Modelling univariate data on constrained supports requires densities that respect fixed boundaries while still allowing asymmetric, boundary-concentrated and multi-peaked shapes. Mixtures and nonparametric smoothers are flexible alternatives, but mixtures introduce latent components and component-number choices, whereas smoothers may not provide a compact parametric interpretation. We propose a transformation-based exponential-family construction for such data. A monotone transformation maps the support to positive intervals; for disconnected supports, ordered components are mapped to disjoint positive intervals so that the inverse transformation is single-valued. On the transformed scale, an exponential-polynomial log-kernel separates endpoint behaviour, determined by the transformation and its Jacobian, from interior modal geometry, determined by a polynomial observation score. The resulting family has explicit normalisability criteria in canonical unit-interval and positive-support cases, a root-based description of stationary points, and the usual regularity properties of minimal exponential families on the interior parameter space. Estimation is developed through likelihood and through a one-dimensional Stein identity. The Stein equations are linear in the natural parameter, avoid evaluation of the normalising constant, yield closed-form estimators under exact identification, and lead to weighted least-squares estimators with a sandwich limit distribution under overidentification. Simulations and empirical studies show that the proposed family can recover genuine multimodality on constrained supports, while predictive validation selects mixtures or support-respecting nonparametric smoothers when those alternatives are better supported by the data.
