Tetsuya Takabatake, Jun Yu, Chen Zhang
arXiv 5 Sep 2025 · Mathematics — Statistics Theory
arXiv:2509.04987 · PDF · DOI · OpenAlex · Extracted main text
This paper proposes a novel exact maximum likelihood (ML) estimation method for general Gaussian processes, where all parameters are estimated jointly. The exact ML estimator (MLE) is consistent and asymptotically normally distributed. We prove the local asymptotic normality (LAN) property of the sequence of statistical experiments for general Gaussian processes in the sense of Le Cam, thereby enabling optimal estimation and facilitating statistical inference. The results rely solely on the asymptotic behavior of the spectral density near zero, allowing them to be widely applied. The established optimality not only addresses the gap left by Adenstedt(1974), who proposed an efficient but infeasible estimator for the long-run mean $\mu$, but also enables us to evaluate the finite-sample performance of the existing method -- the commonly used plug-in MLE, in which the sample mean is substituted into the likelihood. Our simulation results show that the plug-in MLE performs nearly as well as the exact MLE, alleviating concerns that inefficient estimation of $\mu$ would compromise the efficiency of the remaining parameter estimates.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Fukasawa, M. and T. Takabatake (2019) Asymptotically efficient estimators for self-similar stationary Gaussian noises under high frequency observations | 0.928 | 5 | 3 | 80% |
| 2 | Wang, X., W. Xiao, J. Yu, and C. Zhang (2024) Maximum likelihood estimation of fractional Ornstein-Uhlenbeck process with discretely sampled data | 0.909 | 8 | 4 | 75% |
| 3 | Cohen, S., F. Gamboa, C. Lacaux, and J.-M. Loubes (2013) LAN property for some fractional type Brownian motion | 0.909 | 8 | 3 | 75% |
| 4 | Dahlhaus, R (1989) Efficient parameter estimation for self-similar processes | 0.899 | 11 | 4 | 73% |
| 5 | Dahlhaus, R (2006) Correction: Efficient parameter estimation for self-similar processes | 0.874 | 5 | 2 | 100% |
| 6 | Adenstedt, R. K (1974) On large-sample estimation for the mean of a stationary random sequence | 0.860 | 11 | 6 | 64% |
| 7 | Shi, S., J. Yu, and C. Zhang (2024) On the spectral density of fractional Ornstein-Uhlenbeck processes | 0.843 | 4 | 3 | 75% |
| 8 | Cheung, Y.-W. and F. X. Diebold (1994) On maximum likelihood estimation of the differencing parameter of fractionally-integrated noise with unknown mean | 0.811 | 4 | 2 | 100% |
| 9 | Lieberman, O., R. Rosemarin, and J. Rousseau (2012) Asymptotic theory for maximum likelihood estimation of the memory parameter in stationary Gaussian processes | 0.744 | 17 | 4 | 41% |
| 10 | Gatheral, J., T. Jaisson, and M. Rosenbaum (2018) Volatility is rough | 0.737 | 4 | 2 | 75% |
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