Weighted Least Squares WLS Estimation

In this case we minimize a weighted SSE with constant weights, i.e., the user-supplied weighting matrix is kept the same for all experiments, Qj=Q for all 

---

However, in the simpler PC-AR case, the regression model of Eq. 13 is linear, and thus, may be estimated by the linear weighted least squares (WLS) estimator ... 

In order to calculate Vx, and therefore the detection limit, it is necessary first to estimate Vy as a function of concentration and then to use this information to estimate the parameters of the calibration curve using weighted least squares (WLS) fitting. Rigorous application of WLS requires knowledge of relative weights, but the technique is already considered adequate when n 5 (18). 

Assuming that we have measured a series of concentrations over time/ we can define a model structure and obtain initial estimates of the model parameters. The objective is to determine an estimate of the parameters (CLe, Vd) such that the differences between the observed and predicted concentrations are comparatively small. Three of the most commonly used criteria for obtaining a best fit of the model to the data are ordinary least squares (OLS)/ weighted least squares (WLS)/ and extended least squares (ELS) ELS is a maximum likelihood procedure. These criteria are achieved by minimizing the following quantities/... 

Carroll and Ruppert (1988) and Davidian and Gil-tinan (1995) present comprehensive overviews of parameter estimation in the face of heteroscedasticity. In general, three methods are used to provide precise, unbiased parameter estimates weighted least-squares (WLS), maximum likelihood, and data and/or model transformations. Johnston (1972) has shown that as the departure from constant variance increases, the benefit from using methods that deal with heteroscedasticity increases. The difficulty in using WLS or variations of WLS is that additional burdens on the model are made in that the method makes the additional assumption that the variance of the observations is either known or can be estimated. In WLS, the goal is not to minimize the OLS objective function, i.e., the residual sum of squares,... 

Least squares (LS) estimation minimizes the sum of squared deviations, comparing observed values to values predicted by a curve with particular parameter values. Weighted LS (WLS) can take into account differences in the variances of residuals generalized LS (GLS) can take into account covariances of residuals as well as differences in weights. Cases of LS estimation include the following ... 

---