General Discussions on Unbiased Estimation
“Non-regular estimation” literally means the theory of statistical estimation when some or other of the regularity conditions for the “usual” theory fail to hold. The concept itself is purely negative and it seems to be almost self-contradiction to try to establish a “general theory” of non-regular estimation. In small sample and large sample theories of estimation of real parameters, however, there are well established sets of regularity conditions, and it is worth while to examine what may follow if any one of these regularity conditions fails to hold. And there has been accumulated substantial amount of results obtained by many authors, though somewhat sporadic investigations, which can give some insight into the structure of non-regular estimation and can clarify the “meaning” of each of the regularity conditions by showing which part of the theorem fails to hold and how it must be modified if it is not satisfied. The purpose of this chapter is to review those results from some unifying viewpoint and also to point out some problems yet to be solved. Our main interest is, therefore, not to look for some strange looking “pathological” examples, but rather to contribute to the main stream of the theory of statistical estimation by clarifying the “regular” theory from the reverse side. Many of the results given in this chapter are discussed in more detail in the subsequent chapters.
KeywordsMaximum Likelihood Estimator Regularity Condition Unbiased Estimation Unbiased Estimator Prior Density
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