Kantorovich Inequality Research Articles

A Kantorovich type inequality for positive matrices

In this paper, we present a Kantorovich type inequality for positive matrices by using the properties of positive linear functionals on matrices. Based on these, we provide some generalized forms of discrete Kantorovich inequality and some inequalities for spectral radius, numerical radius and spectral norm of the product of positive matrices.

Generalizations of the Kantorovich and Wielandt Inequalities with Applications to Statistics

By utilizing the properties of positive definite matrices, mathematical expectations, and positive linear functionals in matrix space, the Kantorovich inequality and Wielandt inequality for positive definite matrices and random variables are obtained. Some novel Kantorovich type inequalities pertaining to matrix ordinary products, Hadamard products, and mathematical expectations of random variables are provided. Furthermore, several interesting unified and generalized forms of the Wielandt inequality for positive definite matrices are also studied. These derived inequalities are then exploited to establish an inequality regarding various correlation coefficients and study some applications in the relative efficiency of parameter estimation of linear statistical models.

Kantorovich inequality for positive operators on quaternionic Hilbert spaces

Kantorovich inequality for positive operators on quaternionic Hilbert spaces

Non-quadratic extension of homogeneous S-lemma and its applications in optimization

In this paper, we propose a non-quadratic extension of homogeneous S-lemma under some conditions. Then we apply the extended homogeneous S-lemma to reveal the hidden convexity property of the generalized Kantorovich inequality problem and the homogeneously lifted two-sided extended trust-region subproblem.

Sobolev–Kantorovich inequalities under CD(0,∞) condition

We refine and generalize several interpolation inequalities bounding the [Formula: see text] norm of a probability density with respect to the reference measure [Formula: see text] by its Sobolev norm and the Kantorovich distance to [Formula: see text] on a smooth weighted Riemannian manifold satisfying [Formula: see text] condition.

An extension of the operator Kantorovich inequality

In this paper, we present an extension of Kantorovich inequality for two operators on a Hilbert space. Also, the multiple version and a related inequality for positive linear maps are obtained. Moreover, we introduce the concept of Specht's ratio and improve some inequalities related to Specht's ratio.

On Improvements of Kantorovich Type Inequalities

In the paper, we give some new improvements of the Kantorovich type inequalities by using Popoviciu’s, Hölder’s, Bellman’s and Minkowski’s inequalities. These results in special case yield Hao’s, reverse Cauchy’s and Minkowski’s inequalities.

New Refinement of the Operator Kantorovich Inequality

We focus on the improvement of operator Kantorovich type inequalities. Among the consequences, we improve the main result of the paper [H.R. Moradi, I.H. Gümüş, Z. Heydarbeygi, A glimpse at the operator Kantorovich inequality, Linear Multilinear Algebra, doi:10.1080/03081087.2018.1441799].

A glimpse at the operator Kantorovich inequality

ABSTRACTWe show the following result: Let A be a positive operator satisfying for some scalars m, M with and be a normalized positive linear map, then Besides, we prove that the second inequality in the above can be squared.

Extension of the Kantorovich inequality for positive multilinear mappings

It is known that the power function f (t) = t2 is not matrix monotone. Recently, it has been shown that t2 preserves the order in some matrix inequalities. We prove that if A = (A1,...,Ak) and B = (B1,...,Bk) are k-tuples of positive matrices with 0 < m ? Ai; Bi ? M (i = 1,...,k) for some positive real numbers m < M, then ?2 (A-11,...,A-1k) ? (1+vk)2/4vk)2 ?-2(A1,...,Ak) and ?2 (A1+B1/2,..., Ak+Bk/2)? (1+vk)2/4vk)2 ?2 (A1#B1,...Ak#Bk), where ? is a unital positive multilinear mapping and v = M/m is the condition number of each Ai.

Certain integral inequalities involving tensor products, positive linear maps, and operator means

We present a number of integral inequalities involving tensor products of continuous fields of bounded linear operators, positive linear maps, and operator means. In particular, the Kantorovich, Gruss, and reverse Holder-McCarthy integral inequalities are obtained as special cases.

Tuning Parameter Selection in Penalized Frailty Models

The penalized likelihood approach of Fan and Li (2001, 2002) differs from the traditional variable selection procedures in that it deletes the non-significant variables by estimating their coefficients as zero. Nevertheless, the desirable performance of this shrinkage methodology relies heavily on an appropriate selection of the tuning parameter which is involved in the penalty functions. In this work, new estimates of the norm of the error are firstly proposed through the use of Kantorovich inequalities and, subsequently, applied to the frailty models framework. These estimates are used in order to derive a tuning parameter selection procedure for penalized frailty models and clustered data. In contrast with the standard methods, the proposed approach does not depend on resampling and therefore results in a considerable gain in computational time. Moreover, it produces improved results. Simulation studies are presented to support theoretical findings and two real medical data sets are analyzed.

Operator Inequalities Related to Q-Class Functions

Abstract We study the operator Q-class functions, present some Hermite- Hadamard inequalities for operator Q-class functions and give some Kantorovich and Jensen type operator inequalities involving Q-class functions

Tuning parameter selection in penalized generalized linear models for discrete data

In recent years, we have seen an increased interest in the penalized likelihood methodology, which can be efficiently used for shrinkage and selection purposes. This strategy can also result in unbiased, sparse, and continuous estimators. However, the performance of the penalized likelihood approach depends on the proper choice of the regularization parameter. Therefore, it is important to select it appropriately. To this end, the generalized cross‐validation method is commonly used. In this article, we firstly propose new estimates of the norm of the error in the generalized linear models framework, through the use of Kantorovich inequalities. Then these estimates are used in order to derive a tuning parameter selector in penalized generalized linear models. The proposed method does not depend on resampling as the standard methods and therefore results in a considerable gain in computational time while producing improved results. A thorough simulation study is conducted to support theoretical findings; and a comparison of the penalized methods with the L1, the hard thresholding, and the smoothly clipped absolute deviation penalty functions is performed, for the cases of penalized Logistic regression and penalized Poisson regression. A real data example is being analyzed, and a discussion follows. © 2014 The Authors. Statistica Neerlandica © 2014 VVS.

On Bounds for Moments of Unimodal Distributions

We provide a simple basic method to find bounds for higher order moments of unimodal distributions in terms of lower order moments when the random variable takes value in a given finite real interval. The bounds for moments in terms of the geometric mean of the distribution are also derived. Both continuous and discrete cases are considered. The bounds for the ratio and difference of moments are obtained. The special cases provide refinements of several well-known inequalities, such as Kantorovich inequality and Krasnosel'skii and Krein inequality.

Several Kantorovich Inequalities Involving Tracy-Singh Product

The Tracy-Singh product, which can be viewed as generalized Kronecher product, is studied and used widely in matrix theory and statistics. Therefore, some well-known Kantorovich inequalities are generalized to Tracy-Singh product in this article. Furthermore, some applications to statistics are presented.

The efficiency comparisons between OLSE and BLUE in a singular linear model

This paper is mainly concerned with the efficiency comparison between OLSE and BLUE in a singular linear model. We define the efficiencies between OLSE and BLUE by means of the matrix Euclidean norm and prove a matrix Euclidean norm version of the Kantorovich inequality to limit upper or lower bounds of these efficiencies. It relaxes the assumptions that the covariance matrix is positive definite and the design matrix has full column rank. MSC: Primary 62J05; secondary 62H05; 62H20

On an operator Kantorovich inequality for positive linear maps

We improve the operator Kantorovich inequality as follows: Let A be a positive operator on a Hilbert space with 0<m≤A≤M. Then for every unital positive linear map Φ, Φ(A−1)2≤((M+m)24Mm)2Φ(A)−2. As a consequence, Φ(A−1)Φ(A)+Φ(A)Φ(A−1)≤(M+m)22Mm.

Generalization on Kantorovich inequality

In this paper, we provide a new form of upper bound for the converse of Jensen's inequality. Thereby, known estimations of the difference and ratio in Jensen's inequality are es- sentially improved. As an application, we also obtain an improvement of Kantorovich inequality.

Sensitivity analysis of SAR estimators: a numerical approximation

The estimation of a spatial autoregressive (SAR) model depends on the spatial correlation parameter ρ in a highly nonlinear way, and the least squares (LS) estimators for ρ cannot be computed in a closed form. In this paper, we propose two simple LS estimators and compare them by distance and covariance properties in order to study the local sensitivity behaviour of these estimators. Using matrix derivatives we calculate the Taylor approximation of the LS estimator in the SAR model up to the second order. In a next step, we compare the covariance structure of the two estimators by Kantorovich inequalities and derive efficiency comparisons by upper bounds. Finally, we explore the quality of our new approximations by a Monte Carlo simulation study. The simulation results show significant computation time reductions and a good approximation behaviour of the SAR LS estimator in the neighborhood of ρ=0, when using a non-spatial LS estimator. The results are encouraging and can be used for further developments like quick diagnostic tools to explore the sensitivity of spatial estimators with respect to the size of the spatial correlation.