Best linear unbiased estimator

To model the relationship between PLA and PLR, we used each of these in ordinary least squares (OLS) multiple regression to explore the relationship between the dependent variables Mean PLR or Mean PLA and the independent variables (Berry and Feldman, 1985).OLS regression was used because data satisfied OLS assumptions for the model as the best linear unbiased estimator (BLUE). Distribution of errors (residuals) is normal, they are uncorrelated with each other, and homoscedastic (constant variance among residuals), with the mean of 0. We also analyzed predicted values plotted against residuals, as they are a better indicator of non-normality in aggregated data, and found them also to be homoscedastic and independent of one other. 

The kriging system of linear equations is derived so that their solution gives kriging weights such that the kriging estimator is a "best linear unbiased estimator." The estimator is linear because the estimator is a weighted sum. It is unbiased because the system... 

In contrast to optimal design, Hamprecht and co-workers recently introdnced a space-filling design techniqne for compound selection. This stochastic method nses the best linear unbiased estimator, in the form of Kriging, " to constrnct selection designs that optimize the integrated mean-square prediction error, or entropy. This... 

Henderson, C.R. Best linear unbiased estimation and ptediction under a selection model. Biometics 1975,3/ 423-477. 

Fig. 4.19. Schematic comparison of methods of multivariate data analysis using the classical and inverse approach. BLUE, best linear unbiased estimator PRESS, predictive residual error...

In principle both the classical and the inverse approach use a multivariate data set. But in the classical approach the variance is minimised, whereas in the inverse approach one tries to find an equilibrium between bias and variance. Therefore the bias is reduced and by the procedure of predictive receivable error sum of squares either via a singular value decomposition or the bidiagonalisation method estimated values, either according to principle component regression or partial least squares, are found. The multilinear regression on the other hand will find the best linear unbiased estimation as an approach to a true concentration. Besides applications in absorption spectroscopy, fluorescence spectra can also be evaluated [74]. 

OLS is synonymous with the following terms least squares regression, linear least squares regression, multiple least squares regression, multivariate least squares regression. OLS provides the best linear unbiased estimator (BLUE) that has the smallest variance among all linear and unbiased estimators. 

These estimates are best in several senses they are the best linear unbiased estimates of the unknowns, and are also the best least-squares estimates (see refs. 2, 23, 32, 33, 35 and 36). However it is worth mentioning that there are also good arguments in favor of using a biased estimate, such as the James-Stein estimator— see refs. 37-40. 

Kriging After the name of D. G. Krige, this term refers to the procedure of constructing the best linear unbiased estimate of a value at a point or of an average over a volume. 

Ordinary kriging is a best linear unbiased estimate of the parameter. It is linear because its estimates are weighted linear combinations of the available data, unbiased because it tries to have the mean error equal to zero, and best because it aims at minimizing the variance of the error (7). The kriging estimator, Z is described as. 

We next describe the first steps in the derivation of the best linear unbiased predictor (BLUP) of Y x) at an untried input vector x (see, for example, Sacks et al 1989). Similar steps are used in Section 4 to estimate the effects of one, two, or more input variables. It is then apparent how to adapt results and computational methods for predicting Y(x) to the problem of estimating such effects. 

Also under OLS assumptions, the regression parameter estimates have a number of optimal properties. First, 0 is an unbiased estimator for 0. Second, the standard error of the estimates are at a minimum, i.e., the standard error of the estimates will be larger than the OLS estimates given any other assumptions. Third, assuming the errors to be normally distributed, the OLS estimates are also the maximum likelihood (ML) estimates for 0 (see below). It is often stated that the OLS parameter estimates are BLUE (Best Linear Unbiased Predictors) in the sense that best means minimum variance. Fourth, OLS estimates are consistent, which in simple terms means that as the sample size increases the standard error of the estimate decreases and the bias of the parameter estimates themselves decreases. 

Figure 1 shows a block diagram for the perturbed state of a robot, e, subject to both the process noise w and measurement noise y. The actually measured perturbed state is denoted as z. The Kalman filter is the best linear estimator in the sense that it produces unbiased, minimum variance estimates (Kalman and Bucy, 1961 Brown, 1983). Let (t) be the estimated perturbed state and 6eg(t) be the residual which is the difference between the true measured perturbed state, z(t), and the estimated perturbed state based on 6a (t), here denoted as (t). It has already been shown (Lewis, 1986) that Cx satisfies a differential equation which can be schematically represented by the block diagram shown in Fig. 2 where K(t) is a Kalman filter gain. K(t) is to be calculated according to the equation... 

Of the possible estimates for the frequency response function, the HI estimator was chosen because it produces the best unbiased linear model of the system [15] in absence of noise on the input. This characteristic is appropriate in the present study because the input force is specified, with noise due to energy transfer present only on the output. 

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