Parameter estimation of Gumbel distribution using Quasi-Newton Broyden Fletcher Goldfarb Shanno (BFGS) method and its application on data of daily precipitation in Purworejo regency

Abstract

Extreme data on an observation can occur due to rare events in the observation, and therefore they should be examined. One of methods to collect extreme data is block maxima. The distribution of extreme datasets collected using block maxima method is called extreme value distribution. Gumbel distribution is defined as extreme value distribution with two parameters. It is difficult to determine exact values in the parameter estimation of Gumbel distribution using maximum likelihood (ML) method. Therefore, the present research seeks to estimate the parameters using approximate solution of BFGS quasi-Newton method. The BFGS method is one of the most popular members of quasi Newton method. The purpose of this research was to determine the parameter estimation of Gumbel distribution with quasi Newton BFGS method. The quasi Newton BFGS method is a numerical method used for nonlinear function optimization without constraint so that the method can be used for parameter estimation from Gumbel distribution whose distribution function is in the form of exponential double function. The parameter estimation of the Gumbel distribution by numerical approach using the quasi Newton BFGS method is done by calculating the parameter values that make the distribution function maximum. This research is a theory research and application by studying several journals and textbooks. The results of this research we obtained the quasi Newton BFGS algorithm and estimation of Gumbel distribution parameters. Such method was applied to data of daily precipitation in Purworejo regency for the purpose of estimating the distribution parameters. The parameter estimation using ML and the quasi-Newton BFGS results in and the Gumbel distribution for period I, and and for period II, and . This indicates that high intensity and range of precipitation have decreased.