Ancillary statistic

In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no information about them.^[1]^[2]^[3] It is opposed to the concept of a complete statistic which contains no ancillary information. It is closely related to the concept of a sufficient statistic which contains all of the information that the dataset provides about the parameters.

A ancillary statistic is a specific case of a pivotal quantity that is computed only from the data and not from the parameters. They can be used to construct prediction intervals. They are also used in connection with Basu's theorem to prove independence between statistics.^[4]

This concept was first introduced by Ronald Fisher in the 1920s,^[5] but its formal definition was only provided in 1964 by Debabrata Basu.^[6]^[7]

^ Lehmann, E. L.; Scholz, F. W. (1992). "Ancillarity" (PDF). Lecture Notes-Monograph Series. Institute of Mathematical Statistics Lecture Notes - Monograph Series. 17: 32–51. doi:10.1214/lnms/1215458837. ISBN 0-940600-24-2. ISSN 0749-2170. JSTOR 4355624.
^ Ghosh, M.; Reid, N.; Fraser, D. A. S. (2010). "Ancillary statistics: A review". Statistica Sinica. 20 (4): 1309–1332. ISSN 1017-0405. JSTOR 24309506.
^ Mukhopadhyay, Nitis (2000). Probability and Statistical Inference. United States of America: Marcel Dekker, Inc. pp. 309–318. ISBN 0-8247-0379-0.
^ Dawid, Philip (2011), DasGupta, Anirban (ed.), "Basu on Ancillarity", Selected Works of Debabrata Basu, New York, NY: Springer, pp. 5–8, doi:10.1007/978-1-4419-5825-9_2, ISBN 978-1-4419-5825-9
^ Fisher, R. A. (1925). "Theory of Statistical Estimation". Mathematical Proceedings of the Cambridge Philosophical Society. 22 (5): 700–725. Bibcode:1925PCPS...22..700F. doi:10.1017/S0305004100009580. hdl:2440/15186. ISSN 0305-0041.
^ Basu, D. (1964). "Recovery of Ancillary Information". Sankhyā: The Indian Journal of Statistics, Series A (1961-2002). 26 (1): 3–16. ISSN 0581-572X. JSTOR 25049300.
^ Stigler, Stephen M. (2001), Ancillary history, Institute of Mathematical Statistics Lecture Notes - Monograph Series, Beachwood, OH: Institute of Mathematical Statistics, pp. 555–567, doi:10.1214/lnms/1215090089, ISBN 978-0-940600-50-8, retrieved 2023-04-24

[1] Lehmann, E. L.; Scholz, F. W. (1992). "Ancillarity" (PDF). Lecture Notes-Monograph Series. Institute of Mathematical Statistics Lecture Notes - Monograph Series. 17: 32–51. doi:10.1214/lnms/1215458837. ISBN 0-940600-24-2. ISSN 0749-2170. JSTOR 4355624.

[fraser-2] Ghosh, M.; Reid, N.; Fraser, D. A. S. (2010). "Ancillary statistics: A review". Statistica Sinica. 20 (4): 1309–1332. ISSN 1017-0405. JSTOR 24309506.

[:0-3] Mukhopadhyay, Nitis (2000). Probability and Statistical Inference. United States of America: Marcel Dekker, Inc. pp. 309–318. ISBN 0-8247-0379-0.

[4] Dawid, Philip (2011), DasGupta, Anirban (ed.), "Basu on Ancillarity", Selected Works of Debabrata Basu, New York, NY: Springer, pp. 5–8, doi:10.1007/978-1-4419-5825-9_2, ISBN 978-1-4419-5825-9

[5] Fisher, R. A. (1925). "Theory of Statistical Estimation". Mathematical Proceedings of the Cambridge Philosophical Society. 22 (5): 700–725. Bibcode:1925PCPS...22..700F. doi:10.1017/S0305004100009580. hdl:2440/15186. ISSN 0305-0041.

[6] Basu, D. (1964). "Recovery of Ancillary Information". Sankhyā: The Indian Journal of Statistics, Series A (1961-2002). 26 (1): 3–16. ISSN 0581-572X. JSTOR 25049300.

[7] Stigler, Stephen M. (2001), Ancillary history, Institute of Mathematical Statistics Lecture Notes - Monograph Series, Beachwood, OH: Institute of Mathematical Statistics, pp. 555–567, doi:10.1214/lnms/1215090089, ISBN 978-0-940600-50-8, retrieved 2023-04-24

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