A Speed-based Estimator of Signal-to-Noise Ratios

Yuang Song () and Olympia Hadjiliadis ()
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Yuang Song: Columbia University
Olympia Hadjiliadis: CUNY-Hunter College

Methodology and Computing in Applied Probability, 2025, vol. 27, issue 2, 1-15

Abstract: Abstract We present an innovative method to measure the signal-to-noise ratio (SNR) in a Brownian motion model. That is, the ratio of the mean to the standard deviation of the Brownian motion. Our method is based on the method of moments estimation of the drawdown and drawup speeds in a Brownian motion model, where the drawdown process is defined as the current drop of the process from its running maximum and the drawup process is the current rise of the process above its running minimum. The speed of a drawdown of K units (or a drawup of K units) is then the time between the last maximum (or minimum) of the process and the time the drawdown (or drawup) process hits the threshold K. Our estimator only requires the record values of the process and the times at which deviations from the record values exceed a certain threshold, whereas the uniformly minimum-variance unbiased estimator (UMVUE) requires the entire path of the process. We derive the asymptotic distributions of both estimators and compare them.

Keywords: Brownian motion; Drawdown and drawup; Signal-to-noise ratio; 60G35; 62G05 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s11009-025-10150-0

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