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Comparison of Shift Experiments on a Banach Space
Abstract
The deficiency of one shift experiment relative to another on an infinite dimensional Banach space is studied. Under certain conditions, this deficiency is shown to be the supremum of the deficiencies between shift experiments defined by finite dimensional marginal distributions. Further, Boll’s “convolution divisibility” criterion for the ordering “being more informative” is extended to infinite dimensional spaces. More detailed results are given for shift experiments defined by Gaussian measures. In particular, the ordering “being more informative” can be characterized in terms of minimax risks.
The classical Hájek-LeCam convolution theorem assumes that the underlying parameter space is a locally compact group. Extensions to Hilbert spaces with Gaussian measures were given by Moussatat, Millar and von der Vaart. We propose an extension covering cylinder measures subject to a domination restriction on their finite dimensional projections. The proof is complex and leaves open a number of problems.
In a location model on an infinite-dimensional Banach space, we study the problem of estimating the location parameter. This estimation problem exhibits an invariance structure where the group involved is the Banach space itself. Under suitable conditions it is shown that the minimax risk coincides with the minimum risk over all equivariant estimators, thus establishing the minimaxity of minimum risk equivariant estimators. Furthermore, such estimators are shown to be extended Bayes estimators, and least favorable sequences of prior distributions are derived. The proofs rely on a general result for structure models and a concentration condition for probability measures on a Banach space related to reproducing kernel Hilbert spaces of Gaussian measures.
Let
\mathfrakE\mathfrak{E}
(Q) be the statistical experiment based on the observation of an unknown function in the presence of an additive noise process with distributionQ. The (possible) loss of information whenQ is replaced by some other noise distributionP is measured by the deficiency of
\mathfrakE\mathfrak{E}
(P) relative to
\mathfrakE\mathfrak{E}
(Q). This deficiency and its relation to the variational distance ofP andQ are studied mainly for Gaussian noise processes. Gaussian diffusion processes and special set-indexed processes are treated in detail.
In Janssen and Reiss (1988) it was shown that in a location model of a Weibull type sample with shape parameter -1 < a < 1 the k(n) lower extremes are asymptotically local sufficient. In the present paper we show that even global sufficiency holds. Moreover, it turns out that convergence of the given statistical experiments in the deficiency metric does not only hold for compact parameter sets but for the whole real line.
We prove some properties of the shift group of sequences of probability measures on a Banach space. Further, given a probability measure P on a lp-space, 1 ≤ p < ∞, or a Hilbert space, conditions are derived to ensure that P is concentrated on some shift group.
This paper gives a general approach to the statistical problem of efficient non-parametric estimation. The main tool is an abstract (infinite dimensional) convolution theorem of the Hajk-Le Cam type. This result is applied to the problems of efficiently estimating measures, quantile functions, spectral functions (of stationary processes), minimum distance functionals, M functional, and so forth.
Let E be the collection of all experiments (in the sense of LeCam) with the same index set. Using the notion of -informativity a binary relation is introduced in E, which turns out to be equivalent with various other relations of comparison studied previously. For a certain subclass of E a characterization of -informativity via randomisation kernels yields the equivalence to Blackwell's sufficiency and in the case of contraction experiments also to sufficiency in the sense of Halmos and Savage. Furthermore the concept of translation invariance is generalized to large subclasses of E. In the case of invariance the characterizing kernels can also be chosen invariant. Applications of the theory to allocation and testing problems as well as to Shannon- and Fisher-informativity illuminate the general set up. The special case of experiments with only a finite index set appears strongly related to the study of orderings in the set of all positive Radon measures on compact convex subsets of certain topological vector spaces.
In this paper we treat the problem of comparison of translation experiments. The "convolution divisibility" criterion for "being more informative" by Boll (1955) [2] is generalized to a "-convolution divisibility" criterion for -deficiency. We also generalize the "convolution divisibility" criterion of V. Strassen (1965) [13] to a criterion for "-convolution divisibility." It is shown, provided least favorable "-factors" can be found, how the deficiencies actually may be calculated. As an application we determine the increase of information--as measured by the deficiency--contained in an additional number of observations for a few experiments (rectangular, exponential, multivariate normal, one way layout). Finally we consider the problem of convergence for the pseudo distance introduced by LeCam (1964) [8]. It is shown that convergence for this distance is topologically equivalent to strong convergence of the individual probability measures up to a shift.
Limits of experiments
- L Lecam