Squares Method For Estimation Research Articles

Test for the choice of approximative models in nonlinear regression when the variance is unknown

Nonlinear regression models are used in many fields. Often the regression for observation-covariate pairs (X(ti), ti) is modeled as X(ti) =f(ti) + [d](fi), i = 1,. .. , n, where f is the continuous possible nonlinear mean function, while the ([d](ti))i=1.....nare zero mean, i.i.d. random errors having finite variance [d]2. The least squares methods for estimation of f are usually based upon a given parametric form of f. In this article we develop two statistical tests, one for testing that f belongs to a given class of functions possibly discontinuous in their first derivative, and another one for comparing two such classes. This is done by introducing an appropriate estimate of the unknown variance [d]2. The numerical results of a simulation study seem satisfactory.

A new estimator combining the ridge regression and the restricted least squares methods of estimation

It is well-known in the literature on multicollinearity that one of the major consequences of multicollinearity on the ordinary least squares estimator is that the estimator produces large sampling variances, which in turn might inappropriately lead to exclusion of otherwise significant coefficients from the model. To circumvent this problem, two accepted estimation procedures which are often suggested are the restricted least squares method and the ridge regression method. While the former leads to a reduction in the sampling variance of the estimator, the later ensures a smaller mean square error value for the estimator. In this paper we have proposed a new estimator which is based on a criterion that combines the ideas underlying these two estimators. The standard properties of this new estimator have been studied in the paper. It has also been shown that this estimator is superior to both the restricted least squares as well as the ordinary ridge regression estimators by the criterion of mean sauare e...

Analysis of survival data: Estimation and specification tests using asymptotic least squares

In this paper we propose an asymptotic least squares method of estimation and test for proportional hazards and accelerated life models with qualitative covariates. We use Nelson-Aalen estimators of the cumulative hazards on the left-hand side of a regression equation whose coefficients are the covariates effects. In this case, asymptotic least squares reduce to generalized least squares on the corresponding regression equation, giving consistent estimates and specification tests. Examples show that asymptotic least squares and other traditional estimators give very close results.

A least squares method for estimation of Bezier curves and surfaces and its applicability to multivariate analysis

A new least squares estimation method for Bezier polynomial curves and surfaces is described and illustrated. The Cartesian coordinates of the vertices of polygons defining the curves separated well natural populations of Anodonta cygnea L. and human sagittal profiles of different sexes and age classes. Multivariate comparisons of the coordinates of Bezier polygon vertices and Euclidean distance measures showed that the polygon coordinates revealed shape differences better than distances between homologous points on the curves. Further, polygon coordinates separated groups for more than two variables as well as or better than the equivalent number distances. In addition, polygon coordinates permit construction of mean shapes and their variances. Possible applications in trend surface analyses and for illustration in computer-aided identification programs are suggested.

MISUNDERSTANDING THE RASCH MODEL

Whitely and Dawis (1974) recently discussed, in this journal, the nature of objectivity with the Rasch model. They made several good points. However, their paper contained some unfortunate misunderstandings and errors. One important misunderstanding concerns the estimation procedure necessary to calibrate items according to the Rasch model. The authors recommend a two-stage estimation procedure which is actually unnecessary and impractical. As a result they conclude that only huge sample sizes make it possible to apply the Rasch model to real data. Since the Rasch model can be and has been applied productively to sets of data as small as 100 persons, and under those circumstances lead to useful results, it is important to correct that misunderstanding and the incorrect conclusion drawn from it. Whitely and Dawis recommend beginning parameter estimation with a least squares method based on item by score-group frequencies. This crude method was first used by Rasch in the 1950's and presented in his 1960 book. It is neither the most convenient nor the most efficient method of estimation and it is unnecessary as a beginning. Far preferable is to begin directly with the unconditional maximum likelihood procedure described by Wright and Panchapakesan (1969) and Wright and Douglas (1975b). Most researchers applying the Rasch model today are using some version of the computer program for unconditional maximum likelihood estimation written by Wright and Panchapakesan in 1965. The danger to them of being misled by the recommendation of Whitely and Dawis is slight.* However, newcomers to the topic who take the Whitely-Dawis recommendation seriously will be led astray. Perhaps because they focused their attention on the least squares method of estimation, Whitely and Dawis conclude that huge sample sizes are necessary for useful item calibration using the Rasch model. This conclusion is based on their incorrect statement that each and every possible score group has to be inhabited by a substantial number of persons for the estimation procedure to proceed. This is untrue. In fact, a set of items may be calibrated even when all the persons in the calibration sample have earned one and the same score. Even the least squares method allows estimation of the relative difficulties of a set of items from merely the log-odds success that any single score group obtains on each item. No additional score groups are necessary for a set of unbiased least squares estimates. The same, of course, is true for the unconditional maximum likelihood procedure. It is only when the researcher wishes, and wisely so, to test the fit of his data to the Rasch model, that more than one score group is needed. In order to evaluate fit, at least two score groups must be available so that the separate estimates of each item difficulty within each score group can be compared with one another to see if they are

Computation of Zellner-Theil's Three Stage Least Squares Estimates

In their paper on the three stage least squares method of estimation of a simultaneous equation system, Zellner and Theil [5] make the interesting observation that the large sample efficiency of the estimates of the parameters in the group of over-identified equations is unaffected if the three stage method is applied to this subsystem alone, ignoring all the exactly identified equations. It is shown in this paper that the estimates themselvesnot just their large sample efficiency-are unaffected if one follows the above mentioned simplified procedure. This result is established for the general case of a simultaneous equation system subject to linear homogeneous a priori restrictions, whereas many other known properties of the three stage least squares method have been demonstrated only for the special case of simple restrictions on the structural coefficients. An error in the expression giving the estimates for the exactly identified equations in Zellner-Theil's paper is also corrected. The results of this paper have obvious significance for the problem of developing an efficient computer program for finding the three stage least squares estimates. ZELLNER AND THEIL [5] have proposed a three stage least squares method of estimation of the parameters of a simultaneous equation system as an alternative to the full information maximum likelihood method. The three stage least squares method has been shown to possess a number of attractive properties. Rothenberg and Leenders [2] have shown that when the variance-covariance matrix Z of the disturbance terms is unrestricted, the estimates have the same asymptotic variancecovariance matrix as the full information maximum likelihood method. Under the same condition of unrestricted Z matrix, Sargan [3] has shown that the difference between the estimates by the three stage least squares method and the estimates by the full information maximum likelihood method is of stochastic order 1/T, where T is the sample size. Though the three stage least squares method is much simpler computationally than the full information maximum likelihood method, it still involves the inversion of a moment matrix of a very large order and the difficulties of computing the inverse matrix accurately, especially when it may be near singular due to the presence of intercorrelation among the explanatory

The Elasticity of U.S. Import Demand: A Theoretical and Empirical Reappraisal

Much useful research, both empirical and theoretical, has recently been undertaken on the elasticity of demand for imports. While so far the results have been mainly negative, considerable insight into this subject has been gained.' All of the statistically derived demand functions for imports yield elasticity estimates that are exceedingly low, much lower than "informed opinion" would expect them to be. These surprisingly low estimates have stimulated many students to a rigorous examination, in relation to the least squares method of estimation, of the nature of the problem and of the data available for its solution. The main conclusion has been either that an estimate by the least squares method is impossible or that the results obtained are gross underestimates. This paper endeavors to show, however, that, at least for certain cases, such as imports of finished manufactures into the United States, meaningful estimates can be derived by the least squares method.