Elliptically Contoured Models in Statistics and Portfolio Theory

Gupta, Arjun K.; Varga, Tamás; Bodnar, Taras

doi:10.1007/978-1-4614-8154-6

Cited by 115 publications

(80 citation statements)

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“…We note that the first change of variables in [6], y = (x − µ) A −1 , transforms the distribution into a spherical distribution with the same generating function [39] (Corollary 2.1 and Definition 2.2). The spherical distribution is invariant to rotations like the second change of variables, y = zR, where R is the rotation in [6].…”

Section: Accurate Es In the General Elliptical Casementioning

confidence: 99%

Accurate Evaluation of Expected Shortfall for Linear Portfolios with Elliptically Distributed Risk Factors

Dobrev

Nesmith

2017

JRFM

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Abstract:We provide an accurate closed-form expression for the expected shortfall of linear portfolios with elliptically distributed risk factors. Our results aim to correct inaccuracies that originate in Kamdem (2005) and are present also in at least thirty other papers referencing it, including the recent survey by Nadarajah et al. (2014) on estimation methods for expected shortfall. In particular, we show that the correction we provide in the popular multivariate Student t setting eliminates understatement of expected shortfall by a factor varying from at least four to more than 100 across different tail quantiles and degrees of freedom. As such, the resulting economic impact in financial risk management applications could be significant. We further correct such errors encountered also in closely related results in Kamdem (2007 and) for mixtures of elliptical distributions. More generally, our findings point to the extra scrutiny required when deploying new methods for expected shortfall estimation in practice.

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Section: Accurate Es In the General Elliptical Casementioning

confidence: 99%

Accurate Evaluation of Expected Shortfall for Linear Portfolios with Elliptically Distributed Risk Factors

Dobrev

Nesmith

2017

JRFM

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“…In this section the aim is to fit the specific parametric model for the density ϕ g,K,µ to the data by estimating the parameters θ and µ where ϕ g,K,µ is given according to (5) and (4) with f R = f θ . Therefore, the two models [1] and [2] fulfill the condition lim r→0+0 g (r) = 0 which ensures the differentiability of the density ϕ g,K,µ at zero.…”

Section: Parametric Estimatorsmentioning

confidence: 99%

“…The modelling of MSCI data using elliptical models is considered in [5]. The data are depicted in Figure 12.…”

Section: Applicationsmentioning

confidence: 99%

“…The book [4] by Fang and Anderson contains a big chapter about statistical inference of elliptically contoured distributions. The theory of elliptically contoured distributions including applications to portfolio theory is presented in the monograph by [5] Gupta et al On combining the advantages of several estimators, semiparametric density estimators for elliptical distributions were derived in papers [6][7][8] by Stute and Werner, by Cui and He and by Liebscher. In [9] Battey and Linton considered a density estimator for elliptical distributions based on Gaussian mixture sieves.…”

Section: Introductionmentioning

confidence: 99%

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Estimation of Star-Shaped Distributions

Liebscher

Richter

2016

Risks

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Scatter plots of multivariate data sets motivate modeling of star-shaped distributions beyond elliptically contoured ones. We study properties of estimators for the density generator function, the star-generalized radius distribution and the density in a star-shaped distribution model. For the generator function and the star-generalized radius density, we consider a non-parametric kernel-type estimator. This estimator is combined with a parametric estimator for the contours which are assumed to follow a parametric model. Therefore, the semiparametric procedure features the flexibility of nonparametric estimators and the simple estimation and interpretation of parametric estimators. Alternatively, we consider pure parametric estimators for the density. For the semiparametric density estimator, we prove rates of uniform, almost sure convergence which coincide with the corresponding rates of one-dimensional kernel density estimators when excluding the center of the distribution. We show that the standardized density estimator is asymptotically normally distributed. Moreover, the almost sure convergence rate of the estimated distribution function of the star-generalized radius is derived. A particular new two-dimensional distribution class is adapted here to agricultural and financial data sets.

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“…The new distribution is defined as a scale mixture of the matrix variate normal distribution and the uniform distribution. Note that the details about the matrix variate normal distribution and the scale mixture of matrix normal distribution can be found in [10,11]. We give the density function of the proposed distribution and show that the linear transformation of a matrix variate slash distributed random matrix has again a matrix variate slash distribution.…”

Section: Introductionmentioning

confidence: 97%

Matrix variate slash distribution

Bulut

Arslan

2015

Journal of Multivariate Analysis

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Please cite this article as: Y.M. Bulut, O. Arslan, Matrix variate slash distribution, Journal of Multivariate Analysis (2015), http://dx. AbstractIn this paper, we introduce a matrix variate slash distribution as a scale mixture of the matrix variate normal and the uniform distributions. We study some properties of the proposed distribution and give maximum likelihood (ML) estimators of its parameters using EM algorithm. We provide an iteratively reweighting algorithm to compute the ML estimates. Also, we give a small simulation study to show performance of the algorithm.

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Elliptically Contoured Models in Statistics and Portfolio Theory

Cited by 115 publications

References 0 publications

Accurate Evaluation of Expected Shortfall for Linear Portfolios with Elliptically Distributed Risk Factors

Accurate Evaluation of Expected Shortfall for Linear Portfolios with Elliptically Distributed Risk Factors

Estimation of Star-Shaped Distributions

Matrix variate slash distribution

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