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January  2020, 5: 2 doi: 10.1186/s41546-020-00044-z

## Moderate deviation for maximum likelihood estimators from single server queues

 P. G. Department of Statistics, Sambalpur University, Odisha, India

Received  February 26, 2019 Published  March 2020

Consider a single server queueing model which is observed over a continuous time interval (0,T], where T is determined by a suitable stopping rule. Let θ be the unknown parameter for the arrival process and $\hat {\theta }_{T}$ be the maximum likelihood estimator of θ. The main goal of this paper is to obtain a moderate deviation result of the maximum likelihood estimator for the single server queueing model under certain regular conditions.
Citation: Saroja Kumar Singh. Moderate deviation for maximum likelihood estimators from single server queues. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 2-. doi: 10.1186/s41546-020-00044-z
##### References:
 [1] Acharya, S.K. (1999). On normal approximation for Maximum likelihood estimation from single server queues, Queueing Syst. 31, 207–216. [2] Acharya, S.K. and S.K. Singh. (2019). Asymptotic properties of maximum likelihood estimators from single server queues: A martingale approach, Commun. Stat. Theory Methods 48, 3549–3557. [3] Basawa, I.V. and N.U. Prabhu. (1981). Estimation in single server queues, Naval. Res. Logist. Quart. 28, 475–487. [4] Basawa, I.V. and N.U. Prabhu. (1988). Large sample inference from single server queues, Queueing Syst. 3, 289–304. [5] Billingsley, P. (1961). Statistical Inference for Markov Processes, The University of Chicago Press, Chicago. [6] Clarke, A.B. (1957). Maximum likelihood estimates in a simple queue, Ann. Math. Statist 28, 1036–1040. [7] Cox, D.R. (1965). Some problems of statistical analysis connected with congestion (W.L. Smith and W. B. Wilkinson, eds.), University of North Carolina Press, Chapel Hill. [8] Dembo, A. and O. Zeitouni. (1998). Large deviation Techniques and Applications, 2nd edn, Springer, New York. [9] Ellis, R.S. (1984). Large deviations for a general class of random vectors, Ann. Probab. 12, 1–12. [10] Gärtner, J. (1977). On large deviations from the invariant measure, Theory Probab. Appl. 22, 24–39. [11] Gao, F. (2001). Moderate deviations for the maximum likelihood estimator, Stat. Probab. Lett. 55, 345– 352. [12] Goyal, T.L. and C.M. Harris. (1972). Maximum likelihood estimation for queues with state dependent service, Sankhya Ser. A 34, 65–80. [13] Hall, P. and C.C. Heyde. (1980). Martingale Limit Theory and Applications, Academic Press, New York. [14] Miao, Y. and Y.-X. Chen. (2010). Note on moderate deviations for the maximum likelihood estimator, Acta Appl. Math. 110, 863–869. [15] Miao, Y. and Y. Wang. (2014). Moderate deviation principle for maximum likelihood estimator, Statistics 48, 766–777. [16] Singh, S.K. and S.K. Acharya. (2019). Equivalence between Bayes and the maximum likelihood estimator in M/M/1 queue, Commun. Stat.–Theory Methods 48, 4780–4793. [17] Wolff, R.W. (1965). Problems of statistical inference for birth and death queueing models, Oper. Res. 13, 243–357. [18] Xiao, Z. and L. Liu. (2006). Moderate deviations of maximum likelihood estimator for independent not identically distributed case, Stat. Probab. Lett. 76, 1056–1064.

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##### References:
 [1] Acharya, S.K. (1999). On normal approximation for Maximum likelihood estimation from single server queues, Queueing Syst. 31, 207–216. [2] Acharya, S.K. and S.K. Singh. (2019). Asymptotic properties of maximum likelihood estimators from single server queues: A martingale approach, Commun. Stat. Theory Methods 48, 3549–3557. [3] Basawa, I.V. and N.U. Prabhu. (1981). Estimation in single server queues, Naval. Res. Logist. Quart. 28, 475–487. [4] Basawa, I.V. and N.U. Prabhu. (1988). Large sample inference from single server queues, Queueing Syst. 3, 289–304. [5] Billingsley, P. (1961). Statistical Inference for Markov Processes, The University of Chicago Press, Chicago. [6] Clarke, A.B. (1957). Maximum likelihood estimates in a simple queue, Ann. Math. Statist 28, 1036–1040. [7] Cox, D.R. (1965). Some problems of statistical analysis connected with congestion (W.L. Smith and W. B. Wilkinson, eds.), University of North Carolina Press, Chapel Hill. [8] Dembo, A. and O. Zeitouni. (1998). Large deviation Techniques and Applications, 2nd edn, Springer, New York. [9] Ellis, R.S. (1984). Large deviations for a general class of random vectors, Ann. Probab. 12, 1–12. [10] Gärtner, J. (1977). On large deviations from the invariant measure, Theory Probab. Appl. 22, 24–39. [11] Gao, F. (2001). Moderate deviations for the maximum likelihood estimator, Stat. Probab. Lett. 55, 345– 352. [12] Goyal, T.L. and C.M. Harris. (1972). Maximum likelihood estimation for queues with state dependent service, Sankhya Ser. A 34, 65–80. [13] Hall, P. and C.C. Heyde. (1980). Martingale Limit Theory and Applications, Academic Press, New York. [14] Miao, Y. and Y.-X. Chen. (2010). Note on moderate deviations for the maximum likelihood estimator, Acta Appl. Math. 110, 863–869. [15] Miao, Y. and Y. Wang. (2014). Moderate deviation principle for maximum likelihood estimator, Statistics 48, 766–777. [16] Singh, S.K. and S.K. Acharya. (2019). Equivalence between Bayes and the maximum likelihood estimator in M/M/1 queue, Commun. Stat.–Theory Methods 48, 4780–4793. [17] Wolff, R.W. (1965). Problems of statistical inference for birth and death queueing models, Oper. Res. 13, 243–357. [18] Xiao, Z. and L. Liu. (2006). Moderate deviations of maximum likelihood estimator for independent not identically distributed case, Stat. Probab. Lett. 76, 1056–1064.
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