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Properties of the Likelihood Ratio for Counting Processes in the Problem of Estimation of Unknown Parameters

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Abstract

We obtain an asymptotic decomposition of the logarithm of the likelihood ratio for counting processes in the case of similar hypotheses. We establish the properties of the normalized likelihood ratio in the problem of estimation of an unknown parameter.

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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 P. Z. Khas'minskii, Asymptotic Theory of Estimation[in Russian], Nauka, Moscow (1979).

    Google Scholar 

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

  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, “On the behavior of the likelihood ratio of semimartingales,” Teor. Ver. Primen., 29, No.3, 440–451 (1984).

    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).

    Google Scholar 

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

    Google Scholar 

  9. 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 Probability), Vol. 2, TVP, Moscow (1993), pp. 311–335.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  13. Yu. N. Lin'kov and M. S. Diallo, Les Propriétés Asymptotiques de la Densité Locale des Mesures pour les Processus a Accroissement Indépentants, Preprint No. 93.06, Institute of Applied Mathematics and Mechanics, Donetsk (1993).

    Google Scholar 

  14. 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).

    Google Scholar 

  15. Yu. N. Lin'kov and Yu. A. Shevlyakov, “Properties of the likelihood ratio for semimartingales with deterministic triplets in the parametric case,” Ukr. Mat. Zh., 51, No.9, 1172–1180 (1999).

    Google Scholar 

  16. O. A. Nikolaeva, “Semimartingale decompositions of the logarithm of the likelihood ratio for counting processes,” Tr. Inst. Prikl. Mat. Mekh. NAN Ukr., 4, 120–126 (1999).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  20. O. A. Vasilenko, “Large deviations in the problem of discrimination between counting processes,” in: Achievements in Physics and Mathematics of Young Scientists of Higher Schools of Ukraine, Kiev University, Kiev (1995), pp. 13–19.

    Google Scholar 

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Lin'kov, Y.N., Nikolaeva, O.A. Properties of the Likelihood Ratio for Counting Processes in the Problem of Estimation of Unknown Parameters. Ukrainian Mathematical Journal 52, 1439–1452 (2000). https://doi.org/10.1023/A:1010384220108

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