| 1. |
|
|
| 2. |
- Ahmad, M. Rauf, et al.
(författare)
-
A U-classifier for high-dimensional data under non-normality
- 2018
-
Ingår i: Journal of Multivariate Analysis. - Uppsala Univ, Dept Stat, Uppsala, Sweden. KTH, Royal Inst Technol, Dept Math, Stockholm, Sweden. : ELSEVIER INC. - 0047-259X .- 1095-7243. ; 167, s. 269-283
-
Tidskriftsartikel (refereegranskat)abstract
- A classifier for two or more samples is proposed when the data are high-dimensional and the distributions may be non-normal. The classifier is constructed as a linear combination of two easily computable and interpretable components, the U-component and the P-component. The U-component is a linear combination of U-statistics of bilinear forms of pairwise distinct vectors from independent samples. The P-component, the discriminant score, is a function of the projection of the U-component on the observation to be classified. Together, the two components constitute an inherently bias-adjusted classifier valid for high-dimensional data. The classifier is linear but its linearity does not rest on the assumption of homoscedasticity. Properties of the classifier and its normal limit are given under mild conditions. Misclassification errors and asymptotic properties of their empirical counterparts are discussed. Simulation results are used to show the accuracy of the proposed classifier for small or moderate sample sizes and large dimensions. Applications involving real data sets are also included.
|
|
| 3. |
- Ahmad, M. Rauf
(författare)
-
Tests for proportionality of matrices with large dimension
- 2022
-
Ingår i: Journal of Multivariate Analysis. - : Elsevier. - 0047-259X .- 1095-7243. ; 189
-
Tidskriftsartikel (refereegranskat)abstract
- A test for proportionality of two covariance matrices with large dimension, possibly larger than the sample size, is proposed. The test statistic is simple, computationally efficient, and can be used for a large class of multivariate distributions including normality. The properties of the statistic, including asymptotic distribution, are given under high-dimensional set up. Through simulations, the statistic is shown to perform accurately, and outperform its recent competitors, constructed on the basis of similar principles. An extension to the multi-sample case is given.
|
|
| 4. |
- Bauder, David, et al.
(författare)
-
Bayesian inference of the multi-period optimal portfolio for an exponential utility
- 2020
-
Ingår i: Journal of Multivariate Analysis. - : Elsevier BV. - 0047-259X .- 1095-7243. ; 175
-
Tidskriftsartikel (refereegranskat)abstract
- We consider the estimation of the multi-period optimal portfolio obtained by maximizing an exponential utility. Employing the Jeffreys non-informative prior and the conjugate informative prior, we derive stochastic representations for the optimal portfolio weights at each time point of portfolio reallocation. This provides a direct access not only to the posterior distribution of the portfolio weights but also to their point estimates together with uncertainties and their asymptotic distributions. Furthermore, we present the posterior predictive distribution for the investor's wealth at each time point of the investment period in terms of a stochastic representation for the future wealth realization. This in turn makes it possible to use quantile-based risk measures or to calculate the probability of default, i.e the probability of the investor wealth to become negative. We apply the suggested Bayesian approach to assess the uncertainty in the multi-period optimal portfolio by considering assets from the FTSE 100 in the weeks after the British referendum to leave the European Union. The behaviour of the novel portfolio estimation method in a precarious market situation is illustrated by calculating the predictive wealth, the risk associated with the holding portfolio, and the probability of default in each period.
|
|
| 5. |
- Bodnar, Olha, senior lecturer, 1979-, et al.
(författare)
-
Exact test theory in Gaussian graphical models
- 2023
-
Ingår i: Journal of Multivariate Analysis. - : Elsevier. - 0047-259X .- 1095-7243. ; 196
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper, we derive several statistical tests on the precision matrix with application to the determination of the structure of an undirected Gaussian graph. The exact distributions of the test statistics are obtained under the null hypotheses, while the exact distributions of the random matrices, which are used in the construction of the test statistics, are deduced under the alternative hypothesis. Moreover, we present the high-dimensional asymptotic distributions of the test statistics under the null hypothesis. The testing problems that an undirected Gaussian graph possesses a structure that corresponds to the precision matrix of an AR(1) process, to the block-diagonal precision matrix and to the precision of a factor model are discussed in detail. The performance of the proposed statistical tests is further investigated via an extensive simulation study and compared to the benchmark approach.
|
|
| 6. |
- Bodnar, Olha, senior lecturer, 1979-, et al.
(författare)
-
Recent advances in shrinkage-based high-dimensional inference
- 2022
-
Ingår i: Journal of Multivariate Analysis. - : Elsevier. - 0047-259X .- 1095-7243. ; 188
-
Tidskriftsartikel (refereegranskat)abstract
- Recently, the shrinkage approach has increased its popularity in theoretical and applied statistics, especially, when point estimators for high-dimensional quantities have to be constructed. A shrinkage estimator is usually obtained by shrinking the sample estimator towards a deterministic target. This allows to reduce the high volatility that is commonly present in the sample estimator by introducing a bias such that the mean-square error of the shrinkage estimator becomes smaller than the one of the corresponding sample estimator. The procedure has shown great advantages especially in the high-dimensional problems where, in general case, the sample estimators are not consistent without imposing structural assumptions on model parameters.In this paper, we review the mostly used shrinkage estimators for the mean vector, covariance and precision matrices. The application in portfolio theory is provided where the weights of optimal portfolios are usually determined as functions of the mean vector and covariance matrix. Furthermore, a test theory on the mean-variance optimality of a given portfolio based on the shrinkage approach is presented as well.
|
|
| 7. |
- Bodnar, Taras, et al.
(författare)
-
Direct shrinkage estimation of large dimensional precision matrix
- 2016
-
Ingår i: Journal of Multivariate Analysis. - : Elsevier BV. - 0047-259X .- 1095-7243. ; 146, s. 223-236
-
Tidskriftsartikel (refereegranskat)abstract
- In this work we construct an optimal shrinkage estimator for the precision matrix in high dimensions. We consider the general asymptotics when the number of variables p -> infinity and the sample size n -> infinity so that p/n -> c is an element of (0, +infinity). The precision matrix is estimated directly, without inverting the corresponding estimator for the covariance matrix. The recent results from random matrix theory allow us to find the asymptotic deterministic equivalents of the optimal shrinkage intensities and estimate them consistently. The resulting distribution-free estimator has almost surely the minimum Frobenius loss. Additionally, we prove that the Frobenius norms of the inverse and of the pseudo-inverse sample covariance matrices tend almost surely to deterministic quantities and estimate them consistently. Using this result, we construct a bona fide optimal linear shrinkage estimator for the precision matrix in case c < 1. At the end, a simulation is provided where the suggested estimator is compared with the estimators proposed in the literature. The optimal shrinkage estimator shows significant improvement even for non-normally distributed data.
|
|
| 8. |
- Bodnar, Taras, et al.
(författare)
-
Exact and asymptotic tests on a factor model in low and large dimensions with applications
- 2016
-
Ingår i: Journal of Multivariate Analysis. - : Elsevier BV. - 0047-259X .- 1095-7243. ; 150, s. 125-151
-
Tidskriftsartikel (refereegranskat)abstract
- In the paper, we suggest three tests on the validity of a factor model which can be applied for both, small-dimensional and large-dimensional data. The exact and asymptotic distributions of the resulting test statistics are derived under classical and high dimensional asymptotic regimes. It is shown that the critical values of the proposed tests can be calibrated empirically by generating a sample from the inverse Wishart distribution with identity parameter matrix. The powers of the suggested tests are investigated by means of simulations. The results of the simulation study are consistent with the theoretical findings and provide general recommendations about the application of each of the three tests. Finally, the theoretical results are applied to two real data sets, which consist of returns on stocks from the DAX index and on stocks from the S&P 500 index. Our empirical results do not support the hypothesis that all linear dependencies between the returns can be entirely captured by the factors considered in the paper.
|
|
| 9. |
- Bodnar, Taras, et al.
(författare)
-
On the exact and approximate distributions of the product of a Wishart matrix with a normal vector
- 2013
-
Ingår i: Journal of Multivariate Analysis. - : Elsevier. - 0047-259X .- 1095-7243. ; 122, s. 70-81
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper we consider the distribution of the product of a Wishart random matrix and a Gaussian random vector. We derive a stochastic representation for the elements of the product. Using this result, the exact joint density for an arbitrary linear combination of the elements of the product is obtained. Furthermore, the derived stochastic representation allows us to simulate samples of arbitrary size by generating independently distributed chi-squared random variables and standard multivariate normal random vectors for each element of the sample. Additionally to the Monte Carlo approach, we suggest another approximation of the density function, which is based on the Gaussian integral and the third order Taylor expansion. We investigate, with a numerical study, the properties of the suggested approximations. A good performance is documented for both methods.
|
|
| 10. |
- Bodnar, Taras, et al.
(författare)
-
Optimal shrinkage estimator for high-dimensional mean vector
- 2019
-
Ingår i: Journal of Multivariate Analysis. - : Elsevier BV. - 0047-259X .- 1095-7243. ; 170, s. 63-79
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper we derive the optimal linear shrinkage estimator for the high-dimensional mean vector using random matrix theory. The results are obtained under the assumption that both the dimension $p$ and the sample size $n$ tend to infinity in such a way that $p∕n\to c\in(0,\infty)$. Under weak conditions imposed on the underlying data generating mechanism, we find the asymptotic equivalents to the optimal shrinkage intensities and estimate them consistently. The proposed nonparametric estimator for the high-dimensional mean vector has a simple structure and is proven to minimize asymptotically, with probability 1, the quadratic loss when $c\in(0,1)$. When $c\in(1,\infty)$ we modify the estimator by using a feasible estimator for the precision covariance matrix. To this end, an exhaustive simulation study and an application to real data are provided where the proposed estimator is compared with known benchmarks from the literature. It turns out that the existing estimators of the mean vector, including the new proposal, converge to the sample mean vector when the true mean vector has an unbounded Euclidean norm.
|
|
