Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix‐variate location mixture of normal distributions

Bodnar, Taras; Mazur, Stepan; Parolya, Nestor

doi:10.1111/sjos.12383

Cited by 13 publications

(8 citation statements)

References 44 publications

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“…In this section, we present the auxiliary results, which are used in proving our main results of Section 2 and can be applied in the discriminant analysis (see [9]). Let us note that our findings are complementing the existing results obtained in [8,[10][11][12]16,34].…”

Section: Auxiliary Resultssupporting

confidence: 89%

Higher order moments of the estimated tangency portfolio weights

Javed

Mazur

Ngailo

2020

Journal of Applied Statistics

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In this paper, we consider the estimated weights of the tangency portfolio. We derive analytical expressions for the higher order noncentral 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.

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Section: Auxiliary Resultssupporting

confidence: 89%

Higher order moments of the estimated tangency portfolio weights

Javed

Mazur

Ngailo

2020

Journal of Applied Statistics

Self Cite

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“…Consequently, the posterior distribution of w TP can be expressed as the product of the (singular) Wishart matrix and a normal vector. The distributional properties of this product are well studied by Bodnar et al [32][33][34][35].…”

Section: Remark 22mentioning

confidence: 99%

Statistical inference for the tangency portfolio in high dimension

2021

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In this paper, we study the distributional properties of the tangency portfolio (TP) weights assuming a normal distribution of the logarithmic returns. We derive a stochastic representation of the TP weights that fully describes their distribution. Under a highdimensional asymptotic regime, i.e., the dimension of the portfolio, k, and the sample size, n, approach infinity such that k/n → c ∈ (0, 1), we deliver the asymptotic distribution of the TP weights. Moreover, we consider tests about the elements of the TP and derive the asymptotic distribution of the test statistic under the null and alternative hypotheses. In a simulation study, we compare the asymptotic distribution of the TP weights with the exact finite sample density. We also compare the high-dimensional asymptotic test with an exact small sample test. We document a good performance of the asymptotic approximations except for small sample sizes combined with c close to one. In an empirical study, we analyse the TP weights in portfolios containing stocks from the S&P 500 index.

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“…In such cases, one should replace the constants m and M in (4.1) and (4.2) by p κ m and p κ M for some κ > 0. This approach would lead only to minor changes in the expressions of the derived asymptotic covariance matrices in this section, where some terms might disappear (see, e.g., Bodnar et al [15] for a similar discussion).…”

Section: High-dimensional Asymptotic Distributionsmentioning

confidence: 99%

Sampling distributions of optimal portfolio weights and characteristics in small and large dimensions

Bodnar

Dette

Parolya

et al. 2021

Random Matrices: Theory Appl.

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Optimal portfolio selection problems are determined by the (unknown) parameters of the data generating process. If an investor wants to realize the position suggested by the optimal portfolios, he/she needs to estimate the unknown parameters and to account for the parameter uncertainty in the decision process. Most often, the parameters of interest are the population mean vector and the population covariance matrix of the asset return distribution. In this paper, we characterize the exact sampling distribution of the estimated optimal portfolio weights and their characteristics. This is done by deriving their sampling distribution by its stochastic representation. This approach possesses several advantages, e.g. (i) it determines the sampling distribution of the estimated optimal portfolio weights by expressions, which could be used to draw samples from this distribution efficiently; (ii) the application of the derived stochastic representation provides an easy way to obtain the asymptotic approximation of the sampling distribution. The later property is used to show that the high-dimensional asymptotic distribution of optimal portfolio weights is a multivariate normal and to determine its parameters. Moreover, a consistent estimator of optimal portfolio weights and their characteristics is derived under the high-dimensional settings. Via an extensive simulation study, we investigate the finite-sample performance of the derived asymptotic approximation and study its robustness to the violation of the model assumptions used in the derivation of the theoretical results.

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Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix‐variate location mixture of normal distributions

Cited by 13 publications

References 44 publications

Higher order moments of the estimated tangency portfolio weights

Higher order moments of the estimated tangency portfolio weights

Statistical inference for the tangency portfolio in high dimension

Sampling distributions of optimal portfolio weights and characteristics in small and large dimensions

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