In this thesis we develop new statistical theory and apply it to practical problems dealing with mean-variance optimal portfolio selection. More precisely, we derive an exact statistical test for the characterization of the location of the tangency portfolio (TP) on the efficient frontier. Since the construction of the TP involves the product of an (inverse) Wishart matrix and a normal vector, we also study the distributional properties of functions involving such a product. The first paper focuses on the determination of the existence of the TP. Due to problem of parameter uncertainty, specifying the location of the TP on the set of feasible portfolio becomes a difficult task. Assuming that the asset returns are independent and multivariate normally distributed, we propose a finite-sample test on mean-variance efficiency of the TP. We derive the distribution of the proposed test statistic under both hypotheses, using which we assess the power of the test and construct a confidence interval. Furthermore, we conduct the out-of sample performance of the portfolio determined by implementing the proposed test. Through an extensive simulation we show the robustness of the new test towards the violation of the normality assumption. In an empirical study we apply the developed theory to real data.In the second paper we derive a stochastic representation of the product of a singular Wishart matrix and a singular Gaussian vector. The derived stochastic representation is then used to obtain the characteristic function of that product and to prove the asymptotic normality under double asymptotic regime. Moreover, the derived stochastic representation gives an efficient way of how the elements of the product should be simulated. A simulation study shows a good performance of the obtained asymptotic distribution.​