Asymptotics for risk capital allocations based on Conditional Tail Expectation

An investigation of the limiting behavior of a risk capital allocation rule based on the Conditional Tail Expectation (CTE) risk measure is carried out. More specifically, with the help of general notions of Extreme Value Theory (EVT), the aforementioned risk capital allocation is shown to be asymptotically proportional to the corresponding Value-at-Risk (VaR) risk measure. The existing methodology acquired for VaR can therefore be applied to a somewhat less well-studied CTE. In the context of interest, the EVT approach is seemingly well-motivated by modern regulations, which openly strive for the excessive prudence in determining risk capitals.     

Conditional Tail Expectation Decomposition and Conditional Mean Risk Sharing for Dependent and Conditionally Independent Losses

Methodology and Computing in Applied Probability - Conditional tail expectations are often used in risk measurement and capital allocation. Conditional mean risk sharing appears to be effective in...     

Conditional Distributions and Characterizations of Multivariate Stable Distribution

Several characterizations of multivariate stable distribution are given based on identically distributed random vectors and conditional multivariate stable distribution.     

A Formula for the Tail Probability of a Multivariate Normal Distribution and Its Applications

An exact asymptotic formula for the tail probability of a multivariate normal distribution is derived. This formula is applied to establish two asymptotic results for the maximum deviation from the mean: the weak convergence to the Gumbel distribution of a normalized maximum deviation and the precise almost sure rate of growth of the maximum deviation. The latter result gives rise to a diagnostic tool for checking multivariate normality by a simple graph in the plane. Some simulation results are presented.     

Error Bounds for Asymptotic Expansion of the Conditional Variance of the Scale Mixtures of the Multivariate Normal Distribution

Let X= A ^1/2 G be a scale mixture of a multivariate normal distribution with X, G ⁿ , Gis a multivariate normal vector, and A is a positive random variable independent of the multivariate random vector G. This study presents asymptotic results of the conditional variance-covariance, Cov(X ₂|X ₁), X ₁ ^m , m < n, under some moment expressions. A new representation form is also presented for conditional expectation of the scale variable on the random vector X ₁ ^m , m < n. Both the asymptotic expression and the representation are manageable and in computable form. Finally, an example is presented to illustrate how the computations are carried out.     

Products of Conditional Expectation Operators: Convergence and Divergence

Journal of Theoretical Probability - In this paper, we investigate the convergence of products of conditional expectation operators. We show that if $$(\Omega ,\mathcal {F},P)$$ is a probability...     

第二类Stirling数的一些同余式

设k,n为非负整数,S(n,k)表示第二类Stirling数.本文研究了S(n,k)模2的方幂的同余式,首先给出了一类二项式系数模2的同余式,然后利用上述结果得到了S(n,a2~m+b)模2~m的同余式.其表达式均由简单二项式系数组成,其中m≥3,b=0,1,2.这些结果改进了Chan和Manna的结果.     

三参数广义帕累托分布的似然矩估计

广义帕累托分布(GPD)在极值统计的POT模型中常常被用来逼近超过阈值u的超出量X_i-u的分布. 为解决经典估计方法存在的问题, Zhang (Zhang J, Likelihood moment estimation for the generalized Pareto distribution, Aust N Z J Stat, 2007, 49:69--77) 对两参数GPD (GP2)提出一种新的估计方法------似然矩估计(LM), 它容易计算且具有较高的渐近有效性. 本文将此方法从两参数的情形推广到三参数GPD (GP3), 结果表明尺度参数和形状参数估计的渐近性质与以上所提到的文章完全相同. 针对GP3的LM估计也具有总是存在、易于计算以及 对绝大多数的形状参数具有接近于最小的偏差和均方误差的特点.     

Robust and Efficient Estimation for the Generalized Pareto Distribution

In this article we implement the minimum density power divergence estimator (MDPDE) for the shape and scale parameters of the generalized Pareto distribution (GPD). The MDPDE is indexed by a constant 0 that controls the trade-off between robustness and efficiency. As increases, robustness increases and efficiency decreases. For = 0 the MDPDE is equivalent to the maximum likelihood estimator (MLE). We show that for > 0 the MDPDE for the GPD has a bounded influence function. For < 0.2 the MDPDE maintains good asymptotic relative efficiencies, usually above 90%. The results from a Monte Carlo study agree with these asymptotic calculations. The MDPDE is asymptotically normally distributed if the shape parameter is less than (1 + )/(2 + ), and estimators for standard errors are readily computed under this restriction. We compare the MDPDE, MLE, Dupuis optimally-biased robust estimator (OBRE), and Peng and Welshs Medians estimator for the parameters. The simulations indicate that the MLE has the highest efficiency under uncontaminated GPDs. However, for the GPD contaminated with gross errors OBRE and MDPDE are more efficient than the MLE. For all the simulated models that we studied the Medians estimator had poor performance.AMS 2000 Subject Classification. Primary—62F35, Secondary—62G35     

广义Pareto分布参数的经验Bayes检验问题

讨论了独立同分布样本情形广义Pareto分布参数的经验Bayes单侧检验问题,利用核密度函数的核估计构造了参数的经验Bayes检验函数,并在一定的条件下,证明了此经验Bayes检验函数的渐近最优性,获得了其收敛速度.     

Robust Estimation of the Generalized Pareto Distribution

One approach used for analyzing extremes is to fit the excesses over a high threshold by a generalized Pareto distribution. For the estimation of the shape and scale parameters in the generalized Pareto distribution, under some restrictions on the value of the scale parameter, maximum likelihood, method of moments and probability weighted moments' estimators are available. However, these are not robust estimators. In this paper we implement a robust estimation procedure known as the method of medians (He and Fung, 1999) to estimate the parameters in the generalized Pareto distribution. The asymptotic distribution of our estimator is normal for any value of the shape parameter except –1.     

Multivariate risk sharing and the derivation of individually rational Pareto optima

Considering that a natural way of sharing risks in insurance companies is to require risk by risk Pareto optimality, we offer in case of strong risk aversion, a simple computable method for deriving all Pareto optima. More importantly all Individually Rational Pareto optima can be computed according to our method.     

Bi-axial Gegenbauer Functions of the Second Kind

Bi-axially symmetric monogenic generating functions on ^{p + q} have been used recently to define generalisations of Gegenbauer polynomials. These polynomials are orthogonal on the unit ball in ^p. Generalised Cauchy transforms of these polynomials are used to define corresponding bi-axial Gegenbauer functions of the second kind. It is demonstrated that these functions of the second kind satisfy second order differential equations related to those satisfied by the corresponding bi-axial Gegenbauer polynomials.     

多元线性模型中条件最优预测的稳健性

研究了多元线性模型中条件最优线性无偏预测的稳健性问题,得到了条件线性可预测变量的这种预测关于协方差矩阵具有稳健性的充要条件.     

On Stirling Functions of the Second Kind

We continue the study of Stirling functions of the second kind, S(α,k) where α > 0 is real. In particular, we consider integral and fractional derivative representations of the S(α,k), as well as connections with generalized Bernoulli and Stirling polynomials.     

Peeking Inside the Black Box: Visualizing Statistical Learning With Plots of Individual Conditional Expectation

This article presents individual conditional expectation (ICE) plots, a tool for visualizing the model estimated by any supervised learning algorithm. Classical partial dependence plots (PDPs) help visualize the average partial relationship between the predicted response and one or more features. In the presence of substantial interaction effects, the partial response relationship can be heterogeneous. Thus, an average curve, such as the PDP, can obfuscate the complexity of the modeled relationship. Accordingly, ICE plots refine the PDP by graphing the functional relationship between the predicted response and the feature for individual observations. Specifically, ICE plots highlight the variation in the fitted values across the range of a covariate, suggesting where and to what extent heterogeneities might exist. In addition to providing a plotting suite for exploratory analysis, we include a visual test for additive structure in the data-generating model. Through simulated examples and real datasets, we demonstrate how ICE plots can shed light on estimated models in ways PDPs cannot. Procedures outlined are available in the R package ICEbox.     

Pareto分布环境因子的估计及其应用

给出了Pareto分布环境因子的定义,讨论了在定数截尾样本下Pareto分布环境因子的极大似然估计和修正极大似然估计,并尝试把环境因子用于可靠性评估中.最后运用Monte Carlo方法对极大似然估计,修正极大似然估计和可靠性指标的均方误差(MSE),进行了模拟比较,结果表明修正极大似然估计优于极大似然估计且考虑环境因子的可靠性评估结果较好.     

球面上第二类Fredholm积分方程配置方法

球面上第二类 Fredholm积分方程经球坐标变换可化为矩形域 H0 上的问题求解 .用有限元法构造H0 上的插值函数 ,它必须满足在 H0 的左、右两边连续 ,然后用配置方程求方程的近似解