Minimax linear estimation on the half-line via Mellin transform

B. Y. Levit

Queen's University, Canada

Abstract: In the white Gaussian noise (WGN) model on the half-line, $dV(x)=f(x)\,dx+\varepsilon \, dW(x)$, $x>0$, minimax linear estimators of the functional $x^{1/2}_0f(x_0)$ are sought for a given $x_0>0$. Here, $f$ is an unknown signal satisfying a certain restriction $f\in\mathfrak{F}$, $\varepsilon>0$ is a given parameter, and $W(\,{\cdot}\,)$ is the standard Wiener process. With the use of the Mellin transform, it is shown that the above problem is equivalent to a similar (but better known) WGN model on the whole line, $dY(u)=g(u)\,du+\varepsilon \, dW(u)$, $-\infty<u<\infty$, where minimax linear estimators of the functional $g(u_0)$ are sought under a matching restriction $g\in \mathfrak{F}_0$, for $u_0=\log x_0.$ This demonstrates a close connection between the two models, which permits one to translate results on minimax linear estimation easily from one model to the other. In particular, it leads to minimax linear estimators on the half-line for a variety of ellipsoidal and cuboidal function classes $\mathfrak{F}$.

Keywords: white Gaussian noise model on the half-line, minimax linear estimation, Mellin transform.

Received: 09.07.2024
Accepted: 26.11.2024

DOI: 10.4213/tvp5732

 English version: Theory of Probability and its Applications, 2025, 70:2, 185–199

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