4 Appendix S-1 4.1 Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S-1 4.2 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S-4 4.3 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S-8 4.4 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

two multivariate calibration methods are compared for the simultaneous chromatographic determination and separation of sulfamethoxazole (smx) and phthalazine (phz) by high performance liquid chromatography (hplc). multivariate calibration techniques such as classical least squares (cls) and inverse least squares (ils) were introduced into hplc to determine the quantification by using uv detecto...

Classical least squares regression consists of minimizing the sum of the squared residuals. Many authors have produced more robust versions of this estimator by replacing the square by something else, such as the absolute value. In this article a different approach is introduced in which the sum is replaced by the median of the squared residuals. The resulting estimator can resist the effect of...

A linear model based on the spherical coordinate system is employed for tting a spherical surface to a data set obtained by a coordinate measuring machine (CMM). In practice, measurement data may have unequal accuracy on the diierent coordinate axes, thus a liner model with heteroscedastic variance of random errors is considered. A feasible weighted least squares estimate is proposed and its su...

Failure to allow for autocorrelation of the disturbances in a regression model can lead to biased and inconsistent parameter estimates, particularly if the model is autoregressive. While consistent estimation methods are available which allow for autocorrelation, estimation is usually much easier when there is some assurance that autocorrelation is absent. In pursuit of such assurance the prese...

Lemma A.3 Under Assumption 4 (iii), we have (i) P i P 2 ii = o(K), P i 6=j PiiPjj = K 2 + o(K), P i 6=j PijPij = P i 6=j PijPji = K + o(K); (ii) P iMiiPii = o(K), P i 6=jMiiPjj = Ktr(M) + o(K) = O(K), P i 6=jMijPij = P i 6=jMijPji = tr(M) + o(K) = O(K); (iii) P iM 2 ii = O(K), P i 6=jMiiMjj = tr (M) P iM 2 ii = O(K ), P i 6=jMijMij = tr(MM 0) P iM 2 ii = O(K), P i 6=jMijMji = tr(M) P iM 2 ii = ...

Suppose we observe an ergodic Markov chain on the real line, with a parametric model for the autoregression function, i.e. the conditional mean of the transition distribution. If one speciies, in addition, a paramet-ric model for the conditional variance, one can deene a simple estimator for the parameter, the maximum quasi-likelihood estimator. It is robust against misspeciication of the condi...

This work gives a simultaneous analysis of both the ordinary least squares estimator and the ridge regression estimator in the random design setting under mild assumptions on the covariate/response distributions. In particular, the analysis provides sharp results on the “out-of-sample” prediction error, as opposed to the “in-sample” (fixed design) error. The analysis also reveals the effect of ...

We study the asymptotic behavior of the least squares estimators of the unknown parameters of general pth-order bifurcating autoregressive processes. Under very weak assumptions on the driven noise of the process, namely conditional pair-wise independence and suitable moment conditions, we establish the almost sure convergence of our estimators together with the quadratic strong law and the cen...

This is a more serious departure from the assumptions of the classical linear model than was the case for the Generalized Regression model, which maintained E("jX) = 0 but permitted nonconstant variances and/or nonzero correlations across error terms. Unlike the Generalized Regression model, the classical least squares estimator will be inconsistent for if the errors are correlated with the reg...