Median unbiased estimates for M.L.R.-families |
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Authors: | Prof Dr J Pfanzagl |
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Institution: | 1. Mathematisches Institut, Weyertal 86, 5 K?ln-Lindenthal
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Abstract: | Summary
Lehmann p. 83] has shown that some families of probability measures with monotone likelihood ratios (m.l.r.) admit median unbiased
estimates which are optimum in the sense that among all median unbiased estimates they minimize the expected loss for any
loss function which assumes its minimal value zero for the “true” parameter value and is nondecreasing as the parameter moves
away from the true value in either direction. This very strong optimum property was proved under the assumption that all probability
measures of the m.l.r.-family have continuous distribution functions, that they are mutually absolutely continuous and that
each element of the support is the median of somep-measure of the family. This result does therefore not cover important cases such as the binomial families or thePoisson family.
The purpose of the present paper is to show the existence ofrandomized median unbiased estimates with the same optimum property for m.l.r.-families which are closed and connected with respect
to the strong topology. Such families are always dominated. We do, however, neither assume that thep-measures are mutually absolutely continuous nor that the distribution functions are continuous. We remark that the use of
randomized estimates is indispensable here because nonrandomized median unbiased estimates do not always exist in the general
case. |
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Keywords: | |
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