Abstract
Practical application of the power-law regression model with an unknown location parameter can be plagued by non-finite least squares parameter estimates. This presents a serious problem in hydrology, since stream flow data is mainly obtained using an estimated stage–discharge power-law rating curve. This study provides a set of sufficient requirements for the data to ensure the existence of finite least squares parameter estimates for a power-law regression with an unknown location parameter. It is shown that in practice, these requirements act as necessary for having a finite least squares solution, in most cases. Furthermore, it is proved that there is a finite probability for the model to produce data having non-finite least squares parameter estimates. The implications of this result are discussed in the context of asymptotic predictions, inference and experimental design. A Bayesian approach to the actual regression problem is recommended.
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Acknowledgements
The authors would like to thank Prof. I. Helland and Prof. B. Natvig at the Division of Statistics and Insurance Mathematics, University of Oslo, for making helpful comments and validations. They would also like to thank Eng. C. Franzetti at Evaluación de Resoursos S.A, Argentina, for providing the Corrientes data. Adrian Read is also gratefully acknowledged for reading and correcting the manuscript.
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Reitan, T., Petersen-Øverleir, A. Existence of the frequentistic estimate for power-law regression with a location parameter, with applications for making discharge rating curves. Stoch Environ Res Ris Assess 20, 445–453 (2006). https://doi.org/10.1007/s00477-006-0037-6
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DOI: https://doi.org/10.1007/s00477-006-0037-6