Estimating copula densities, using model selection techniques

Recently a new way of modeling dependence has been introduced considering a sequence of parametric copula models, covering more and more dependency aspects and thus giving a closer approximation to the true copula density. The method uses contamination families based on Legendre polynomials. It has been shown that in general after a few steps accurate approximations are obtained. In this paper selection of the adequate number of steps is considered, and estimation of the unknown parameters within the chosen contamination family is established, thus obtaining an estimator of the unknown copula density. There should be a balance between the complexity of the model and the number of parameters to be estimated. High complexity gives a low model error, but a large stochastic or estimation error, while a very simple model gives a small stochastic error, but a large model error. Techniques from model selection are applied, thus letting the data tell us which aspects are important enough to capture into the model. Natural and simple estimators of the involved Fourier coefficients complete the procedure. Theoretical results show that the expected quadratic error is reduced by the selection rule to the same order of magnitude as in a classical parametric problem. The method is applied on a real data set, illustrating that the new method describes the data set very well: the error involved in the classical Gaussian copula density is reduced with no fewer than 50%.     

Hyperbolic wavelet thresholding methods and the curse of dimensionality through the maxiset approach

In this paper we compute the maxisets of some denoising methods (estimators) for multidimensional signals based on thresholding coefficients in hyperbolic wavelet bases. That is, we determine the largest functional space over which the risk of these estimators converges at a chosen rate. In the unidimensional setting, refining the choice of the coefficients that are subject to thresholding by pooling information from geometric structures in the coefficient domain (e.g., vertical blocks) is known to provide ‘large maxisets’. In the multidimensional setting, the situation is less straightforward. In a sense these estimators are much more exposed to the curse of dimensionality. However we identify cases where information pooling has a clear benefit. In particular, we identify some general structural constraints that can be related to compound functional models and to a minimal level of anisotropy.     

A quantile-copula approach to conditional density estimation

A new kernel-type estimator of the conditional density is proposed. It is based on an efficient quantile transformation of the data. The proposed estimator, which is based on the copula representation, turns out to have a remarkable product form. Its large-sample properties are considered and comparisons in terms of bias and variance are made with competitors based on nonparametric regression. A comparative simulation study is also provided.     

A copula entropy approach to correlation measurement at the country level

The entropy optimization approach has widely been applied in finance for a long time, notably in the areas of market simulation, risk measurement, and financial asset pricing. In this paper, we propose copula entropy models with two and three variables to measure dependence in stock markets, which extend the copula theory and are based on Jaynes’s information criterion. Both of them are usually applied under the non-Gaussian distribution assumption. Comparing with the linear correlation coefficient and the mutual information, the strengths and advantages of the copula entropy approach are revealed and confirmed. We also propose an algorithm for the copula entropy approach to obtain the numerical results. With the experimental data analysis at the country level and the economic circle theory in international economy, the validity of the proposed approach is approved; evidently, it captures the non-linear correlation, multi-dimensional correlation, and correlation comparisons without common variables. We would like to make it clear that correlation illustrates dependence, but dependence is not synonymous with correlation. Copulas can capture some special types of dependence, such as tail dependence and asymmetric dependence, which other conventional probability distributions, such as the normal p.d.f. and the Student’s t p.d.f., cannot.     

Bayesian methods in global optimization

This paper reviews methods which have been proposed for solving global optimization problems in the framework of the Bayesian paradigm.     

On the compound Poisson risk model with dependence based on a generalized Farlie-Gumbel-Morgenstern copula

In this paper we consider an extension to the classical compound Poisson risk model in which we introduce a dependence structure between the claim amounts and the interclaim time. This structure is embedded via a generalized Farlie-Gumbel-Morgenstern copula. In this framework, we derive the Laplace transform of the Gerber-Shiu discounted penalty function. An explicit expression for the Laplace transform of the time of ruin is given for exponential claim sizes.     

Pointwise consistency of the hermite series density estimate

The Hermite series estimate of a density f?L_p, p > 1, convergessin the mean square to f (x) for almost all x? |R, ifN (n) → ∞ and N (n) / n² → ) as n → ∞, where N is the number of the Hermite functions in the estimate while n is the number of observations. Moreover, the mean square and weak consistency are equivalent. For m times differentiable densities, the mean squares convergence rate is O(n^?(2m?1)/2m). Results for complete convergence are also given.     

Applying copula models to individual claim loss reserving methods

The estimation of loss reserves for incurred but not reported (IBNR) claims presents an important task for insurance companies to predict their liabilities. Recently, individual claim loss models have attracted a great deal of interest in the actuarial literature, which overcome some shortcomings of aggregated claim loss models. The dependence of the event times with the delays is a crucial issue for estimating the claim loss reserving. In this article, we propose to use semi-competing risks copula and semi-survival copula models to fit the dependence structure of the event times with delays in the individual claim loss model. A nonstandard two-step procedure is applied to our setting in which the associate parameter and one margin are estimated based on an ad hoc estimator of the other margin. The asymptotic properties of the estimators are established as well. A simulation study is carried out to evaluate the performance of the proposed methods.     

Minimax estimation and the least squares method

This paper is devoted to the problem of minimax estimation of parameters in linear regression models with uncertain second order statistics. The solution to the problem is shown to be the least squares estimator corresponding to the least favourable matrix of the second moments. This allows us to construct a new algorithm for minimax estimation closely connected with the least squares method. As an example, we consider the problem of polynomial regression introduced by A. N. Kolmogorov     

Ruin measures for a compound Poisson risk model with dependence based on the Spearman copula and the exponential claim sizes

This paper is devoted to an extension to the classical compound risk model. We relax the independence assumption of claim amounts and interclaim times. The dependent structure between these random variables is described by the Spearman copula. We study the Laplace transform of the discounted penalty function and we give the explicit expression of it for the exponential claim size.     

A note on the conditional density estimate in the single functional index model

In this paper, we consider the estimation of the conditional density of a scalar response variable Y, given a Hilbertian random variable X when the observations are linked with a single-index structure. We establish the pointwise and the uniform almost complete convergence (with the rate) of the kernel estimate of this model. As an application, we show how our result can be applied in the prediction problem via the conditional mode estimate. Finally, the estimation of the functional index via the pseudo-maximum likelihood method is also discussed but not tackled.     

Multi-symplectic methods for the Ito-type coupled KdV equation

In this paper, we find that the Ito-type coupled KdV equation can be written as a multi-symplectic Hamiltonian partial differential equation (PDE). Then, multi-symplectic Fourier pseudospectral method and multi-symlpectic wavelet collocation method are constructed for this equation. In the numerical experiments, we show the effectiveness of the proposed methods. Some comparisons between the proposed methods are also made with respect to global conservation properties.     

Asymptotic normality of wavelet density estimator under censored dependent observations

In this paper, we discuss the asymptotic normality of the wavelet estimator of the density function based on censored data, when the survival and the censoring times form a stationary ??-mixing sequence. To simulate the distribution of estimator such that it is easy to perform statistical inference for the density function, a random weighted estimator of the density function is also constructed and investigated. Finite sample behavior of the estimator is investigated via simulations too.     

Wavelet linear estimations of density derivatives from a negatively associated stratified size-biased sample

We define a wavelet linear estimator for density derivative in Besov space based on a negatively associated stratified size-biased random sample. We provide two upper bounds of wavelet estimations on L^p （1 ≤ p 〈 ∞） risk.     

关于Camassa-Holm方程动量密度的紧支集在半轴上的估计

主要考虑在半轴上Camassa-Holm方程解的动量密度紧支集大小的估计,方法是根据区间长度与区间特征值的关系,通过估计第一Dirichlet特征值来估计动量密度紧支集的长度.因为知道动量密度紧支集外解的性态,所以通过估计动量密度支集的大小可以得到方程解的更多信息.     

Smooth density and its short time estimate for jump process determined by SDE

We study a nondegenerate jump process on Euclidean space determined by SDE. We show the existence of the smooth density $p(s,x;t,y)$ of its transition probability and its short time asymptotics as $t?s\to 0$. Assumptions required for these facts are relaxed considerably from past works by Picard and Ishikawa–Kunita. We show these facts using Malliavin calculus on Poisson space. Our calculus is simpler and more efficient than previous works.     

Integrating DEA-oriented performance assessment and target setting using interactive MOLP methods

Data envelopment analysis (DEA) and multiple objective linear programming (MOLP) are tools that can be used in management control and planning. Whilst these two types of model are similar in structure, DEA is directed to assessing past performances as part of management control function and MOLP to planning future performance targets. This paper is devoted to investigating equivalence models and interactive tradeoff analysis procedures in MOLP, such that DEA-oriented performance assessment and target setting can be integrated in a way that the decision makers’ preferences can be taken into account in an interactive fashion. Three equivalence models are investigated between the output-oriented dual DEA model and the minimax reference point formulations, namely the super-ideal point model, the ideal point model and the shortest distance model. These models can be used to support efficiency analysis in the same way as the conventional DEA model does and also support tradeoff analysis for setting target values by individuals or groups. A case study is conducted to illustrate how DEA-oriented efficiency analysis can be conducted using the MOLP methods and how such performance assessment can be integrated into an interactive procedure for setting realistic target values.     

Multiple solutions for semilinear elliptic boundary value problems with double resonance

In this paper we study the multiplicity of nontrivial solutions of semilinear elliptic boundary value problems which may be double resonance near infinity between two consecutive eigenvalues of −Δ with zero Dirichlet boundary data. The methods we use here are Morse theory, minimax methods and bifurcation theory.     

The density of infinitely differentiable functions in Sobolev spaces with mixed boundary conditions

We present a detailed proof of the density of the set in the space of test functions V ⊂ H ¹ (Ω) that vanish on some part of the boundary ∂Ω of a bounded domain Ω. This work was supported by the grants GAČR 201/03/0570 and MSM 262100001.