R -optimal designs for trigonometric regression models
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R-optimal designs for trigonometric regression models Lei He1 · Rong-Xian Yue1,2 Received: 22 December 2017 / Revised: 31 May 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract This paper is concerned with the problem of constructing R-optimal designs for trigonometric regression models with different orders. More precisely, explicit Roptimal designs for the first-order trigonometric regression model on a partial cycle are derived by using the idea of complete class approach. The relative R-efficiency of the equidistant sampling method is then discussed. Moreover, when the explanatory variable varies in a complete cycle, the R-optimal designs for estimating the specific pairs of the coefficients in the trigonometric regression of larger order are obtained by invoking the equivalence theorem. Several examples are presented for illustration. Keywords R-optimal designs · Equivalence theorem · Complete class · Trigonometric regression models Mathematics Subject Classification 62K05
1 Introduction Fourier or trigonometric regression is widely used in applications to characterize cyclic phenomena that arise in the engineering (McCool 1979), medical sciences (Kitsos et al. 1988) and biological systems (Lestrel 1997). It is well-known that, with a careful choice of design, the efficiency of the statistical analysis can be improved substantially, and a series of studies to date have been developed on constructing optimal designs for Fourier regression models. The existing literature on this direction are mainly concerned with Kiefer’s Φq criteria (see Pukelsheim 1993, Chap. 9), in which the E-, A-, and D-optimality criteria correspond to q = −∞, −1 and 0, respectively. Karlin and Studden (1966) obtained
This work was supported by NSFC Grant 11471216
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Rong-Xian Yue [email protected]
1
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2
Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
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L. He, R.-X. Yue
the D-optimal designs for estimating all the coefficients corresponding to the cosinus functions on a semi-circle. Hill (1978) derived the D-optimal designs for the first-order trigonometric regression model on the design space [−π/2, π/2]. Wu (2002) provided a thorough treatment of the first-order trigonometric regression model on a partial cycle to construct some optimal exact designs and Φq -optimal designs (including E-, A- and D-optimality). Dette et al. (2009) discussed the problem of finding L-optimal designs for estimating certain pairs of the coefficients in the trigonometric regression models on a complete cycle, and subsequently, Dette et al. (2011) made further developments for estimating more general classes of linear combinations of the coefficients. When the objective is to predict response, Xu and Shang (2014) studied the classical Q-optimal and minimax robust designs for a Fourier regression model with a given order m, where the Q-optimal designs can be constructed by minimizing the average variance of the pr
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