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We introduce a new class of`standardized' optimality criteria which depend on`standardized' covariances of the least squares estimators and provide an alternative to the commonly used criteria in design theory. Besides a nice statistical interpretation the new criteria satisfy an extremely useful invariance property which allows an easy calculation of optimal designs on many linearly transformed design spaces.

July 28, 2010 AbstractIn this paper we investigate the problem of measuring deviations from stationarity in locally stationary time series. Our approach is based on a direct estimate of the L 2 -distance between the spectral density of the locally stationary process and its best approximation by a spectral density of a stationary process. An explicit expression of the minimal distance is derived, which depends only on integrals of the spectral density of the stationary process and its square. These integrals can be estimated directly without estimating the spectral density, and as a consequence, the estimation of the measure of stationarity does not require the specification of smoothing parameters. We show weak convergence of an appropriately standardized version of the statistic to a standard normal distribution. The results are used to construct confidence intervals for the measure of stationarity and to develop a new test for the hypothesis of stationarity which does not require regularization. Finally, we investigate the finite sample properties of the resulting confidence intervals and tests by means of a small simulation study and illustrate the methodology in three data examples.

The exact mean-squared error (MSE) of estimators of the variance in nonparametric regression based on quadratic forms is investigated. In particular, two classes of estimators are compared: Hall, Kay and Titterington's optimal difference-based estimators and a class of ordinary difference-based estimators which generalize methods proposed by Rice and Gasser, Sroka and Jennen-Steinmetz. For small sample sizes the MSE of the ®rst estimator is essentially increased by the magnitude of the integrated ®rst two squared derivatives of the regression function. It is shown that in many situations ordinary difference-based estimators are more appropriate for estimating the variance, because they control the bias much better and hence have a much better overall performance. It is also demonstrated that Rice's estimator does not always behave well. Data-driven guidelines are given to select the estimator with the smallest MSE.