Thinning Pearson type 3 distribution and modelling time series with Pearson type 3 marginal
1 : Laboratoire d'Analyse Numérique Informatique et Biomathématiques
(LANIBIO)
-
Site web
03 BP 7021 Ouagadougou 03 Burkina Faso -
Burkina Faso
2 : Laboratoire de Mathématiques et de leurs Applications de Pau
(LMA-PAU)
-
Site web
CNRS : UMR5142, Université de Pau et des Pays de l'Adour
Bâtiment I.P.R.A Avenue de l'Université BP 1155 64013 Pau cedex -
France
The Pearson type 3 distribution also called three-parameters gamma distribution is widely
used in various scientific fields such as in reliability and hydrology. We focused our investiga-
tions on its applications in hydrologic frequency analysis for modelling annuals maximum
flows.
Stationary first order Markov AR(1) time series model is proposed for the case where the obser-
vation or response variable comes from Pearson type 3 distribution. The process is constructed
by means of the thinning operation of Joe (1996). For the case of Pearson type 3 margin, convex
thinning operator is defined and we proved that it is still distributed according to a Pearson
type 3 where the innovations is independent and identically distributed according to gamma
distribution. From this definition, we state an hierarchical auto-regressive model for time series
with Pearson type 3 margin. Stationary property of the model is proved and the autocorrelation
function is also evaluated. The EM algorithm is used to estimate the parameters of the likelihood
function of the model.The real dificulty for the implementation of the EM-algorithm is that the
E-step cannot be computed directly. An approximation to the E-step is obtained by using the
Gauss-Legendre Quadrature and the Monte Carlo method. We demonstrate experimentally the
efficiency of this algorithm on a variety of data.
used in various scientific fields such as in reliability and hydrology. We focused our investiga-
tions on its applications in hydrologic frequency analysis for modelling annuals maximum
flows.
Stationary first order Markov AR(1) time series model is proposed for the case where the obser-
vation or response variable comes from Pearson type 3 distribution. The process is constructed
by means of the thinning operation of Joe (1996). For the case of Pearson type 3 margin, convex
thinning operator is defined and we proved that it is still distributed according to a Pearson
type 3 where the innovations is independent and identically distributed according to gamma
distribution. From this definition, we state an hierarchical auto-regressive model for time series
with Pearson type 3 margin. Stationary property of the model is proved and the autocorrelation
function is also evaluated. The EM algorithm is used to estimate the parameters of the likelihood
function of the model.The real dificulty for the implementation of the EM-algorithm is that the
E-step cannot be computed directly. An approximation to the E-step is obtained by using the
Gauss-Legendre Quadrature and the Monte Carlo method. We demonstrate experimentally the
efficiency of this algorithm on a variety of data.
