Pui Lam Leung

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

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.

A mathematical programming approach to clusterwise regression model and its extensions

Abstract The clusterwise regression model is used to perform cluster analysis within a regression framework. While the traditional regression model assumes the regression coefficient (β) to be identical for all subjects in the sample, the clusterwise regression model allows β to vary with subjects of different clusters. Since the cluster membership is unknown, the estimation of the clusterwise regression is a tough combinatorial optimization problem. In this research, we propose a “Generalized Clusterwise Regression Model” which is formulated as a mathematical programming (MP) problem. A nonlinear programming procedure (with linear constraints) is proposed to solve the combinatorial problem and to estimate the cluster membership and β simultaneously. Moreover, by integrating the cluster analysis with the discriminant analysis, a clusterwise discriminant model is developed to incorporate parameter heterogeneity into the traditional discriminant analysis. The cluster membership and discriminant parameters are estimated simultaneously by another nonlinear programming model.

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

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 et al. (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→∞ and p/n→y∈(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 ρ, 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.

Estimating the city-block two-dimensional scaling model with simulated annealing

Abstract Two-dimensional Scaling is a technique to represent dissimilarities among n objects in a two-dimensional space so that the interpoint distances can best approximate the observed dissimilarities between pairs of objects. The coordinates are found by minimizing the STRESS function. It is well known that the number of local minima of the STRESS function increase with n. In this paper, we present a new approach for finding the global minimum of the STRESS function for the city-block two-dimensional scaling model. The proposed method consists of two stages. While the least square regression is used to obtain the local minimum of the STRESS function in stage 1, simulated annealing is applied to search for the global minimum in stage 2. Real and simulated examples (n=30, 50, 70) are used to assess the performance of the proposed algorithm. Results show that the coordinates can be quite accurately recovered by the proposed method.

Estimation of the scale matrix and its eigenvalues in the wishart and the multivariate F distributions

In this paper, the problem of estimating the scale matrix and their eigenvalues in a Wishart distribution and in a multivariate F distribution (which arise naturally from a two-sample setting) are considered. A new class of estimators which shrink the eigenvalues towards their arithmetic mean are proposed. It is shown that the new estimator which dominates the usual unbiased estimator under the squared error loss function. A simulation study was carried out to study the performance of these estimators.

Estimation of eigenvalues of the scale matrix of the multivariate f distribution

Let F have the multivariate F distribution with a scale matrix Δ. In this paper, the problem of estimating the eigenvalues of the scale matrix Δ is considered. New class of estimators are obtained which dominate the best linear estimator of the form cF. Simulation study is also carried out to compare the performance of these estimators.

Improved estimation of a covariance matrix in an elliptically contoured matrix distribution

In this paper, the problem of estimating the covariance matrix of the elliptically contoured distribution (ECD) is considered. A new class of estimators which shrink the eigenvalues towards their arithmetic mean is proposed. It is shown that this new estimator dominates the unbiased estimator under the squared error loss function. Two special classes of ECD, namely, the multivariate-elliptical t distribution and the e-contaminated normal distribution are considered. A simulation study is carried out and indicates that this new shrinkage estimator provides a substantial improvement in risk under most situations.

Combining ordinal forecasts with an application in a financial market

The literature on combining forecasts has almost exclusively focused on combining point forecasts. The issues and methods of combining ordinal forecasts have not yet been fully explored, even though ordinal forecasting has many practical applications in business and social research. In this paper, we consider the case of forecasting the movement of the stock market which has three possible states (bullish, bearish and sluggish). Given the sample of states predicted by different forecasters, several statistical and operation research methods can be applied to determine the optimal weight assigned to each forecaster in combining the ordinal forecasts. The performance of these methods is examined using Hong Kong stock market forecasting data, and their accuracies are found to be better than the consensus method and individual forecasts.

An identity for the noncentral wishart distribution with application

This paper generalizes an identity for the Wishart distribution (derived independently by C. Stein and L. Haff) to the noncentral Wishart distribution. As an application of this noncentral Wishart identity, we consider the problem of estimating the noncentrality matrix of a noncentral Wishart distribution. This noncentral Wishart identity is used to develop a class of orthogonally invariant estimators which dominate the usual unbiased estimator.

Estimating functions of canonical correlation coefficients

Let ρ21,…,ρ2p be the squares of the population canonical correlation coefficients from a normal distribution. This paper is concerned with the estimation of the parameters δ1,…,δp, where δi = ρ2i(1 − ρ2i), i = 1,…,p, in a decision theoretic way. The approach taken is to estimate a parameter matrix Δ whose eigenvalues are δ1,…,δp, given a random matrix F whose eigenvalues have the same distribution as r2i(1 − r2i), i = 1,…,p, where r1,…,rp are the sample canonical correlation coefficients.