Chronis Moyssiadis

The exact D-optimal first order saturated design with 17 observations

Abstract The exact D-optimal first order saturated design with 17 observations is given. The upper bound of the determinant of the information matrix is established and a design attaining this value is constructed. The information matrix is proved to be unique and the optimal design contains the B.I.B. design (16, 16, 6, 6, 2).

The maximum determinant of 21×21 (+1, −1)-matrices and d-optimal designs

The 21×21 (+1, −1)-matrix R∗ with the maximum determinant is given. An algorithm is developed to find all 21×21 matrices M with determinant the square of an integer and ⩾(det R∗)2, where M=(mij), symmetric, positive-definite, mii=21, mij≡1 mod 4, i≠j. There are found, besides M∗=R∗TR∗, two more such matrices M1, M2 and then the non-existence of Ri such that R1TR1 = M1, R2TR2 = M2 is proved.

Optimal exact experimental designs with correlated errors through a simulated annealing algorithm

Abstract Simulated annealing (SA) is a stochastic optimization method with principles taken from the physical process called “annealing” which aims to bring a solid to its ground state or a state of minimum energy. SA is known as a simple heuristic tool suitable for providing direct or approximate solutions to a wide variety of combinatorial problems. This paper is concerned with the problem of determining optimal exact experimental designs with n observations and k two-level factors assuming the existence of correlated errors with a known correlation structure. A simulated annealing algorithm has been developed and applied for the search of D- and A-optimal designs. An extensive discussion regarding the right choices of the initial parameters is presented and a method of self-improvement of the algorithm is suggested via a series of repeated executions. Finally, a version of the SA algorithm is used to find optimal exact designs in the case of continuous observations with known covariance function.

Construction of optimal fractional factorial resolution V designs with N ≡ 2 mod 16 observations

Abstract It is shown that if there exists an orthogonal array of size N −2, with k constraints, 2 levels and strength 4, then there exists an N -observation 2 k fractional factorial design of resolution V which is optimal with respect to a large class of optimality criteria. These optimal designs are obtained by adding two runs to an orthogonal array of size N −2. Simple rules for choosing the two additional runs are given.

Some D-optimal odd-equireplicated designs for a covariate model

Abstract A linear model with one treatment at V levels and the first order regression on K continuous covariates with values on the K -cube is considered. The main interest is restricted to the subclass of odd-equireplicated designs, i.e. designs with equal number R ≡ 1 mod 2 of observations per treatment level. Lopes Troya (1982) has constructed families of designs d which attain the upper bound of the determinant of the information matrix M ( d ). In this paper another series of D-optimal designs is constructed for which M ( d ) has different structure from the known ones, in the cases where V R is an integer. Also, for N = VR ≡ i mod 4 observations and V > iR , new D-optimal designs are constructed for which the maximum number of covariates is V − iR , where i = 1, 2.

A-optimal incomplete block designs with unequal block sizes for comparing test treatments with a control

Abstract A-optimal designs for comparing v test treatments with a control in b blocks of unequal sizes are considered. A new class of designs is defined, namely the BTIUB designs, which can be considered as an extension of the BTIB designs of Bechhofer and Tamhane in the case of blocks with unequal sizes. Some conditions for the existence and construction of BTIUB designs are given. Finally an algorithm for the construction of A-optimal BTIUB designs is developed and it is applied in the case of two block sizes. Tables of some A-optimal BTIUB designs are given.

Alternative m-estimators of location and their linear convex combinations

Some alternative M estimators of location are studied. Their ψ-functions are defined by a single algebraic expression for all values of the independent variable and they combine good robustness and performance properties. A re-descending M-estiriiator and an M-estimator with monotone ψ are introduced and their linear convex combinations are considered. As a result a class of estimators decreasing to a positive number is obtained with 50% breakdown point. The breakdown point of this class is calculated via a generalization of a Hubers theorem. A simulation study was performed in order to examine the finite sample behavior of the proposed estimators.

A-optimization of exact first-order saturated designs for N ≡ 1 mod 4 observations

Abstract The case of exact first-order saturated designs with N ≡ 1 mod 4 observations [ N ≠ 2 s ( s + 1) + 1, s = 1,2,…] is considered. Conditions are found for the A- optimality of such designs, when the first r rows of the corresponding design matrix have a known structure. By modifying the method of Sathe and Shenoy (1989) for the case N ≡ 1 mod 4, the structure of the information matrix of the A- optimal design is found. Then, by exhaustive search, it is proved that the known D- optimal saturated design for N = 9 (Ehlich, 1964) is also A- optimal.

Exact d-optimal n observations 2k designs of resolution III, when n≏ 1 or 2 mod 4

Exact D-optimal N observations designs of resolution III are given when each of the k factors enters at two levels. The upper bound of the determinant of the information matrix is established and designs attaining this value are constructed. These designs cover almost all situations likely to occur in a practical problem. The saturated D-optimal designs are not always balanced.

Robustness of confidence intervals for scale parameters based on m-estimators

The robustness of confidence intervals for a scale parameter based on M-esimators is studied, especially in small size samples. The coverage probablity is used as measure of robustness. A theorem for a lower bound of the minimum coverage probability of M-estimators is presented and it is applied in order to examine the behavior of the standard deviation and the median absolute deviation, as interval estimators. This bound can confirm the robustness of any other scale M-estimator in interval estimation. The idea of stretching is used to formulate the family of distributions that are considered as underlying. Critical values for the confidence interval are computed where it is needed, that is for the median absolute deviation in the Normal, Uniform and Cauchy distribution and for the standard deviation in the Uniform and Cauchy distribution. Simulation results have been achieved for the estimation of the coverage probabilities and the critical values.