Monographs on statistics and applied probability

http://perso.ens-lyon.fr/patrick.flandrin/LedoitWolf_JMA2004.pdf

A well-conditioned estimator for large-dimensional covariance matrices

Inferring large-scale covariance matrices from sparse genomic data is an ubiquitous problem in bioinformatics. Clearly, the widely used standard covariance and correlation estimators are ill-suited for this purpose. As statistically efficient and computationally fast alternative we propose a novel shrinkage covariance estimator that exploits the Ledoit-Wolf (2003) lemma for analytic calculation of the optimal shrinkage intensity. Subsequently, we apply this improved covariance estimator (which has guaranteed minimum mean squared error, is well-conditioned, and is always positive definite even for small sample sizes) to the problem of inferring large-scale gene association networks. We show that it performs very favorably compared to competing approaches both in simulations as well as in application to real expression data.

/pdf/a-shrinkage-approach-to-large-scale-covariance-matrix-4hqooogmqu.pdf

A Shrinkage Approach to Large-Scale Covariance Matrix Estimation and Implications for Functional Genomics

The Bootstrap and Edgeworth Expansion

The commutation matrix $K$ is defined as a square matrix containing only zeroes and ones. Its main properties are that it transforms vecA into vecA', and that it reverses the order of a Kronecker product. An analytic expression for $K$ is given and many further properties are derived. Subsequently, these properties are applied to some problems connected with the normal distribution. The expectation is derived of $\varepsilon' A\varepsilon\cdot\varepsilon' B\varepsilon\cdot\varepsilon'C\varepsilon$, where $\varepsilon \sim N(0, V)$, and $A, B, C$ are symmetric. Further, the expectation and covariance matrix of $x \otimes y$ are found, where $x$ and $y$ are normally distributed dependent variables. Finally, the variance matrix of the (noncentral) Wishart distribution is derived.

The commutation matrix: Some properties and applications

Extending the work of Jobson and Korkie (1981), Lo (2002) and Memmel (2003), this paper applies the technique of the repeated measures design to develop the Multiple Sharpe ratio test statistic to test the hypothesis of the equality of the multiple Sharpe ratios. We also work out the asymptotic distribution of the statistic and its properties. To demonstrate the superiority of our proposed statistic over the traditional pair-wise Sharpe ratio test, we illustrate our approach by testing the equality of Sharpe ratios for the eighteen iShares. Whereas the pair-wise Sharpe ratio test show that the performance of all the 18 iShares are indistinguishable, our test results reject the equality of the Sharpe ratios in each year as well as in the entire sample; implying that the 18 iShares perform differently in each year as well as in the entire sample, with some outperforming others in the market. The test in our paper provides investors with a tool to evaluate their portfolio performances and enables them to make wiser decisions in their investments.

On Testing the Equality of the Multiple Sharpe Ratios, with Application on the Evaluation of Ishares

A mathematical programming approach to clusterwise regression model and its extensions

Using the Markowitz mean-variance portfolio optimization theory, researchers have shown that the traditional estimated return greatly overestimates the theoretical optimal return, especially when the dimension to sample size ratio p/n is large. Bai, Liu, and Wong (2009) propose a bootstrap-corrected estimator to correct the overestimation, but there is no closed form for their estimator. To circumvent this limitation, this paper derives explicit formulas for the estimator of the optimal portfolio return. We also prove that our proposed closed-form return estimator is consistent when n\rightarrow \infty and p/n \rightarrow y\in (0,1). Our simulation results show that our proposed estimators dramatically outperform traditional estimators for both the optimal return and its corresponding allocation under different values of p/n ratios and different inter-asset correlations p, especially when p/n is close to 1. We also find that our proposed estimators perform better than the bootstrap-corrected estimators for both the optimal return and its corresponding allocation. Another advantage of our improved estimation of returns is that we can also obtain an explicit formula for the standard deviation of the improved return estimate and it is smaller than that of the traditional estimate, especially when p/n is large. In addition, we illustrate the applicability of our proposed estimate on the US stock market investment.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID2006227_code341156.pdf?abstractid=1968889&mirid=4

An Improved Estimation to Make Markowitz's Portfolio Optimization Theory Users Friendly and Estimation Accurate with Application on the US Stock Market Investment

This paper applies the mean-variance portfolio optimization (PO) approach and the stochastic dominance (SD) test to examine preferences for international diversification versus domestic diversification from American investors’ viewpoints. Our PO results imply that the domestic diversification strategy dominates the international diversification strategy at a lower risk level and the reverse is true at a higher risk level. Our SD analysis shows that there is no arbitrage opportunity between international and domestic stock markets; domestically diversified portfolios with smaller risk dominate internationally diversified portfolios with larger risk and vice versa; and at the same risk level, there is no difference between the domestically and internationally diversified portfolios. Nonetheless, we cannot find any domestically diversified portfolios that stochastically dominate all internationally diversified portfolios, but we find some internationally diversified portfolios with small risk that dominate all the domestically diversified portfolios.

https://www.mdpi.com/1911-8074/7/2/45/pdf

Pui Lam Leung

Papers

On Testing the Equality of the Multiple Sharpe Ratios, with Application on the Evaluation of Ishares

On Testing the Equality of the Multiple Sharpe Ratios, with Application on the Evaluation of Ishares

A mathematical programming approach to clusterwise regression model and its extensions

An Improved Estimation to Make Markowitz's Portfolio Optimization Theory Users Friendly and Estimation Accurate with Application on the US Stock Market Investment

International Diversification Versus Domestic Diversification: Mean-Variance Portfolio Optimization and Stochastic Dominance Approaches