Hausdorff moment problem and fractional moments

Gzyl, Henryk; Tagliani, Aldo

doi:10.1016/j.amc.2010.04.059

Cited by 33 publications

(7 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…7]. In a recent interesting study, Gzyl and Tagliani concluded that thirty-two was "the maximum allowable moments before incurring numerical instability, unless one conducts the calculation with high accuracy [50]. They also assert that "the additional information introduced by using the (M + 1)-order moment is 'visible' only after the 0.6M-th decimal digit".…”

Section: Cumulative Distribution Function Calculationsmentioning

confidence: 99%

Hilbert-Schmidt Orthogonality of det(rho) and det(rho^{PT}) over the Two-Rebit Systems rho and Further Determinantal Moment Analyses

Slater

2010

Preprint

View full text Add to dashboard Cite

A complete description of the multitudinous ways in which quantum particles can be entangled requires the use of high-dimensional abstract mathematical spaces. We report here a particularly interesting feature of the nine-dimensional convex set-endowed with Hilbert-Schmidt (Euclidean/flat) measure-composed of two-rebit (4 × 4) density matrices (ρ). To each ρ is assigned the product of its (nonnegative) determinant |ρ| and the determinant of its partial transpose |ρ P T |negative values of which, by the results of Peres and Horodecki, signify the entanglement of ρ.Integrating this product, |ρ||ρ P T | = |ρρ P T |, over the nine-dimensional space, using the indicated (HS) measure, we obtain the result zero. The two determinants, thus, form a pair of multivariate orthogonal polynomials with respect to HS measure. However, orthogonality does not hold, we find, if the symmetry of the nine-dimensional two-rebit scenario is broken slightly, nor with the use of non-flat measures, such as the prominent Bures (minimal monotone) measure-nor in the full HS extension to the 15-dimensional convex set of two-qubit density matrices. We discuss relationsinvolving the HS moments of |ρ P T |-to the long-standing problem of determining the probability that a generic pair of rebits/qubits is separable.

show abstract

Section: Cumulative Distribution Function Calculationsmentioning

confidence: 99%

Hilbert-Schmidt Orthogonality of det(rho) and det(rho^{PT}) over the Two-Rebit Systems rho and Further Determinantal Moment Analyses

Slater

2010

Preprint

View full text Add to dashboard Cite

show abstract

“…In this paper, we employ the MaxEnt PDF estimation, which is widely concerned in recent years [7][8][9], for the realization of the PDF estimation module, and the deduced algorithm is described in the next subsection.…”

Section: Basic Rationale For the Blind-cfar Detectormentioning

confidence: 99%

A new CFAR detector based on PDF estimation of clutter using maximum entropy method

Wang

2011

SPIE Proceedings

View full text Add to dashboard Cite

A new CFAR (constant false alarm rate) algorithm is proposed for the Blind-CFAR detector which is proposed for CFAR detection in various types of clutter. The new algorithm employs the maximum entropy (MaxEnt) method to estimate the PDF (probability density function) of clutter and thus the CFAR detection thresholds based on the samples of reference cells and some moment constraints. Performance of the new algorithm is proved to converge to the performance of the ideal detector as the numbers of the reference cells and the moment constraints approach to infinity. With finite reference cells and moment constraints, performance of the new algorithm is compared with that of the maximum likelihood (ML) CFAR detector in uniform Weibull clutter with known and mismatched shape parameters. The results show that, the Blind-CFAR detector employing the new algorithm performs as well as the ML-CFAR detector in case of known shape parameters and it outperforms the ML-CFAR detector in case of mismatched shape parameters.

show abstract

“…Recently, a moment‐type estimation technique, namely a procedure that recovers the underlying distribution from its assigned moments (estimated moments), has been developed and applied both in univariate and in multivariate cases: see Mnatsakanov (2008a; 2008b) and Gzyl & Tagliani (2010) for the univariate case; and Mnatsakanov & Li (2010) and Mnatsakanov (2011) for the multivariate case. Hence, to estimate the entropy H ( f ) we propose to use the moment‐recovered (MR) approximations (Mnatsakanov 2008a; 2011) of corresponding distributions supported by a unit sphere and by a unit cube.…”

Section: Introductionmentioning

confidence: 99%

Estimation of Multivariate Shannon Entropy Using Moments

Mnatsakanov¹,

Li²,

Harner³

2011

Australian &Amp; New Zealand Journal of Statistics

View full text Add to dashboard Cite

Three new entropy estimators of multivariate distributions are introduced. The two cases considered here concern when the distribution is supported by a unit sphere and by a unit cube. In the former case, the consistency and the upper bound of the absolute error for the proposed entropy estimator are established. In the latter one, under the assumption that only the moments of the underlying distribution are available, a non-traditional estimator of the entropy is suggested. We also study the practical performances of the constructed estimators through simulation studies and compare the estimators based on the moment-recovered approaches with their counterparts derived by using the histogram and kth nearest neighbour constructions. In addition, one worked example is briefly discussed.

show abstract

Hausdorff moment problem and fractional moments

Cited by 33 publications

References 10 publications

Hilbert-Schmidt Orthogonality of det(rho) and det(rho^{PT}) over the Two-Rebit Systems rho and Further Determinantal Moment Analyses

Hilbert-Schmidt Orthogonality of det(rho) and det(rho^{PT}) over the Two-Rebit Systems rho and Further Determinantal Moment Analyses

A new CFAR detector based on PDF estimation of clutter using maximum entropy method

Estimation of Multivariate Shannon Entropy Using Moments

Contact Info

Product

Resources

About