Abstract
At the core of this paper is a simple geometric object, namely the risk set of a statistical testing problem on the one hand and f-divergences, which were introduced by Csiszár (1963) on the other hand. f-divergences are measures for the hardness of a testing problem depending on a convexreal valued function f on the interval [0,?). The choice of this parameter f can be adjusted so as to match the needs for specific applications.One of these adjustments of the parameter f is exemplified in Section 3 of this paper. There it is illustrated that the appropriate choice of f for the construction of least favourable distributions in robust statistics is the convex function f(u) =?(1 + u^2) ?(1+u)/?2 yielding the perimeter of the risk setof a testing problem.After presenting the definition, mentioning the basic properties of a risk set and giving the integral geometric representation of f-divergences the paper will focus on the perimeter of the risk set.All members of the class of f-divergences of perimeter-type introduced and investigated in Österreicher and Vajda (2003) and Vajda (2009) turn out to be metric divergences corresponding to a class of entropies introduced by Arimoto (1971).Without essential loss of insight we restrict ourselves to discrete probability distributions and note that the extension to the general case relies strongly on the Lebesgue-Radon-Nikodym Theorem.
Highlights
What is basic for this paper is a testing problem (P, Q), which is a pair of probability distributions P and Q defined on a set Ω = {x1, x2, . . . } of at least two elements
The above approach to define a family of measures of the ’hardness’ of a testing problem, which stresses modelling, relies on the following representation theorem for so-called f -divergences If (Q, P ) given by Feldman and Osterreicher (1981)
Let us concentrate on those properties of the convex function f which allows for metric divergences
Summary
What is basic for this paper is a (simple versus simple) testing problem (P, Q), which is a pair of probability distributions P and Q defined on a set Ω = {x1, x2, . A ⊆ Ω, of the probabilities P (A) and Q(Ac) of type I and type II error satisfying P (A) + Q(Ac) ≤ 1 is the risk set R(P, Q) of the testing problem (P, Q). Motivation 1: The most widely used measure of the deviation of two probability distributions in statistics is Pearson’s χ2-divergence χ2(Q, P ) = ∑ (q(x) − p(x)) .
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