## Constructing

### BibTeX

@MISC{Mcvinish_constructing,

author = {R. Mcvinish and P. K. Pollett},

title = {Constructing},

year = {}

}

### OpenURL

### Abstract

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### Citations

328 |
Brownian motion and Stochastic Calculus. 2 nd edition
- SHREVE
- 1991
(Show Context)
Citation Context ...nable approximation to the conditional moments of X(t). The conditional mean and conditional variance of the OU process can be evaluated using standard results (see section 5.6 of Karatzas and Shreve =-=[17]-=-). Let φ(t,z) = E(Z(t)|Z(0) = z) and let ψ(t,x) = E((Z(t) − φ(t,x)) 2 |Z(0) = z). The functions φ(t,z) and ψ(t,z) are given by the solutions to the initial value problems dφ(t,z) = Aφ(t,z), φ(0,z) = z... |

212 | Introduction to Stochastic Search and Optimization - Spall, C - 2001 |

120 |
Martingale estimating functions for discretely observed diffusion processes. Bernouilli
- Bibby, Sørensen
- 1995
(Show Context)
Citation Context ...e consistent. Accurate evaluation of a conditional moment can be achieved by simulating a large number of sample paths and taking the appropriate average. This approach was used by Bibby and Sorenson =-=[7]-=- to evaluate the estimating functions for diffusion processes. An alternate approach is to use a stochastic root finding algorithm as described in Spall [22, chapter 4] to solve the estimating equatio... |

71 |
Quasi-Likelihood and Its Application
- Heyde
- 1997
(Show Context)
Citation Context ...e (c > 40). Secondly, the proposed confidence sets appear to have rather poor coverage properties (see Figure 1 of [21]). In order to address these problems, we use the theory of estimating equations =-=[14]-=- in constructing our estimators. The OU approximation is still used as a guide in the formulation of the estimating equations. As a consequence, our estimators should perform well for many queuing mod... |

55 |
Limit theorems for sequences of jump Markov processes approximating ordinary differential processes
- Kurtz
- 1971
(Show Context)
Citation Context ...a result, likelihood based estimation is far from routine. Given this difficulty, Ross et al. [21] proposed an approximate maximum likelihood estimator for M/M/c queues. Based on the results of Kurtz =-=[19]-=- and Barbour [3], they proposed using the likelihood function of a (Gaussian) OrnsteinUhlenbeck (OU) process as an approximation of the true likelihood function, hereafter referred to as the OU likeli... |

35 | A diffusion approximation for a Markovian queue with reneging. Queueing Syst
- WARD, GLYNN
- 2003
(Show Context)
Citation Context ...∗ t (θ) based on a diffusion approximation to the queue length process. The literature on queueing processes contains a number of diffusion approximations that are valid under a variety of conditions =-=[16,10,1,24]-=-. While these approximations could serve as a suitable starting point for constructing estimating equations, we take as our starting point the class of density dependent Markov processes and their app... |

30 | Estimating functions for discretely sampled diffusion-type models. Working paper
- Bibby, Jacobsen, et al.
- 2004
(Show Context)
Citation Context ...θ)〉 T = T ∑ t=1 ( at(θ)Eθ mt(θ)mt(θ) ′ ) |Ft−1 at(θ) ′ . Let θ0 denote the true parameter value. Assuming that X(t) is ergodic and certain moments of X(t) are finite, the conditions of Theorem 2.3 of =-=[8]-=- hold and GT (θ0) converges in distribution to a normal random variable as T → ∞, 〈G(θ0)〉 −1/2 T GT (θ0) d → N(0,I). (2) It follows from (2) that estimating functions in G satisfy T −1 GT (θ) P θ → 0.... |

27 |
Limiting diffusion approximations for the many server queue and the repairman problem
- Iglehart
- 1965
(Show Context)
Citation Context ...∗ t (θ) based on a diffusion approximation to the queue length process. The literature on queueing processes contains a number of diffusion approximations that are valid under a variety of conditions =-=[16,10,1,24]-=-. While these approximations could serve as a suitable starting point for constructing estimating equations, we take as our starting point the class of density dependent Markov processes and their app... |

19 | Markov Processes, Characterization and Convergence. Wiley series in probability and mathematical statistics - Ethier, Kurtz - 1986 |

12 |
Quasi-stationary distributions in Markov population processes
- Barbour
- 1976
(Show Context)
Citation Context ...estimating function. If the convergence in (2) can be strengthened to stable convergence, then an asymptotic (1−α) confidence set for θ is given by { θ : G ′ T (θ)〈GT (θ)〉 −1 GT (θ) ≤ χ 2 } p;(1−α) , =-=(4)-=- where χ 2 p;(1−α) is the 1 − α percentile of the χ2 distribution with p degrees of freedom.5 2.3 Numerical solution The optimal MEE involves quantities that are not available in closed form for most... |

11 | Non-linear regression models for approximate bayesian computation
- Blum, François
- 2009
(Show Context)
Citation Context ...was able to compute the maximum likelihood estimate. Two simulation based inferential procedures have also been proposed for this problem: indirect inference [13] and approximate Bayesian computation =-=[9]-=-. Basawa et al. [6] demonstrated how Research supported by the Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems. R. McVinish · P.K. Pollett Department... |

9 |
Estimating functions in indirect inference
- Heggland, Frigessi
- 2004
(Show Context)
Citation Context ...tly using a filtering algorithm and hence was able to compute the maximum likelihood estimate. Two simulation based inferential procedures have also been proposed for this problem: indirect inference =-=[13]-=- and approximate Bayesian computation [9]. Basawa et al. [6] demonstrated how Research supported by the Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Syste... |

8 |
On a functional central limit theorem for Markov population processes
- Barbour
- 1974
(Show Context)
Citation Context ...ood based estimation is far from routine. Given this difficulty, Ross et al. [21] proposed an approximate maximum likelihood estimator for M/M/c queues. Based on the results of Kurtz [19] and Barbour =-=[3]-=-, they proposed using the likelihood function of a (Gaussian) OrnsteinUhlenbeck (OU) process as an approximation of the true likelihood function, hereafter referred to as the OU likelihood. The OU lik... |

6 | Existence condition for the diffusion approximations of multiclass priority queueing networks, Queueing Systems Theory Appl
- Chen, Ye
- 2001
(Show Context)
Citation Context ...∗ t (θ) based on a diffusion approximation to the queue length process. The literature on queueing processes contains a number of diffusion approximations that are valid under a variety of conditions =-=[16,10,1,24]-=-. While these approximations could serve as a suitable starting point for constructing estimating equations, we take as our starting point the class of density dependent Markov processes and their app... |

5 |
Bayesian estimation for the M/G/1 queue using a phase type approximation
- Auśın, Wiper, et al.
- 2004
(Show Context)
Citation Context ...atics Subject Classification (2000) 62M05 · 60K25 · 60J28 1 Introduction Much of the literature on parameter estimation for queueing processes concerns continuous observation of the entire queue (see =-=[2]-=- among others). However, there has been some recent interest in parameter estimation based on other types of data. The problem of parameter estimation from inter-departure times for M/G/1 queues has b... |

5 |
Estimation for queues from queue length data”. Queueing Syst. Theory Appl
- Ross, Taimre, et al.
- 2007
(Show Context)
Citation Context ...queueing systems, the transition probabilities for the queue length process are not easily calculated. As a result, likelihood based estimation is far from routine. Given this difficulty, Ross et al. =-=[21]-=- proposed an approximate maximum likelihood estimator for M/M/c queues. Based on the results of Kurtz [19] and Barbour [3], they proposed using the likelihood function of a (Gaussian) OrnsteinUhlenbec... |

4 |
Diffusion approximation in overloaded switching queueing models, Queueing Systems
- Anisimov
- 2002
(Show Context)
Citation Context |

3 | Quasi-likelihood and generalizing the EM algorithm - Heyde, Morton - 1996 |

2 |
Filtering recursion for calculating likelihoods for queues based on inter-departure data
- Fearnhead
- 2004
(Show Context)
Citation Context ...nterest in parameter estimation based on other types of data. The problem of parameter estimation from inter-departure times for M/G/1 queues has been studied using a variety of techniques. Fearnhead =-=[12]-=- showed that the corresponding likelihood function can be computed exactly using a filtering algorithm and hence was able to compute the maximum likelihood estimate. Two simulation based inferential p... |

2 |
Determining an adequate probe separation for estimating the arrival rate in an M/D/1 queue using single-packet probing. Queueing Syst
- Novak, Watson
- 2009
(Show Context)
Citation Context ...E-mail: pkp@maths.uq.edu.au2 to compute the maximum likelihood estimate for a GI/G/1 queue based on waiting time data, adjusted for any idle times, using the EM algorithm. Recently, Novak and Watson =-=[20]-=- devised a method of moments estimator for an M/D/1 queue based on delay measurements from single packet probing. In the present paper, we focus on the problem of parameter estimation from discrete ti... |

2 | A fork-join queueing model" Diffusion approximation, integral representations and asymptotics, Queueing Systems 22 - Tan, Knessl - 1996 |

1 |
Parameter estimation using partial information with applications to queueing and related models, Statist
- Basawa, Bhat, et al.
- 2008
(Show Context)
Citation Context ... the maximum likelihood estimate. Two simulation based inferential procedures have also been proposed for this problem: indirect inference [13] and approximate Bayesian computation [9]. Basawa et al. =-=[6]-=- demonstrated how Research supported by the Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems. R. McVinish · P.K. Pollett Department of Mathematics, Un... |