Abstract
In all setups when the structure of UMVUEs is known, there exists a subalgebra \(\mathcal {U}\) (MVE-algebra) of the basic s-algebra such that all \(\mathcal {U}\)-measurable statistics with finite second moments are UMVUEs. It is shown that MVE-algebras are, in a sense, similar to the subalgebras generated by complete sufficient statistics. Examples are given when these subalgebras differ, in these cases a new statistical structure arises.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bahadur R. R. (1957). On unbiased estimates of uniformly minimum variance. Sankhyā Ser. A 18, 211–224.
Bondesson L. (1983). On uniformly minimum variance unbiased estimation when no complete sufficient statistics exist. Metrika 30, 49–54.
Fisher R. A. (1936). Uncertain inference. Proc. American Academy of Arts and Sciences 72, 245–258.
Fisher R. A. (1973). Statistical Methods and Scientific Inference, 3rd edn. Hafner Press.
Fraser D. A. S. (1957). Nonparametric Methods in Statistics. Wiley, New York.
Kagan A. M. and Konikov M. (2006). The structure of the UMVUEs from categorical data. Theor. Probab. Appl. 50, 466–473.
Kagan A. M. and Malinovsky Y. (2013). On the Nile problem by Sir Ronald Fisher. Electron. J. Stat. 7, 1968–1982.
Kagan A. M., Malinovsky Y. and Mattner L. (2014). Partially complete sufficient statistics are jointly complete. Theor. Probab. Appl. 59, 542–561.
Landers D. and Rogge L. (1976). A note on completeness. Scand. J. Stat. 3, 139.
Lehmann E. and Scheffé H. (1950). Completness, similar regions, and unbisased estimation: part I. Sankhyā 10, 305–340.
Lehmann E. and Casella G. (1998). Theory of Point Estimation, 2nd edn. Springer, New York.
Nayak T. K. and Sinha B. (2012). Some aspects of minimum variance unbiased estimation in presence of ancillary statistics. Stat. Prob. Lett. 82, 1129–1135.
Plachky D. (1977). A characterization of bounded completeness in the undominated case. Transactions of the 7th Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the 1974 European Meeting of Statisticians 7A, 477–480.
Rao C. R. (1952). Some theorems on minimum variance estimation. Sankhyā Ser. A 12, 27–42.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kagan, A.M., Malinovsky, Y. On the Structure of UMVUEs. Sankhya A 78, 124–132 (2016). https://doi.org/10.1007/s13171-015-0076-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13171-015-0076-5