The telegraph process X(t), t>0, (Goldstein, 1951) and the geometric telegraph process S(t) = s_0 exp{(mu -1/2 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,... , 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 -> 0, n Delta_n = T< infty as n -> 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 $n$.
Parametric estimation for the standard and geometric telegraph process observed at discrete times / S.M. Iacus, A. De Gregorio. - [s.l] : null, 2006 Jul.
Parametric estimation for the standard and geometric telegraph process observed at discrete times
S.M. IacusPrimo
;
2006
Abstract
The telegraph process X(t), t>0, (Goldstein, 1951) and the geometric telegraph process S(t) = s_0 exp{(mu -1/2 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,... , 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 -> 0, n Delta_n = T< infty as n -> 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 $n$.Pubblicazioni consigliate
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