This work explores extensions in portfolio optimization from a statistical perspective, focusing on how investors allocate wealth among various risky assets using historical price data. This data, while informative, is only partially reliable for predicting future returns due to changing economic conditions. The first study compares different asset allocation paradigms both theoretically and empirically, utilizing a broad range of common stocks from the U.S. market. The second study quantitatively measures the diversification effect among risky assets, presenting and comparing various approaches, including a new measure of diversification. This measure allows for the quantification of estimation error based on distributional assumptions of the price process. An extensive empirical analysis spans the last five decades, examining overall market variation and the behavior of different measures. The final study advances portfolio optimization methodology by employing recent data analysis techniques without assuming a specific distribution for the price-generating process. It reveals that using meaningful risk measures allows for a reformulation of the investment risk problem in a more descriptive and computationally manageable manner. Detailed explanations and geometric illustrations clarify the statistical and methodological properties, supported by both simulated and historical price data applications.
