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On universal estimates for binary renewal processes. (English) Zbl 1158.62053
In many applications the occurrences of a zero, which represent the failure times of some system which is renewed after each failure, are of importance and so the problem arises of estimating when the next failure will occur. The classical binary renewal process is a stochastic process \(\{X_n\}\) taking values in \(\{0, 1\}\), where the lengths of the runs of 1’s between successive zeros are independent.
The authors of this paper investigate the possibility of giving a universal estimator at time \(n\) for the residual waiting time to the next zero in the binary renewal process \(\{X_n\}\). Let \(\{p_k\}_{k=0}^{\infty}\) be the conditional probability that a run of \(k\) 1’s follows a given 0 event. This distribution describes completely the renewal process as a two-sided stationary process. If the distribution of the process is known, then after observing \(X_{0},X_{1},\ldots,X_n\) one may give a consistent estimator for the expected value of residual waiting time to the occurrence of the next zero as \[ \mu_L={\sum_{k=L}^{\infty} (k-L)p_k}/ {\sum_{k=L}^{\infty}p_k} \] if there is at least one zero among the values of \(X_{0},X_{1},\ldots,X_n\), and the last zero occurs at the moment \(X_{n-L}=0\). This \(L\) is denoted by \(\tau(X_{0},X_{1},\ldots,X_n)\). From the stationarity it follows that \(P(\tau=L)\) is proportional to \(\sum_{k=L}^{\infty}p_k\).
For the estimator itself it is most natural to use the empirical distribution observed in the data segment \(X_{0},X_{1},\ldots,X_n\). However, if there is an insufficient number of occurrences of 1-blocks of length at least \(\tau(X_{0},X_{1},\ldots,X_n)\), then we do not expect the empirical distribution to be close to the true distribution. For this reason estimates only along stopping times \(\lambda_1,\lambda_2,\dots\) are considered and the main positive result is that there is a sequence of universally defined stopping times \(\lambda_n\) with density 1 and estimators \(h_n(X_{0},X_{1},\ldots,X_n)\) which are almost surely converging to \(\mu_{\tau}(X_{0},X_{1},\ldots,X_{\lambda_n})\). Estimators \(\hat{p}_l(X_{0},X_{1},\ldots,X_{\lambda_n})\) are also defined which are almost surely converging in the variation metric to the conditional distribution of the residual waiting time. A variety of different estimates are proposed, including universal estimates for the expected time to renewal as well as estimates for the conditional distribution of the time to renewal.

MSC:
62M05 Markov processes: estimation; hidden Markov models
60K05 Renewal theory
62M20 Inference from stochastic processes and prediction
62G30 Order statistics; empirical distribution functions
60G25 Prediction theory (aspects of stochastic processes)
62L15 Optimal stopping in statistics
60K25 Queueing theory (aspects of probability theory)
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