Estimation in discretely observed diffusions killed at a threshold

TL;DR

This paper addresses the challenge of estimating parameters in discretely observed diffusion processes that are terminated at a threshold, highlighting bias issues and proposing methods for bias correction and consistent estimation.

Contribution

It introduces likelihood calculations for killed diffusions, analyzes bias in single trajectory estimation, and demonstrates bias correction techniques and asymptotic properties with multiple trajectories.

Findings

Likelihood functions for killed diffusions are derived or approximated.

Bias in parameter estimates from single trajectories can be large and skewed.

Bias correction via parametric bootstrap improves estimation accuracy.

Abstract

Parameter estimation in diffusion processes from discrete observations up to a first-hitting time is clearly of practical relevance, but does not seem to have been studied so far. In neuroscience, many models for the membrane potential evolution involve the presence of an upper threshold. Data are modeled as discretely observed diffusions which are killed when the threshold is reached. Statistical inference is often based on the misspecified likelihood ignoring the presence of the threshold causing severe bias, e.g. the bias incurred in the drift parameters of the Ornstein-Uhlenbeck model for biological relevant parameters can be up to 25-100%. We calculate or approximate the likelihood function of the killed process. When estimating from a single trajectory, considerable bias may still be present, and the distribution of the estimates can be heavily skewed and with a huge variance. Parametric bootstrap is effective in correcting the bias. Standard asymptotic results do not apply, but consistency and asymptotic normality may be recovered when multiple trajectories are observed, if the mean first-passage time through the threshold is finite. Numerical examples illustrate the results and an experimental data set of intracellular recordings of the membrane potential of a motoneuron is analyzed.