The Annals of Statistics

On the approximate maximum likelihood estimation for diffusion processes

Jinyuan Chang and Song Xi Chen

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Abstract

The transition density of a diffusion process does not admit an explicit expression in general, which prevents the full maximum likelihood estimation (MLE) based on discretely observed sample paths. Aït-Sahalia [J. Finance 54 (1999) 1361–1395; Econometrica 70 (2002) 223–262] proposed asymptotic expansions to the transition densities of diffusion processes, which lead to an approximate maximum likelihood estimation (AMLE) for parameters. Built on Aït-Sahalia’s [Econometrica 70 (2002) 223–262; Ann. Statist. 36 (2008) 906–937] proposal and analysis on the AMLE, we establish the consistency and convergence rate of the AMLE, which reveal the roles played by the number of terms used in the asymptotic density expansions and the sampling interval between successive observations. We find conditions under which the AMLE has the same asymptotic distribution as that of the full MLE. A first order approximation to the Fisher information matrix is proposed.

Article information

Source
Ann. Statist., Volume 39, Number 6 (2011), 2820-2851.

Dates
First available in Project Euclid: 24 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.aos/1327413770

Digital Object Identifier
doi:10.1214/11-AOS922

Mathematical Reviews number (MathSciNet)
MR3012393

Zentralblatt MATH identifier
1246.62181

Subjects
Primary: 62M05: Markov processes: estimation
Secondary: 62F12: Asymptotic properties of estimators

Keywords
Asymptotic expansion asymptotic normality consistency discrete time observation maximum likelihood estimation

Citation

Chang, Jinyuan; Chen, Song Xi. On the approximate maximum likelihood estimation for diffusion processes. Ann. Statist. 39 (2011), no. 6, 2820--2851. doi:10.1214/11-AOS922. https://projecteuclid.org/euclid.aos/1327413770


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