A minimax linear estimator for linear parameters under restrictions in form of inequalities

Linear Regression

“…2.5 In admissibility of t he Ordinary Least Squa res Estim at or 69 The use of alternatives to the or dina ry least squares est imator can be beneficial when the least squares est imator is unreliable or can uniformly be outperformed.…”

“…According to t he above theorem, we can obtain a linear minim ax solution by minimi zing th e expression with respect to th e p x n matrix A. Unfort unately, the explicit solut ion is rather complicated in general, see also [69]. As shown by the following theorem, we can determine the maximal weighted squa red error risk of an ar bitrary Ay E £ (f3 ) wit h respect to all param et er vectors f3 E 0~.…”

“…If, in addition, we compute the corresponding lower quas i For the special case W = T = I p , this solut ion is not only minimax wit hin t he set 5 ({3 ) but also wit hin t he set £ ({3), see [69]. In Example 3.23 we have seen th at for 30 generated realizatio ns y in t he linear regression model from Example 3.22, th e average observed loss of 1.1765 of t he ordina ry least squa res esti mator is distinct ly greater th~n the average observed loss 0.3783 of the upp er quasi minimax esti mator {3~2T" .…”

Partial Minimax Estimation in Regression Analysis

“…matrix, & is the known center of the ellipsoid, and the positive scalar c is also known. This prior information may be combined with the data set (y,X) of the linear regression model to obtain a minimax estimator of the parameter vector which minimizes the maximal risk (KUKS AND OLMAN, 1972;BUNKE, 1975 a,b,c;LAUTER, 1975; HOFFMANN, 1979;TOUTENBURG, 1982;TRENKLER and STAHLECKER, 1984).…”

К Теории Минимаксно-Байесовского Оценивания

“…Интересно, что в этом случае апостериорная точность Ер|0* -в\ 2 будет одной и той же для всех распределений Р £ 2? 0 с одинаковой величи ной априорной точности Ер|0 -#о| 2 И что в этом случае оценка (3.9) оптимальна в совсем иной задаче [13]. …”