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Parametric estimation for the standard and the geometric telegraph process observed at discrete times
Stefano Iacus, Department of Economics, Business and Statistics, University of Milan, IT
Alessandro De Gregorio, Department of Statistics, University of Padova
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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. $\Delta_n \to 0$, $n\Delta = T<\infty$ as $n \to \infty$, with $T>0$ 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 $n$.
SUGGESTED CITATION:
Stefano Iacus and Alessandro De Gregorio,
"Parametric estimation for the standard and the geometric telegraph process observed at discrete times"
(July 2006).
UNIMI - Research Papers in Economics, Business, and Statistics.
Statistics and Mathematics.
Working Paper 14.
http://services.bepress.com/unimi/statistics/art14
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