Halfplane Trimming for Bivariate Distributions

J. C. Masse and R. Theodorescu

Journal of Multivariate Analysis, 1994, vol. 48, issue 2, 188-202

Abstract: Let [mu] be a probability measure on R2 and let u [set membership, variant] (0, 1). A bivariate u-trimmed region D(u), defined as the intersection of all halfplanes whose [mu]-probability measure is at least equal to u, is studied. It is shown that D(u) is not empty for u sufficiently close to 1 and that D(u) satisfies some natural continuity properties. Limit behavior is also considered, the main result being that the weak convergence of a sequence of probability measures entails the pointwise convergence with respect to Hausdorff distance of the associated trimmed regions; this is then applied to derive asymptotics of the empirical trimmed regions. A brief discussion of the extension of the results to higher dimensions is also given.

Date: 1994
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