Least squares estimate statistical properties

The least squares estimates d and 62 have several important statistical properties. For example. 

Since a and b are statistical estimates based on a sample, they have an error term (the standard error of the estimate) associated with them. Among all estimates A and B, the least squares estimates have the smallest error term. This is the third important property of the least squares estimates. 

Dynamics of the dissipative two-state system. Reviews of Modern Physics 59 (l) l-85. Louisell, W. H. 1973. Quantum Statistical Properties of Radiation. New York John WUey. Marquardt, D. W. 1963. An algorithm for least-squares estimation of nonlinear parameters. 

The statistical properties for the least squares estimate 6 of the FSF model parameters given in Equation (4.26) are used in this section to develop statistical confidence bounds for frequency response and step response models estimated using the FSF model structme. We begin by stating the key assumptions on which all of this analysis is based and then summarize the key properties of the least squares estimator, which may be found in Ljung (1987). 

Quantitative Structure - Activity Relationships (QSARs) are estimation methods developed and used to predict certain effects or properties of chemical substances, which are primarily based on the structure of the chemicals. The development of QSARs often relies on the application of statistical methods such as multiple linear regression (MLR) or partial least squares regression (PLS). However, since toxicity data often include uncertainties and measurements errors, when the aim is to point out the more toxic and thus hazardous chemicals and to set priorities, order models can be used as alternative to statistical methods such as multiple linear regression. 

Once a general equation of state involving Wq versus P and T such as (15.50) is available, it can be used to calculate a statistically smoothed solvus. This is the solvus that provides the best least-squares fit to the experimental data. With caution, it can also be used to extrapolate beyond the T and P range of the experimental data (this is safer if the excess properties such as Wh and Wv are derived directly from real data and not calculated from Wg or otherwise estimated). First, an equation for the total free energy of the system as a function of T, P, and concentration is derived using (15.50) for and the relation... 

A correct knowledge of the error structure is needed in order to have a correct summary of the statistical properties of the estimates. This is a difficult task. Measurement errors are usually independent, and often a known distribution, for example, Gaussian, is assumed. Many properties of least squares hold approximately for a wide class of distributions if weights are chosen optimally, that is, equal to the inverse of the variances of the measurement errors, or at least inversely proportional to them if variances are known up to a proportionality constant, that is, is equal or proportional to Zy, the N x N covariance matrix of the measurement error v. Under these circumstances, an asymptotically correct approximation of the covariance matrix of the estimation error 0 = 0 — 0 can be used to evaluate the precision of parameter estimates ... 

It is notable that such kinds of error sources are fairly treated using the concept of measurement uncertainty which makes no difference between random and systematic . When simulated samples with known analyte content can be prepared, the effect of the matrix is a matter of direct investigation in respect of its chemical composition as well as physical properties that influence the result and may be at different levels for analytical samples and a calibration standard. It has long since been suggested in examination of matrix effects [26, 27] that the influence of matrix factors be varied (at least) at two levels corresponding to their upper and lower limits in accordance with an appropriate experimental design. The results from such an experiment enable the main effects of the factors and also interaction effects to be estimated as coefficients in a polynomial regression model, with the variance of matrix-induced error found by statistical analysis. This variance is simply the (squared) standard uncertainty we seek for the matrix effects. 

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