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Properties of the likelihood ratio for semimartingales with deterministic triplets in the parametric case

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Abstract

We consider semimartingales with deterministic discontinuous triplets. We obtain properties of the like-lihood ratio for the parametric case in terms of the Hellinger processes.

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References

  1. L. Le Cam, “Locally asymptotically normal families of distributions,”Univ. Calif. Publ. Statist.,3, No. 2, 37–98 (1960).

    Google Scholar 

  2. I. A. Ibragimov and R. Z. Khas'minskii,Asymptotic Theory of Estimation [in Russian], Nauka, Moscow (1979).

    MATH  Google Scholar 

  3. K. O. Dzhaparidze,Estimate of Parameters and Tests of Hypotheses in the Spectral Analysis of Stationary Time Series [in Russian], Tbilisi University, Tbilisi (1981).

    Google Scholar 

  4. Yu. A. Kutoyants,Estimation of Parameters of Random Processes [in Russian], Armenian Academy of Sciences, Erevan (1980).

    Google Scholar 

  5. Yu. N. Lin'kov,Asymptotic Methods of Statistics of Random Processes [in Russian], Naukova Dumka, Kiev (1993).

    Google Scholar 

  6. A. F. Taraskin, “Behavior of the likelihood ratio of semimartingales,”Teor. Veroyatn. Primen.,29, No. 3, 440–451 (1984).

    MathSciNet  Google Scholar 

  7. Y. Ogata, “The asymptotic behaviour of maximum likelihood estimators for stationary point processes,”Ann. Inst. Statist. Math.,30, No. 2, 243–261 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  8. Yu. N. Lin'kov, “Asymptotic discrimination of counting processes,”Ukr. Mat. Zh.,45, No. 7, 972–979 (1993).

    MathSciNet  Google Scholar 

  9. Yu. N. Lin'kov, “Large deviations in the problem of discrimination of counting processes,”Ukr. Mat. Zh.,45, No. 11, 1514–1521 (1993).

    MathSciNet  Google Scholar 

  10. Yu. N. Lin'kov, “Asymptotic properties of the local density of measures for counting processes,” in:Evolutionary Stochastic Systems in Physics and Biology (Frontiers in Pure and Applied Mathematics, Vol. 2), TVP, Moscow (1993), pp. 311–335.

    Google Scholar 

  11. Yu. N. Lin'kov, “Limit theorems for the local density of measures in the hypothesis-testing problems of counting processes,” in:Probability Theory and Mathematical Statistics, Proceedings of the Sixth Vilnius Conference (Vilnius, June 28–July 3, 1993), TEV, Vilnius/VSP, Utrecht (1994), pp. 497–515.

    Google Scholar 

  12. Yu. N. Lin'kov, “Limit theorems for the local density of measures of counting processes and some statistical applications,” in:Proceedings of the 2nd Ukrainian-Hungarian Conference on New Trends in Probability and Mathematical Statistics (Mukachevo, September 28–October 2, 1992), TVIMS, Kiev (1995), pp. 143–161.

    Google Scholar 

  13. Yu. N. Lin'kov and Munir al Shahf, “Asymptotic discrimination of renewal processes,”Ukr. Mat. Zh.,44, No. 10, 1382–1388 (1992).

    Article  MATH  Google Scholar 

  14. M. G. Akritas, “Asymptotic theory for estimating the parameters of a Levy process,”Ann. Inst. Statist. Math.,34, No. 2, 259–280 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  15. M. G. Akritas and R. A. Johnson, “Asymptotic inference in Levy processes of the discontinuous type,”Ann. Statist.,9, No. 3, 604–614 (1981).

    MATH  MathSciNet  Google Scholar 

  16. T. Komatsu, “Statistics of stochastic processes with jumps,”Lect. Notes Math.,550, 276–289 (1976).

    MathSciNet  Google Scholar 

  17. Yu. N. Lin'kov and M. S. Diallo,Les Propriétés Asimptotiques de la Densité Locale des Mesures pour les Processus á Accroissements Indépendants, Preprint No. 93.06, Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk (1993).

    Google Scholar 

  18. Yu. N. Lin'kov and Yu. A. Shevlyakov, “Properties of the likelihood ratio for processes with independent increments,”Random Oper. Stochast. Equat.,5, No. 3, 237–252 (1997).

    Article  MathSciNet  Google Scholar 

  19. R. Sh. Liptser and A. N. Shiryaev,The Theory of Martingales [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  20. J. Jacod, “Calcul stochastique et problèmes de martingales,”Lect. Notes Math.,714, 1–539 (1979).

    Article  MathSciNet  Google Scholar 

  21. J. Jacod and A. N. Shiryaev,Limit Theorems for Stochastic Processes, Springer, Berlin (1987).

    MATH  Google Scholar 

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Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1172–1180, September, 1999.

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Lin'kov, Y.N., Shevlyakov, Y.A. Properties of the likelihood ratio for semimartingales with deterministic triplets in the parametric case. Ukr Math J 51, 1321–1329 (1999). https://doi.org/10.1007/BF02592999

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  • DOI: https://doi.org/10.1007/BF02592999

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