2,809 research outputs found
Quantile-based optimization under uncertainties using adaptive Kriging surrogate models
Uncertainties are inherent to real-world systems. Taking them into account is
crucial in industrial design problems and this might be achieved through
reliability-based design optimization (RBDO) techniques. In this paper, we
propose a quantile-based approach to solve RBDO problems. We first transform
the safety constraints usually formulated as admissible probabilities of
failure into constraints on quantiles of the performance criteria. In this
formulation, the quantile level controls the degree of conservatism of the
design. Starting with the premise that industrial applications often involve
high-fidelity and time-consuming computational models, the proposed approach
makes use of Kriging surrogate models (a.k.a. Gaussian process modeling).
Thanks to the Kriging variance (a measure of the local accuracy of the
surrogate), we derive a procedure with two stages of enrichment of the design
of computer experiments (DoE) used to construct the surrogate model. The first
stage globally reduces the Kriging epistemic uncertainty and adds points in the
vicinity of the limit-state surfaces describing the system performance to be
attained. The second stage locally checks, and if necessary, improves the
accuracy of the quantiles estimated along the optimization iterations.
Applications to three analytical examples and to the optimal design of a car
body subsystem (minimal mass under mechanical safety constraints) show the
accuracy and the remarkable efficiency brought by the proposed procedure
Dependence on the Dimension for Complexity of Approximation of Random Fields
We consider an \eps-approximation by n-term partial sums of the
Karhunen-Lo\`eve expansion to d-parametric random fields of tensor product-type
in the average case setting. We investigate the behavior, as d tends to
infinity, of the information complexity n(\eps,d) of approximation with error
not exceeding a given level \eps. It was recently shown by M.A. Lifshits and
E.V. Tulyakova that for this problem one observes the curse of dimensionality
(intractability) phenomenon. The aim of this paper is to give the exact
asymptotic expression for the information complexity n(\eps,d).Comment: 18 pages. The published in Theory Probab. Appl. (2010) extended
English translation of the original paper "Zavisimost slozhnosti approximacii
sluchajnyh polej ot rasmernosti", submitted on 15.01.2007 and published in
Theor. Veroyatnost. i Primenen. 54:2, 256-27
Affiliation, equilibrium existence and the revenue ranking of auctions
We consider private value auctions where biddersâ types are dependent, a case usually
treated by assuming affiliation. We show that affiliation is a restrictive assumption in
three senses: topological, measure-theoretic and statistical (affiliation is a very
restrictive characterization of positive dependence). We also show that affiliationâs
main implications do not generalize for alternative definitions of positive dependence.
From this, we propose new approaches to the problems of pure strategy equilibrium
existence in first-price auctions (PSEE) and the characterization of the revenue ranking
of auctions. For equilibrium existence, we slightly restrict the set of distributions
considered, without loss of economic generality, and offer a complete characterization
of PSEE. For revenue ranking, we obtain a characterization of the expected revenue
differences between second and first price auctions with general dependence of types
Optimal inference in a class of regression models
We consider the problem of constructing confidence intervals (CIs) for a
linear functional of a regression function, such as its value at a point, the
regression discontinuity parameter, or a regression coefficient in a linear or
partly linear regression. Our main assumption is that the regression function
is known to lie in a convex function class, which covers most smoothness and/or
shape assumptions used in econometrics. We derive finite-sample optimal CIs and
sharp efficiency bounds under normal errors with known variance. We show that
these results translate to uniform (over the function class) asymptotic results
when the error distribution is not known. When the function class is
centrosymmetric, these efficiency bounds imply that minimax CIs are close to
efficient at smooth regression functions. This implies, in particular, that it
is impossible to form CIs that are tighter using data-dependent tuning
parameters, and maintain coverage over the whole function class. We specialize
our results to inference on the regression discontinuity parameter, and
illustrate them in simulations and an empirical application.Comment: 39 pages plus supplementary material
On the Necessity of Five Risk Measures
The banking systems that deal with risk management depend on underlying risk
measures. Following the Basel II accord, there are two separate methods by
which banks may determine their capital requirement. The Value at Risk measure
plays an important role in computing the capital for both approaches. In this
paper we analyze the errors produced by using this measure. We discuss other
measures, demonstrating their strengths and shortcomings. We give examples,
showing the need for the information from multiple risk measures in order to
determine a bank's loss distribution. We conclude by suggesting a regulatory
requirement of multiple risk measures being reported by banks, giving specific
recommendations.Comment: 23 pages, 9 figure
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