The statistical modelling of precipitation data for a
given portion of territory is fundamental for the monitoring of
climatic conditions and for Hydrogeological Management Plans
(HMP). This modelling is rendered particularly complex by the
changes taking place in the frequency and intensity of precipitation,
presumably to be attributed to the global climate change. This paper
applies the Wakeby distribution (with 5 parameters) as a theoretical
reference model. The number and the quality of the parameters
indicate that this distribution may be the appropriate choice for
the interpolations of the hydrological variables and, moreover, the
Wakeby is particularly suitable for describing phenomena producing
heavy tails. The proposed estimation methods for determining the
value of the Wakeby parameters are the same as those used for
density functions with heavy tails. The commonly used procedure
is the classic method of moments weighed with probabilities
(probability weighted moments, PWM) although this has often shown
difficulty of convergence, or rather, convergence to a configuration
of inappropriate parameters. In this paper, we analyze the problem of
the likelihood estimation of a random variable expressed through its
quantile function. The method of maximum likelihood, in this case,
is more demanding than in the situations of more usual estimation.
The reasons for this lie, in the sampling and asymptotic properties of
the estimators of maximum likelihood which improve the estimates
obtained with indications of their variability and, therefore, their
accuracy and reliability. These features are highly appreciated in
contexts where poor decisions, attributable to an inefficient or
incomplete information base, can cause serious damages.
 Brachetti P. and Ciccoli, M. and Di Pillo, G. and Lucidi, S. “A new
version of the Price’s algorithm for global optimization” Journal of Global
Optimization 10, 165 – 184 (1997).
 Dempster, A. P. and Laird, N. M. and Rubin, D. B. “Maximum likelihood
from incomplete data via the EM Algorithm” Journal of the Royal
Statistical Society. Series B (Methodological), 39, 1–38 (1977).
 Gilchrist, W. G., “Modeling and fitting quantile distributions and
regressions”, Sheffield Hallam University (2006).
 Griffiths, G. A. “A theoretically based Wakeby distribution for
annual flood series” Hydrological Sciences - Journal - des Sciences
Hydrologiques 34, 231–248 (1989).
 Honaker, J. and King, G. and Blackwell, M. “Amelia II: A Program for
Missing Data” Journal of Statistical Software 45, 1–47 (2011).
 Jones, R. A. and Scholz, F. W. and Ossiander, M. and Shorack G. R.
“Tolerance bounds for log gamma regression models Technometrics 27,
109 – 118 (1985).
 Lawton, W. H. and Sylvestre, E. A., “Elimination of linear parameters in
nonlinear regression” Technometrics 13, 461 – 467 (1971).
 Parzen, E. “Nonparametric statistical data modelling” Journal of the
American Statistical Association 74,105 – 131 (1979).
 Su, B. and Kundzewicz, Z. W. and Jiang, T. “Simulation of extreme
precipitation over the Yangtze River Basin using Wakeby distribution”
Theoretical and Applied Climatology 96, 209–219 (2009).
 Tarsitano, A. “Fitting Wakeby model using maximum likelihood”
Convegno intermedio SIS 2005: Statistica e Ambiente,Messina, 21-23
September, 2005, 253–256 (2005).