Replicationfree optimal designs in regression analysis
Replicationfree optimal designs in regression analysis
复制标题
回归分析中的无复制优化设计
DOI：
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
E. Boer
中科院分区：
文献类型：
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作者：
D. Rasch;E. Hendrix;E. Boer
让
$$matrix{ {{y_i} = fleft( {{x_i}, heta }
ight) + {e_i},} & {i = 1,...,n,} & {{x_i} in B subset {{
m{R}}^1}} cr } $$
是具有回归函数f和误差项I的回归模型。未知参数8可以有p≤n个分量，即θ T = (θ1，…θ p)∈Ω∧R p。我们假设渐近最小二乘理论的通常条件已经满足（见Rasch， 1995，第16章）。我们用^θ表示θ的最小二乘估计，用V表示^θ的（渐近）协方差矩阵，它可能依赖于也可能不依赖于θ。设Φ为任意V随n单调递减的泛函，作为x i∈B （i=l，…，n）的最优性准则；我选择的x被称为精确设计。如果定义设计的x i集合在大小为n的B中所有可能的设计中使函数Φ最小，则我们称大小为n的B中的设计（局部或全局）Φ-optimal。如果设计依赖于θ，则该设计为局部最优，否则该设计为全局最优。大小为n且有r个支撑点的设计称为大小为n的精确r点设计，可表示为
$$left( {matrix{ {{x_1}} hfill & {{x_2}} hfill & ldots hfill & {{x_r}} hfill cr {{n_1}} hfill & {{n_2}} hfill & ldots hfill & {{n_r}} hfill cr } }
ight),,sumlimits_{i = 1}^r {{n_i} = n,,{n_i},{
m{integer}}{
m{.}}} $$
(1）
更多信息请参见Pukelsheim（1994）。
Let
$$matrix{ {{y_i} = fleft( {{x_i}, heta }
ight) + {e_i},} & {i = 1,...,n,} & {{x_i} in B subset {{
m{R}}^1}} cr } $$
be a regression model with a regression function f and i.i.d. error terms e i . The unknown parameter 8 may possess p ≤n components i.e. θ T = (θ1,…θ p ) ∈ Ω ⊂R p . We assume that the usual condition for the asymptotic least squares theory are fulfilled (see Rasch, 1995, chapter 16). By ^θ we denote the least squares estimator of θ and by V the (asymptotic) covariance matrix of ^θ which may or may not be dependent on θ. Let Φ be any functional of V monotonically decreasing with n which is used as an optimality criterion for an optimal choice of the x i ∈B (i=l,…,n); the x i chosen are called an exact design. We call a design (locally or globally) Φ-optimal in B of size n if the set of the x i defining the design is minimizing the functional Φ amongst all possible designs in B of size n. The design is locally optimal, if it depends on θ, otherwise the design is globally optimal. A design of size n with r support points is called an exact r-point design of size n and can be written as
$$left( {matrix{ {{x_1}} hfill & {{x_2}} hfill & ldots hfill & {{x_r}} hfill cr {{n_1}} hfill & {{n_2}} hfill & ldots hfill & {{n_r}} hfill cr } }
ight),,sumlimits_{i = 1}^r {{n_i} = n,,{n_i},{
m{integer}}{
m{.}}} $$
(1)
See for more information Pukelsheim (1994).