二元反應實驗(binary response experiments)是一種被廣泛運用在各種領域裡面的實驗方法。很多論文都討論過各種不同模型下的最適設計，也有很多論文研究該使用何種設計來區分模型。這篇文章的主要目的是當實驗者有兩個來自對稱尺度族(symmetric location and scale families)的可能模型時，應當使用何種設計使得使用錯誤模型所造成的最大機率誤差達到最小。在這篇文章中我們主要探討這樣的兩點設計，稱為最小偏誤兩點設計(minimum bias two-points design)，或簡稱為mB2設計。我們將會探討以及比較mB2設計和D-最適設計(D-optimal design)、A-最適設計(A-optimal design)在正確模型下的D-效率(D-efficiency)以及A-效率(A-efficiency)，還有在錯誤模型下的偏誤(biases)和平方均誤(mean square errors)。

The binary response experiments are often used in many areas. In many investigations, different kinds of optimal designs are discussed under an assumed model. There are also some discussions on optimal designs for discriminating models. The main goal in this work is to find an optimal design with two support points which minimizes the maximal probability differences between possible models from two types of symmetric location and scale families. It is called the minimum bias two-points design, or the $mB_2$ design in short here. D- and A-efficiencies of the $mB_2$ design obtained here are evaluated under an assumed model. Furthermore, when the assumed model is incorrect, the biases and the mean square errors in evaluating the true probabilities are computed and compared with that by using the D- and A-optimal designs for the incorrectly assumed model.

1 Introduction 1
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Optimization criterion . . . . . . . . . . . . . . . . . . . . 3
2 The min-max results for two models 9
2.1 The probit and logit case . . . . . . . . . . . . . . . . . . . 9
2.2 General cases . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Efficiencies and biases comparisons 20
3.1 The probit and logit case . . . . . . . . . . . . . . . . . . . 21
3.2 The probit and double reciprocal case . . . . . . . . . . . . 23
4 Discussions and conclusions 24
Appendix 28
A The convergence of MLEs for two-points designs with a misspecified link model 28
B Properties of the scale function and the distance function 30
C Figures of difference between two models 37
D Tables for probit being the true model with logit link function 39
E Tables for double-reciprocal being the true model with probit link 40
F Some further works about $mB_3$ design for the probit and double reciprocal case 41

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