Parameter estimation in Markov processes from discrete observations up to the first-hitting time of an upper threshold is clearly of practical relevance, but does not seem to have been studied so far, except for a few special models with discrete state space. In neuroscience, many models for the membrane potential evolution involve the presence of a threshold. Data are modeled as discretely observed trajectories of some process, which is killed as soon as the threshold is reached. Examples are the Ornstein-Uhlenbeck or other diffusion processes, random walk type models, or processes with jumps. From the statistical analysis of such data, the need for a consistent inferential method has emerged. For example, the bias incurred in the drift parameters of the Ornstein-Uhlenbeck model for biological relevant parameters is 25--100%, when ignoring the presence of the threshold, as is custumary in applications. Thus, it is of paramount importance to address this estimation problem. We calculate or approximate the likelihood function of the killed process. However, when estimating from a single trajectory, considerable bias may be encountered and the distribution of the estimates can be heavily skewed and with a huge variance. The 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 how the results can be applied for specific models.
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