Intensive Numerical and Symbolic Computing in Parametric Test Theory
- 213 Downloads
In the construction of multiparameter significance tests intensive computing may be involved at two levels: numerical computing in the construction of critical regions with particular optimality properties, symbolic computing to obtain a better understanding of a statistical model e.g. by giving the geometry of the model. Each level is explained in the construction of two tests for a simple null hypothesis in two lifetime models.
The g-LMMPUMα test maximizes the local mean power with respect to a metric g under all unbiased tests. This test is obtained for the two-parameter gamma family. The construction of the critical region involves the solution of a system of non-linear equations, two-dimensional numerical quadrature, numerical Fourier inversion and interpolation. The NAG Fortran library is a basic tool.
The geodesic test uses the Rao distance (information distance) between the ML-estimators and the null hypothesis as test statistic. This test is obtained for the two-parameter Weibull family. The symbolic computation of tensor components and connection symbols requires essentially the partial derivatives of the loglikelihood and expectations. The Mathematica language allows direct symbolic computation of partial derivatives. For the expectations a package within Mathematica is constructed.
Key Wordsgamma family geodesic test locally most mean power test Weibull family.
Unable to display preview. Download preview PDF.
- Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Wiley, New York.Google Scholar
- Amari, S.-I., Barndorff-Nielsen, O. E., Kass, R. E. Lauritzen, S. L., and Rao, C. R. (1987) Differential geometry in statistical inference. Lecture Notes - Monograph Series 10. Institute of Mathematical Statistics, Hayward (California).Google Scholar
- Burbea, J. and Oller, J. M. (1989) On Rao distance asymptotic distribution. Mathematical Preprint Series No. 67. Universitat de Barcelona.Google Scholar
- Jensen, J. L. (1986) Inference for the mean of a gamma distribution with unkwown shape parameter. Scandinavian Journal of Statistics, 13, 135–151.Google Scholar
- Nandi, S. B. The Numerical Algorithms Group Ltd (1987) The NAG Fortran Library Manual - Mark 12. NAG Central Office, Oxford.Google Scholar
- Sen Gupta, A. and Vermeire, L. (1982) Locally optimal tests for multiparameter hypotheses. Technical Report 671, Department of Statistics, Univ. of Wisconsin, Madison.Google Scholar
- Sen Gupta, A. and Vermeire, L. (1986) Locally optimal tests for multiparameter hypotheses. Journal of the American Statistical Association, 81, 397, 818–825.Google Scholar
- Van Lindt, D. (1984) Statistical Tests for Multiparameter hypotheses in a Differential Geometric Framework. (In Dutch). Doctoral thesis, K.U. Leuven, Supervisor L. Vermeire.Google Scholar
- Vermeire, L. and Wauters, D. (1988) The g — LMMPUa test for the inverse Gaussian family with a simple null hypothesis. Technical Report, K.U. Leuven, Campus Kortrijk.Google Scholar
- Wauters, D. and Vermeire, L. (1992) Geometric structure and multiparameter tests for the Weibull family. Technical Report, K.U. Leuven Campus Kortrijk.Google Scholar
- Wolfram, S. (1991) Mathematica: A System for Doing Mathematics by Computer. Second edition. Addison-Wesley, Redwood City, California.Google Scholar
- Yoshizawa, T. (1971) A geometrical interpretation of location and scale parameters. Memorandum TYH-3, University of Tokyo, Japan.Google Scholar