Project/Area Number  09640267 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  KOBE UNIVERSITY 
Principal Investigator 
SHIRAKURA Teruhiro Kobe University Faculty of Human Development Professor, 発達科学部, 教授 (30033913)

CoInvestigator(Kenkyūbuntansha) 
TAZAWA Shinsei Kinki University Faculty of Science and Technology Professor, 理工学部, 教授 (80098657)
KAKIUCHI Itsuro Kobe University Faculty of Technology Associate Professor, 工学部, 助教授 (90091248)
INABA Taichi Kobe University Faculty of Human Development Lecturer, 発達科学部, 講師 (80176403)
TAKAHASHI Tadashi Kobe University Faculty of Human Development Associate Professor, 発達科学部, 助教授 (30179494)
OHMORI Hiroyuki Ehime University Faculty of Education Professor, 教育学部, 教授 (20036370)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥2,500,000 (Direct Cost : ¥2,500,000)
Fiscal Year 1998 : ¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 1997 : ¥1,500,000 (Direct Cost : ¥1,500,000)

Keywords  Fractional factorial design / Search design / Search linear model / Unknown parameter vector / Sum of squares due to error / 一部実施要因計画 / 検索可能計画 / 検索線形モデル / 未知母数ベクトル / 残差平方和 
Research Abstract 
Consider a search design with m factors, and consider two distinct vectors xi_1 and xi_2 of factorial effects. It is assumed that the effects of xi_1 are completely unknown. It is known that at most k effects of xi_2 are nonzero and the remaining effects are zero (k is a given small number). However it is not known which are the nonzero effects. The problem is to search the k nonzero effects and estimate them along with xi_1 by using the observation vector y based on a search design. Suppose zeta(kx1) is a nonzero vector of xi_2. Then two procedures for searching them are considered : (1) Compute S(zeta)^2 the sum of squares due to error (SSE) based, on an estimate y'"*"(zeta) obtained from estimates "xi"_2 and "zeta". Find a vector zeta_1 for which S(zeta_1)^2 turns out to be a minimum for all possible zeta of 3xi_2. Then take zeta_1 as the possibly nonzero vector of xi_2. (2) Compute the estimate "gamma"(zeta)^2 of "*"zeta"*"^2. Find a vector zeta_2 for which "gamma"(zeta_2)^2 turns
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out to be a maximum. Then take zeta_2 as the possibly nonzero vector of xi_2. In this study, as a generalization of (2) we proposed a new procedure : (3) Compute the estimate tau"("zeta)^2 of tau="*"Qzeta"*"^2, where Q is a certain positive definite matrix. Find a vector zeta_3 for which tau"("zeta_3)^2 turns out to be a maximum. Then take zeta_3 as the possibly nonzero vector of xi_2. We gave a geometrical property for the (3) and a relation to a statistical testing (H_0 : zeta = 0, H_1 : zeta "*" 0). That is, the maximaization of tau"("zeta_3)^2 is equivalent to that of the test Fstatistics. As a result, it was shown that this procedure (3) is equivalent to (1), i.e., xi_1 = xi_3. In case of k=l, for search designs which had been obtained up to the present, we calculated searching probabilities for nonzero effects for values (delta^2 =1.0, 1.5 ..., 6.0) of delta^2 ="*"Qzeta_0"*"^2/siguma^2 by using a computer, where zeta_0 is a true nonzero vector of xi_2. As a byproduct of a construction of search design, we presented positive useful results on the problems of the classification of weighing matrix, enumeration of certain graphs and optimality of fractional factorial designs in case of correlated errors. Less
