COMPARISON OF SEARCHING PROBABILITIES FOR SEARCH DESIGNS

考虑一个有m个因子的搜索设计，并考虑两个不同的因子效应向量xi_1和xi_2。假设xi_1的作用是完全未知的。已知xi_2最多有k个效应是非零的，其余效应为零（k是一个给定的小数）。然而，目前尚不清楚哪些是非零影响。问题是搜索k个非零效应，并通过使用基于搜索设计的观察向量y来估计它们与xi_1。假设（kx1）是xi_2的非零向量。然后考虑了搜索它们的两种方法：(1)根据由“xi”_2和“zeta”估计得到的估计y“*”（zeta）计算误差平方和（SSE） S(zeta)^2。求一个向量S(zeta_1)^2是所有可能的（3xi_2）最小值。然后取_1作为x_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，其中“gamma”（zeta_2）^2变成…More是一个最大值。然后取(2)作为(2)的非零向量。在本文中，作为(2)的推广，我们提出了一个新的过程：(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，它的τ (zeta_3)^2是最大值。然后取zeta_3作为xi_2可能的非零向量。我们给出了(3)的几何性质和与统计检验的关系（H_0: zeta = 0, H_1: zeta“*”0）。也就是说，tau (zeta_3)^2的最大化等价于检验f统计量的最大化。结果表明，该过程(3)等价于(1)，即xi_1 = xi_3。在k=l的情况下，对于到目前为止已经获得的搜索设计，我们计算了值（delta^2 =1.0, 1.5…）时非零效应的搜索概率。， 6.0)的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副产品，我们在权重矩阵的分类、某些图的枚举以及在相关误差情况下分数因子设计的最优性问题上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