Statistical inference across time scales

TL;DR

This paper explores how statistical inference varies across different time scales using a toy model, revealing a smooth transition from Poissonian to Gaussian regimes and highlighting the impact of sampling rate on information loss.

Contribution

It introduces a detailed analysis of inference across time scales in a compound Poisson process, showing the quadratic variation estimator's efficiency and limitations.

Findings

Quadratic variation estimator is efficient at microscopic and macroscopic scales.

Significant information loss occurs at intermediate scales due to sampling rate.

Smooth transition from Poissonian to Gaussian regimes across time scales.

Abstract

We investigate statistical inference across time scales. We take as toy model the estimation of the intensity of a discretely observed compound Poisson process with symmetric Bernoulli jumps. We have data at different time scales: microscopic, intermediate and macroscopic. We quantify the smooth statistical transition from a microscopic Poissonian regime to a macroscopic Gaussian regime. The classical quadratic variation estimator is efficient in both microscopic and macroscopic scales but surprisingly shows a substantial loss of information in the intermediate scale that can be explicitly related to the sampling rate. We discuss the implications of these findings beyond this idealised framework.