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Statistical Analysis of Wiener Process Based on Partial Observations
Hrochová, Magdalena ; Hlubinka, Daniel (advisor) ; Omelka, Marek (referee)
Wiener process-a random process with continuous time-plays an important role in mathematics, physics or economy. It is often good to know whether it contains any deterministic part, e.g. drift or scale. However, it is nearly impossible either observe the whole trajectory of the process or preserve its full history. This thesis deals with a statistical analysis based on partial observations, namely passage times through some given barriers. We propose several statistical methods for testing hypotheses about drift or scale using these observations. As supporting methods, we consider the maximum likelihood theory, non-parametric test against a trend, and binomial test. For testing the value of scale in the model with no drift and constant scale we recommend maximum likelihood theory. We derive the estimate and related tests in the case of observing only three barriers. The simulation study suggested observing more barriers for testing monotony of scale in a model with linear drift, or testing monotone and convex/concave drift in a model with constant scale. 1
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