Modeling Count Distributions via Skewness-Kurtosis Orthogonal Expansions

초록

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

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