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Parametric estimation for the standard and geometric telegraph process observed at discrete times
Statistics Theory · Submitted 25 Jul 2006
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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 $t_i=i\Delta_n,i=0,1,..., n$. 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 $n$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $t_i=i\Delta_n,i=0,1,..., n$. | background 0.80 | 0.14 | 0.02 | asserted 0.90 |
| s1 | For both models $\lambda$, the underlying rate of the Poisson process, is a parameter to be estimated. | background 0.79 | 0.17 | 0.04 | asserted 1.00 |
| s2 | In the geometric case, also $\sigma>0$ has to be estimated. | background 0.74 | 0.20 | 0.03 | asserted 1.00 |
| s3 | 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. | contribution 0.70 | 0.88 | 0.10 | asserted 0.92 |
| s4 | The process $X(t)$ in non markovian, non stationary and not ergodic thus we use approximation arguments to derive estimators. | method 0.79 | 0.43 | 0.04 | asserted 1.00 |
| s5 | Given the complexity of the equations involved only estimators on the first model can be studied analytically. | limitation 0.99 | 0.49 | 0.05 | asserted 1.00 |
| s6 | Therefore, we run an extensive Monte Carlo analysis to study the performance of the proposed estimators also for small sample size $n$. | method 0.83 | 0.95 | 0.09 | asserted 0.98 |
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Original 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 $t_i=i\Delta_n,i=0,1,..., n$. 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 $n$.
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