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Biometrika 1982 69(2):287-296; doi:10.1093/biomet/69.2.287
© 1982 by Biometrika Trust
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Bounds on the efficiency of estimates based on overlapping data

C. L. MALLOWS and Y. VARDI

Bell Laboratories, Murray Hilly New Jersey, U.S.A.

If we have several estimates of the same quantity, based on partially overlapping sets of data, we can contemplate combining them, linearly, to give a single estimate. We give bounds, which depend only on the degrees of overlap but not on the individual variances, for the efficiency of the best linear unbiased combination of these estimates. with and without covariance information, relative to the overall best linear unbiased estimate. In particular, if the sets have no more than pairwise overlap, each of these ‘partial information’ estimates has efficiency at least 8/9. This result holds also in the multivariate case, and applies in an asymptotic sense when maximum likelihood estimates based on overlapping sets of data are combined linearly, relative to the overall maximum likelihood estimate. The result is applied to the problem of estimating a renewal distribution from time windowed realizations of several independent processes.

Key Words: Asymptotic relative efficiency • Combination of estimates • Renewal theory


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