Optimal designs for quadratic regression with random block effects: The case of block size two

A new known modification for simulation of annealing to search the global extremum of the functions of many variables uses the fact that the function whenn→ ¥ converges to the δ-function concentrated at the point of global maximum off(x). The case when the function has many equal extrema is discussed in detail. Problems of this type are often present, particularly in the design of regression experiments. Here we introduce the reader to an extremum search method that is effective in solving a wide range of applied problems, and also illustrate the use of the method in some of the simplest problems of designing the regression experiments. The proposed modification of simulated annealing uses quasi-random search at the intermediate stages. This is not the most rapid, but very reliable method which provide a complete exploring of the function domain. When solving numerical examples, the so-called exactD-optimal designs are constructed, which are very difficult to be obtained by other methods. Although with the increase in the number of variables, the complexity of the method (as well as the complexity of other well-known methods) increases dramatically due to an increase in the order of the determinant, the proposed algorithm is simple, reliable, and easily parallelized. It is known that the gain from using optimal designs in some cases can justify any computational costs of developing those designs. Using the described technique, the reader will be able to construct (even using the laptop capacity) the optimal designs in different areas at moderate values of the parameters (for example, for quadratic regression for s variables in variables fors≤ 10).

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Some optimal designs of block size two

Journal of Statistical Planning and Inference ◽

10.1016/0378-3758(93)90093-l ◽

1993 ◽

Vol 37 (2) ◽

pp. 245-253 ◽

Cited By ~ 5

Author(s):

Sunanda Bagchi ◽

Ching-Shui Cheng

Keyword(s):

Block Size ◽

Optimal Designs

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D-Optimal Designs for Paired Observations in Quadratic Regression with Mixed Effects

Communications in Statistics - Simulation and Computation ◽

10.1080/03610918.2012.625826 ◽

2012 ◽

Vol 41 (7) ◽

pp. 1107-1119

Author(s):

Tobias Mielke

Keyword(s):

Mixed Effects ◽

Optimal Designs ◽

Quadratic Regression

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Largest eigenvalue of large random block matrices: A combinatorial approach

Random Matrices Theory and Application ◽

10.1142/s2010326317500083 ◽

2017 ◽

Vol 06 (02) ◽

pp. 1750008

Author(s):

Debapratim Banerjee ◽

Arup Bose

Keyword(s):

Block Structure ◽

Extreme Points ◽

Block Size ◽

Relative Growth ◽

Largest Eigenvalue ◽

Block Matrices ◽

Variance Formula ◽

The Matrix ◽

Random Block ◽

Symmetric Block

We study the largest eigenvalue of certain block matrices where the number of blocks and the block size both increase with suitable conditions on their relative growth. In one of them, we employ a symmetric block structure with large independent Wigner blocks and in the other we have the Wigner block structure with large independent symmetric blocks. The entries are assumed to be independent and identically distributed with mean [Formula: see text] variance [Formula: see text] with an appropriate growth condition on the moments. Under our conditions the limit spectral distribution of these matrices is the standard semi-circle law. It is natural to ask if the extreme eigenvalues converge to the extreme points of its support, namely [Formula: see text]. We exhibit models where this indeed happens as well as models where the spectral norm converges to [Formula: see text]. Our proofs are based on combinatorial analysis of the behavior of the trace of large powers of the matrix.

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