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In the presence of non-normally distributed asset returns, optimal portfolio selection techniques require estimates for variance-covariance parameters, along with estimates for higher-order moments and comoments of the return distribution. This is a formidable challenge that severely exacerbates the dimensionality problem already present with mean-variance analysis. This paper extends the existing literature, which has mostly focused on the covariance matrix, by introducing improved estimators for the coskewness and cokurtosis parameters. We find that the use of these enhanced estimates generates a significant improvement in investors' welfare. We also find that it is only when improved estimators are used that portfolio selection with higher-order moments dominates mean-variance analysis from an out-of-sample perspective. We would like to thank the editor, Joel Hasbrouck, as well as an anonymous referee for their very useful comments. We also thank René Garcia, Patrick Rousseau, Pierre Batteau, Patrick Navatte, and seminar participants at EDHEC Business school and University of Marseille-Aix III for very helpful suggestions. We further acknowledge financial support from Newedge Prime Brokerage, the sponsor of the Research Chair in "Advanced Modelling for Alternative Investments" at EDHEC Risk

In this paper we extend Hasanhodzic and Lo (2007) by assessing the out-ofsample performance of various non-linear and conditional hedge fund replication models. We find that going beyond the linear case does not necessarily enhance the replication power. On the other hand, we find that selecting factors on the basis on an economic analysis allows for a substantial improvement in out-of-sample replication quality, whatever the underlying form of the factor model. Overall, we confirm the findings in Hasanhodzic and Lo (2007) that the performance of the replicating strategies is systematically inferior to that of the actual hedge funds.

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