The most basic problem in numerical analysis (methods) is the root finding problem. In this paper we are interesting in new numerical method which based on the Modified Bisection Method(MBM) referred to by Tanakan superficially and he didn't know her as a numerical method for finding the roots of a function. Hence in this study we define a new numerical method base on MBM with error bound and the number iterations necessary. Finally we present our new MBM for multi-roots with the R software.
The general Pareto distribution (GPD) has been widely used a lot in the extreme value for example to model exceedance over a threshold. Feature of The GPD that when applied to real data sets depends substantially and clearly on the parameter estimation process. Mostly the estimation is preferred by maximum likelihood because have a consistent estimator with lowest bias and variance. The objective of the present study is to develop efficient estimation methods for the maximum likelihood estimator for the shape parameter or extreme value index. Which based on the numerical methods for maximizing the log-likelihood by introduce an algorithm for computing maximum likelihood estimate of The GPD parameters. Finally, a numerical examples are given to illustrate the obtained results, they are carried out to investigate the behavior of the method.
In this article, we introduce adaptive estimators for parameters of the (GPD) Generalized Pareto Distribution under censored data via the KIB-estimator. The KIB-estimator is based on the Maximum Likelihood Estimates (MLE) by the exceedances over the threshold t under random censoring which was developed by [1]. Hence, it was proved that the KIB-estimator is Maximum Likelihood (ML) estimator with the uncensored case. We use the standardized MLE based on the exceedances on the uncensored situation which converge to a centered bivariate normal distribution. Whose found by [2] to standardized our adaptive KIB estimator of the GPD parameters under random censorship. As an application, we establish the asymptotic normality of an estimator of the excess-of- loss reinsurance premium for heavy-tailed distribution, through the adapted KIB estimator of GPD under censored data.
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