Bayesian estimation of the efficient frontier

In this paper, we consider the estimation of the three determining parameters of the efficient frontier, the expected return, and the variance of the global minimum variance portfolio and the slope parameter, from a Bayesian perspective. Their posterior distribution is derived by assigning the diffuse and the conjugate priors to the mean vector and the covariance matrix of the asset returns and is presented in terms of a stochastic representation. Furthermore, Bayesian estimates together with the standard uncertainties for all three parameters are provided, and their asymptotic distributions are established. All obtained findings are applied to real data, consisting of the returns on assets included into the S&P 500. The empirical properties of the efficient frontier are then examined in detail.     

Generalized Estimating Equations for Symmetric Distributions Observed with Admixture

A semiparametric two-component mixture model is considered, in which the distribution of one (primary) component is unknown and assumed symmetric. The distribution of the other component (admixture) is known. Generalized estimating equations are constructed for the estimation of the mixture proportion and the location parameter of the primary component. Asymptotic normality of the estimates is demonstrated and the lower bound for the asymptotic covariance matrix is obtained. An adaptive estimation technique is proposed to obtain the estimates with nearly optimal asymptotic variances.     

Generalized estimating equations for mixtures with varying concentrations

A finite mixture model is considered in which the mixing probabilities vary from observation to observation. A parametric model is assumed for one mixture component distribution, while the others are nonparametric nuisance parameters. Generalized estimating equations (GEE) are proposed for the semi‐parametric estimation. Asymptotic normality of the GEE estimates is demonstrated and the lower bound for their dispersion (asymptotic covariance) matrix is derived. An adaptive technique is developed to derive estimates with nearly optimal small dispersion. An application to the sociological analysis of voting results is discussed. The Canadian Journal of Statistics 41: 217–236; 2013 © 2013 Statistical Society of Canada     

Assessment of vague and noninformative priors for Bayesian estimation of the realized random effects in random-effects meta-analysis

Random-effects meta-analysis has become a well-established tool applied in many areas, for example, when combining the results of several clinical studies on a treatment effect. Typically, the inference aims at the common mean and the amount of heterogeneity. In some applications, the laboratory effects are of interest, for example, when assessing uncertainties quoted by laboratories participating in an interlaboratory comparison in metrology. We consider the Bayesian estimation of the realized random effects in random-effects meta-analysis. Several vague and noninformative priors are examined as well as a proposed novel one. Conditions are established that ensure propriety of the posteriors for the realized random effects. We present extensive simulation results that assess the inference in dependence on the choice of prior as well as mis-specifications in the statistical model. Overall good performance is observed for all priors with the novel prior showing the most promising results. Finally, the uncertainties reported by eleven national metrology institutes and universities for their measurements on the Newtonian constant of gravitation are assessed.     

How risky is the optimal portfolio which maximizes the Sharpe ratio?

In this paper, we investigate the properties of the optimal portfolio in the sense of maximizing the Sharpe ratio (SR) and develop a procedure for the calculation of the risk of this portfolio. This is achieved by constructing an optimal portfolio which minimizes the Value-at-Risk (VaR) and at the same time coincides with the tangent (market) portfolio on the efficient frontier which is related to the SR portfolio. The resulting significance level of the minimum VaR portfolio is then used to determine the risk of both the market portfolio and the corresponding SR portfolio. However, the expression of this significance level depends on the unknown parameters which have to be estimated in practice. It leads to an estimator of the significance level whose distributional properties are investigated in detail. Based on these results, a confidence interval for the suggested risk measure of the SR portfolio is constructed and applied to real data. Both theoretical and empirical findings document that the SR portfolio is very risky since the corresponding significance level is smaller than 90 % in most of the considered cases.     

CUSUM control schemes for monitoring the covariance matrix of multivariate time series

Modified cumulative sum (CUSUM) control charts and CUSUM schemes for residuals are suggested to detect changes in the covariance matrix of multivariate time series. Several properties of these schemes are derived when the in-control process is a stationary Gaussian process. A Monte Carlo study reveals that the proposed approaches show similar or even better performance than the schemes based on the multivariate exponentially weighted moving average (MEWMA) recursion. We illustrate how the control procedures can be applied to monitor the covariance structure of developed stock market indices.     

Multivariate autoregressive extreme value process and its application for modeling the time series properties of the extreme daily asset prices

Abstract

In this article we suggest a new multivariate autoregressive process for modeling time-dependent extreme value distributed observations. The idea behind the approach is to transform the original observations to latent variables that are univariate normally distributed. Then the vector autoregressive DCC model is fitted to the multivariate latent process. The distributional properties of the suggested model are extensively studied. The process parameters are estimated by applying a two-stage estimation procedure. We derive a prediction interval for future values of the suggested process. The results are applied in an empirically study by modeling the behavior of extreme daily stock prices.     

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

In this paper, we consider the asymptotic distributions of functionals of the sample covariance matrix and the sample mean vector obtained under the assumption that the matrix of observations has a matrix‐variate location mixture of normal distributions. The central limit theorem is derived for the product of the sample covariance matrix and the sample mean vector. Moreover, we consider the product of the inverse sample covariance matrix and the mean vector for which the central limit theorem is established as well. All results are obtained under the large‐dimensional asymptotic regime, where the dimension p and the sample size n approach infinity such that p/n→c ∈ [0, + ∞) when the sample covariance matrix does not need to be invertible and p/n→c ∈ [0,1) otherwise.     

Goodness-of-fit tests for centralized Wishart processes

Abstract

In this paper we present several goodness-of-fit tests for the centralized Wishart process, a popular matrix-variate time series model used to capture the stochastic properties of realized covariance matrices. The new test procedures are based on the extended Bartlett decomposition derived from the properties of the Wishart distribution and allows to obtain sets of independently and standard normally distributed random variables under the null hypothesis. Several tests for normality and independence are then applied to these variables in order to support or to reject the underlying assumption of a centralized Wishart process. In order to investigate the influence of estimated parameters on the suggested testing procedures in the finite-sample case, a simulation study is conducted. Finally, the new test methods are applied to real data consisting of realized covariance matrices computed for the returns on six assets traded on the New York Stock Exchange.