While univariate nonparametric estimation methods have been developed for estimating returns in mean-downside risk portfolio optimization, the problem of handling possible cross-correlations in a vector of asset returns has not been addressed in portfolio selection. We present a novel multivariate nonparametric portfolio optimization procedure using kernel-based estimators of the conditional mean and the conditional median. The method accounts for the covariance structure information from the full set of returns. We also provide two computational algorithms to implement the estimators. Via the analysis of 24 French stock market returns, we evaluate the in-sample and out-of-sample performance of both portfolio selection algorithms against optimal portfolios selected by classical and univariate nonparametric methods for three highly different time periods and different levels of expected return. By allowing for cross-correlations among returns, our results suggest that the proposed multivariate nonparametric method is a useful extension of standard univariate nonparametric portfolio selection approaches.

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均值-方差和均值-半方差投资组合选择：多元非参数方法

虽然已经开发了单变量非参数估计方法来估计平均下行风险投资组合优化中的回报，但在投资组合选择中尚未解决处理资产回报向量中可能的互相关的问题。我们提出了一种新的多元非参数投资组合优化程序，它使用条件均值和条件中位数的基于核的估计量。该方法考虑了来自完整回报集的协方差结构信息。我们还提供了两种计算算法来实现估计器。通过对24个法国股市回报的分析，我们针对三个高度不同的时间段和不同的预期回报水平，针对经典和单变量非参数方法选择的最佳投资组合，评估了两种投资组合选择算法的样本内和样本外性能。通过考虑回报之间的互相关，我们的结果表明，所提出的多元非参数方法是标准单变量非参数投资组合选择方法的有用扩展。