D- and A-optimal designs of experiments for trigonometric regression with heteroscedastic observations

Abstract

Herein for the regression function $y(x)=\theta_{1}+\displaystyle\sum_{s=1}^{k}(\theta_{2s}\cos{sx}+\theta_{2s+1} \sin{sx})$ representing a trigonometrical sum of an $k$ order, we constructed continuous $D$- and $A$-optimal designs of experiments $\varepsilon_{n}^{0}= \begin{Bmatrix} x_{1}^{0},\dots, x_{n}^{0}\\ \frac{1}{n},\dots, \frac{1}{n} \end{Bmatrix}$ with points of a spectrum $x_{i}^{0}=\frac{2\pi(i-1)}{n}+ \varphi, i=\overline{1,n}, n\geq 2k+1$, where $\varphi$ is an arbitrary angle $(\varphi\geq 0)$, for which the determinant of the information matrix of the experiment design is not equal to zero. These designs of experiments are constructed for heteroscedastic observations with variances
$\mathrm d (x)\geq \sigma^{2}, \mathrm d (x_{i}^{0})= \sigma^{2}, \sigma\neq 0,i=\overline{1,n}$. For a special case of the considered regression function $(k=1)$, we constructed the saturated designs of experiments for observations with unequal accuracy and dispersions accepting various values in the points of a spectrum of such plans.