Summary
The problem considered in this paper is that of evaluating the performance of a forecaster who predicts the intensity of a point process or the dirft and diffusion rates of a continuous process. It is shown that we can evaluate this performance in a “prequential” manner, without the usual assumption that the forecasts are generated in accordance with some probability distribution. Technically, the results in this paper are prequential counterparts of the Dambis-Dubins-Schwarz reduction of a continuous martingale, via a change of time, to a Wiener process, and the Papangelou-Meyer reduction of a counting process to a Poisson process.
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References
Dambis, K. E. (1965). On the decomposition of continuous submartingales.theory Probab. Appl. 10, 401–410.
Dawid, A. P. (1984). Statistical theory: the prequential approach.J. Roy. Statist. Soc. A 147, 278–292, (with discussion).
Dawid, A. P. (1985). Calibration-based empirical probability.Ann. Statist. 13 1251–1273, (with discussion).
Dellacherie, C. and Meyer, P. A. (1982).Probabilities and Potential B. Amsterdam: North-Holland.
Dempster, A. P. (1967). Upper and lower probabilities induced by a multivalued mapping.Ann. Math. Statist. 38, 325–339.
Doob, J. L. (1953).Stochastic Processes. Chichester: Wiley.
Dubins, L. E. and Schwarz, G. (1965). On continuous martingales.Proc. Nat. Acad. Sci. USA 53, 913–916.
Elliott, R. J. (1982).Stochastic Calculus and Applications. Berlin: Springer.
De Finetti, B. (1964). Foresight: its logical laws, its subjective sources.Studies in Subjective Probability (H. E. Kyburg and H. E. Smokler, eds.). Chichester: Wiley, 93–158.
De Finetti, B. (1975).Theory of Probability. Chichester: Wiley.
Jacod, J. and Shiryaev, A. N. (1987).Limit Theorems for Stochastic Processes. Berlin: Springer.
Kolmogorov, A. N. (1950).Foundations of the Theory of Probability. New York: Chelsea.
Lévy, P. (1948).Processus Stochastiques et Mouvement Brownien. Paris: Gauthier-Villars.
Meyer, P. A. (1971). Démonstration simplifée d'un théorème de Knight.Lect. Notes Math. 191, 191–195.
Papangelou, F. (1972). Integrability of expected increments of point process and a related random change of scale.Trans. Amer. Math. Soc. 165, 486–506.
Seillier-Moiseiwitsch, F. and Dawid, A. P. (1993). On testing the validity of sequential probability forecasts.J. Amer. Statist. Assoc. 88, 355–359.
Shafer, G. (1976).A Mathematical Theory of Evidence. Princeton: University Press.
Shafer, G. (1990a). Perspectives on the theory and practice of belief functions.Internat. J. Approx. Reasoning 4, 323–362.
Shafer, G. (1990b). The unity of probability.Acting under Uncertainty: Multidisciplinary Conceptions (G. von Furstenberg, ed.), New York: Kluwer, 95–126.
Shafer, G. (1992). Can the various meanings of probability be reconciled?A Handbook for Data Analysis in the Behavioral Sciences: Methodological Issues (G. Keren and C. Lewis, eds.) Hillsdale: Lawrence Erlbaum, 165–196.
Vovk, V. G. (1993). A logic of probability, with application to the foundations of statistics.J. Roy. Statist. Soc. B 55, 317–351, (with discussion).
Watanabe, S. (1964). On discontinuous additive functionals and Lévy measures of a Markov process.Jap. J. Math.,34, 53–79.
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Vovk, V.G. Forecasting point and continuous processes: Prequential analysis. Test 2, 189–217 (1993). https://doi.org/10.1007/BF02562675
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DOI: https://doi.org/10.1007/BF02562675