The use of relative residues in non-linear regression

A non-linear curve-fitting model is presented which minimizes the sum of squares of relative residues, and expressions are derived for the fit parameters and their respective errors. A detailed comparison is made between the new general relative least squares model (GRLS) and other non-linear regression models available in the literature, using two sets of data representing fluid mechanics problems encountered in many engineering applications. The results showed that GRLS was the best model for fitting non-linear functions in the case of experimental data spanning several orders of magnitude, indicating its potential as a tool for data analysis.     

The use of relative residues in fitting experimental data: an example from fluid mechanics

A curve fitting model is presented which minimizes the sum of squares of relative residues and expressions for the fit coefficients and their respective errors are derived. The new model is compared to the normal least squares model, using as an example the Reynolds number-drag coefficient data for a sphere. The results show that the best fit was obtained with the new model, indicating it may provide a useful tool for data analysis.     

Asymptotics of R-, MD- and LAD-estimators in linear regression models with long range dependent errors

Summary This paper establishes the uniform closeness of a weighted residual empirical process to its natural estimate in the linear regression setting when the errors are Gaussian, or a function of Gaussian random variables, that are strictly stationary and long range dependent. This result is used to yield the asymptotic uniform linearity of a class of rank statistics in linear regression models with long range dependent errors. The latter result, in turn, yields the asymptotic distribution of the Jaeckel (1972) rank estimators. The paper also studies the least absolute deviation and a class of certain minimum distance estimators of regression parameters and the kernel type density estimators of the marginal error density when the errors are long range dependent.Research of this author was partly supported by the NSF grant: DMS-9102041     

The blue and minque in Gauss-Markoff model with linear transformation of the observable variables

For a singular linear model A = (y, Xβ, σ2 V) and its transformed model AF = (Fy, FXβ, σ2FVF'), where V is nonnegative definite and X can be rank-deficient,the expressions for the differences of the estimates for the vector of FXβ and the variance factor σ2 are given. Moreover, the necessary and sufficient conditions for the equalities of the estimates for the vector of FXβ and the variance factor σ2 are also established. In the meantime, works in Baksalary and Kala (1981) are strengthened and consequences in Puntanen and Nurhonen (1992), and Puntanen (1996) are extended.     

The rate of convergence for the least squares estimator in nonlinear regression model with dependent errors

We study the parameter estimation in a nonlinear regression model with a general error's structure,strong consistency and strong consistency rate of the least squares estimator are obtained.     

The rate of convergence of the least squares estimator in a non-linear regression model with dependent errors

The rate of convergence of the least squares estimator in a non-linear regression model with errors forming either a φ-mixing or strong mixing process is obtained. Strong consistency of the least squares estimator is obtained as a corollary.