## Optimal Design and Related Areas in Optimization and StatisticsLuc Pronzato, Anatoly Zhigljavsky The present volume is a collective monograph devoted to applications of the optimal design theory in optimization and statistics. The chapters re?ect the topics discussed at the workshop “W-Optimum Design and Related Statistical Issues” that took place in Juan-les-Pins, France, in May 2005. The title of the workshop was chosen as a light-hearted celebration of the work of Henry Wynn. It was supported by the Laboratoire I3S (CNRS/Universit ́ e de Nice, Sophia Antipolis), to which Henry is a frequent visitor. The topics covered partly re?ect the wide spectrum of Henry’s research - terests. Algorithms for constructing optimal designs are discussed in Chap. 1, where Henry’s contribution to the ?eld is acknowledged. Steepest-ascent - gorithms used to construct optimal designs are very much related to general gradientalgorithmsforconvexoptimization. Inthelasttenyears,asigni?cant part of Henry’s research was devoted to the study of the asymptotic prop- ties of such algorithms. This topic is covered by Chaps. 2 and 3. The work by Alessandra Giovagnoli concentrates on the use of majorization and stoch- tic ordering, and Chap. 4 is a hopeful renewal of their collaboration. One of Henry’s major recent interests is what is now called algebraic statistics, the application of computational commutative algebra to statistics, and he was partly responsible for introducing the experimental design sub-area, reviewed in Chap. 5. One other sub-area is the application to Bayesian networks and Chap. 6 covers this, with Chap. 7 being strongly related. |

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### Contents

WIterations and Ripples Therefrom | 1 |

13 Derivatives and Optimality Conditions | 2 |

14 Algorithms | 3 |

15 A SteepestAscent Algorithm | 9 |

References | 11 |

Studying Convergence of Gradient Algorithms Via Optimal Experimental Design Theory | 13 |

22 Renormalized Version of Gradient Algorithms | 14 |

23 A Multiplicative Algorithm for Optimal Design | 16 |

55 Indicator Function for Complex Coded Designs | 118 |

56 Indicator Function vs Grobner Basis | 121 |

57 Mixture Designs | 126 |

58 Conclusions | 130 |

References | 131 |

The Geometry of Causal Probability Trees that are Algebraically Constrained | 133 |

62 Manifest Probilities and Solution Spaces | 137 |

63 Expressing Causal Effects Through Algebra | 139 |

which Correspond to a Given Gradient Algorithm | 18 |

25 Optimum Design Gives the Worst Rate of Convergence | 19 |

26 Some Special Cases | 20 |

27 The SteepestDescent Algorithm with Relaxation | 23 |

28 SquareRoot Algorithm | 30 |

29 AOptimality | 32 |

210 αRoot Algorithm and Comparisons | 33 |

References | 36 |

A DynamicalSystem Analysis of the Optimum sGradient Algorithm | 39 |

for the Minimization of a Quadratic Function | 40 |

33 Asymptotic Behaviour of the Optimum sGradient Algorithm in Rd | 52 |

34 The Optimum 2Gradient Algorithm in Rd | 55 |

35 Switching Algorithms | 66 |

References | 79 |

Bivariate Dependence Orderings for Unordered Categorical Variables | 81 |

42 Dependence Orderings for Two Nominal Variables | 83 |

43 InterRaters Agreement for Categorical Classiﬁcations | 90 |

44 Conclusions and Further Research | 94 |

References | 95 |

Methods in Algebraic Statistics for the Design of Experiments | 97 |

52 Background | 98 |

53 Generalized Confounding and Polynomial Algebra | 102 |

54 Models and Monomials | 113 |

64 From Models to Causal ACTs to Analysis | 142 |

65 Equivalent Causal ACTs | 147 |

66 Conclusions | 151 |

References | 154 |

Bayes Nets of Time Series Stochastic Realizations and Projections | 155 |

Stochastic Realization and Conditional Independence | 160 |

73 LCOLCI Time Series | 163 |

74 TDAG as Generalized Time | 165 |

References | 166 |

Asymptotic Normality of Nonlinear Least Squares under Singular Experimental Designs | 167 |

82 The Convergence of the Design Sequence to a Design Measure | 171 |

83 Consistency of Estimators | 176 |

84 On the Geometry of the Model Under the Design Measure | 179 |

85 The Regular Asymptotic Normality of h0N | 182 |

86 Estimation of a Multidimensional Function H0 | 186 |

References | 190 |

Robust Estimators in Nonlinear Regression Models with LongRange Dependence | 192 |

92 Main Results | 196 |

93 Auxiliary Assertions | 207 |

94 Proofs | 215 |

References | 217 |

222 | |

### Other editions - View all

Optimal Design and Related Areas in Optimization and Statistics Luc Pronzato,Anatoly Zhigljavsky No preview available - 2008 |

Optimal Design and Related Areas in Optimization and Statistics Luc Pronzato,Anatoly Zhigljavsky No preview available - 2011 |

### Common terms and phrases

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