COMPARISON OF SEARCHING PROBABILITIES FOR SEARCH DESIGNS

考虑具有m个因子的搜索设计，并考虑因子效应的两个不同向量xi_1和xi_2。假设xi_1的影响完全未知。已知xi_2的最多k个效应不为零，其余效应为零（k是给定的小数字）。然而，我们并不知道哪些是非零效应。问题是搜索k个非零效应，并通过使用基于搜索设计的观测向量y来沿着xi_1估计它们。设zeta（kx 1）是xi_2的非零向量。然后考虑两种搜索它们的方法：（1）根据从估计值“Xi”_2和“zeta”得到的估计值y“*"（zeta）计算S（zeta）^2，即误差平方和（SSE）。求一个矢量zeta_1，使得S（zeta_1）^2对于所有可能的zeta为3xi_2的情况都是最小的。然后取zeta_1作为xi_2的可能非零向量。(2)计算“*“zeta”*"^2的估计值“gamma”（zeta）^2。找到一个向量zeta_2，“gamma”（zeta_2）^2为该向量旋转 ...更多信息 这是一个最大的。然后取zeta_2作为xi_2的可能非零向量。在本研究中，作为（2）的推广，我们提出了一个新的方法：（3）计算tau="*“Qzeta”*"^2的估计tau”（“zeta）^2，其中Q是某个正定矩阵。找到一个向量zeta_3，其中τ”（“zeta_3）^2是最大值。然后取zeta_3作为xi_2的可能非零向量。我们给出了（3）的几何性质和与统计检验（H_0：zeta = 0，H_1：zeta“*”0）的关系。也就是说，τ”（“zeta_3）^2的极大化等价于检验F-统计量的极大化。结果表明，该过程（3）等价于（1），即，xi_1 = xi_3。在k=l的情况下，对于到目前为止获得的搜索设计，我们计算了值（delta^2 =1.0，1.5...，6.0)用计算机求出delta^2 ="*“Qzeta_0”*"^2/siguma^2，其中zeta_0是xi_2的真非零向量。作为搜索设计构造的一个副产品，我们对加权矩阵的分类、某些图的计数和相关误差情况下部分因子设计的最优性等问题给出了积极有用的结果。少

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 … More 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 F-statistics. 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 by-product 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