Abstract.
Suppose that (X
i
,Y
i
),i=1,2, ... ,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 ρ1 and ρ2 is impossible if the density f
ρ
of F
ρ
is square integrable and the second largest singular value of the linear operator
is strictly less than 1 for ρ=ρ1 and ρ2. We also consider this result from the perspective of a bivariate empirical process which contains information equivalent to that of the broken sample.
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Dedicated to Professor Xiru Chen on His 70th Birthday
Mathematics Subject Classification (2000): primary: 60F99, 62F12
Research supported by NSFC Grant 201471000 and the NUS Grant R-155-000-040-112.
Research supported by the Texas Advanced Research Program.
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Bai, Z., Hsing, T. The broken sample problem. Probab. Theory Relat. Fields 131, 528–552 (2005). https://doi.org/10.1007/s00440-004-0384-5
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DOI: https://doi.org/10.1007/s00440-004-0384-5
Keywords
- Consistent estimation
- Empirical process
- Gaussian process
- Kulback-Leibler information