Διαστήματα Εμπιστοσύνης Για Εκατοστημόρια Σε Στάσιμες Arma Διαδικασίες: Μία Εμπειρική Εφαρμογή Σε Περιβαλλοντικά Δεδομένα[Confidence intervals for percentiles in stationary ARMA processes: An application using environmental data]

Percentiles estimation plays an important role at the stage of making decisions in many scientific fields. However, the up-to-now research on developing estimation methods for percentiles has been based on the assumption that the data in the sample are formed independently. In the current paper we suppress this restrictive assumption by assuming that the values of the variable under study are formed according to the general linear process. After deriving the asymptotic distribution of the Maximum Likelihood estimator for the 100×Pth percentile, we give the general form of the corresponding asymptotic confidence interval. Then, the performance of the estimated asymptotic confidence interval is evaluated in finite samples from the stationary AR(1) and ARMA(1,1) through Monte-Carlo simulations by computing two statistical criteria: (a) the actual confidence level, (b) the expected half-length as percentage of the true value of the percentile. Simulation results show that the validity of the estimated asymptotic confidence interval depends upon the sample size, the size of the 1st order theoretical autocorrelation coefficient, and the true cumulative probability P related to the percentile. Finally, an application example is given using the series of the CO2 annual emissions intensity in Greece (kg per kg of oil equivalent energy use) for the period 1961-2010. Confidence intervals for percentiles are constructed on this series and discussion about the validity of the estimation procedure follows according to the findings from the simulation experiments regarding the values of the aforementioned criteria.