Friedrich Liese

University of Rostock · Faculty of Mathematics and Natural Sciences

The paper studies the relations between ϕ-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 ϕ-divergences already established in [F. Österreicher and D. Feldman, “Divergenzen von Wahrscheinlichk...

Quantitative trait loci (QTLs) may affect not only the mean of a trait but also its variability. A special aspect is the variability between multiple measured traits of genotyped animals, such as the within-litter variance of piglet birth weights. The sample variance of repeated measurements is assigned as an observation for every genotyped individ...

We consider quantizations of observations represented by finite partitions of observation spaces. Partitions usually decrease the sensitivity of observations to their probability distributions. A sequence of quantizations is considered to be asymptotically sufficient for a statistical problem if the loss of sensitivity is asymptotically negligible....

The paper deals with the f-divergences of Csiszar generalizing the discrimination information of Kullback, the total variation distance, the Hellinger divergence, and the Pearson divergence. All basic properties of f-divergences including relations to the decision errors are proved in a new manner replacing the classical Jensen inequality by a new...

The problem of selecting the best of k populations is studied for data which are incomplete as some of the values have been deleted randomly. This situation is met in extreme value analysis where only data exceeding a threshold are observable. For increasing sample size we study the case where the probability that a value is observed tends to zero,...

The convergence rates of empirical Bayes estimation in the exponential family are studied in this paper. We first develop an approach for obtaining the lower bound of empirical Bayes estimators. As an application of the approach, we demonstrate that O(n−1) is the lower bound rate for priors with bounded compact support. Second, we construct an empi...

For ergodic ARCH processes, we introduce a one-parameter family of Lp-estimators. The construction is based on the concept of weighted M-estimators. Under weak assumptions on the error distribution, the consistency is established. The asymptotic normality is proved for the special cases p=1 and 2. To prove the asymptotic normality of the L1-estimat...

The paper presents relatively simple verifiable conditions for n-consistency and asymptotic normality of M-estimators of vector parameters in a wide class of statistical models. The conditions are established for the M-estimators with absolutely continuous ρ-functions of locally bounded variation, and for the class of models including, e.g., linear...

Local polynomials are used to construct estimators for the value m(xo) of the regression function m and the values of the derivatives D 1 m(xo) in a general class of nonparametric regression models. The covariables are allowed to be random or non-random. Only asymp-totic conditions on the average distribution of the covariables are used as smoothne...

From consistency results on estimators asymptotically minimizing a general criterion, we derive necessary and sufficient conditions for the convergence of approximate M-estimators in nonlinear regression models with nonstationary and/or dependent errors. Special attention is paid to models with errors satisfying the law of large numbers. The condit...

For a general class of mixed models which includes the (Γ,γ)-model introduced by Shiryaev and Spokoiny (1993) we prove the minimaxity of a Pitman type estimator. This minimaxity is closely related to the asymptotic minimaxity of a sequence of Bayes estimators which is a consequence of the asymptotic shift invariance of the priors. Such priors are c...

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 S S. Gupta and T. Liang [Sankhyā, Ser. B 61, No. 2, 289–304 (1999; Zbl 0972.62003)] who proved that the regret risk converges to zero with rate O(n -λ/2 ), 0<λ≤2. The aim of...

For a sample taken from an i.i.d. sequence of Poisson point processes with not necessarily finite unknown intensity measure the arithmetic mean is shown to be an estimator which is consistent uniformly on certain classes of functions. The method is a reduction to the case of finite intensity measure, which in turn can be dealt with using empirical...

Real valued M-estimators \(\hat \theta _n : = \min \sum\limits_1^n {\varrho \left( {Y_i - \tau \left(\theta \right)} \right)} \) in a statistical model 1 with observations \(Y_i \sim F_{{\theta }_0 }\) are replaced by \(\mathbb{R}^p \)-valued M-estimators \(\hat \beta _n : = \min \sum\limits_1^n {\varrho \left( {Y_i - \tau \left( {u\left( {z_i^T \b...

We establish local asymptotic normality of thinned empirical point processes, based on n i.i.d. random elements, if the probability an{\alpha }_{n} of thinning satisfies an ® n ® ¥ 0,nan ® n ® ¥ ¥{\alpha }_n \to _{n \to \infty } 0,n{\alpha }_n \to _{n \to \infty } \infty . It turns out that the central sequence is determined by the limit of the c...

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 the logarithm of the likelihood ratio is shown to be infinitely divisible. The canonical measure is expressed in terms of the...

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 rate of the optimal decision rule is determined and expressed in terms of the normalized limit of moment generating functions of likelihood ratios. Nece...

For i.i.d. Poisson point processes with intensity measure [Lambda] an estimator for [theta][infinity]([Lambda]) = [integral operator] [infinity] d[Lambda] is introduced. Consistency as well as rates for the convergence are established. An Edgeworth-type expansion for the distribution function is obtained. The estimator is asymptotically efficient i...

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 (1996) who proved that the regret risk ER(n) converges to zero with rate 0(n(exp -lambda/2), 0 < lambda less than or equal 2. The aim of this p...

For a class of histogram based distribution estimators it is proved the consistency in χ2-divergence and expected χ2-divergence. For one of these estimators, introduced formerly by Barron, is also evaluated the rate of consistency in the expected χ2-divergence. These results are stronger than similar results formerly established in the literature f...

Using L. Rüschendorf’s approach [“Asymptotische Statistik.” (1988; Zbl 0661.62001)] to the uniform weak compactness lemma for tests under the LAN-condition a simplified approach to the Hajek-Le Cam bound is given if the decision space is compact.

Let E n be a sequence of experiments weakly converging to a limit experiment E. One of the basic objectives of asymptotic decision theory is to derive asymptotically “best” decisions in E n from optimal decisions in the limit experiment E. A central statement in this context is the Hájek-LeCam bound which is an asymptotic lower bound for the maximu...

Assume k independent populations are given which are distributed according to Qϑ1, …, Qϑk. Taking samples of size n the population with the largest ϑ-value is to be selected. For the sequence of models localized at ϑ0 and any sequence of selection rules an asymptotic upper bound is derived for the minimum probability of correct selection in the ind...

The Hellinger transform for stochastic processes is studied. The Hellinger transform is calculated for multidimensional Gaussian autoregressive processes and for Ornstein-Uhlenbeck processes.

GeneralizedM-estimates (minimum contrast estimates) and their asymptotically equivalent approximate versions are considered. A relatively simple condition is found which is equivalent with consistency of all approximateM-estimates under wide assumptions about the model. This condition is applied in several directions. (i) A more easily verifiable c...

Introduction The lower Hajek-LeCam bound is a central statement of asymptotic decision theory. The traditional way (see Strasser [4], LeCam [1]) to establish this statement is carried out in the following way. Using the concept of "-deficiency, a metric is introduced which describes the so-called strong convergence of statistical experiments. A fir...

The distribution of an inhomogeneous Wiener process is determined by the mean function m(t)=EW(t) and the variance function b(t)=V(W(t)) which depend on unknown parameter ϑ∈Θ. Observations are assumed to be in discrete time points where the sample size tends to infinity. Using the general theory of I. A. Ibragimow and R. Z. Khas’minskij [Asymptotic...

We investigate the problem of giving conditions for the strong convergence, contiguity and entire separation of laws corresponding to sequences of diffusion processes where W is a Wiener process and the functions an are measurable. The conditions are purely analytical and formulated explicitly in terms of the functions an.

The consistency and asymptotic normality of minimum contrast estimation (which includes the maximum likelihood estimation as a special case) is established if the sample is from a renewal process and the observation time tends to infinity. It is shown, that the conditions for consistency and asymptotic normality for maximum likelihood estimation ar...

This paper extends the results of Chen and Wu [1] concerning consistency of M-estimators in the linear regression model. We consider M-estimators defined by [formula] in the general regression model yi = f(xi,[theta] ) + [epsilon]i, where f(x, [theta]) is continuous on a separable metric space X [circle times operator] [Theta], (x1, x2, ... ) is a...

For a diffusion type process dXt = dWi + a(t, X)dt and a sequence (fn) of nonnegative functions necessary and sufficient conditions to the fn are established which guarantee the a.s. convergence of fn(Xt)dt to zero. This result is applied to derive simple necessary and sufficient conditions for the strong convergence of distributions of diffusion p...

It can be shown that the sequence of spontaneous movements in the active sleep of healthy neonates can be described as a Poisson process. This statistical model is the basis of the modelling of movement-related heart-rate reactions (MRHR) as a secondary process. Modelling these processes gives essential information about possible diagnostic paramet...

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Diffusion processes defined by stochastic differential equations are considered. For the BELLINGER integrals of the distribution laws two side estimations are obtained. These inequalities are used to establish necessary and sufficient conditions for the conver¬gence in variational distance and for the contiguity of sequences of diffusion processes....

A hypothesis testing problem of discriminating between two diffusion processes ξi (t), 0 ≤dξi = ai(t,ξi) dt + dW (t), 0 ≤ t ≤ T, is considered. The errors of first and second kind can be estimated in terms of Hellinger integrals. In the present paper upper bounds for the Hellinger integrals are obtained.

This paper deals with f -divergences of probability measures considered in the same general form as e.g. in [12] or [45], where f is an arbitrary (not necessarily differentiable) convex function. Important particular cases or subclasses are mentioned, including those introduced by Bhattacharyya [3], Kakutani [32], Shannon [61] and Kull-back with Le...