Exact maximum likelihood inference for drifted multi-sub-fractional Brownian motion at discrete observation
Abstract
Sub-fractional Brownian motion is self-similar and long-range dependent but has no stationary increments, so the increment covariance is not Toeplitz and no spectral density is available. We show that a complete finite-sample likelihood theory survives nonetheless. The model is a linear trend observed at N equidistant times through a superposition of m independent sub-fractional Brownian motions with known Hurst indices and a common scale. Nondegeneracy follows from realising the process as the even part of a two-sided fractional Brownian motion, and the maximum likelihood estimators of the trend and of the scale are explicit. The statistics on which inference rests are pivotal, their laws depending on the sample size alone, so intervals and tests of exact level are available at every N≥2, together with complete sufficiency, minimum variance unbiasedness and attainment of the Cramér--Rao bound. An explicit variance bound gives strong consistency and asymptotic normality, and simulations confirm the exact coverage and the predicted effect of a misspecified Hurst vector.
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BibTeX
@misc{alharby2026exact,
title = {{Exact maximum likelihood inference for drifted multi-sub-fractional Brownian motion at discrete observation}},
author = {Afrah Al-Harby and Ezzedine Mliki and Manal Al-Ohali},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22617},
eprint = {2609.22617},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22617},
}
RIS
TY - UNPB
TI - Exact maximum likelihood inference for drifted multi-sub-fractional Brownian motion at discrete observation
AU - Afrah Al-Harby
AU - Ezzedine Mliki
AU - Manal Al-Ohali
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22617
UR - https://arxiv.org/abs/2609.22617
AB - Sub-fractional Brownian motion is self-similar and long-range dependent but has no stationary increments, so the increment covariance is not Toeplitz and no spectral density is available. We show that a complete finite-sample likelihood theory survives nonetheless. The model is a linear trend observed at N equidistant times through a superposition of m independent sub-fractional Brownian motions with known Hurst indices and a common scale. Nondegeneracy follows from realising the process as the even part of a two-sided fractional Brownian motion, and the maximum likelihood estimators of the trend and of the scale are explicit. The statistics on which inference rests are pivotal, their laws depending on the sample size alone, so intervals and tests of exact level are available at every N≥2, together with complete sufficiency, minimum variance unbiasedness and attainment of the Cramér--Rao bound. An explicit variance bound gives strong consistency and asymptotic normality, and simulations confirm the exact coverage and the predicted effect of a misspecified Hurst vector.
ER -