An asymptotic study of the joint maximum likelihood estimation of the regularity and the amplitude parameters of a periodized Mat{\'e}rn model

S\'ebastien J Petit (LNE)

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An asymptotic study of the joint maximum likelihood estimation of the regularity and the amplitude parameters of a periodized Mat{\'e}rn model

Sébastien J Petit (LNE)

Statistics Theory · Submitted 16 Sep 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The mean integrated squared error is also analyzed with fixed and estimated parameters, showing that maximum likelihood estimation yields asymptotically the same error as if the ground truth was known.

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arXiv ID2209.07791 (version 7)

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CitationS\'ebastien J Petit (LNE). "An asymptotic study of the joint maximum likelihood estimation of the regularity and the amplitude parameters of a periodized Mat{\'e}rn model." arXiv:2209.07791 (2022). https://pith.science/paper/KDXCBIW6

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This work considers parameter estimation for Gaussian process interpolation with a periodized version of the Mat{\'e}rn covariance function introduced by Stein. Convergence rates are studied for the joint maximum likelihood estimation of the regularity and the amplitude parameters when the data are sampled according to the model. The mean integrated squared error is also analyzed with fixed and estimated parameters, showing that maximum likelihood estimation yields asymptotically the same error as if the ground truth was known. Finally, the case where the observed function is a fixed deterministic element of a Sobolev space of continuous functions is also considered, suggesting that a joint estimation does not select the regularity parameter as if the amplitude were fixed.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	This work considers parameter estimation for Gaussian process interpolation with a periodized version of the Mat{\'e}rn covariance function introduced by Stein.	contribution 0.48	0.46	0.03	asserted 0.94
s1	Convergence rates are studied for the joint maximum likelihood estimation of the regularity and the amplitude parameters when the data are sampled according to the model.	contribution 0.92	0.84	0.12	asserted 0.92
s2	The mean integrated squared error is also analyzed with fixed and estimated parameters, showing that maximum likelihood estimation yields asymptotically the same error as if the ground truth was known.	result 0.98	0.91	0.41	asserted 1.00
s3	Finally, the case where the observed function is a fixed deterministic element of a Sobolev space of continuous functions is also considered, suggesting that a joint estimation does not select the regularity parameter as if the amplitude were fixed.	result 0.71	0.78	0.20	hedged possibility 1.00

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Original abstract

This work considers parameter estimation for Gaussian process interpolation with a periodized version of the Mat{\'e}rn covariance function introduced by Stein. Convergence rates are studied for the joint maximum likelihood estimation of the regularity and the amplitude parameters when the data are sampled according to the model. The mean integrated squared error is also analyzed with fixed and estimated parameters, showing that maximum likelihood estimation yields asymptotically the same error as if the ground truth was known. Finally, the case where the observed function is a fixed deterministic element of a Sobolev space of continuous functions is also considered, suggesting that a joint estimation does not select the regularity parameter as if the amplitude were fixed.

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Compares with it 1 paper

Although it is a rather strict assumption, quasi-uniformity is more general than the uniformity assumptions used in [ref] and [ref] .

Smoothness Estimation for Whittle-Mat\'ern Processes on Closed Riemannian Manifolds · Introduction · 0.97

Cites it as background 1 paper

[ref] has studied the ML and LOO-CV estimators for the smoothness parameter in the Matérn model; see also [ref] .

Comparing Scale Parameter Estimators for Gaussian Process Interpolation with the Brownian Motion Prior: Leave-One-Out Cross Validation and Maximum Likelihood · Related work · 0.97

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The following result is a minor adaptation of the reasoning used by [ref] and [ref] , showing that the conjecture is verified more generally if $ϕ$ is bounded.

Asymptotic Bounds for Smoothness Parameter Estimates in Gaussian Process Interpolation · The deterministic case · 0.86

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