Reliabilities of χ2- and F Tests in Gauss-Markov Model
Publication: Journal of Surveying Engineering
Volume 125, Issue 3
Abstract
Power and level breakdown points are used in robust statistics to determine the global reliability of a test. A method for calculating the finite-sample versions of these for tests such as χ2 or F is given here when observations have the Gauss-Markov model. Outliers are compared with the generating model using a coordinate transformation and simulated samples. Outliers are considered in two groups, namely, “doubtful” and “clear.” The reliability of a χ2- or F test changes with the magnitude, number, and kind of outliers (i.e., random outliers or influential outliers), and also with the level of significance, α. The χ2- and F tests are generally insensitive to both doubtful and clear outliers. The χ2 test can reliably reject the null hypothesis at α = 0.05, when the observations contain multiple clear random outliers, only if the magnitude of one of them is at least greater than 5σ. Also, the F test can reliably reject the null hypothesis at α = 0.05, when the observations contain multiple clear random outliers, only if the magnitude of one of them is at least greater than 6σ. The greater the magnitude and number of clear outliers that are contained in the observations, the more successful is the χ2- or F test in rejecting the null hypothesis. The reliabilities of the χ2- and F tests decrease as the number of unknowns increases. For this case, the power and level finite-sample breakdown points of the χ2- and F tests are estimated here to be approximately 1/2.
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References
1.
Baarda, W. (1968). “A testing procedure for use in geodetic networks.” Netherlands Geodetic Commission, New Series, Delft, The Netherlands, 2(5).
2.
Chatterjee, S., and Hadi, A. S. (1988). Sensitivity analysis in linear regression. Wiley, New York.
3.
Cooper, M. A. R. (1987). Control surveys in civil engineering. Collins, London.
4.
Donoho, D. L., and Huber, P. J. ( 1983). “The notion of breakdown point.” A festschrift for Erich L. Lehmann, P. J. Bickel, K. A. Doksum, and J. L. Hodges, eds., Wadsworth, Belmont, Calif., 157–184.
5.
Hampel, F., Ronchetti, E., Rousseeuw, P., and Stahel, W. (1986). Robust statistics: The approach based on influence functions. Wiley, New York.
6.
He, X. (1991). “A local breakdown property of robust tests in linear regression.” J. Multivariate Anal., 38, 294–305.
7.
He, X., Simpson, D. G., and Portnoy, S. L. (1990). “Breakdown robustness of tests.” J. Am. Statistical Assn., 85, 446–452.
8.
Hekimoğlu, Ş. ( 1997a). “Reliabilities of χ2- and F tests.” Second Turkish German Jointly Geodetic Days, Altan and Gründig, eds., 27–29 May, Berlin, 1997, 229–250.
9.
Hekimoğlu, Ş. (1997b). “The finite sample breakdown points of the conventional iterative outlier detection procedures.”J. Surv. Engrg., ASCE, 123(1), 15–31.
10.
Huber, P. J. (1981). Robust statistics. Wiley, New York.
11.
Koch, K. R. (1988). Parameter estimation and hypothesis testing in linear models. Springer, New York.
12.
Markatou, M., and He, X. (1994). “Bounded influence and high breakdown point testing procedures in linear models.” J. Am. Statistical Assn., 89, 543–549.
13.
Mikhail, E. M. (1976). Observations and least squares. IEP, New York.
14.
Niemeier, W. ( 1985). “Deformationsanalyse.” Geodatische Netze in Landes-und Ingenieurvermessung II, Kontaktstudium, Herausg. von H. Pelzer, Kontrad Wittwer, Stuttgart, Germany (in German).
15.
Pelzer, H. (1971). Zur Analyse geodaetischer Deformationsmessungen. DGK Reihe C, 164, (in German).
16.
Pope, A. J. (1976). “The statistics of residuals and the detection of outliers.” NOAA Tech. Rep., NOS 65 NGS 1, U.S. Dept. of Commerce, Rockville, Md.
17.
Rousseeuw, P. J., and Leroy, A. M. (1987). Robust regression and outlier detection. Wiley, New York.
18.
Shorack, G. R. (1969). “Testing and estimating ratios of scale parameters.” J. Am. Statistical Assn., 64, 999–1013.
19.
Vanicek, P., and Krakiwsky, J. (1986). Geodesy: The concepts. Elsevier Scientific, Amsterdam.
20.
Wolf, H. (1975). Ausgleichungsrechnung, Formeln zur praktischen Anwendung. Dümlers Verlag, Bonn, Germany (in German).
21.
Ylvisaker, D. (1977). “Test resistance.” J. Am. Statistical Assn., 72, 551–557.
Information & Authors
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Published In
Journal of Surveying Engineering
Volume 125 • Issue 3 • August 1999
Pages: 109 - 135
History
Received: Mar 9, 1998
Published online: Aug 1, 1999
Published in print: Aug 1999
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