Abstract
This paper considers multiple regression model with multivariate spherically symmetric errors to determine optimal β-expectation tolerance regions for the future regression vector (FRV) and future residual sum of squares (FRSS) by using the prediction distributions of some appropriate functions of future responses. The prediction distribution of the FRV, conditional on the observed responses, is multivariate Student-t distribution. Similarly, the prediction distribution of the FRSS is a beta distribution. The optimal β-expectation tolerance regions for the FRV and FRSS have been obtained based on the F -distribution and beta distribution, respectively. The results in this paper are applicable for multiple regression model with normal and Student-t errors.
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References
Aitchison J and Dunsmore IR (1975). Statistical prediction analysis. Cambridge University Press, Cambridge
Aitchison J and Sculthorpe D (1965). Some problems of statistical prediction. Biometrika 55: 469–483
Anderson TW (1993). Nonnormal multivariate distributions: inference based on elliptically contoured distributions. In: Rao, CR (eds) Multivariate analysis: future directions, pp 1–24. North-Holland, Amsterdam
Bishop J (1976) Parametric tolerance regions. Unpublished PhD thesis, Dept. of Statistics, University of Toronto, Canada
Chew V (1966). Confidence, prediction and tolerance regions for multivariate normal distribution. J Am Stat Assoc 62: 605–617
Chmielewski MA (1981). Elliptically symmetric distributions. Int Stat Rev 49: 67–74
Eaton ML (1983). Multivariate statistics—vector space approach. Wiley, New York
Fang KT and Anderson TW (1990). Statistical inference in elliptically contoured and related distributions. Allerton Press Inc., New York
Fang KT and Zhang Y (1990). Generalized multivariate analysis. Science Press and Springer, New York
Fang KT, Kotz S and Ng KW (1990). Symmetric multivariate and related distributions. Chapman and Hall, New York
Fisher RA (1956). Statistical methods in scientific inference. Oli & Boyd, London
Fraser DAS (1968). The structure of inference. Wiley, New York
Fraser DAS (1979). Inference and linear models. McGraw-Hill, New York
Fraser DAS and Guttman I (1956). Tolerance regions. Ann Math Stat 27: 162–179
Fraser DAS and Haq MS (1969). Structural probability and prediction for the multilinear model. J R Stat Soc B 31: 317–331
Fuchs C and Kenett S (1987). Multivariate tolerance regions and F-tests. J Qual Technol 19: 122–131
Geisser S (1993). Predictive inference: an introduction. Chapman & Hall, London
Guttman I (1970a). Statistical tolerance regions. Griffin, London
Guttman I (1970b). Construction of β-content tolerance regions at confidence level γ for large samples for k-variate normal distribution. Ann Math Stat 41: 376–400
Haq MS and Khan S (1990). Prediction distribution for a linear regression model with multivariate Student-t error distribution. Commun Stat Theory Methods 19(12): 4705–4712
Haq MS and Rinco S (1976). β-expectation tolerance regions for a generalized multilinear model with normal error variables. J Multivariate Anal 6: 414–21
Khan S (1996). Prediction inference for heteroscedastic multiple regression model with class of spherical error. Aligarh J Stat 15&16: 1–17
Khan S (2004). Predictive distribution of regression vector and residual sum of squares for normal multiple regression model. Commun Stat Theory Methods 33(10): 2423–2443
Khan S (2005) Optimal tolerance regions for some functions of multiple regression model with normal error, Working Paper Series, SC-MC-0504. Faculty of Sciences, University of Southern Queensland, Australia
Khan S (2006). Optimal tolerance regions for some functions of multiple regression model with student-t errors. J Stat Manage Syst 9(3): 699–715
Khan S and Haq MS (1994). Prediction inference for multilinear model with errors having multivariate Student-t distribution and first-order autocorrelation structure. Sankhya Part B Indian J Stat 56: 95–106
Moore D (2003). The basic practice of statistics, 3rd edn. Freeman, New York
Ng VM (2000). A note on predictive inference for multivariate elliptically contoured distributions. Commun Stat Theory Methods 29(3): 477–483
Prucha IR and Kelejian HH (1984). The structure of simultaneous equation estimators: a generalization towards non-normal disturbances. Econometrica 52: 721–736
Zellner A (1976). Bayesian and non-Bayesian analysis of the regression model with multivariate Student-t error term. J Am Stat Assoc 60: 601–616
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Khan, S. Optimal tolerance regions for future regression vector and residual sum of squares of multiple regression model with multivariate spherically contoured errors. Stat Papers 50, 511–525 (2009). https://doi.org/10.1007/s00362-007-0095-y
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DOI: https://doi.org/10.1007/s00362-007-0095-y
Keywords
- Multiple regression model
- Prediction distribution
- Optimal β − expectation tolerance region
- Invariant differential
- Non-informative prior
- Spherical/elliptical distributions
- Multivariate Student-t, beta and F-distributions