Minimax adjustment technique in a parameter restricted linear model

We consider an approach yielding a minimax estimator in the linear regression model with a priori information on the parameter vector, e.g., ellipsoidal restrictions. This estimator is computed directly from the loss function and can be motivated by the general Pitman nearness criterion. It turns out that this approach coincides with the projection estimator which is obtained by projecting an initial arbitrary estimate on the subset defined by the restrictions.     

Estimation using the linear regression model with incomplete ellipsoidal restrictions

Using the linear regression model with incomplete ellipsoidal restrictions, it is shown that the known Kuks-Olman estimator is still an appropriate choice.     

Minimax estimator of regression coefficient in normal distribution under balanced loss function

This article investigates linear minimax estimators of regression coefficient in a linear model with an assumption that the underlying distribution is a normal one with a nonnegative definite covariance matrix under a balanced loss function. Some linear minimax estimators of regression coefficient in the class of all estimators are obtained. The result shows that the linear minimax estimators are unique under some conditions.     

All admissible linear estimators of a regression coefficient under a balanced loss function

Admissibility of linear estimators of a regression coefficient in linear models with and without the assumption that the underlying distribution is normal is discussed under a balanced loss function. In the non-normal case, a necessary and sufficient condition is given for linear estimators to be admissible in the space of homogeneous linear estimators. In the normal case, a sufficient condition is provided for restricted linear estimators to be admissible in the space of all estimators having finite risks under the balanced loss function. Furthermore, the sufficient condition is proved to be necessary in the normal case if additional conditions are assumed.     

Analysis of the accuracy of the linear regression model for a change in the number of parameters

An exact linear regression model and an approximate one for recovering an unknown functional dependence in the presence of random perturbations, are studied. It is shown that the accuracy of the model increases for a decrease in the number of its parameters.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 266–271, February, 1990.     

Best estimator of the regression parameter in a reversing model

A mathematical model of data processing for frequency sensors is proposed. Minimum-variance linear unbiased estimator of the constant signal is determined in the presence of white noise and unknown nonstochastic drift by reversal of observations. The reversal time is taken into account. It is shown that if the drift is from a finite-dimensional space, the estimation process reduces to solving a system of linear equations. An example is considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 94–99, 1986.     

Estimators of scale parameters in linear regression

This note discusses the asymptotic distribution of two scale and location invariant estimators of two scale parameters in the multiple linear regression model. Both of these estimators need an initial estimator of the regression parameter vector. The asymptotic distribution of one of these estimators does not depend on this initial estimator. Both of these estimators are useful in the computation of scale and translation invariant adaptive estimators and M-estimators of the regression parameter vector.     

Theory of estimation of parameters in a linear regression scheme

Fairly simple asymptotically optimal equivariant polynomial estimators of any degree k are constructed for the parameters of a standard linear regression scheme whose design matrix satisfies a certain additional condition. These estimators depend on the error distribution function only in terms of its first 2K moments. An explicit equation for an optimal equivariant quadratic estimator of parameters is also presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 53, pp. 118–129, 1975.The author express his gratitude to A. M. Kagan for posing the problem and for his interest.     

Bootstrap approximation of wavelet estimates in a semiparametric regression model

The inference for the parameters in a semiparametric regression model is studied by using the wavelet and the bootstrap methods. The bootstrap statistics are constructed by using Efron＇s resampling technique, and the strong uniform convergence of the bootstrap approximation is proved. Our results can be used to construct the large sample confidence intervals for the parameters of interest. A simulation study is conducted to evaluate the finite-sample performance of the bootstrap method and to compare it with the normal approximation-based method.     

Strong convergence of LAD estimates in a censored regression model

A regression model with a nonnegativity constraint on the dependent variable, known as censored median regression model, is considered. Under some mild conditions, the LAD estimate of the regression coefficient is shown to be strongly consistent. Furthermore, its convergence rate and Bahadur strong representation are also obtained.     

Central limit theorem of linear regression model under right censorship

In this paper,the estimation of joint dlstribution F(y,z)of(Y,Z)and the estimation in thelinear regression model Y=b'Z+εfor complete data are extended to that of the right censored data.Theregression parameter estimates of b and the variance of ε are weighted least square estimates with randomweights. The central limit theorems of the estimators are obtained under very weak conditions and the derivedasymptotic variance has a very simple form.     

Bayesian inferences on nonlinear functions of the parameters in linear regression

A variety of statistical problems (e.g. the x-intercept in linear regression, the abscissa of the point of intersection of two simple linear regression lines or the point of extremum in quadratic regression) can be viewed as questions of inference on nonlinear functions of the parameters in the general linear regression model. In this paper inferences on the threshold temperatures and summation constants in crop development will be made. A Bayesian approach for the general formulation of this problem will be developed. By using numerical integration, credibility intervals for individual functions as well as for linear combinations of the functions of the parameters can be obtained. The implementation of an odds ratio procedure is facilitated by placing a proper prior on the ratio of the relevant parameters.Financially supported by the University of the Orange Free State Research Fund.     

Estimation of the nuisance parameter for a seminartingale regression model

A nuisance parameter is introduced to the semimartingale regression model proposed by Aalen (1980), and we construct two estimators for this nuisance parameter based on the results of parametric estimation which were given by Mckeague (1986) using the method of sieves. The consistency of the estimators is also provided.