The Convergence Rate of Fréchet Distribution under the Second-Order Regular Variation Condition

In this article we consider the asymptotic behavior of extreme distribution with the extreme value index γ>0 . The rates of uniform convergence for Fréchet distribution are constructed under the second-order regular variation condition.

ON THE REGULARITY OF THE MINIMUM SOLUTION OF THE RESTRAINED VARIATIONAL PROBLEMS

This paper is concerned with the regularity of minimum solution u of the following functional L(u) = integral(Omega) a alpha(beta)(x)g(ij)(u)D alpha u(i)D(beta)upsilon(i)dx on the restraint E = {u is an element of W-0(1,2) (Omega, R(N))\parallel to u parallel to L(D) = 1}. Under appropriate conditions, the bounded minimum solution u of the above functional is proved to be nothing but Holder continuous.

Variational regularization method of solving the Cauchy problem for Laplace's equation: Innovation of the Grad–Shafranov(GS) reconstruction

The simplified linear model of Grad-Shafranov （GS） reconstruction can be reformulated into an inverse boundary value problem of Laplace＇s equation. Therefore, in this paper we focus on the method of solving the inverse boundary value problem of Laplace＇s equation. In the first place, the variational regularization method is used to deal with the ill- posedness of the Cauchy problem for Laplace＇s equation. Then, the ＇L-Curve＇ principle is suggested to be adopted in choosing the optimal regularization parameter. Finally, a numerical experiment is implemented with a section of Neumann and Dirichlet boundary conditions with observation errors. The results well converge to the exact solution of the problem, which proves the efficiency and robustness of the proposed method. When the order of observation error δ is 10-1, the order of the approximate result error can reach 10-3.

Finite Time Ruin Probability with Variable Interest Rate and Extended Regular Variation

Consider an insurance risk model, in which the surplus process satisfies a recursive equationU n =U n?1(1+r n )?X n forn≥1, whereU 0=x≥0 is the initial surplus, {r n ;n≥1} the interest rate sequence, {X n ;n≥1} the sequence of i. i. d. real-valued random variables with common distribution functionF, which denotes the gross loss during thenth year. We investigate the ruin probability within a finite time horizon and give the asymptotic result asx→∞. Key words variable interest rate - extend regular variation - finite time ruin probability CLC number O 211.9 Foundation item: Supported by the National Natural Science Foundation of China (10071058, 70273029)Biography: WEI Xiao (1979-), female, Ph. D candidate, research direction: large deviations and its applications, insurance mathematics.

Identification for the Low-Contrast Image Signal with Regularized Variational Term and Dynamical Saturating Nonlinearity

In recent years,image processing based on stochastic resonance(SR)has received more and more attention.In this paper,a new model combining dynamical saturating nonlinearity with regularized variational term for enhancement of low contrast image is proposed.The regularized variational term can be setting to total variation(TV),second order total generalized variation(TGV)and non-local means(NLM)in order to gradually suppress noise in the process of solving the model.In addition,the new model is tested on a mass of gray-scale images from standard test image and low contrast indoor color images from Low-Light dataset(LOL).By comparing the new model and other traditional image enhancement models,the results demonstrate the enhanced image not only obtain good perceptual quality but also get more excellent value of evaluation index compared with some previous methods.

First and Second Order Asymptotics of the Spectral Risk Measure for Portfolio Loss Under Multivariate Regular Variation

In the context of multivariate regular variation,the authors establish the first-order asymptotics of the spectral risk measure of portfolio loss.Furthermore,by the notion of second-order regular variation,the second-order asymptotics of the spectral risk measure of portfolio loss is also presented.In order to illustrate the derived results,a numerical example with Monte Carlo simulation is carried out.

Total Variation Based Parameter-Free Model for Impulse Noise Removal

We propose a new two-phase method for reconstruction of blurred im-ages corrupted by impulse noise.In the first phase,we use a noise detector to iden-tify the pixels that are contaminated by noise,and then,in the second phase,we reconstruct the noisy pixels by solving an equality constrained total variation mini-mization problem that preserves the exact values of the noise-free pixels.For images that are only corrupted by impulse noise(i.e.,not blurred)we apply the semismooth Newton’s method to a reduced problem,and if the images are also blurred,we solve the equality constrained reconstruction problem using a first-order primal-dual algo-rithm.The proposed model improves the computational efficiency(in the denoising case)and has the advantage of being regularization parameter-free.Our numerical results suggest that the method is competitive in terms of its restoration capabilities with respect to the other two-phase methods.

General Regular Variation of n-th Order and the 2nd Order Edgeworth Expansion of the Extreme Value Distribution (Ⅰ)

In Part Ⅰ the concept of the general regular variation of n-th order is proposed and its construction is discussed. The uniqueness of the standard expression and the higher order regularity of the auxiliary functions are proved.

Lipschitz and Total-Variational Regularization for Blind Deconvolution

In[3],Chan and Wong proposed to use total variational regularization for both images and point spread functions in blind deconvolution.Their experimental results show that the detail of the restored images cannot be recovered.In this paper,we consider images in Lipschitz spaces,and propose to use Lipschitz regularization for images and total variational regularization for point spread functions in blind deconvolution.Our experimental results show that such combination of Lipschitz and total variational regularization methods can recover both images and point spread functions quite well.

The Ruin Probability in the Presence of Extended Regular Variation and Optimal Investment

Considering the classical model with risky investment, we are interested in the ruin probability that is minimized by a suitably chosen investment strategy for a capital market index. For claim sizes with common distribution of extended regular variation, starting from an integro-differential equation for the maximal survival probability, we find that the corresponding ruin probability as a function of the initial surplus is also extended regular variation.

General Regular Variation of the n-th Order and 2nd Order Edgeworth Expansions of the Extreme Value Distribution （Ⅱ）

In this part II the fundamental inequality of the third order general regular variation is proved and the second order Edgeworth expansion of the distribution of the extreme values is discussed.

Tail Distortion Risk Measure for Portfolio with Multivariate Regularly Variation

For the multiplicative background risk model,a distortion-type risk measure is used to measure the tail risk of the portfolio under a scenario probability measure with multivariate regular variation.In this paper,we investigate the tail asymptotics of the portfolio loss ∑_(i=1)^(d)R_(i)S,where the stand-alone risk vector R=(R_(1),...,R_(d))follows a multivariate regular variation and is independent of the background risk factor S.An explicit asymptotic formula is established for the tail distortion risk measure,and an example is given to illustrate our obtained results.

ESTIMATION AND UNCERTAINTY QUANTIFICATION FOR PIECEWISE SMOOTH SIGNAL RECOVERY

This paper presents an application of the sparse Bayesian learning(SBL)algorithm to linear inverse problems with a high order total variation(HOTV)sparsity prior.For the problem of sparse signal recovery,SBL often produces more accurate estimates than maximum a posteriori estimates,including those that useℓ1 regularization.Moreover,rather than a single signal estimate,SBL yields a full posterior density estimate which can be used for uncertainty quantification.However,SBL is only immediately applicable to problems having a direct sparsity prior,or to those that can be formed via synthesis.This paper demonstrates how a problem with an HOTV sparsity prior can be formulated via synthesis,and then utilizes SBL.This expands the class of problems available to Bayesian learning to include,e.g.,inverse problems dealing with the recovery of piecewise smooth functions or signals from data.Numerical examples are provided to demonstrate how this new technique is effectively employed.

Moderate Deviations for Random Sums of Heavy-Tailed Random Variables

Let {Xn;n≥ 1} be a sequence of independent non-negative random variables with common distribution function F having extended regularly varying tail and finite mean μ = E（X1） and let {N（t）; t ≥0} be a random process taking non-negative integer values with finite mean λ（t） = E（N（t）） and independent of {Xn; n ≥1}. In this paper, asymptotic expressions of P（（X1 ＋… ＋XN（t）） -λ（t）μ 〉 x） uniformly for x ∈[γb（t）, ∞） are obtained, where γ〉 0 and b（t） can be taken to be a positive function with limt→∞ b（t）/λ（t） = 0.

The Asymptotic Behavior of the Ruin Probability within a Random Horizon

Subject to the assumption that the common distribution of claim sizes belongs to the extendedregular variation class,the present work obtains a simple asymptotic formula for the ruin probability within arandom or nonrandom horizon in the renewal model.

An Alternating Direction Approximate Newton Algorithm for Ill-Conditioned Inverse Problems with Application to Parallel MRI

Analternating direction approximateNewton(ADAN)method is developed for solving inverse problems of the form min{φ(Bu)+(1/2)||Au−f||^(2)_(2)},whereφis convex and possibly nonsmooth,and A and B arematrices.Problems of this form arise in image reconstruction where A is the matrix describing the imaging device,f is the measured data,φis a regularization term,and B is a derivative operator.The proposed algorithm is designed to handle applications where A is a large dense,ill-conditioned matrix.The algorithm is based on the alternating direction method of multipliers(ADMM)and an approximation to Newton’s method in which a term in Newton’s Hessian is replaced by aBarzilai–Borwein(BB)approximation.It is shown thatADAN converges to a solution of the inverse problem.Numerical results are provided using test problems from parallel magnetic resonance imaging.ADAN was faster than a proximal ADMM scheme that does not employ a BB Hessian approximation,while it was more stable and much simpler than the related Bregman operator splitting algorithm with variable stepsize algorithm which also employs a BB-based Hessian approximation.

Stroke-Based Surface Reconstruction

In this paper,we present a surface reconstruction via 2D strokes and a vector field on the strokes based on a two-step method.In the first step,from sparse strokes drawn by artists and a given vector field on the strokes,we propose a nonlinear vector interpolation combining total variation(TV)and H1 regularization with a curl-free constraint for obtaining a dense vector field.In the second step,a height map is obtained by integrating the dense vector field in the first step.Jump discontinuities in surface and discontinuities of surface gradients can be well reconstructed without any surface distortion.We also provide a fast and efficient algorithm for solving the proposed functionals.Since vectors on the strokes are interpreted as a projection of surface gradients onto the plane,different types of strokes are easily devised to generate geometrically crucial structures such as ridge,valley,jump,bump,and dip on the surface.The stroke types help users to create a surface which they intuitively imagine from 2D strokes.We compare our results with conventional methods via many examples.

Precise Large Deviations for Sums of Claim-size Vectors in a Two-dimensional Size-dependent Renewal Risk Model

Consider a two-dimensional renewal risk model,in which the claim sizes{Xk;k≥1}form a sequence of i.i.d.copies of a non-negative random vector whose two components are dependent.Suppose that the claim sizes and inter-arrival times form a sequence of i.i.d.random pairs,with each pair obeying a dependence structure via the conditional distribution of the inter-arrival time given the subsequent claim size being large.Then a precise large-deviation formula of the aggregate amount of claims is obtained.

Extensions of Breiman’s Theorem of Product of Dependent Random Variables with Applications to Ruin Theory

We consider the tail behavior of the product of two dependent random variables X andΘ.Motivated by Denisov andZwart(JAppl Probab 44:1031-1046,2007),we relax the condition of the existing α+ε th moment ofΘin Breiman’s theorem to the existingαth moment and obtain the similar result as Breiman’s theorem of the dependent product XΘ,while X andΘfollow a copula function.As applications,we consider a discrete-time insurance risk model with dependent insurance and financial risks and derive the asymptotic tail behaviors for the(in)finite-time ruin probabilities.

Second-order Asymptotics on Distributions of Maxima of Bivariate Elliptical Arrays

Let {（ξni, ηni）, 1 ≤ i ≤ n, n ≥ 1} be a triangular array of independent bivariate elliptical random vectors with the same distribution function as （S1,ρnS1 ＋ √1- ρ2nS2）, ρn ∈（0, 1）, where （S1,S2） is a bivariate spherical random vector. For the distribution function of radius√S12 ＋ S22 belonging to the max-domain of attraction of the Weibull distribution, the limiting distribution of maximum of this triangular array is known as the convergence rate of p~ to 1 is given. In this paper, under the refinement of the rate of convergence of p~ to 1 and the second-order regular variation of the distributional tail of radius, precise second-order distributional expansions of the normalized maxima of bivariate elliptical triangular arrays are established.