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.

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.

On effects of discretization on estimators of drift parameters for diffusion processes

1996 ◽  
Vol 33 (04) ◽  
pp. 1061-1076 ◽  
Author(s):  
P. E. Kloeden ◽  
E. Platen ◽  
H. Schurz ◽  
M. Sørensen
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Diffusion Processes ◽  
Maximum Likelihood Estimators ◽  
Statistical Properties ◽  
Stochastic Integrals

In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.

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.

Pearson’s chi-squared statistics: approximation theory and beyond

Biometrika ◽  
2019 ◽  
Vol 106 (3) ◽  
pp. 716-723
Author(s):  
Mengyu Xu ◽  
Danna Zhang ◽  
Wei Biao Wu
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Approximation Theory ◽  
Quadratic Forms ◽  
Degrees Of Freedom ◽  
Goodness Of Fit ◽  
High Dimensional ◽  
Goodness Of Fit Test ◽  
Chi Squared

Summary We establish an approximation theory for Pearson’s chi-squared statistics in situations where the number of cells is large, by using a high-dimensional central limit theorem for quadratic forms of random vectors. Our high-dimensional central limit theorem is proved under Lyapunov-type conditions that involve a delicate interplay between the dimension, the sample size, and the moment conditions. We propose a modified chi-squared statistic and introduce an adjusted degrees of freedom. A simulation study shows that the modified statistic outperforms Pearson’s chi-squared statistic in terms of both size accuracy and power. Our procedure is applied to the construction of a goodness-of-fit test for Rutherford’s alpha-particle data.

Discrete time methods for simulating continuous time Markov chains

1976 ◽  
Vol 8 (04) ◽  
pp. 772-788 ◽  
Author(s):  
Arie Hordijk ◽  
Donald L. Iglehart ◽  
Rolf Schassberger
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Markov Chains ◽  
Confidence Intervals ◽  
Discrete Time ◽  
Continuous Time ◽  
Statistical Efficiency ◽  
Numerical Examples ◽  
The Central Limit Theorem ◽  
Continuous Time Markov Chains

This paper discusses several problems which arise when the regenerative method is used to analyse simulations of Markov chains. The first problem involves calculating the variance constant which appears in the central limit theorem used to obtain confidence intervals. Knowledge of this constant is very helpful in evaluating simulation methodologies. The second problem is to devise a method for simulating continuous time Markov chains without having to generate exponentially distributed holding times. Several methods are presented and compared. Numerical examples are given to illustrate the computional and statistical efficiency of these methods.

THE FUNCTIONAL CENTRAL LIMIT THEOREM AND WEAK CONVERGENCE TO STOCHASTIC INTEGRALS I

2000 ◽  
Vol 16 (5) ◽  
pp. 621-642 ◽  
Author(s):  
Robert M. de Jong ◽  
James Davidson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Central Limit ◽  
Unit Root ◽  
Functional Central Limit Theorem ◽  
Stochastic Integrals ◽  
Mixing Processes ◽  
Unit Root Testing ◽  
Functional Central Limit

This paper gives new conditions for the functional central limit theorem, and weak convergence of stochastic integrals, for near-epoch-dependent functions of mixing processes. These results have fundamental applications in the theory of unit root testing and cointegrating regressions. The conditions given improve on existing results in the literature in terms of the amount of dependence and heterogeneity permitted, and in particular, these appear to be the first such theorems in which virtually the same assumptions are sufficient for both modes of convergence.

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