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Mazur, S., Otryakhin, D. & Podolskij, M. (2020). Estimation of the linear fractional stable motion. Bernoulli, 26(1), 226-252
Open this publication in new window or tab >>Estimation of the linear fractional stable motion
2020 (English)In: Bernoulli, ISSN 1350-7265, E-ISSN 1573-9759, Vol. 26, no 1, p. 226-252Article in journal (Other academic) Published
Abstract [en]

In this paper, we investigate the parametric inference for the linear fractional stable motion in high and low frequency setting. The symmetric linear fractional stable motion is a three-parameter family, which constitutes a natural non-Gaussian analogue of the scaled fractional Brownian motion. It is fully characterised by the scaling parameter σ>0, the self-similarity parameter H∈(0,1) and the stability index α∈(0,2) of the driving stable motion. The parametric estimation of the model is inspired by the limit theory for stationary increments Lévy moving average processes that has been recently studied in (Ann. Probab. 45 (2017) 4477–4528). More specifically, we combine (negative) power variation statistics and empirical characteristic functions to obtain consistent estimates of (σ,α,H). We present the law of large numbers and some fully feasible weak limit theorems.

Place, publisher, year, edition, pages
The International Statistical Institute, 2020
Keywords
Fractional processes, limit theorems, parametric estimation, stable motion
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:oru:diva-65216 (URN)10.3150/19-BEJ1124 (DOI)000499083900008 ()2-s2.0-85076579187 (Scopus ID)
Note

Funding Agencies:

Project "Ambit fields: probabilistic properties and statistical inference" - Villum Fonden  

Danmarks Grundforskningsfond

Örebro University  

Project "Models for macro and financial economics after the financial crisis" - Jan Wallander and Tom Hedelius Foundation  P18-0201

Available from: 2018-02-25 Created: 2018-02-25 Last updated: 2020-03-17Bibliographically approved
Javed, F., Mazur, S. & Ngailo, E. (2020). Higher order moments of the estimated tangency portfolio weights. Journal of Applied Statistics
Open this publication in new window or tab >>Higher order moments of the estimated tangency portfolio weights
2020 (English)In: Journal of Applied Statistics, ISSN 0266-4763, E-ISSN 1360-0532, , p. 18Article in journal (Other academic) Epub ahead of print
Abstract [en]

In this paper, we consider the estimated weights of the tangency portfolio. We derive analytical expressions for the higher order non-central and central moments of these weights when the returns are assumed to be independently and multivariate normally distributed. Moreover, the expressions for mean, variance, skewness and kurtosis of the estimated weights are obtained in closed forms. Later, we complement our results with a simulation study where data from the multivariate normal and t-distributions are simulated, and the first four moments of estimated weights are computed by using the Monte Carlo experiment. It is noteworthy to mention that the distributional assumption of returns is found to be important, especially for the first two moments. Finally, through an empirical illustration utilizing returns of four financial indices listed in NASDAQ stock exchange, we observe the presence of time dynamics in higher moments.

Place, publisher, year, edition, pages
Taylor & Francis, 2020. p. 18
Keywords
Tangency portfolio, higher order moments, Wishart distribution
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:oru:diva-57933 (URN)10.1080/02664763.2020.1736523 (DOI)000518525900001 ()
Funder
The Jan Wallander and Tom Hedelius Foundation, P18-0201
Note

Funding Agency:

Örebro University

Available from: 2017-06-07 Created: 2017-06-07 Last updated: 2020-03-20Bibliographically approved
Gulliksson, M. & Mazur, S. (2019). An Iterative Approach to Ill-Conditioned Optimal Portfolio Selection. Computational Economics
Open this publication in new window or tab >>An Iterative Approach to Ill-Conditioned Optimal Portfolio Selection
2019 (English)In: Computational Economics, ISSN 0927-7099, , p. 21Article in journal (Refereed) Epub ahead of print
Abstract [en]

Covariance matrix of the asset returns plays an important role in the portfolioselection. A number of papers is focused on the case when the covariance matrixis positive definite. In this paper, we consider portfolio selection with a singu-lar covariance matrix. We describe an iterative method based on a second orderdamped dynamical systems that solves the linear rank-deficient problem approxi-mately. Since the solution is not unique, we suggest one numerical solution that canbe chosen from the iterates that balances the size of portfolio and the risk. The nu-merical study confirms that the method has good convergence properties and givesa solution as good as or better than the constrained least norm Moore-Penrose solu-tion. Finally, we complement our result with an empirical study where we analyzea portfolio with actual returns listed in S&P 500 index.

Place, publisher, year, edition, pages
Springer, 2019. p. 21
Keywords
Mean-variance portfolio, singular covariance matrix, linear ill-posed problems, second order damped dynamical systems
National Category
Probability Theory and Statistics Economics Other Mathematics
Research subject
Statistics; Economics; Mathematics
Identifiers
urn:nbn:se:oru:diva-74364 (URN)10.1007/s10614-019-09943-6 (DOI)
Available from: 2019-05-22 Created: 2019-05-22 Last updated: 2019-11-18Bibliographically approved
Bodnar, T., Mazur, S. & Parolya, N. (2019). Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix-variate location mixture of normal distributions. Scandinavian Journal of Statistics, 46(2), 636-660
Open this publication in new window or tab >>Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix-variate location mixture of normal distributions
2019 (English)In: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 46, no 2, p. 636-660Article in journal (Refereed) Published
Abstract [en]

In this paper we consider the asymptotic distributions of functionals of the sample covariance matrix and the sample mean vector obtained under the assumption that the matrix of observations has a matrix-variate location mixture of normal distributions. The central limit theorem is derived for the product of the sample covariance matrix and the sample mean vector. Moreover, we consider the product of the inverse sample covariance matrix and the mean vector for which the central limit theorem is established as well. All results are obtained under the large-dimensional asymptotic regime where the dimension p and the sample size n approach to infinity such that p/n → c ∈ [0, +∞) when the sample covariance matrix does not need to be invertible and p/n → c ∈ [0, 1) otherwise.

Place, publisher, year, edition, pages
John Wiley & Sons, 2019
Keywords
Normal mixtures, skew normal distribution, large dimensional asymptotics, stochas- tic representation, random matrix theory
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:oru:diva-61246 (URN)10.1111/sjos.12383 (DOI)000465606900012 ()2-s2.0-85061927074 (Scopus ID)
Funder
Swedish Research Council, 2013-5180Riksbankens Jubileumsfond, P13-1024: 1
Available from: 2017-10-04 Created: 2017-10-04 Last updated: 2019-06-19Bibliographically approved
Bodnar, T., Mazur, S., Ngailo, E. & Parolya, N. (2019). Discriminant analysis in small and large dimensions. Theory of Probability and Mathematical Statistics, 100, 24-42
Open this publication in new window or tab >>Discriminant analysis in small and large dimensions
2019 (English)In: Theory of Probability and Mathematical Statistics, ISSN 1547-7363, Vol. 100, p. 27p. 24-42Article in journal (Other academic) Published
Abstract [en]

We study the distributional properties of the linear discriminant function under the assumption of normality by comparing two groups with the same covariance matrix but different mean vectors. A stochastic representation for the discriminant function coefficients is derived, which is then used to obtain their asymptotic distribution under the high-dimensional asymptotic regime. We investigate the performance of the classification analysis based on the discriminant function in both small and large dimensions. A stochastic representation is established, which allows to compute the error rate in an efficient way. We further compare the calculated error rate with the optimal one obtained under the assumption that the covariance matrix and the two mean vectors are known. Finally, we present an analytical expression of the error rate calculated in the high-dimensional asymptotic regime. The finite-sample properties of the derived theoretical results are assessed via an extensive Monte Carlo study.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2019. p. 27
Keywords
discriminant function, stochastic representation, large-dimensional asymp- totics, random matrix theory, classification analysis
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:oru:diva-61248 (URN)000493468200003 ()
Note

Funding Agencies:

Swedish International Development Cooperation Agency (SIDA) through the TZ-Sweden Programme for Research, Higher Education and Institutional Advancement  

Örebro University  

project "Models for macro and financial economics after the financial crisis" - Jan Wallander and Tom Hedelius Foundation  P18-0201

Available from: 2017-10-04 Created: 2017-10-04 Last updated: 2019-11-21Bibliographically approved
Karlsson, S. & Mazur, S. (2019). Flexible Fat-tailed BVARs. In: : . Paper presented at 10th European Seminar on Bayesian Econometrics, St Andrews, Scotland, September 2-3, 2019.
Open this publication in new window or tab >>Flexible Fat-tailed BVARs
2019 (English)Conference paper, Oral presentation only (Refereed)
Abstract [en]

We propose a general class of fat-tailed distributions which includes the t,Cauchy, Laplace and slash distributions as well as the normal distribution as spe-cial cases. Full conditional posterior distributions for the Bayesian VAR-model arederived and used to construct a MCMC-sampler for the joint posterior distribution.The framework allows for selection of a specic special case as the distribution forthe error terms in the VAR if the evidence in the data is strong while at the sametime allowing for considerable exibility and more general distributions than oeredby any of the special cases.

National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:oru:diva-76718 (URN)
Conference
10th European Seminar on Bayesian Econometrics, St Andrews, Scotland, September 2-3, 2019
Funder
The Jan Wallander and Tom Hedelius Foundation, P18-0201
Available from: 2019-09-24 Created: 2019-09-24 Last updated: 2019-09-30Bibliographically approved
Mazur, S. & Otryakhin, D. (2019). Linear Fractional Stable Motion with the RLFSM R Package. Örebro, Sweden: Örebro University, School of Business
Open this publication in new window or tab >>Linear Fractional Stable Motion with the RLFSM R Package
2019 (English)Report (Other academic)
Abstract [en]

Linear fractional stable motion is a type of a stochastic integral driven by symmetric alpha-stable Levy motion. The integral could be considered as a non-Gaussian analogue of the fractional Brownian motion. The present paper discusses R package rlfsm created for numerical procedures with the linear fractional stable motion. It is a set of tools for simulation of these processes as well as performing statistical inference and simulation studies on them. We introduce: tools that we developed to work with that type of motions as well as methods and ideas underlying them. Also we perform numerical experiments to show finite-sample behavior of certain estimators of the integral, and give an idea of how to envelope workflow related to the linear fractional stable motion in S4 classes and methods. Supplementary materials, including codes for numerical experiments, are available online. rlfsm could be found on CRAN and gitlab.

Place, publisher, year, edition, pages
Örebro, Sweden: Örebro University, School of Business, 2019. p. 32
Series
Working Papers, School of Business, ISSN 1403-0586 ; 9/2019
Keywords
Fractional processes, limit theorems, parametric estimation, stochastic simulation, stable motion
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:oru:diva-77935 (URN)
Available from: 2019-11-18 Created: 2019-11-18 Last updated: 2019-11-18Bibliographically approved
Bodnar, T., Mazur, S., Podgorski, K. & Tyrcha, J. (2019). Tangency portfolio weights for singular covariance matrix in small and large dimensions: estimation and test theory. Journal of Statistical Planning and Inference, 201, 40-57
Open this publication in new window or tab >>Tangency portfolio weights for singular covariance matrix in small and large dimensions: estimation and test theory
2019 (English)In: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171, Vol. 201, p. 28p. 40-57Article in journal (Refereed) Published
Abstract [en]

In this paper we derive the finite-sample distribution of the estimated weights of the tangency portfolio when both the population and the sample covariance matrices are singular. These results are used in the derivation of a statistical test on the weights of the tangency portfolio where the distribution of the test statistic is obtained under both the null and the alternative hypotheses. Moreover, we establish the high-dimensional asymptotic distribution of the estimated weights of the tangency portfolio when both the portfolio dimension and the sample size increase to infinity. The theoretical findings are implemented in an empirical application dealing with the returns on the stocks included into the S&P 500 index. 

Place, publisher, year, edition, pages
Elsevier, 2019. p. 28
Keywords
Tangency portfolio, singular Wishart distribution, singular covariance matrix, high-dimensional asymptotics, hypothesis testing
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:oru:diva-63498 (URN)10.1016/j.jspi.2018.11.003 (DOI)000459528700004 ()2-s2.0-85058549449 (Scopus ID)
Funder
Swedish Research Council, 2008-5382
Note

Funding Agencies:

Örebro University, Sweden  

Project "Models for macro and financial economics after the financial crisis" - Jan Wallander and Tom Hedelius Foundation, Sweden  P18-0201 

Available from: 2017-12-20 Created: 2017-12-20 Last updated: 2019-06-18Bibliographically approved
Bauder, D., Bodnar, T., Mazur, S. & Okhrin, Y. (2018). Bayesian inference for the tangent portfolio. International Journal of Theoretical and Applied Finance, 21(8), Article ID 1850054.
Open this publication in new window or tab >>Bayesian inference for the tangent portfolio
2018 (English)In: International Journal of Theoretical and Applied Finance, ISSN 0219-0249, Vol. 21, no 8, p. 25article id 1850054Article in journal (Other academic) Published
Abstract [en]

In this paper we consider the estimation of the weights of tangent portfolios from the Bayesian point of view assuming normal conditional distributions of the logarithmic returns. For diffuse and conjugate priors for the mean vector and the covariance matrix, we derive stochastic representations for the posterior distributions of the weights of tangent portfolio and their linear combinations. Separately we provide the mean and variance of the posterior distributions, which are of key importance for portfolio selection. The analytic results are evaluated within a simulation study, where the precision of coverage intervals is assessed. 

Place, publisher, year, edition, pages
World Scientific Publishing Co. Pte. Ltd., 2018. p. 25
Keywords
Asset allocation, tangent portfolio, Bayesian analysis, diffuse and conjugate priors, stochastic representation
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:oru:diva-63496 (URN)10.1142/S0219024918500541 (DOI)000455592700006 ()2-s2.0-85056810852 (Scopus ID)
Funder
Swedish Research CouncilThe Jan Wallander and Tom Hedelius Foundation, P18-0201
Note

Funding Agency:

German Science Foundation (DFG)  BO 3521/3-1  SCHM 859/13-1

Available from: 2017-12-20 Created: 2017-12-20 Last updated: 2019-01-29Bibliographically approved
Bodnar, T., Mazur, S., Muhinyuza, S. & Parolya, N. (2018). On the product of a singular Wishart matrix and a singular Gaussian vector in high dimensions. Theory of Probability and Mathematical Statistics, 99, 37-50
Open this publication in new window or tab >>On the product of a singular Wishart matrix and a singular Gaussian vector in high dimensions
2018 (English)In: Theory of Probability and Mathematical Statistics, ISSN 0094-9000, Vol. 99, p. 37-50Article in journal (Refereed) Published
Abstract [en]

In this paper we consider the product of a singular Wishart random matrix and a singular normal random vector. A very useful stochastic representation is derived for this product, in using which the characteristic function of the product and its asymptotic distribution under the double asymptotic regime are established. The application of obtained stochastic representation speeds up the simulation studies where the product of a singular Wishart random matrix and a singular normal random vector is present. We further document a good performance of the derived asymptotic distribution within a numerical illustration. Finally, several important properties of the singular Wishart distribution are provided.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2018
Keywords
Singular Wishart distribution, singular normal distribution, stochastic representation, high-dimensional asymptotics
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:oru:diva-61249 (URN)000493467200004 ()
Available from: 2017-10-04 Created: 2017-10-04 Last updated: 2019-11-15Bibliographically approved
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Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-1395-9427

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