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An approximation of the fractional Brownian motion based on the Ornstein-Uhlenbeck process is used to obtain an asymptotic likelihood function. Two estimators of the Hurst index are then presented in the likelihood approach. The first estimator is produced according to the observed values of the sample path; while the second one employs the likelihood function of the incremental process. We also employ visual roughness of realization to restrict the parameter space and to obtain prior information in Bayesian approach. The methods are then compared with three contemporary estimators and an experimental data set is studied.

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In some pro le monitoring applications, the independency assumption of consecutive binary response values within each pro le is violated. To the best of our knowledge, estimating the time of a change in the parameters of an autocorrelated binary pro le is neglected in the literature. In this paper, two maximum likelihood estimators are proposed to estimate the real time of step changes and drift in Phase II monitoring of binary pro les in the case of within-pro le autocorrelation. Our proposed estimators identify the change point not only in the autocorrelated logistic regression parameters, but also in autocorrelation coe cient. The performance of the proposed estimators to identify the time of change points either in regression parameters or autocorrelation coe cient is evaluated through simulation studies. The results, in terms of the accuracy and precision criteria, show the satisfactory performance of the proposed estimators under both step changes and drift. Moreover, a numerical example is given to illustrate the application of the proposed estimators.

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