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Linear estimators for Gaussian random variables in Hilbert spaces

Stefan Tappe

TL;DR

The paper addresses statistical inference for Gaussian models valued in a separable Hilbert space with unknown mean in a subspace $U$ and trace-class covariance $Q$, observing $Y=\zeta+\sigma\varepsilon$ where $\varepsilon\sim N(0,Q)$. It develops linear estimators based on orthogonal projections, derives their unbiasedness and risk properties, and extends to functionals and finite-dimensional truncations to enable learning in infinite dimensions. Confidence intervals are constructed for both known and unknown variance, with unbiased variance estimators and independence results that support valid inference. The work also delivers Fisher-based hypothesis tests for subspace inclusion and a comprehensive infinite-dimensional linear regression framework, including practical ML perspectives and Wiener/Brownian-bridge examples to illustrate the methodology.

Abstract

We study a statistical model for infinite dimensional Gaussian random variables with unknown parameters. For this model we derive linear estimators for the mean and the variance of the Gaussian distribution. Furthermore, we construct confidence intervals and perform hypothesis testing. A linear regression problem in infinite dimensions and some perspectives to statistical and machine learning are presented as well.

Linear estimators for Gaussian random variables in Hilbert spaces

TL;DR

The paper addresses statistical inference for Gaussian models valued in a separable Hilbert space with unknown mean in a subspace and trace-class covariance , observing where . It develops linear estimators based on orthogonal projections, derives their unbiasedness and risk properties, and extends to functionals and finite-dimensional truncations to enable learning in infinite dimensions. Confidence intervals are constructed for both known and unknown variance, with unbiased variance estimators and independence results that support valid inference. The work also delivers Fisher-based hypothesis tests for subspace inclusion and a comprehensive infinite-dimensional linear regression framework, including practical ML perspectives and Wiener/Brownian-bridge examples to illustrate the methodology.

Abstract

We study a statistical model for infinite dimensional Gaussian random variables with unknown parameters. For this model we derive linear estimators for the mean and the variance of the Gaussian distribution. Furthermore, we construct confidence intervals and perform hypothesis testing. A linear regression problem in infinite dimensions and some perspectives to statistical and machine learning are presented as well.
Paper Structure (13 sections, 34 theorems, 136 equations, 1 figure)

This paper contains 13 sections, 34 theorems, 136 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The statistical model
  3. Estimation of the mean
  4. Perspectives to statistical and machine learning
  5. Estimation of functionals of the mean
  6. Confidence intervals with known variance
  7. Estimation of the variance
  8. Confidence intervals with unknown variance
  9. Hypothesis testing
  10. Linear regression in infinite dimensions
  11. Linear operators in Hilbert spaces
  12. Gaussian random variables in Hilbert spaces
  13. The Student's $t$-distribution and the Fisher distribution

Key Result

Lemma 2.4

For each $\vartheta \in \Theta$ we have the decomposition In particular, if $T(Y)$ is an unbiased estimator, then for each $\vartheta \in \Theta$ we have

Figures (1)

  • Figure 1: Observation of the random variable $Y$ in case $k=4$.

Theorems & Definitions (73)

  • Remark 2.1
  • Example 2.2: Wiener process
  • Example 2.3: Brownian bridge
  • Lemma 2.4
  • proof
  • Proposition 3.1
  • Lemma 3.2
  • Theorem 3.3
  • proof
  • Proposition 4.1
  • ...and 63 more