Parametric estimation for the standard and the geometric telegraph process observed at discrete times

Stefano Iacus () and Alessandro De Gregorio
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Alessandro De Gregorio: Department of Statistics, University of Padova

No unimi-1033, UNIMI - Research Papers in Economics, Business, and Statistics from Universitá degli Studi di Milano

Abstract: The telegraph process $X(t)$, $t>0$, (Goldstein, 1951) and the geometric telegraph process $S(t) = s_0 \exp\{(\mu -\frac12\sigma^2)t + \sigma X(t)\}$ with $\mu$ a known constant and $\sigma>0$ a parameter are supposed to be observed at $n+1$ equidistant time points $t_i=i\Delta_n,i=0,1,\ldots, n$. For both models $\lambda$, the underlying rate of the Poisson process, is a parameter to be estimated. In the geometric case, also $\sigma>0$ has to be estimated. We propose different estimators of the parameters and we investigate their performance under the high frequency asymptotics, i.e. $\Delta_n \to 0$, $n\Delta = T 0$ fixed. The process $X(t)$ in non markovian, non stationary and not ergodic thus we use approximation arguments to derive estimators. Given the complexity of the equations involved only estimators on the first model can be studied analytically. Therefore, we run an extensive Monte Carlo analysis to study the performance of the proposed estimators also for small sample size $n$.

Keywords: telegraph process; discretely observed process; inference for stochastic processes (search for similar items in EconPapers)
Date: 2006-07-25
Note: oai:cdlib1:unimi-1033
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Journal Article: Parametric estimation for the standard and geometric telegraph process observed at discrete times (2008)
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