived and discussed. Finally, general methods to account for error variance in bridge studies by simultaneously accounting for both conventional and linking sources of error are recommended. Report Number: ETS RR–19‐42 [ABSTRACT FROM AUTHOR]

A topic of ongoing debate in survey research is whether items measuring different constructs should be grouped by construct or intermixed. Previous researchers have failed to reach a consensus on whether grouping or intermixing measurement items leads to better construct validity. The inconsistency of their findings has to a large extent resulted from a tendency to confound true variance and systematic error variance. Drawing on measurement and cognitive theories, we propose that for established measures with satisfactory psychometric properties, grouped items capture less unsystematic and systematic error variance and more true variance than their intermixed counterparts. This argument is supported by the findings of a field experiment with 853 students in 142 project teams, who reported their teams' relationship and task conflict on grouped and intermixed items. The theoretical and practical implications of these findings for researchers who use survey instruments are discussed. [ABSTRACT FROM AUTHOR]

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Summary: Measurement error in the continuous covariates of a model generally yields bias in the estimators. It is a frequent problem in practice, and many correction procedures have been developed for different classes of models. However, in most cases, some information about the measurement error distribution is required. When neither validation nor auxiliary data (e.g., replicated measurements) are available, this specification turns out to be tricky. In this article, we develop a flexible likelihood‐based procedure to estimate the variance of classical additive error of Gaussian distribution, without additional information, when the covariate has compact support. The performance of this estimator is investigated both in an asymptotic way and through finite sample simulations. The usefulness of the obtained estimator when using the simulation extrapolation (SIMEX) algorithm, a widely used correction method, is then analyzed in the Cox proportional hazards model through other simulations. Finally, the whole procedure is illustrated on real data. [ABSTRACT FROM AUTHOR]

Global Positioning System (GPS) radio occultation (RO) and radiosonde (RS) observations are two major types of observations assimilated in numerical weather prediction (NWP) systems. Observation error variances are required input that determines the weightings given to observations in data assimilation. This study estimates the error variances of global GPS RO refractivity and bending angle and RS temperature and humidity observations at 521 selected RS stations using the three-cornered hat method with additional ERA-Interim reanalysis and Global Forecast System forecast data available from 1 January 2016 to 31 August 2019. The global distributions, of both RO and RS observation error variances, are analyzed in terms of vertical and latitudinal variations. Error variances of RO refractivity and bending angle and RS specific humidity in the lower troposphere, such as at 850 hPa (3.5 km impact height for the bending angle), all increase with decreasing latitude. The error variances of RO refractivity and bending angle and RS specific humidity can reach about 30 N-unit2, 3 × 10−6 rad2, and 2 (g kg−1)2, respectively. There is also a good symmetry of the error variances of both RO refractivity and bending angle with respect to the equator between the Northern and Southern Hemispheres at all vertical levels. In this study, we provide the mean error variances of refractivity and bending angle in every 5°-latitude band between the equator and 60°N, as well as every interval of 10 hPa pressure or 0.2 km impact height. The RS temperature error variance distribution differs from those of refractivity, bending angle, and humidity, which, at low latitudes, are smaller (less than 1 K2) than those in the midlatitudes (more than 3 K2). In the midlatitudes, the RS temperature error variances in North America are larger than those in East Asia and Europe, which may arise from different radiosonde types among the above three regions.

Quantitative genetic analyses require extensive measurements of phenotypic traits, a task that is often not trivial, especially in wild populations. On top of instrumental measurement error, some traits may undergo transient (i.e., nonpersistent) fluctuations that are biologically irrelevant for selection processes. These two sources of variability, which we denote here as measurement error in a broad sense, are possible causes for bias in the estimation of quantitative genetic parameters. We illustrate how in a continuous trait transient effects with a classical measurement error structure may bias estimates of heritability, selection gradients, and the predicted response to selection. We propose strategies to obtain unbiased estimates with the help of repeated measurements taken at an appropriate temporal scale. However, the fact that in quantitative genetic analyses repeated measurements are also used to isolate permanent environmental instead of transient effects requires that the information content of repeated measurements is carefully assessed. To this end, we propose to distinguish "short‐term" from "long‐term" repeats, where the former capture transient variability and the latter help isolate permanent effects. We show how the inclusion of the corresponding variance components in quantitative genetic models yields unbiased estimates of all quantities of interest, and we illustrate the application of the method to data from a Swiss snow vole population. [ABSTRACT FROM AUTHOR]

This note extends the results in the 2016 article by Raykov, Marcoulides, and Li to the case of correlated errors in a set of observed measures subjected to principal component analysis. It is shown that when at least two measures are fallible, the probability is zero for any principal component--and in particular for the first principal component--to be error-free. In conjunction with the findings in Raykov et al., it is concluded that in practice no principal component can be perfectly reliable for a set of observed variables that are not all free of measurement error, whether or not their error terms correlate, and hence no principal component can practically be error-free. [ABSTRACT FROM AUTHOR]

A fundamental strategy to diminish variations in manufacturing process urged the practitioners to characterize the quality of a process by a relationship between the response variable and one or more explanatory variables instead of a single quality characteristic; this state is known as a profile or a function. Profile monitoring mainly aims to test the stability of this relationship. Many researches have been carried out to study the different sampling techniques in the performance of linear profile under the maximum likely hood (MLE) estimation strategy, whereas using different estimation strategy has not been discussed so far. This paper is dedicated to introduce Bayesian estimation strategies with a proposal of novel control charts for jointly monitoring the linear profile. We considered restricted and pretest estimators, besides the estimation of distrust probability under the null hypothesis. Analytical and numerical results showed that the proposed estimators outperformed the MLE method. The proposed control charts have been used to monitor the two-phase flow in the oil industry to control the relationship between the flow rate and the pressure difference between two points. [ABSTRACT FROM AUTHOR]

A summary of the formal expressions of the classes UΘ and ΘU of the estimator (A,U,Θ) is presented where two cases are distinguished: the drift ε(P) is known a priori; and the drift should be determined because is unknown. The classic expressions for evaluate the error of interpolation, the error variance for Kriging and the interpolation variance of J. K. Yamamoto are analized and consequently a new expression valid to evaluate the error of any estimation (A,U,Θ) is proposed. [ABSTRACT FROM AUTHOR]