### Estimates of Hellinger integrals of infinitely divisible distributions

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The paper studies the relations between $\phi $-divergences and fundamental concepts of decision theory such as sufficiency, Bayes sufficiency, and LeCam’s deficiency. A new and considerably simplified approach is given to the spectral representation of $\phi $-divergences already established in Österreicher and Feldman [28] under restrictive conditions and in Liese and Vajda [22], [23] in the general form. The simplification is achieved by a new integral representation of convex functions in terms of elementary...

Local polynomials are used to construct estimators for the value $m\left({x}_{0}\right)$ of the regression function $m$ and the values of the derivatives ${D}_{\gamma}m\left({x}_{0}\right)$ in a general class of nonparametric regression models. The covariables are allowed to be random or non-random. Only asymptotic conditions on the average distribution of the covariables are used as smoothness of the experimental design. This smoothness condition is discussed in detail. The optimal stochastic rate of convergence of the estimators is established. The results...

Real valued $M$-estimators ${\widehat{\theta}}_{n}:=min{\sum}_{1}^{n}\rho ({Y}_{i}-\tau \left(\theta \right))$ in a statistical model with observations ${Y}_{i}\sim {F}_{{\theta}_{0}}$ are replaced by ${\mathbb{R}}^{p}$-valued $M$-estimators ${\widehat{\beta}}_{n}:=min{\sum}_{1}^{n}\rho ({Y}_{i}-\tau \left(u\left({z}_{i}^{T}\phantom{\rule{0.166667em}{0ex}}\beta \right)\right))$ in a new model with observations ${Y}_{i}\sim {F}_{u\left({z}_{i}^{t}{\beta}_{0}\right)}$, where ${z}_{i}\in {\mathbb{R}}^{p}$ are regressors, ${\beta}_{0}\in {\mathbb{R}}^{p}$ is a structural parameter and $u:\mathbb{R}\to \mathbb{R}$ a structural function of the new model. Sufficient conditions for the consistency of ${\widehat{\beta}}_{n}$ are derived, motivated by the sufficiency conditions for the simpler “parent estimator” ${\widehat{\theta}}_{n}$. The result is a general method of consistent estimation in a class of nonlinear (pseudolinear) statistical problems. If...

Using the concept of Hellinger integrals, necessary and sufficient conditions are established for the contiguity of two sequences of distributions of Poisson point processes with an arbitrary state space. The distribution of logarithm of the likelihood ratio is shown to be infinitely divisible. The canonical measure is expressed in terms of the intensity measures. Necessary and sufficient conditions for the LAN-property are formulated in terms of the corresponding intensity measures.

Classical goodness of fit tests are no longer asymptotically distributional free if parameters are estimated. For a parametric model and the maximum likelihood estimator the empirical processes with estimated parameters is asymptotically transformed into a time transformed Brownian bridge by adding an independent Gaussian process that is suitably constructed. This randomization makes the classical tests distributional free. The power under local alternatives is investigated. Computer simulations...

For a sequence of statistical experiments with a finite parameter set the asymptotic behavior of the maximum risk is studied for the problem of classification into disjoint subsets. The exponential rates of the optimal decision rule is determined and expressed in terms of the normalized limit of moment generating functions of likelihood ratios. Necessary and sufficient conditions for the existence of adaptive classification rules in the sense of Rukhin [Ru1] are given. The results are applied to...

In this paper empirical Bayes methods are applied to construct selection rules for the selection of all good exponential distributions. We modify the selection rule introduced and studied by Gupta and Liang [10] who proved that the regret risk converges to zero with rate $O\left({n}^{-\lambda /2}\right),0<\lambda \le 2$. The aim of this paper is to study the asymptotic behavior of the conditional regret risk ${\mathcal{R}}_{n}$. It is shown that $n{\mathcal{R}}_{n}$ tends in distribution to a linear combination of independent ${\chi}^{2}$-distributed random variables. As an application we...

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