Estimation of the confidence limits for the quadratic forms in normal variables using a simple Gaussian distribution approximation. (English) Zbl 1091.62004
The distribution of a quadratic form \(Q=\sum_{k=1}^n d_k y_k^2\), where \(y_k\) are i.i.d. standard Gaussian variables, is approximated by \({\mathcal N}(\sum_{k=1}^n d_k, 2\sum_{k=1}^n d_k^2)\). The related approximation for quantiles of \(Q\) (one-sided confidence bounds) is analysed via simulations. E.g., it is noted that for \(n=100\) the relative error of such approximation is less then 7%.
Reviewer: R. E. Maiboroda (Kyïv)
MSC:
| 62E17 | Approximations to statistical distributions (nonasymptotic) |
| 62-08 | Computational methods for problems pertaining to statistics |
| 62E20 | Asymptotic distribution theory in statistics |
| 62F25 | Parametric tolerance and confidence regions |
References:
| [1] | Box, G. E. P. (1954),‘Some theorems on quadratic forms applied in the study of analysis of variance problems: Effect of inequality of variance in one-way classification’The Annals of Mathematical Statistics25, 290-302. · Zbl 0055.37305 · doi:10.1214/aoms/1177728786 |
| [2] | Durrett, R. (1996),Probability: Theory and Applications, Wadsworth, Belmont. |
| [3] | Imhof, J. P. (1961),‘Computing the distribution of quadratic forms in normal variables’Biometrika48 (4), 419-426. · Zbl 0136.41103 · doi:10.1093/biomet/48.3-4.419 |
| [4] | Jackson, J.E. & Mudholkar, G. S. (1979),‘Control procedures for residuals associated with principal Component Analysis’Technometrics21 (3), 341-349. · Zbl 0439.62039 · doi:10.1080/00401706.1979.10489779 |
| [5] | Jensen, D.R. & Solomon, H. (1972),‘A Gaussian approximation to the distribution of a definite quadratic Form’Journal of American Statistical Association67 (340), 898-902. · Zbl 0254.62013 |
| [6] | Juricic, D. & Žele, M. (2002),‘Robust detection of sensor faults by means of a statistical test’Automatica (Oxf.)38, 737-742. · Zbl 1006.94036 · doi:10.1016/S0005-1098(01)00256-4 |
| [7] | Kotz, S., Johnson, N. L. & Boyd, D. W. (1967),‘Series representation distribution of quadratic forms in normal variables: I. Central case’The Annals of Mathematical Statistics38, 832-848. · Zbl 0146.40906 |
| [8] | Nomikos, P. & MacGregor, J. F. (1995),‘Multivariate SPC Charts for monitoring batch processes’Technometrics37 (1), 41-59. · Zbl 0825.62740 · doi:10.1080/00401706.1995.10485888 |
| [9] | Rohatgi, V.K. (1976)An Introduction to Probability Theory and Mathematical Statistics, New York: John Wiley and Sons. · Zbl 0354.62001 |
| [10] | Wise, B. M., Ricker, N. L., Veltkamp, D. F. & Kowalski, B. R. (1990),‘A theoretical basis for the use of principal component models for monitoring multivariate processes’Process and control quality1, 41-51. |
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