Shape optimal design criterion in linear models

Abstract

Abstract. Within the framework of classical linear regression model optimal design criteria of stochastic nature are considered. The particular attention is paid to the shape criterion. Also its limit behaviour is established which generalizes that of the distance stochastic optimality criterion. Examples of the limit maximin criterion are considered and optimal designs for the line fit model are found.

Integral stochastic optimal design criteria in linear models

“…In view of Theorem 1 from Zaigraev (2002) it can be shown, similarly as it is done in Theorem 2 here, that the ISS1 r ðeÞ-criterion is equivalent to the D-criterion as e ! 0.…”

“…It is worth noting that in the case k ¼ 2 it is possible to find necessary and su‰cient conditions characterizing design domination relative to the DScriterion (see Lemma 1 in Zaigraev 2002). Namely, a design x 1 is at least as good as x 2 with respect to the DS-criterion for the LSE of b if and only if ðlog l 1 ; log l 2 Þ 0 w ðlog m 1 ; log m 2 Þ;…”

“…Investigation of the stochastic criteria through generalizing the DS-criterion has been started in Liski and Zaigraev (2001) and continued in Zaigraev (2002). In the first paper the stochastic convex (SC) criterion is introduced.…”

“…Another criterion, namely the shape stochastic (SS) criterion, is investigated in Zaigraev (2002). Let the class A be of the form…”

“…is given in Zaigraev (2002). The function p à ðdÞ increases monotonically from p 0 ¼ 0:5 (D-optimal design) to p y ¼ 0:6 (E-optimal design).…”