A Note on the Modified Chebyshev Design in the Polynomial Regressions

In the polynomial regression of a given order m, the unique G-optimal design is well known. However, it crucially depends on the order m ; consequently, it can be used only when m is given a priori. From the modern perspective, m is hardly ever exactly known. In this aspect, the Chebyshev design which does not require the knowledge of m, is very efficient, because it is almost optimal in the interior of the interval [-1, 1]. On the other hand, its risk is almost twice bigger than that of the optimal design near the end-points. In this paper we introduce the modified Chebyshev design which improves the Chebyshev design by giving some weight to the end-points. The numerical results show that it improves the G-efficiency of the Chebyshev design roughly by 65%.