On the shrinkage estimation of variance

Biau, Gérard; Yatracos, Yannis G.

On the shrinkage estimation of variance
[Sur l’estimation de la variance par “shrinkage”]

Biau, Gérard ; Yatracos, Yannis G.

Journal de la société française de statistique, Tome 153 (2012) no. 1, pp. 5-21.

Résumé
Abstract

Nous montrons dans cet article que certains estimateurs de la variance $σ^{2}$ et de l’écart type $σ$ sont plus souvent proches de leur cible au sens de Pitman que les estimateurs correspondants obtenus par “shrinkage”, pourtant connus pour améliorer l’erreur quadratique moyenne. Nos résultats sont valables asymptotiquement et pour une grande famille de lois de probabilité. Ils indiquent en particulier que le critère de proximité de Pitman, malgré sa nature controversée, devrait être envisagé comme un outil utile à l’évaluation de la qualité des estimateurs de $σ^{2}$ et de $σ$ .

For a large class of distributions and large samples, it is shown that estimates of the variance $σ^{2}$ and of the standard deviation $σ$ are more often Pitman closer to their target than the corresponding shrinkage estimates which improve the mean squared error. Our results indicate that Pitman closeness criterion, despite its controversial nature, should be regarded as a useful and complementary tool for the evaluation of estimates of $σ^{2}$ and of $σ$ .

MR Zbl

Keywords: variance estimation, standard deviation, “shrinkage”, Pitman closeness, mean squared error
Mot clés : estimation de la variance, écart type shrinkage, proximité de Pitman, erreur quadratique moyenne

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Biau, Gérard; Yatracos, Yannis G. On the shrinkage estimation of variance. Journal de la société française de statistique, Tome 153 (2012) no. 1, pp. 5-21. https://www.numdam.org/item/JSFS_2012__153_1_5_0/

Bibliographie
Cité par

[1] Arnold, B.C. Inadmissibility of the usual scale estimate for a shifted exponential distribution, Journal of the American Statistical Association, Volume 65 (1970), pp. 1260-1264 | MR | Zbl

[2] Brown, L. Inadmissibility of the usual estimators of scale parameters in problems with unknown location and scale parameters, The Annals of Mathematical Statistics, Volume 39 (1968), pp. 29-48 | MR | Zbl

[3] Goodman, L.A. A simple method for improving some estimators, The Annals of Mathematical Statistics, Volume 22 (1953), pp. 114-117 | MR | Zbl

[4] Goodman, L.A. A Note on the Estimation of Variance, Sankhyā, Volume 22 (1960), pp. 221-228 | Zbl

[5] Hall, P. The Bootstrap and Edgeworth Expansion, Springer-Verlag, New York, 1992 | MR | Zbl

[6] Johnson, N.L.; Kotz, S.; Balakrishnan, N. Continuous Univariate Distributions. Vol. 1. Second Edition, John Wiley & Sons, New York, 1994 | Zbl

[7] Khattree, R. On comparison of estimates of dispersion using generalized Pitman nearness criterion, Communications in Statistics-Theory and Methods, Volume 16 (1987), pp. 263-274 | MR | Zbl

[8] Khattree, R. Comparing estimators for population variance using Pitman nearness, The American Statistician, Volume 46 (1992), pp. 214-217

[9] Khattree, R. Estimation of guarantee time and mean life after warranty for two-parameter exponential failure model, Australian Journal of Statistics, Volume 34 (1992), pp. 207-215 | MR | Zbl

[10] Khattree, R. On the estimation of $σ$ and the process capability indices $C_{p}$ and $C_{p m}$ , Probability in the Engineering and Informational Sciences, Volume 13 (1999), pp. 237-250 | MR | Zbl

[11] Lehmann, E.L.; Casella, G. Theory of Point Estimation. Second Edition, Springer-Verlag, New York, 1998 | MR | Zbl

[12] Maatta, J.M.; Casella, G. Developments in decision-theoretic variance estimation, Statistical Science, Volume 5 (1990), pp. 90-120 | MR | Zbl

[13] Pitman, E.J.G. The “closest” estimates of statistical parameters, Proceedings of the Cambridge Philosophical Society, Volume 33 (1937), pp. 212-222 | JFM | Zbl

[14] Robert, C.P.; Hwang, J.T.G.; Strawderman, W.E. Is Pitman closeness a reasonable criterion? With comment and rejoinder., Journal of the American Statistical Association, Volume 88 (1993), pp. 57-76 | MR | Zbl

[15] Rukhin, A.L. How much better are better estimators of a normal variance, Journal of the American Statistical Association, Volume 82 (1987), pp. 925-928 | MR

[16] Stein, C. Inadmissibility of the usual estimate for the variance of a normal distribution with unknown mean, Annals of the Institute of Statistical Mathematics, Volume 16 (1964), pp. 155-160 | MR | Zbl

[17] van der Vaart, A.W. Asymptotic Statistics, Cambridge University Press, Cambridge, 1999 | MR | Zbl

[18] Yatracos, Y.G. Artificially augmented samples, shrinkage, and mean squared error reduction, Journal of the American Statistical Association, Volume 100 (2005), pp. 1168-1175 | MR | Zbl

[19] Yatracos, Y.G. Pitman’s closeness criterion and shrinkage estimates of the variance and the s.d. (2011) ( http://ktisis.cut.ac.cy/handle/10488/5009 )