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.
Similar content being viewed by others
REFERENCES
L. Le Cam, “Locally asymptotically normal families of distributions,” Univ. Calif. Publ. Statist., 3, No.2, 37–98 (1960).
I. A. Ibragimov and P. Z. Khas'minskii, Asymptotic Theory of Estimation[in Russian], Nauka, Moscow (1979).
K. O. Dzhaparidze, Estimation of Parameters and Verification of Hypotheses in the Spectral Analysis of Stationary Time Series[in Russian], Tbilisi University (1981).
Yu. A. Kutoyants, Estimation of Parameters of Random Processes[in Russian], Armenian Academy of Sciences, Erevan (1980).
Yu. N. Lin'kov, Asymptotic Methods of Statistics of Random Processes[in Russian], Naukova Dumka, Kiev (1993).
A. F. Taraskin, “On the behavior of the likelihood ratio of semimartingales,” Teor. Ver. Primen., 29, No.3, 440–451 (1984).
Y. Ogata, “The asymptotic behaviour of maximum-likelihood estimators for stationary point processes,” Ann. Inst. Statist. Math., 30, No.2, 243–261 (1978).
Yu. N. Lin'kov, “Asymptotic distinction of counting processes,” Ukr. Mat. Zh., 45, No.7, 972–979 (1993).
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.
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.
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.
Yu. N. Lin'kov and Munir al Shahf, “Asymptotic discrimination between renewal processes,” Ukr. Mat. Zh., 44, No.10, 1382–1388 (1992).
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).
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).
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).
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).
P. Sh. Liptser and A. N. Shiryaev, Theory of Martingales[in Russian], Nauka, Moscow (1986).
J. Jacod, “Calcul stochastique et problèmes de martingales,” Lect. Notes Math., 714, 1–539 (1979).
J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer, Berlin (1987).
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.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
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
Issue Date:
DOI: https://doi.org/10.1023/A:1010384220108