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On the Product of a Singular Wishart Matrix and a Singular Gaussian Vector in High Dimension
Stockholm University, Faculty of Science, Department of Mathematics.ORCID iD: 0000-0001-7855-8221
Stockholm University, Faculty of Science, Department of Mathematics.
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 of this product is derived, using which its characteristic function and asymptotic distribution under the double asymptotic regime are established. We further document a good finite sample performance of the obtained high-dimensional asymptotic distribution via an extensive Monte Carlo study.

Place, publisher, year, edition, pages
2018. Vol. 99, p. 37-50
Keywords [en]
Singular Wishart distribution, singular normal distribution, stochastic representation, high-dimensional asymptotics
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:su:diva-164884ISI: 000493467200004OAI: oai:DiVA.org:su-164884DiVA, id: diva2:1280632
Available from: 2019-01-20 Created: 2019-01-20 Last updated: 2020-04-27Bibliographically approved
In thesis
1. Statistical Inference of Tangency Portfolio in Small and Large Dimension
Open this publication in new window or tab >>Statistical Inference of Tangency Portfolio in Small and Large Dimension
2020 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis considers statistical test theory in portfolio theory. It analyses the asymptotic behavior of the considered tests in the high-dimensional setting, meaning k/n c ∈ (0, ∞) as n → ∞, where k and n are portfolio size and sample size, respectively. It also considers the high-dimensional asymptotic of the product of components involved in the computation of the optimal portfolio. The thesis comprises four manuscripts:

Paper I is concerned with the test on the location of the tangency portfolio on the set of feasible portfolios. Considering the independent and normally multivariate asset returns, we propose a finite-sample test on the mean-variance efficiency of the tangency portfolio (TP). We derive the distribution of the proposed test statistic under both the null and alternative hypotheses, using which we assess the power of the test and construct a confidence interval. The out-of-sample performance of the portfolio determined by the proposed test is conducted and through an extensive simulation study, we show the robustness of the developed test towards the violation of the normality assumptions. We also apply the developed test to real data in the empirical study.

Paper II extends the results of paper I. It is concerned with the study of the asymptotic distributions of the test on the existence of efficient frontier (EF) and the efficiency of the tangency portfolio in the mean-variance space in the high-dimension setting under both the null and alternative hypotheses. Finite-sample performance and robustness of the proposed tests are studied through an extensive simulation study.

In paper III, we study the distributional properties of the TP weights under the assumption of normally distributed logarithmic returns. The distribution of the weights of the TP is given under the form of a stochastic representation (SR). Using the derived SR we deliver the asymptotic distribution of the TP weights under a high-dimensional asymptotic regime. Besides, we consider tests about the elements of the TP weights and derive the asymptotic distribution of the test statistic under the null and alternative hypotheses. In a simulation study, we compare the power function of the high-dimensional asymptotic and the exact tests. Moreover, in an empirical study, we apply the developed theory in analysing the TP weights in a portfolio made of stocks from the S&P 500 index.

In paper IV, we derive a stochastic representation of the product of a singular Wishart matrix and a singular Gaussian vector. We then use the derived SR in the obtention of the characteristic function of that product and in proving the asymptotic normality under the double asymptotic regime. The performance of the obtained asymptotic is shown in the simulation study.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2020. p. 31
Keywords
Tangency portfolio, Mean-variance portfolio, High-dimensional asymptotics, Test theory
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-180914 (URN)978-91-7911-112-0 (ISBN)978-91-7911-113-7 (ISBN)
Public defence
2020-06-10, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript. Paper 2: Accepted. Paper 3: Manuscript.

Available from: 2020-05-18 Created: 2020-04-23 Last updated: 2020-05-26Bibliographically approved

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arXiv:1611.03042

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