The broken sample problem

Abstract

Suppose that (X i Y i )i=12 ... n are iid. random vectors with uniform marginals and a certain joint distribution F ρ where ρ is a parameter with ρ=ρ o corresponds to the independence case. However the X's and Y's are observed separately so that the pairing information is missing. Can ρ be consistently estimated? This is an extension of a problem considered in (1980) which focused on the bivariate normal distribution with ρ being the correlation. In this paper we show that consistent discrimination between two distinct parameter values ρ 1 and ρ 2 is impossible if the density f ρ of F ρ is square integrable and the second largest singular value of the linear operator [InlineMediaObject not available: see fulltext.] is strictly less than 1 for ρ=ρ 1 and ρ 2. We also consider this result from the perspective of a bivariate empirical process which contains information equivalent to that of the broken sample. © Springer-Verlag 2004.