scholarly journals The Moment-SOS hierarchy and the Christoffel-Darboux kernel

2021 ◽  
Author(s):  
Jean Bernard Lasserre
Keyword(s):  
Reproducing Kernel ◽  
Upper Bounds ◽  
Global Optimum ◽  
Global Minimizer ◽  
Dirac Measure ◽  
Compact Set ◽  
Christoffel Function ◽  
Reference Measure ◽  
Optimal Polynomial ◽  
The Moment

Abstract We consider the global minimization of a polynomial on a compact set B. We show that each step of the Moment-SOS hierarchy has a nice and simple interpretation that complements the usual one. Namely, it computes coefficients of a polynomial in an orthonormal basis of L 2 (B,μ) where μ is an arbitrary reference measure whose support is exactly B. The resulting polynomial is a certain density (with respect to μ) of some signed measure on B. When some relaxation is exact (which generically takes place) the coefficients of the optimal polynomial density are values of orthonormal polynomials at the global minimizer and the optimal (signed) density is simply related to the Christoffel-Darboux (CD) kernel and the Christoffel function associated with μ. In contrast to the hierarchy of upper bounds which computes positive densities, the global optimum can be achieved exactly as integration against a polynomial (signed) density because the CD-kernel is a reproducing kernel, and so can mimic a Dirac measure (as long as finitely many moments are concerned).

scholarly journals THE DECAY OF THE WALSH COEFFICIENTS OF SMOOTH FUNCTIONS

2009 ◽  
Vol 80 (3) ◽  
pp. 430-453 ◽  
Author(s):  
JOSEF DICK
Keyword(s):  
Lower Bound ◽  
Fractional Order ◽  
Bounded Variation ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Upper Bounds ◽  
Reproducing Kernel Hilbert Spaces ◽  
Smooth Functions ◽  
Kernel Hilbert Spaces

AbstractWe give upper bounds on the Walsh coefficients of functions for which the derivative of order at least one has bounded variation of fractional order. Further, we also consider the Walsh coefficients of functions in periodic and nonperiodic reproducing kernel Hilbert spaces. A lower bound which shows that our results are best possible is also shown.

scholarly journals Optimal Polynomial Prediction Measures and Extremal Polynomial Growth

2020 ◽  
Author(s):  
L. Bos ◽  
N. Levenberg ◽  
J. Ortega-Cerdà
Keyword(s):  
Least Squares ◽  
Related Problem ◽  
Polynomial Growth ◽  
Compact Set ◽  
Extremal Polynomial ◽  
Real Polynomials ◽  
Optimal Polynomial

Abstract We show that the problem of finding the measure supported on a compact set $$K\subset \mathbb {C}$$ K ⊂ C such that the variance of the least squares predictor by polynomials of degree at most n at a point $$z_0\in \mathbb {C}^d\backslash K$$ z 0 ∈ C d \ K is a minimum is equivalent to the problem of finding the polynomial of degree at most n,  bounded by 1 on K,  with extremal growth at $$z_0.$$ z 0 . We use this to find the polynomials of extremal growth for $$[-1,1]\subset \mathbb {C}$$ [ - 1 , 1 ] ⊂ C at a purely imaginary point. The related problem on the extremal growth of real polynomials was studied by Erdős (Bull Am Math Soc 53:1169–1176, 1947).

scholarly journals ON THE ASYMPTOTIC EFFICIENCY OF GMM

2013 ◽  
Vol 30 (2) ◽  
pp. 372-406 ◽  
Author(s):  
Marine Carrasco ◽  
Jean-Pierre Florens
Keyword(s):  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Asymptotic Variance ◽  
Inner Product ◽  
Method Of Moment ◽  
Conditional Moment ◽  
Moment Conditions ◽  
Gmm Estimator ◽  
The Moment ◽  
Semiparametric Efficiency Bound

The efficiency of the generalized method of moment (GMM) estimator is addressed by using a characterization of its variance as an inner product in a reproducing kernel Hilbert space. We show that the GMM estimator is asymptotically as efficient as the maximum likelihood estimator if and only if the true score belongs to the closure of the linear space spanned by the moment conditions. This result generalizes former ones to autocorrelated moments and possibly infinite number of moment restrictions. Second, we derive the semiparametric efficiency bound when the observations are known to be Markov and satisfy a conditional moment restriction. We show that it coincides with the asymptotic variance of the optimal GMM estimator, thus extending results by Chamberlain (1987,Journal of Econometrics34, 305–33) to a dynamic setting. Moreover, this bound is attainable using a continuum of moment conditions.

Inequalities with application in economic risk analysis

1972 ◽  
Vol 9 (2) ◽  
pp. 441-444 ◽  
Author(s):  
Robert A. Agnew
Keyword(s):  
Risk Analysis ◽  
Lower Bounds ◽  
Generating Function ◽  
Random Variable ◽  
Moment Generating Function ◽  
Upper Bounds ◽  
Economic Risk ◽  
The Moment

Two sharp lower bounds for the expectation of a function of a non-negative random variable are obtained under rather weak hypotheses regarding the function, thus generalizing two sharp upper bounds obtained by Brook for the moment generating function. The application of these bounds to economic risk analysis is discussed.

Inequalities with application in economic risk analysis

1972 ◽  
Vol 9 (02) ◽  
pp. 441-444 ◽  
Author(s):  
Robert A. Agnew
Keyword(s):  
Risk Analysis ◽  
Lower Bounds ◽  
Generating Function ◽  
Random Variable ◽  
Moment Generating Function ◽  
Upper Bounds ◽  
Economic Risk ◽  
The Moment

Two sharp lower bounds for the expectation of a function of a non-negative random variable are obtained under rather weak hypotheses regarding the function, thus generalizing two sharp upper bounds obtained by Brook for the moment generating function. The application of these bounds to economic risk analysis is discussed.

Analysis of planar anisotropy by means of the Steiner compact

1989 ◽  
Vol 26 (3) ◽  
pp. 490-502 ◽  
Author(s):  
Jan Rataj ◽  
Ivan Saxl
Keyword(s):  
Graphical Method ◽  
Upper Bounds ◽  
Compact Set ◽  
Planar Anisotropy

A graphical method for the estimation of the anisotropy of planar fibre systems based on the Steiner compact set is proposed and discussed. Upper bounds for the deviation in probability of the graphical estimate of the Steiner compact are given and a consistency theorem is proved.

scholarly journals Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies

2018 ◽  
Vol 18 (3) ◽  
pp. 345-354 ◽  
Author(s):  
Silouanos Brazitikos ◽  
Apostolos Giannopoulos ◽  
Dimitris-Marios Liakopoulos
Keyword(s):  
Convex Bodies ◽  
Upper Bounds ◽  
Compact Set ◽  
Uniform Cover ◽  
Cover Inequalities ◽  
Projections Of Convex Bodies ◽  
Lower Dimensional

AbstractThe classical Loomis–Whitney inequality and the uniform cover inequality of Bollobás and Thomason provide upper bounds for the volume of a compact set in terms of its lower dimensional coordinate projections. We provide further extensions of these inequalities in the setting of convex bodies. We also establish the corresponding dual inequalities for coordinate sections; these uniform cover inequalities for sections may be viewed as extensions of Meyer’s dual Loomis–Whitney inequality.

scholarly journals GENERALIZATION OF GMM TO A CONTINUUM OF MOMENT CONDITIONS

2000 ◽  
Vol 16 (6) ◽  
pp. 797-834 ◽  
Author(s):  
Marine Carrasco ◽  
Jean-Pierre Florens
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Diffusion Processes ◽  
Generalized Method Of Moments ◽  
Efficient Estimation ◽  
Conditional Moment ◽  
Cross Sectional ◽  
Moment Conditions ◽  
Optimal Estimator ◽  
The Moment

This paper proposes a version of the generalized method of moments procedure that handles both the case where the number of moment conditions is finite and the case where there is a continuum of moment conditions. Typically, the moment conditions are indexed by an index parameter that takes its values in an interval. The objective function to minimize is then the norm of the moment conditions in a Hilbert space. The estimator is shown to be consistent and asymptotically normal. The optimal estimator is obtained by minimizing the norm of the moment conditions in the reproducing kernel Hilbert space associated with the covariance. We show an easy way to calculate this estimator. Finally, we study properties of a specification test using overidentifying restrictions. Results of this paper are useful in many instances where a continuum of moment conditions arises. Examples include efficient estimation of continuous time regression models, cross-sectional models that satisfy conditional moment restrictions, and scalar diffusion processes.

scholarly journals Optimal sup norm bounds for newforms on GL2 with maximally ramified central character

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Félicien Comtat
Keyword(s):  
Number Theory ◽  
Lower Bound ◽  
Modular Forms ◽  
Upper Bound ◽  
Upper Bounds ◽  
Central Character ◽  
Ramanujan Conjecture ◽  
The Moment ◽  
Level Aspect

AbstractRecently, the problem of bounding the sup norms of {L^{2}}-normalized cuspidal automorphic newforms ϕ on {\mathrm{GL}_{2}} in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character χ of ϕ is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general χ. If the level N is a square, our result reduces to\|\phi\|_{\infty}\ll N^{\frac{1}{4}+\epsilon},at least under the Ramanujan Conjecture. In particular, when χ has conductor N, this improves upon the previous best known bound {\|\phi\|_{\infty}\ll N^{\frac{1}{2}+\epsilon}} in this setup (due to [A. Saha, Hybrid sup-norm bounds for Maass newforms of powerful level, Algebra Number Theory 11 2017, 1009–1045]) and matches a lower bound due to [N. Templier, Large values of modular forms, Camb. J. Math. 2 2014, 1, 91–116], thus our result is essentially optimal in this case.

scholarly journals A study of positive linear operators by the method of moments

1984 ◽  
Author(s):  
Γεώργιος Αναστασίου
Keyword(s):  
Method Of Moments ◽  
Upper Bounds ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Dirac Measure ◽  
Spline Functions ◽  
Identity Operator ◽  
Optimal Functions ◽  
Finite Measures ◽  
Second Modulus Of Continuity

Let Q a compact and convex subset of R('k), k (GREATERTHEQ) 1 and let {L(,j)}(,j(ELEM)(, )) be a sequence of positive linear operators from C('n)(Q), (n (ELEM) ('+)) to C(Q). The convergence of L(,j) to the identity operator I is closely related to the weak convergence of a sequence of finite measures (mu)(,j) to the unit (Dirac) measure (delta)(,x0), x(,0) (ELEM) Q.New estimates are given for the remainder (VBAR)(INT)(,Q)f d(mu)(,j) -- f(x(,0))(VBAR), where f (ELEM) C('n)(Q). Using moments methods, Shisha-Mond type best or nearly best upper-bounds are established for various choices of k, Q, n and given moments of (mu)(,j). Some of them lead to attainable inequalities. The optimal functions/measures are typically spline functions and finitely supported measures. The corresponding inequalities involve various measures of smoothness of f such as the first or second modulus of continuity of f('(n)), the Peetre K-functional of f or certain modifications and generalizations.Finally some miscellaneous sharp inequalities are obtained. These lead to Korovkin type convergence theorems relative to ratios of Fourier-Stieltjes hyperbolic coefficients.

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