Sankhya: The Indian Journal of Statistics

2002, Volume 64, Series B, Pt. 3, 322--338

Some Criterion-Robust Optimal Designs For The Dual Problem Of Model Discrimination And Parameter Estimation

By

MEI-MEI ZEN, and MIN-HSIAO TSAI

National Cheng-Kung University, Tainan 70101, Taiwan, R.O.C.

SUMMARY. Consider the problem of discriminating between the polynomial regression models of degree $k-1$ and $k$ on $[-1,1]$. For the dual problem, objective (1), to discriminate the degree, is one which we must pay attention to, and either objective (2) or (3), to make inferences, is the other one. For the multiple-objective consideration, Pukelsheim and Rosenberger (1993) presented several designs as $k=3$. Tsai and Zen (2000a) proposed a multiple-objective selection criterion (namely, $M$-criterion) which puts equal weights on all three objectives for any $k$. In this work, we take a more general selection criterion, called $M_{\gamma}$-criterion, which puts weight $\gamma\,(0\le\gamma\le{1})$ for objective (1) and $(1-\gamma)/2$ for both objectives (2) and (3), and the corresponding $M_{\gamma}$-optimal design is derived in terms of canonical moments. The behavior of the $M_{\gamma}$-optimal designs is investigated under different weighted selection criterion. The extreme value of the minimum $M_{\gamma}$-efficiency of any $M_{\gamma'}$-optimal design is obtained at $\gamma'=\gamma^{*}$, which results in the $M_{\gamma^{*}}$-optimal design as a criterion-robust optimal design for the problem.

AMS (1991) subject classification}. 62K05.

Key words and phrases. Multiple-objective, selection criterion, $M_{\gamma}$-optimal design, canonical moments, efficiency.

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