Maximum likelihood estimation of stable Paretian models

Stable Paretian distributions have attractive properties for empirical modeling in finance, because they include the normal distribution as a special case but can also allow for heavier tails and skewness. A major reason for the limited use of stable distributions in applied work is due to the facts that there are, in general, no closed-form expressions for its probability density function and that numerical approximations are nontrivial and computationally demanding. Therefore, Maximum Likelihood (ML) estimation of stable Paretian models is rather difficult and time consuming. Here, we study the problem of ML estimation using fast Fourier transforms to approximate the stable density functions. The performance of the ML estimation approach is investigated in a Monte Carlo study and compared to that of a widely used quantile estimator. Extensions to more general distributional models characterized by time-varying location and scale are discussed.     

The impact of fat tailed returns on asset allocation

This paper analyzes the asset allocation problem of an investor who can invest in equity and cash when there is time variation in expected returns on the equity. The solution methodology is multistage stochastic asset allocation problem with decision rules. The uncertainty is modeled using economic scenarios with Gaussian and stable Paretian non-Gaussian innovations. The optimal allocations under these alternative hypothesis are compared. Our computational results suggest that asset allocation may be up to 20% different depending on the utility function and the risk aversion level of the investor. Certainty equivalent return can be increased up to .13% and utility can be improved up to .72% by switching to the stable Paretian model.     

基于贝叶斯极值估计的商业银行内部欺诈风险度量研究

内部欺诈风险是我国商业银行面临的一个重大风险来源。本文针对内部欺诈具有的低频率高损失的特点,采用不同分布分段刻画其损失统计分布规律,对于低于和高于门限值的样本点,采用Box-Cox变换和全Paretian分布模型进行分析,然后采用贝叶斯估计对全Paretian分布模型的参数进行估计,接着在此基础上对建立了一个内部欺诈风险度量模型,然后使用所构建的风险度量模型对操作风险在险风险值、经济资本和最大可能损失进行了测算,最后提出了防范操作风险的政策建议。     

Computing the probability density function of the stable Paretian distribution

The practical implementation of a stable Paretian model is a nontrivial task, because—with the exception of a few special cases—its probability density function cannot be expressed analytically. Here, we present an algorithm for calculating the probability density function of the asymmetric stable Paretian distribution. Due to the use of the Fast Fourier Transform, the algorithm is computationally efficient, easily implemented, and of similar accuracy as existing algorithms.     

Geometric stable laws: Estimation and applications

Geometric stable laws constitute a class of limiting distributions of appropriately normalized random sums of i.i.d. random variables. We consider the problem of estimation of the parameters of univariate and multivariate geometric stable laws. Our estimation technique is based on the method of moments and yields consistent and asymptotically normal estimators. We apply our estimators to a currency exchange data and show that the geometric stable dominates Paretian stable and normal models.     

关于二项分布的极限分布的一个注记

讨论了用泊松分布和正态分布近似表示二项分布的精确程度问题,对于泊松分布,指出了它对二项分布B(n,p)的概率值的近似精确与否基本上只依赖于参数p而不依赖于n,并说明了经验条件“np≤5”的不确切.     

A Capital Asset Pricing Model Under Stable Paretian Distributions in a Pure Exchange Economy

In this paper, we established a Capital Asset Pricing Model (CAPM) subject to the assumption that the asset return rates obey symmetric stable Paretian distributions. This assumption seems to be closer to reality than the standard ones such as normality or finite variance. Conclusion similar to the original CAPM formula is drawn in this paper.     

Modeling fat tails in stock returns: a multivariate stable-GARCH approach

In this paper a new multivariate volatility model is proposed. It combines the appealing properties of the stable Paretian distribution to model the heavy tails with the GARCH model to capture the volatility clustering. Returns on assets are assumed to follow a sub-Gaussian distribution, which is a particular multivariate stable distribution. In this way the characteristic function of the fitted returns has a tractable expression and the density function can be recovered by numerical methods. A multivariate GARCH structure is then adopted to model the covariance matrix of the Gaussian vectors underlying the sub-Gaussian system. The model is applied to a bivariate series of daily U.S. stock returns. Value-at-risk for long and short positions is computed and compared with the one obtained using the multivariate normal and the multivariate Student’s t distribution. Finally, exploiting the recent developments in the vast dimensional time-varying covariances modeling, possible feasible extensions of our model to higher dimensions are suggested and an illustrative example using the Dow Jones index components is presented.     

Method-of-moments estimators of stable distribution parameters

A method is developed for estimating the four parameters of stable Paretian distributions. Based on a procedure proposed by an earlier researcher but never developed, the method proves to be mathematically simple and easy to apply. The method is extensively tested and sampling properties of the estimators are studied, providing approximate confidence intervals for the estimators.     

Extended anonymity and Paretian relations on infinite utility streams

We examine the range of anonymity that is compatible with a Paretian social welfare relation (SWR) on infinite utility streams. Three alternative coherence properties of an SWR are considered, namely, acyclicity, quasi-transitivity, and Suzumura consistency. For each case, we show that a necessary and sufficient condition for a set of permutations to be the set of permissible permutations of some Paretian SWR is given by the cyclicity of permutations and a weakening of group structure. Further, for each case of coherence property, we show that the extended Pareto rule is the least element of the class of anonymous Paretian SWRs.     

两个未知均值方差混合模型有限制下对数极大似然比的极限分布

本文讨论了检验样本是来自一个正态总体还是两个未知均值和方差的正态的混合分布,采用对数极大似然比的检验,如果不加限制,Ｈａｒｔｉｎｇａｎｍ曾指出不是寻找的、Ｘ＾２分布,我们在混合的中了一点后得到了其极限分布产工给出了分位点数值表。     

A simple estimator for the characteristic exponent of the stable Paretian distribution

A new estimator for the index α of symmetric stable Paretian distributions is proposed. In addition to being numerically simple, fast, and reliable, simulations suggest that even in small samples, the estimator is unbiased for α ∈ [1, 2], and almost exactly normally distributed. Standard errors are also given, the magnitudes of which are shown to depend only on sample size and not on the true value of α.     

W.F. Sheppard's correspondence with Karl Pearson and the development of his tables and moment estimates

As a new statistician, W.F. Sheppard wrote a series of letters to the leading statistician of the day, Karl Pearson, for his advice. Written a century ago and spanning three decades, the letters provide a glimpse into the development of two significant contributions to statistics: the normal probability tables, and the corrections of moment estimates. Sheppard's normal probability tables were the first set of modern tables for the standard normal distribution and have been widely used since the twentieth century. We provide an examination of the statistical research carried out by Sheppard and Pearson in the context of their correspondence.     

威布尔分布或极值分布可靠度置信下限的回归表示

对于定数截尾样本，给出了基于极值分布的位置和尺度参数的最好线性无偏估计（BLUE），获得了威布尔分布的可靠度的点估计和置信限之间的回归模型，从而可由威布 尔可靠度的点估计根据回归方程得到可靠度的置信下限，省去了大量的用表，为实际工作者带来了极大的方便，计算结果表明，回归方程有很高的精度。     

虚拟企业收益分配问题博弈研究

收益分配是虚拟企业的一个敏感而关键的问题，制订合理的收益分配方案对虚拟企业无疑是重要的。本分析了虚拟企业收益分配过程应考虑的因素，在此基础上应用博弈论建立了虚拟企业收益分配的博弈模型，并进行了相关分析，所得出的结论可用于虚拟企业的收益分配策略的制订，最后，论证了在虚拟企业中进行合作机制设计和应用的必要性。     

On bounds for the mode and median of the generalized hyperbolic and related distributions

Except for certain parameter values, a closed form formula for the mode of the generalized hyperbolic (GH) distribution is not available. In this paper, we exploit results from the literature on modified Bessel functions and their ratios to obtain simple but tight two-sided inequalities for the mode of the GH distribution for general parameter values. As a special case, we deduce tight two-sided inequalities for the mode of the variance-gamma (VG) distribution, and through a similar approach we also obtain tight two-sided inequalities for the mode of the McKay Type I distribution. The analogous problem for the median is more challenging, but we conjecture some monotonicity results for the median of the VG and McKay Type I distributions, from we which we conjecture some tight two-sided inequalities for their medians. Numerical experiments support these conjectures and also lead us to a conjectured tight lower bound for the median of the GH distribution.     

Incomplete Dirichlet integrals with applications to ordered uniform spacings

Sobel, Uppuluri, and Frankowski, (Selected Tables in Math. Statistics, Vol. IV. Amer. Math. Soc., Providence, R. I., 1977) consider an incomplete Dirichlet integral of type I with several interesting applications connected with the multinomial distribution and provide tables of this integral along with other useful tables. Two incomplete Dirichlet integrals are discussed here along with some useful recurrence relations, providing simple methods of deriving the distribution theory of ordered uniform spacings.     

Fast and Accurate Computation of the Exact Confidence Limits for the Common Odds Ratio in Several 2 × 2 Tables

Abstract

Mehta, Patel, and Gray proposed a network algorithm for the exact confidence limits of the common odds ratio in several 2 × 2 tables. Their algorithm was implemented in the StatXact and Egret statistical packages and further discussed by Vollset, Hirji, and Elashoff. The need to evaluate polynomials of potentially very high degrees, however, poses some numerical difficulties. This article presents a method that cuts the degree of polynomials by at least one half. Two other modifications to further stabilize and speed the computation are also proposed.