Abstract
In this paper we show that Samuelson's [11] inequality is essentially due to Gauss [6] whilst a more general result of the same type is due to Aitken [1, 2]. We also show that the adding-up constraint on the deviations from sample means implicit in Trenkler and Puntanen's [14] multivariate generalisation of Samuelson's Inequality can be regarded as a special case of a more general formulation involving a set of linear constraints on the deviations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aitken, A.C.: On least squares and linear combination of observations. Proc.Roy. Soc. Edinb. A 55, 42–47 (1935)
Aitken, A.C.: On the estimation of many statistical parameters. Proc. Roy. Soc.Edinb. A 62, 33–37 (1948)
Farebrother, R.W.: Some early statistical contributions to the theory and practice of linear algebra. Linear Algebra Appl. 237, 205–224 (1996)
Farebrother, R.W.: Fitting Linear Relationships: A History of the Calculus of Observations 1750–1900. Springer, New York (1999)
Gauss, C.F.: Disquisitio de elementis ellipticis Palladis ex oppositionibus an-norum 1803, 1804, 1805, 1807, 1808, 1809. In: Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores 1, pp. 2–26 (1811). Reprinted in his We r k e , vol. 6, pp. 1–24, Göttingen (1874)
Gauss, C.F.: Theoria combinationis observationum erroribus minimis obnox-iae: pars posterior. In: Commentationes Societatis Regiae Scientiarum Gottin-gensis Recentiores 5, pp. 63–90 (1823). Reprinted in his We r k e , vol. 4, pp. 57–93, Göttingen (1880). Reprinted with an English translation by Stewart (1995,pp. 50–97). [For details of translations into other languages, see Farebrother (1999).]
Jensen, S.T.: The Laguerre-Samuelson inequality with extensions and applications in statistics and matrix theory. M.Sc. Thesis, McGill University, Mon-tréal, Canada (1999)
Laguerre, E.N.: Sur une méthode pour obtenir par approximation les racines d'une équation algébrique qui a toutes ses racines réelles. Nouvelles Annales de Mathématiques (Paris) 2e Série 19, 161–171 and 193–202 (1880)
Laplace, P.S.: Premier Supplément to Théorie Analytique des Probabilités (1816), Mme. Courcier, Paris, 1820. Reprinted in his Oeuvres Complètes,vol. 7, Imprimerie Royale, Paris (1847) and Gauthier-Villars et Fils, Paris (1886)
Olkin, I.: A matrix formulation on how deviant an observation can be. Am.Stat. 46, 205–209 (1992)
Samuelson, P.A.: How deviant can you be? J. Am. Stat. Assoc. 63, 1522–1525(1968)
Stewart, G.W.: Theory of the Combination of Observations Least Subject to Errors. SIAM Publ., Philadelphia, Pennsylvania (1995)
Thompson, W.R.: On a criterion for the rejection of observations and the distribution of the ratio of deviation to sample standard deviation. Ann. Math.Stat. 6, 214–219 (1935)
Trenkler, G., Puntanen, S.J.: A multivariate version of Samuelson's Inequality.Linear Algebra Appl. 410, 143–149 (2006)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Physica-Verlag Heidelberg
About this chapter
Cite this chapter
Farebrother, R.W. (2009). Further Results on Samuelson's Inequality. In: Schipp, B., Kräer, W. (eds) Statistical Inference, Econometric Analysis and Matrix Algebra. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2121-5_20
Download citation
DOI: https://doi.org/10.1007/978-3-7908-2121-5_20
Publisher Name: Physica-Verlag HD
Print ISBN: 978-3-7908-2120-8
Online ISBN: 978-3-7908-2121-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)