Construction of a New Distribution Based on the Standardized Student-t Distribution and Its Application

We propose a flexible framework for modeling financial returns using orthogonal expansions of the standardized Student-t density. The Skew-Kurtotic t Expansion (SKTE) captures skewness and excess kurtosis while preserving heavy tails. Orthogonal polynomials are constructed with the t kernel, while a fourth-order truncation provides a parsimonious and interpretable parameterization. Squaring and normalization ensure a valid density. Truncation accuracy analysis shows that the fourth-order approximation balances computational efficiency and tail fidelity, with higher-order terms offering minimal improvement. Empirical analysis of daily returns for the Hang Seng Index, S&P 500, front-month natural gas futures, and EUR/USD exchange rate shows that SKTE outperforms the classical Gram–Charlier expansion, particularly in tail modeling. Maximum likelihood estimation confirms stable skewness and kurtosis adjustments, while location and scale parameters align the distribution to each asset. These results indicate that SKTE provides a robust and flexible framework for modeling asymmetric and heavy-tailed stock return distributions, which is highly relevant for understanding extreme market behavior.