Summary
We study the Hellinger type distances\(\rho _p (P_T ,\tilde P_T )\) on a filtered space. Herep≧2 is an arbitrary number andP T and\(\tilde P_T \) are two probability measures stopped at a random timeT. We give lower and upper bounds for\(\rho _p (P_T ,\tilde P_T )\) in predictable terms.
References
Burkholder, D.L.: Distribution function inequalities for martingales, Ann. Probab.1, 19–42 (1973)
Dzhaparidze, K.: Parameter estimation and hypothesis testing in spectral analysis of stationary time series. Berlin Heidelberg New York: Springer 1986
Ibragimov, I.A., Has'minskii, R.Z.: Statistical estimation-asymptotic theory. Berlin Heidelberg New York: Springer 1981
Jacod, J.: Calcul stochastique et problèmes de Martingales, (Lect. Notes Math., vol. 714) Berlin Heidelberg New York: Springer 1979
Jacod, J., Shiryaev, A.N.: Limit theorems for stochastic processes. Berlin Heidelberg New York: Springer 1987
Kutoyants, Yu. A.: Parameter estimation for stochastic processes. Berlin: Heldermann 1984
Lenglart, E., Lepingle, D., Pratelli, M.: Presentation unifiee de certaines inegalites de la theorie des Martingales. (Lect. Notes Math., vol. 784, pp. 26–48). Berlin Heidelberg New York: Springer 1980
Liese, F., Vajda, I.: Convex statistical distances. Leipzig: Teubner 1987
Liptser, R.Sh., Shiryaev, A.N.: Theory of Martingales. Dordrecht: Kluwer 1989
Novikov, A.A.: On discontinuous martingales. Theory Probab. Appl.20, 11–26 (1975)
Valkeila, E., Vostrikova, L.: An integral representation for the Hellinger distance. Math. Scand.58, 239–254 (1986)
Valkeila, E., Vostrikova, L.: On predictable criteria of (c n )-consistency of estimates. Theory Probab. Appl.32, 477–489 (1987)
Vostrikova, L.: OnF-processes and their applications. In: Kalashnikov, V.V. Penkov, B., Zolotarev, V.M. (eds.) Stability problems for stochastic models. (Lect. Notes Math., vol. 1233, pp. 190–203) Berlin Heidelberg New York: Springer 1987
Vostrikova, L.: On the weak convergence of likelihood ratio processes of general statistical parametric models. Stochastics23, 277–298 (1988)
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Dzhaparidze, K., Valkeila, E. On the Hellinger type distances for filtered experiments. Probab. Th. Rel. Fields 85, 105–117 (1990). https://doi.org/10.1007/BF01377632
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DOI: https://doi.org/10.1007/BF01377632
Keywords
- Stochastic Process
- Probability Measure
- Probability Theory
- Mathematical Biology
- Arbitrary Number