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Asymptotics for doubly flexible logspline response models. (English) Zbl 0785.62034
Author’s summary: Consider a $${\mathcal Y}$$-valued response variable having a density function $$f(\cdot | x)$$ that depends on an $${\mathcal X}$$-valued input variable $$x$$. It is assumed that $${\mathcal X}$$ and $${\mathcal Y}$$ are compact intervals and that $$f(\cdot \mid \cdot)$$ is continuous and positive on $${\mathcal X} \times {\mathcal Y}$$. Let $$F(\cdot \mid x)$$ denote the distribution function of $$f(\cdot \mid x)$$ and let $$Q(\cdot \mid x)$$ denote its quantile function.
A finite-parameter exponential family model based on tensor-product $$B$$- splines is constructed. Maximum likelihood estimation of the parameters of the model based on independent observations of the response variable at fixed settings of the input variable yields estimates of $$f(\cdot \mid \cdot)$$, $$F(\cdot \mid \cdot)$$, and $$Q(\cdot \mid \cdot)$$. Under mild conditions, if the number of parameters suitably tends to infinity as $$n \to \infty$$, these estimates have optimal rates of convergence. The asymptotic behavior of the corresponding confidence bounds is also investigated.

MSC:
 62G05 Nonparametric estimation 62G20 Asymptotic properties of nonparametric inference 62F12 Asymptotic properties of parametric estimators
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