Modeling Count Distributions via Skewness–Kurtosis Orthogonal Expansions

We develop a semi-parametric framework for representing discrete probability mass functions through orthogonal polynomial representations. Classical count models, such as the Poisson and negative binomial distributions, impose restrictive structural assumptions that often fail to accommodate empirical features including heavy overdispersion, multimodality, and nonstandard tail behavior. To address these limitations, we introduce a linear-tilt model constructed from orthonormal polynomial systems associated with Poisson and negative binomial baselines, namely the Charlier and Meixner families. The proposed representation improves the baseline distribution using additional information from empirical moments. This allows the distribution to flexibly adjust its shape, capturing differences in skewness and kurtosis. We establish theoretical properties of the expansion within a weighted Hilbert space formulation, where the coefficients arise as orthogonal projections that can be expressed as expectations of the corresponding polynomial basis functions. In addition, we analyze approximation behavior and provide numerical bounds on the resulting numerical error and convergence properties of truncated approximations. The practical relevance of the proposed methodology is illustrated through applications to several empirical datasets, demonstrating its ability to capture complex distributional structures while preserving a tractable semi-parametric form.