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M-estimation for Gaussian processes with time-inhomogeneous drifts from high-frequency data

Yasutaka Shimizu

TL;DR

This work develops a contrast-based inference framework for Gaussian processes with time-inhomogeneous drifts observed at high frequency. By built from local increments, the local-Gauss contrast yields an M-estimator that avoids full covariance-matrix inversions, and is supported by consistency and asymptotic normality results under ergodicity; the drift parameter exhibits a nonstandard convergence rate due to direct Riemann integrability, while kernel parameters may be recovered via moment-based corrections or mollification for non-smooth kernels. The authors also provide concrete examples (Gaussian, OU, Rational Quadratic) and numerical simulations illustrating identifiability issues and finite-sample behavior, plus strategies to improve estimation stability in practice. The approach offers a simple, scalable alternative to likelihood-based methods, with potential hybrids with frequency-domain techniques and extensions to higher-order kernel information through moment estimators.

Abstract

We propose a contrast-based estimation method for Gaussian processes with time-inhomogeneous drifts, observed under high-frequency sampling. The process is modeled as the sum of a deterministic drift function and a stationary Gaussian component with a parametric kernel. Our method constructs a local contrast function from adjacent increments, which avoids inversion of large covariance matrices and allows for efficient computation. We prove consistency and asymptotic normality of the resulting estimators under general ergodicity conditions. A distinctive feature of our approach is that the drift estimator attains a nonstandard convergence rate, stemming from the direct Riemann integrability of the drift density. This highlights a fundamental difference from standard estimation regimes. Furthermore, when the local contrast fails to identify all parameters in the covariance kernel, moment-based corrections can be incorporated to recover identifiability. The proposed framework is simple, flexible, and particularly well suited for high-frequency inference with time-inhomogeneous structure.

M-estimation for Gaussian processes with time-inhomogeneous drifts from high-frequency data

TL;DR

This work develops a contrast-based inference framework for Gaussian processes with time-inhomogeneous drifts observed at high frequency. By built from local increments, the local-Gauss contrast yields an M-estimator that avoids full covariance-matrix inversions, and is supported by consistency and asymptotic normality results under ergodicity; the drift parameter exhibits a nonstandard convergence rate due to direct Riemann integrability, while kernel parameters may be recovered via moment-based corrections or mollification for non-smooth kernels. The authors also provide concrete examples (Gaussian, OU, Rational Quadratic) and numerical simulations illustrating identifiability issues and finite-sample behavior, plus strategies to improve estimation stability in practice. The approach offers a simple, scalable alternative to likelihood-based methods, with potential hybrids with frequency-domain techniques and extensions to higher-order kernel information through moment estimators.

Abstract

We propose a contrast-based estimation method for Gaussian processes with time-inhomogeneous drifts, observed under high-frequency sampling. The process is modeled as the sum of a deterministic drift function and a stationary Gaussian component with a parametric kernel. Our method constructs a local contrast function from adjacent increments, which avoids inversion of large covariance matrices and allows for efficient computation. We prove consistency and asymptotic normality of the resulting estimators under general ergodicity conditions. A distinctive feature of our approach is that the drift estimator attains a nonstandard convergence rate, stemming from the direct Riemann integrability of the drift density. This highlights a fundamental difference from standard estimation regimes. Furthermore, when the local contrast fails to identify all parameters in the covariance kernel, moment-based corrections can be incorporated to recover identifiability. The proposed framework is simple, flexible, and particularly well suited for high-frequency inference with time-inhomogeneous structure.

Paper Structure

This paper contains 32 sections, 20 theorems, 298 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Models and Assumptions
  3. Gaussian processes with time-inhomogeneous drifts
  4. M-estimation for sampled Gaussian processes
  5. Local-Gauss contrast
  6. Moment estimators
  7. Examples and simulations
  8. Drifted Gaussian processes with Gaussian Kernels
  9. Ornstein-Uhlenbeck processes
  10. The Rational Quadratic kernel
  11. Numerical experiments
  12. Drifted Gaussian processes with Gaussian Kernels
  13. Discussion of numerical experiments
  14. Concluding remarks
  15. Proofs of main theorems
  16. ...and 17 more sections

Key Result

Theorem 3.1

Suppose the assumptions Aas:2--Aas:4 and Bas:K2--Bas:id. Then the $M$-estimator $\widehat{\theta}_n$ is consistent to $\theta_0$: under the sampling sampling.

Figures (3)

  • Figure 1: Normal QQ plots for Case (I): Scaled estimators $h_n^{-1/2}(\widehat{\xi}_n - \xi_0)$ and $n^{1/2}(\widehat{\alpha}_n - \alpha_0)$, $n^{1/2}(\widehat{\beta}_n - \beta_0)$ over 500 replications. The plots show good agreement with the theoretical normal distribution.
  • Figure 2: Normal QQ plots for Case (II): Scaled estimators from joint estimation with $h_n = n^{-0.4}$. Upward bias in $\widehat{\alpha}_n$ and downward bias in $\widehat{\beta}_n$ are accompanied by departures from normality, especially for $\widehat{\alpha}_n$ although $\widehat{\xi}_n$ still seems to be asymptotically normal.
  • Figure 3: Normal QQ plots for Case (III): Scaled estimators with $\xi$ fixed at its true value and $h_n = n^{-0.8}$. While bias is minimal, the slow growth of $T_n:=nh_n=n^{0.2}$ results in clear deviations from normality compared to Case (I), where $T_n=n^{0.6}$, reflecting insufficient mixing in finite samples.

Theorems & Definitions (50)

  • Definition 1.1
  • Definition 1.2
  • Example 2.1
  • Example 2.2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Remark 3.2: Relation to composite likelihood methods
  • Theorem 3.1
  • ...and 40 more