The parameter estimation of radiation pollution process in the case of geometrical distributed periods between observations (Q2703315)

Let \(x_{t}\), \(t=1,2,\ldots\), be the level of radiation at time \(t\). This paper deals with the mathematical model for \(x_{t}\): NEWLINE\[NEWLINEx_{t}=e^{-\mu} x_{t-1}+\int\limits_{t-1}^{t}e^{-\mu(t-u)}dA(u),NEWLINE\]NEWLINE where \(A(t)=\sum\limits_{k=1}^{\nu(t)}\xi_{k}\) is a generalized Poisson process, \(\nu(t)\) is a Poisson process with parameter \(\lambda\), \(\xi_{k}\) are jumps of the process, \(E\xi_{k}=\alpha\) and \(E\xi_{k}^2=\beta\). The observation process has the form \(y_{t}=z_{t}x_{t}\), \(t=1,2,\ldots\), where \(P\{z_{t}=0\}=p\), \(P\{z_{t}=1\}=1-p\), and the process \(z_{t}\) is independent on the process \(x_{t}\). The author proposes a consistent, asymptotically normal and unbiased estimate \(\mu_{T}^{*}=Tp\lambda\alpha/\sum_{t=1}^{T}y_{t}\) for the parameter \(\mu\).