Lower partial moments for skew elliptical distributions

Robust modeling using skewed distributions are essential in risk management, since many real life examples do not accept the hypothesis the randomness can be modeled by symmetric distributions. This work closes the gap of derivation of explicit representations for lower partial moments of arbitrary powers $ n \geq 1 $ of normal, skew normal and skew-t distributions that are vital in risk analysis. To the best of our experience, there has been no work in lower partial moment representations using skewed family of distributions. Extensive numerical studies are conducted to statistically examine, whether daily stock prices of the prespecified companies from different sectors can be fitted to these families of distributions. It is verified that for short enough time intervals, it can not be rejected that the stock price data is drawn from some or all of these three families. Furthermore, different portfolios are compared by calculating their LPM's, and it is concluded which of the portfolios is less risky than the other. Our findings suggest that this work closes this gap both theoretically in terms of explicit representations of lower partial moments for skewed family of distributions, and practically in terms of calibration of historical data to the derived operators for risk management and robust portfolio formation.