In this paper, we are interested in the rate of convergence for the central limit theorem of the maximum likelihood estimator of the drift coefficient for a stochastic partial differential equation based on continuous time observations of the Fourier coefficients ui(t),i=1,…,N of the solution, over some finite interval of time [0,T]. We provide explicit upper bounds for the Wasserstein distance for the rate of convergence when N→∞ and/or T→∞. In the case when T is fixed and N→∞, the upper bounds obtained in our results are more efficient than those of the Kolmogorov distance given by the relevant papers of Mishra and Prakasa Rao, and Kim and Park.
This paper deals with the rate of convergence for the central limit theorem of estimators of the drift coefficient, denoted θ, for the Ornstein-Uhlenbeck process $X := \{X_{t},t\geq 0\}$ X : = { X t , t ≥ 0 } observed at high frequency. We provide an approximate minimum contrast estimator and an approximate maximum likelihood estimator of θ, namely $\widetilde{\theta}_{n}:= {1}/{ (\frac{2}{n} \sum_{i=1}^{n}X_{t_{i}}^{2} )}$ θ ˜ n : = 1 / ( 2 n ∑ i = 1 n X t i 2 ) , and $\widehat{\theta}_{n}:= -{\sum_{i=1}^{n} X_{t_{i-1}} (X_{t_{i}}-X_{t_{i-1}} )}/{ (\Delta _{n} \sum_{i=1}^{n} X_{t_{i-1}}^{2} )}$ θ ˆ n : = − ∑ i = 1 n X t i − 1 ( X t i − X t i − 1 ) / ( Δ n ∑ i = 1 n X t i − 1 2 ) , respectively, where $t_{i} = i \Delta _{n}$ t i = i Δ n , $i=0,1,\ldots , n $ i = 0 , 1 , … , n , $\Delta _{n}\rightarrow 0$ Δ n → 0 . We provide Wasserstein bounds in the central limit theorem for $\widetilde{\theta}_{n}$ θ ˜ n and $\widehat{\theta}_{n}$ θ ˆ n .
Abstract:Private sector operators of response services such as ambulance, fire or police etc. are often regulated by targets on the distribution of response times. This may result in inefficient overstaffing to ensure those targets are met. In this paper, we use a network chain of M/M/K queues to model the arrival and completion of jobs on the system so that quantities such as the expected total time waiting for all jobs can be calculated. The Markov nature enables us to evoke the Hamilton Jacobi Bellman equation (HJB) principle to optimize the required number of staff whilst still meeting targets.
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