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Preserving Log‐Concavity Under Convolution: Comment
Author(s) -
Miravete Eugenio J.
Publication year - 2002
Publication title -
econometrica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 16.7
H-Index - 199
eISSN - 1468-0262
pISSN - 0012-9682
DOI - 10.1111/1468-0262.00327
Subject(s) - citation , computer science , information retrieval , convolution (computer science) , library science , mathematical economics , operations research , mathematics , artificial intelligence , artificial neural network
In arguing that their results are broadly applicable BMR claim that log–concavity of the convolution is not a very restrictive condition, since it is implied by the log–concavity of either the density of θ1 or that of θ2. They formally state this as Proposition 16 in the appendix to their paper and provide a ‘proof’. Unfortunately, this result is false. Assume that f1(θ1) is the density function of a uniform random variable on the unit interval, and f2(θ2) is the density of a beta distribution with parameters p = 0.4 and q = 0.5, also defined on the unit interval. Both distributions are defined on bounded supports and the uniform density is log–concave as required by BMR. However, this beta density is not log–concave and thus, there are regions in the support of θ0 where the uniform–beta convolution distribution F0(θ0) and the corresponding survival function S0(θ) = 1−F0(θ0) are not log–concave.