Predicting Future Ranked Statistics and Recorded Values for Some Statistical Distributions
Authors: Ajeel Nidhal Khaleel 
Published: November 2021

Regional frequency analysis (AFR) brings together a variety of statistical methods aimed at predicting the behavior of extreme hydrological variables at ungauged sites. Regression techniques, geostatistical methods and classification are among the statistical tools frequently encountered in the literature. Methodologies based on these tools lead to regional models that offer a simple, but very useful description of the relationship between extreme hydrological variables and physiometeorological characteristics of a site. These regional models then make it possible to predict the behavior of variables of interest at places where no hydrological information is available. These methods are generally based on restrictive theoretical assumptions, including linearity and normality. These do not reflect the reality of natural phenomena. The general objectives of this paper are to identify the methods affected by these hypotheses, evaluate their impacts and propose improvements aimed at obtaining more realistic and fairer representations. Projection pursuit regression is a non-parametric method similar to generalized additive models and artificial neural networks that are considered in AFR to take into account the non-linearity of hydrological processes. In a comparative study, this paper shows that regression with revealing directions makes it possible to obtain more parsimonious models while preserving the same predictive power as the other nonparametric methods. Canonical Correlation Analysis (ACC) is used to create neighborhoods within which a model (e.g. multiple regression) is used to predict hydrologic variables at ungagged sites on the other hand, ACC strongly depends on the assumptions of normality and linearity. A new methodology for delineating neighborhoods is proposed in this paper and uses revealing direction regression to predict a reference point representing hydrological and physiometeorological information that is relevant to these groupings. The results show that the new methodology generalizes that of ACC, improves the homogeneity of neighborhoods and leads to better performance. In AFR, kriging techniques on transformed spaces are suggested in order to predict extreme hydrological variables. However, a transformation is required so that the hydrological variables of interest derive approximately from a multidimensional normal distribution. This transformation introduces a bias and leads to suboptimal predictions. Solutions have been proposed, but have not been tested in AFR. This paper proposes the approach of spatial copulas and shows that this approach provides satisfactory solutions to the problems encountered with kriging techniques. Max-stable processes are a theoretical formalization of spatial extremes and correspond to a more faithful representation of hydrological processes on the other hand; their characterization of extreme dependence poses technical problems which slow down their adoption. In this paper, the approximate Bayesian calculus is examined as a solution. The results of a simulation study show that the approximate Bayesian computation is superior to the standard approach of compound likelihood. In addition, this approach is more appropriate in order to take into account specification errors.