🤖 AI Summary
This study addresses statistical inference for distributional models lacking classical density functions or finite moments by establishing a unified theoretical framework. The authors generalize Godambe’s inference functions to the space of distributions and introduce an observation operator to formally characterize diverse data-generating mechanisms, including point observations, interval censoring, and convolutional measurements. For the first time, inference functionals are defined in a distributional sense. Leveraging Schwartz’s theory of generalized functions, the Hájek–Le Cam convolution theorem, and Bhapkar–Godambe projections, the paper develops rigorous results on consistency, asymptotic normality, and optimality. A central contribution is the identification of a three-tier information hierarchy: Fisher information bounds the information captured by the observation operator, which in turn bounds the information attainable by any inference functional. The framework’s validity is demonstrated through applications to heavy-tailed distributions, interval-censored location models, and elliptical contour models.
📝 Abstract
This paper generalises inference functions (Godambe, 1960) to distributional statistical models, in which each probability measure is represented by a distribution--kernel pair $(T_θ, \varphi) \in \mathcal S'(\mathbb R) \times \mathcal S(\mathbb R)$. The generalisation is strategically motivated: the key properties of maximum likelihood estimation-consistency and asymptotic normality -derive not from maximising the likelihood but from the MLE being the root of a regular inference function. Extending inference functions to the distributional setting provides an optimality theory for models lacking classical densities or finite moments.
The extension requires enlarging the notion of observation. We introduce observation operators $\mathcal O : \mathcal S'(\mathbb R) \to \mathcal Y$ mapping distributional models to an observation space, and define inference functionals as estimating equations composed with these operators. The framework encompasses classical point observations, interval-censored data, convolutional measurements, and transform-based statistics.
We establish asymptotic theory (consistency, asymptotic normality, Godambe optimality) under mild conditions and derive a hierarchy of information bounds -- classical Fisher information dominates the information available through the observation operator, which in turn dominates the information captured by any inference functional -- via the Hájek--Le~Cam convolution theorem. The two gaps quantify distinct sources of information loss: the observation mechanism and the choice of inference functional. Examples include sinusoidal inference functions for heavy-tailed distributions, interval-censored location inference, elliptically contoured models, and nuisance parameters via the Bhapkar--Godambe projection.