## 1. Introduction

[2] The percentage of time for which the given streamflow was equaled or exceeded over a historical period can be estimated by the so-called period-of-record flow duration curves (FDCs). FDCs are the survival functions of streamflows with a given time resolution (e.g., daily, weekly, or monthly) [*Vogel and Fennessey*, 1994]. Recalling that quantiles *x*_{P} are values of a variable *X* (e.g., discharge) exceeded with a fixed probability *P*, an FDC can be described simply as a plot of *x*_{P} versus *P*, where *P* is given by the complement of the distribution function *F* of *X*: *P* = 1 − *F*(*X* ≤ *x*_{P}).

[3] FDCs are widely used in a number of applications [e.g., *Smakhtin*, 2001]. However, their use is often criticized as their interpretation depends on the period of record, and also, no procedures for computing theoretical CIs are available [e.g., *Vogel and Fennessey*, 1994, 1995]. To overcome these drawbacks, *Vogel and Fennessey* [1994] suggested reinterpreting FDCs on an annual base by considering *N* annual FDCs (AFDCs), each corresponding to one of the *N* years of data. For daily data, each curve is a sequence of *n* = 365 values *X*_{i}, with *i* = 1*,..,n*, arranged in ascending order *X*_{1:n} ≤ *X*_{2:n} ≤ .. ≤ *X*_{n:n}, where *X*_{i:n} is the *i*th order statistic [e.g., *Kottegoda and Rosso* 2008]. AFDCs summarize the distribution functions of the *n* order statistics *X*_{i:n} from the annual minima *X*_{1:n} to the annual maxima *X*_{n:n}. Taking the median (or average) of the *N* values available for each *X*_{i:n}, it is possible to build a median (or average) AFDC, which represents a typical year wherein the interpretation is not affected by abnormal observations during the period of records [*Vogel and Fennessey*, 1994]. Moreover, other percentiles as well as the median can be taken into account to provide *α* percentiles of ACDFs which can be used for constructing CIs for the median [e.g., *Castellarin et al.*, 2007].

[4] To develop a mathematical model of the relationship between FDCs and AFDCs for obtaining regional models of AFDC to be applied at ungauged sites, *Castellarin et al.* [2004a] adopted an index flow method similar to that used in regional flood frequency analysis [e.g., *Dalrymple*, 1960; *Hosking and Wallis*, 1997]. The basic assumption of the model is that the daily streamflow *X* is the product of two random variables:

where the index flow *Q* summarizes the interannual precipitation variability and is often assumed to be the mean annual flow, whereas the dimensionless streamflow *X*′ and its distribution function *F*_{X′} describe the hydrologic behavior of the basin. Under the hypothesis of independence of *Q* and *X*′, the distribution function of *X* can be written as [e.g., *Kottegoda and Rosso*, 2008, pp. 137–139]

where *f*_{X′} is the probability density function of *X*′ and *F*_{Q} is the distribution function of *Q*. The integral is computed over the domain Ω of the variable *X*′, and the theoretical FDC is given by 1*-F*(*x*). The same index flow representation holds for the order statistics, which are simply observations arranged in ascending (or descending) order. Thus, the *i*th order statistic *X*_{i:n} can be expressed as the product

Irrespective of the serial correlation of daily streamflow observations, *Castellarin et al.* [2004a] assume that the dimensionless daily discharges *X*′ are independent and identically distributed (iid), as this hypothesis does not influence FDCs and AFDCs. Moreover, under iid assumption, the theoretical distribution of *X*_{i:n}′ can be deduced by the distribution of *X*′ as follows:

where *I*(*z; a, b*) = ∫_{0}^{z}t^{a−}^{1}(1 *− t*)^{b−1} d*t/B*(*a, b*) is the beta cumulative distribution function. From equations (2), (3), and (4), it follows that the distribution function of *X*_{i:n} is formally similar to the distribution of *X*:

where is the probability density function of *X*_{i:n}′. The percentiles of the AFDCs corresponding to a given probability *α* can be obtained by inverting equation (5) for *i* = 1*,..,n*.

[5] The above mentioned framework provides a method for linking FDCs and AFDCs. However, this model yields CIs only for AFDCs and not for FDCs. Alternatively, unlike AFDCs, FDCs can be used directly for filling gaps and for extending daily streamflow series or generating streamflow series at ungauged river basins [*Castellarin et al.*, 2004b]. Thus, an ideal model should provide FDCs which are useful for simulation and are complemented with CIs. The theory of nonparametric CIs for quantiles and fractional order statistics allows the introduction of analytical CIs for FDCs in an index flow framework. In section 2, we discuss the mathematical arguments that lead to defining CIs for quantiles which follow a generic distribution function. Subsequently, the results are used to define parametric and semiparametric quantile-based CIs for index flow AFDCs and FDCs. Therefore, these CIs are compared with CIs obtained by equation (5), Monte Carlo simulations, and bootstrap resampling. Finally, discussion and conclusions are provided to complete the study.