However, if the target is the estimation of a vector of values (such as the CDF or the return level function evaluated in a grid of points, for instance), more accurate than computing the corresponding pointwise CIs for each one of the quantities of interest is to construct simultaneous CIs (also called a confidence band). In other words, a single 0.95-CI for each one of the quantities of interest is computed and, in this sense, they are called pointwise CIs. Given a high confidence level (95%, for instance), each one of these CIs will contain the corresponding unknown quantile with a high probability (0.95, for example). In hydrology, CIs are usually computed for given quantiles corresponding to fixed return levels. Other types of uncertainty include the randomness of the natural process or the uncertainty depending on the choice of a particular model and are excluded from this study. CI estimation for quantiles was previously considered in, among others. On the contrary, underestimation of return leves could lead to increase damages and associated costs, with the consequent losses for insurance companies. Return level overestimation clearly supposes an increase of the costs related with monitoring and life protection. The proper estimation of these CIs provides hydrology experts with a robust tool in the risk evaluation and management strategies. This can be carried out calculating confidence intervals (CIs) for these quantiles or return levels. In the present research, we are mainly interested in the study of the statistical uncertainty related with the prediction of statistical measures of interest (quantiles) of certain random variables. Some significant research on uncertainty analysis in a hydrological context has been performed in, for example,, and references therein. In both cases (parametric and nonparametric estimation), the evaluation of the estimator variations can be made by computing an assessment of their uncertainty. Applications of the CDF estimation in this setting are presented, for instance, in and. These kind of proposals are normally well adapted to the more irregular shapes found in practice for hydrological variables. Nonparametric estimation uses the information provided by the recorded data to construct directly estimators of the functions of interest, supposing only general assumptions about the random variable in study. On the other hand, nonparametric techniques have been employed to deal with hydrological problems in the last decades. Note that to apply these parametric approaches, a statistical model for the variable under study has to be assumed, which can represent an important drawback of this methodology if model assumptions are violated. This parametric scheme has been widely investigated in the specialized literature and, in recent years, efforts have been made in developing more accurate and robust estimation techniques. Then, the corresponding unknown parameters have to be estimated. Usually, given a data set, the CDF is fitted assuming that this function belongs to a known parametric family. It appears naturally as a secondary factor in the estimation problem of F, the cumulative distribution function (CDF) of the variable X. Quantile estimation is a valuable issue in decision making in water-related models. Formal definitions of the return level and return period concepts are given in the following section. If the variable X measures the river flow, the return level for a given period T is the quantile x T with probability 1/ T. Quantile and return level notions are closely related. In a hydrological context, this quantity may correspond to design values of environmental loads (waves, snow), river discharges and flood levels. A p-quantile x p of a random variable X is defined as the value which is exceeded with probability p, that is P( X > x p) = p. ![]() Some of the most handled methods are based in the estimation of quantiles. In order to increase the accuracy of the estimates, several procedures of flood frequency analysis employing all the geographical and hydrological information available in the region of interest have been proposed. ![]() Usually, there is a considerable uncertainty in measuring extreme values. Many times, the amount, accuracy and representativeness of the records are not enough to achieve an adequate reliability in results. Studies about river flows or, in general, hydrological processes are based on available records.
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