A robustness type optimality criterion for experimental design

This dissertation is made up of three different parts. The principal object of part I is to demonstrate the overriding influence which the design matrix of a regression model has on the sensitivity of statistical tests to the assumption of non-normality. This sensitivity is studied through the dependence of the tests' critical values on the regression variables, when the data originate from a non-normal parent. It is proved that the robustness of the tests' critical values to the assumption of non-normality can be improved through an adequate choice of the design matrix. Furthermore, a proof is given for the existence of a design matrix that is optimal in making the size of the test statistics closest to that advertised under normal theory. A general search procedure for such optimal designs is given. This procedure is demonstrated in two numerical examples in which a U-shaped Pearson parent distribution is assumed.

In Part II, a study is carried out concerning the distribution of quadratic forms in non-normal variables. A general formula is proved for the characteristic function of a quadratic form which is not positive (or negative) semi-definite, and in which the random vector is not normally distributed. This formula is of value for finding the distribution of the ratio of two quadratic forms in non-normal variables.

The theory is used for finding numerically the distribution of a ratio of quadratic forms in a three-dimensional random vector having a stable distribution. The result is compared with a more conventional approximate approach based on assuming normality, and making use of the Cornish-Fisher expansion.

The purpose of Part III is to introduce a new stopping rule procedure for the method of steepest ascent used in response surface experiments. The new procedure is designed to provide an increase in the efficiency of the existing stopping rules. It is based on a sequential hypothesis testing controlled by two stopping bounds. These bounds are determined so that the efficiency of the procedure is maximized.