On J.M. Keynes' The principal averages and the laws of error which lead to them: refinement and generalisation
AbstractKeynes (1911) derived general forms of probability density functions for which the “most probable value” is given by the arithmetic mean, the geometric mean, the harmonic mean, or the median. His approach was based on indirect (i.e., posterior) distributions and used a constant prior distribution for the parameter of interest. It was therefore equivalent to maximum likelihood (ML) estimation, the technique later introduced by Fisher (1912). Keynes' results suffer from the fact that he did not discuss the supports of the distributions, the sets of possible parameter values, and the normalising constants required to make sure that the derived functions are indeed densities. Taking these aspects into account, we show that several of the distributions proposed by Keynes reduce to well-known ones, like the exponential, the Pareto, and a special case of the generalised inverse Gaussian distribution. Keynes' approach based on the arithmetic, the geometric, and the harmonic mean can be generalised to the class of quasi-arithmetic means. This generalisation allows us to derive further results. For example, assuming that the ML estimator of the parameter of interest is the exponential mean of the observations leads to the most general form of an exponential family with location parameter introduced by Dynkin (1961) and Ferguson (1962, 1963). --
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Bibliographic InfoPaper provided by Friedrich-Alexander-Universität Erlangen-Nürnberg, Institut für Wirtschaftspolitik und Quantitative Wirtschaftsforschung (IWQW) in its series IWQW Discussion Paper Series with number 07/2008.
Date of creation: 2008
Date of revision:
ML estimator; criterion function; median; quasi-arithmetic mean; exponential family;
Find related papers by JEL classification:
- C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
- C16 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Econometric and Statistical Methods; Specific Distributions
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Blog mentionsAs found by EconAcademics.org, the blog aggregator for Economics research:
- Bayes Estimators, Loss Functions, and J. M. Keynes
by Dave Giles in Econometrics Beat: Dave Giles' Blog on 2012-05-11 17:20:00
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