The relative efficiency of direct and maximum likelihood estimates of mean waiting time in the simple queue, M/M/1

1972 ◽  
Vol 9 (02) ◽  
pp. 396-403 ◽  
Author(s):  
John H. Jenkins
Keyword(s):  
Maximum Likelihood ◽  
Waiting Time ◽  
Continuous Time ◽  
Relative Efficiency ◽  
Asymptotic Variance ◽  
Complete Information ◽  
Maximum Likelihood Estimates ◽  
Time Study ◽  
Direct Estimate ◽  
Mean Waiting Time

A problem of estimating waiting time in the statistical analysis of queues is investigated. The continuous time study of the M/M/1 queue made by Bailey is adapted to obtain the asymptotic variance of a direct estimate of waiting time as obtained under conditions of incomplete information. This is then compared with the asymptotic variance of the maximum likelihood estimate as obtained under conditions of complete information and based on the results of Clarke.

The relative efficiency of direct and maximum likelihood estimates of mean waiting time in the simple queue, M/M/1

1972 ◽  
Vol 9 (2) ◽  
pp. 396-403 ◽  
Author(s):  
John H. Jenkins
Keyword(s):  
Maximum Likelihood ◽  
Waiting Time ◽  
Continuous Time ◽  
Relative Efficiency ◽  
Asymptotic Variance ◽  
Complete Information ◽  
Maximum Likelihood Estimates ◽  
Time Study ◽  
Direct Estimate ◽  
Mean Waiting Time

A problem of estimating waiting time in the statistical analysis of queues is investigated. The continuous time study of the M/M/1 queue made by Bailey is adapted to obtain the asymptotic variance of a direct estimate of waiting time as obtained under conditions of incomplete information. This is then compared with the asymptotic variance of the maximum likelihood estimate as obtained under conditions of complete information and based on the results of Clarke.

Asymptotic likelihood theory for diffusion processes

1975 ◽  
Vol 12 (2) ◽  
pp. 228-238 ◽  
Author(s):  
B. M. Brown ◽  
J. I. Hewitt
Keyword(s):  
Central Limit Theorem ◽  
Maximum Likelihood ◽  
Limit Theorem ◽  
Continuous Time ◽  
Diffusion Processes ◽  
Maximum Likelihood Estimates ◽  
Stochastic Integrals ◽  
Ratio Test ◽  
Chi Squared ◽  
Asymptotic Likelihood

We investigate the large-sample behaviour of maximum likelihood estimates (MLE's) of the parameters of a diffusion process, which is observed throughout continuous time. The results (limit normal distribution for the MLE and an asymptotic chi-squared likelihood ratio test) correspond exactly to classical asymptotic likelihood results, and follow easily from a central limit theorem for stochastic integrals.

Limit theorem for continuous-time random walks with two time scales

2004 ◽  
Vol 41 (2) ◽  
pp. 455-466 ◽  
Author(s):  
Peter Becker-Kern ◽  
Mark M. Meerschaert ◽  
Hans-Peter Scheffler
Keyword(s):  
Random Walks ◽  
Time Scales ◽  
Waiting Time ◽  
Continuous Time ◽  
Fractional Partial Differential Equations ◽  
Finite Dimensional ◽  
Mean Waiting Time ◽  
Continuous Time Random Walks ◽  
Random Jumps ◽  
The Mean

Continuous-time random walks incorporate a random waiting time between random jumps. They are used in physics to model particle motion. A physically realistic rescaling uses two different time scales for the mean waiting time and the deviation from the mean. This paper derives the scaling limits for such processes. These limit processes are governed by fractional partial differential equations that may be useful in physics. A transfer theorem for weak convergence of finite-dimensional distributions of stochastic processes is also obtained.

Limit theorem for continuous-time random walks with two time scales

2004 ◽  
Vol 41 (02) ◽  
pp. 455-466 ◽  
Author(s):  
Peter Becker-Kern ◽  
Mark M. Meerschaert ◽  
Hans-Peter Scheffler
Keyword(s):  
Random Walks ◽  
Time Scales ◽  
Waiting Time ◽  
Continuous Time ◽  
Fractional Partial Differential Equations ◽  
Finite Dimensional ◽  
Mean Waiting Time ◽  
Continuous Time Random Walks ◽  
Random Jumps ◽  
The Mean

Continuous-time random walks incorporate a random waiting time between random jumps. They are used in physics to model particle motion. A physically realistic rescaling uses two different time scales for the mean waiting time and the deviation from the mean. This paper derives the scaling limits for such processes. These limit processes are governed by fractional partial differential equations that may be useful in physics. A transfer theorem for weak convergence of finite-dimensional distributions of stochastic processes is also obtained.

The Estimation of Open Higher-Order Continuous Time Dynamic Models with Mixed Stock and Flow Data

1986 ◽  
Vol 2 (3) ◽  
pp. 350-373 ◽  
Author(s):  
A. R. Bergstrom
Keyword(s):  
Maximum Likelihood ◽  
Continuous Time ◽  
Dynamic Models ◽  
Higher Order ◽  
Maximum Likelihood Estimates ◽  
Order System ◽  
Exogenous Variables ◽  
Time Dynamic ◽  
Mixed Stock ◽  
Order Continuous

This article extends recent work on the Gaussian or quasi-maximum likelihood estimation of the parameters of a closed higher-order continuous time dynamic model by introducing exogenous variables into the model The method presented yields exact maximum likelihood estimates when the innovations are Gaussian and the exogenous variables are polynomials in time of degree not exceeding two, and it can be expected to yield very good estimates under more general conditions. It is applicable, in principle, to a system of any order with mixed stock and iow data. The precise formulas for its implementation are derived, in this article, for a second-order system in which both the endog-enous and exogenous variables are a mixture of stock and flow variables.

Asymptotic likelihood theory for diffusion processes

1975 ◽  
Vol 12 (02) ◽  
pp. 228-238 ◽  
Author(s):  
B. M. Brown ◽  
J. I. Hewitt
Keyword(s):  
Central Limit Theorem ◽  
Maximum Likelihood ◽  
Limit Theorem ◽  
Continuous Time ◽  
Diffusion Processes ◽  
Maximum Likelihood Estimates ◽  
Stochastic Integrals ◽  
Ratio Test ◽  
Chi Squared ◽  
Asymptotic Likelihood

We investigate the large-sample behaviour of maximum likelihood estimates (MLE's) of the parameters of a diffusion process, which is observed throughout continuous time. The results (limit normal distribution for the MLE and an asymptotic chi-squared likelihood ratio test) correspond exactly to classical asymptotic likelihood results, and follow easily from a central limit theorem for stochastic integrals.

scholarly journals Simulation and Calibration Between Parameters of Continuous Time Random Walks and Subdifusive Model

2017 ◽  
Vol 18 (2) ◽  
pp. 0305 ◽  
Author(s):  
Ana Paula De Paiva Pereira ◽  
João Paulo Fernandes ◽  
Allbens Picardi Faria Atman ◽  
José Luiz Acebal
Keyword(s):  
Theoretical Model ◽  
Random Walks ◽  
Waiting Time ◽  
Continuous Time ◽  
Fractional Derivatives ◽  
Full Range ◽  
Theoretical Models ◽  
Jump Length ◽  
Mean Waiting Time ◽  
Continuous Time Random Walks

We address the problem of subdiusion or normal diusion to perform a calibration between the parameters used in simulation and the parameters of a subdifusive model. The theoretical model is written as a generalized diusion equation with fractional derivatives in time. The data is generated by simulations consisting of continuous-time random walks with controlled mean waiting time and jump length variance to provide a full range of cases between subdiusion andnormal diusion. From the simulations, we compare the accuracy of two methods to obtain the diusion constant, the order of fractional derivatives: the analysis of the dispersion of the variance in time and the optimization tting of theoretical model solutions to histogram of positions. We highlight the connection between the parameters of the simulations the parameters of the theoretical models.

scholarly journals Directional Selection and the Site-Frequency Spectrum

Genetics ◽  
2001 ◽  
Vol 159 (4) ◽  
pp. 1779-1788 ◽  
Author(s):  
Carlos D Bustamante ◽  
John Wakeley ◽  
Stanley Sawyer ◽  
Daniel L Hartl
Keyword(s):  
Maximum Likelihood ◽  
High Power ◽  
Negative Selection ◽  
Likelihood Estimation ◽  
Directional Selection ◽  
Likelihood Ratio Tests ◽  
Maximum Likelihood Estimates ◽  
Likelihood Methods ◽  
Ancestral States ◽  
Asymptotic Variances

Abstract In this article we explore statistical properties of the maximum-likelihood estimates (MLEs) of the selection and mutation parameters in a Poisson random field population genetics model of directional selection at DNA sites. We derive the asymptotic variances and covariance of the MLEs and explore the power of the likelihood ratio tests (LRT) of neutrality for varying levels of mutation and selection as well as the robustness of the LRT to deviations from the assumption of free recombination among sites. We also discuss the coverage of confidence intervals on the basis of two standard-likelihood methods. We find that the LRT has high power to detect deviations from neutrality and that the maximum-likelihood estimation performs very well when the ancestral states of all mutations in the sample are known. When the ancestral states are not known, the test has high power to detect deviations from neutrality for negative selection but not for positive selection. We also find that the LRT is not robust to deviations from the assumption of independence among sites.

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