Generalized Confidence Interval Of Parameters In Gumbel Distribution

The Gumbel distribution is one of the main types of extremum distribution,one of the main purposes of extremal analysis is to estimate the quantile xp.In the hydrological statistics,xp is the return level of T= 1/1-p;In risk management,it is the VaR of the estimated portfolio;In the actuarial calculation,it is estimated that the initial reserve of the year T.So there are important applications in real life.The xp quantile of Gumbel distribution is:xp = ?-? ln[-ln(p)],we know from the quantile's expression,The accuracy of estimation method of the location parameter and scale parameter of the distribution function is directly affected.Therefore,It is of great theoretical significance and practical value to study the distribution parameters of Gumbel.This paper is based on the minimum risk equivariant estimators of the Gumbel distribution's location-scale parameter,we give the generalized pivotal quantity(GPQ)of location parameter,scale parameter and quantile,then we can find the generalized confidence interval of the quantile.At the same time,two conclusions about the difference between two location parameters of Gumbel are given,and the paper is divided into five chapters.The first chapter gives the existing domestic and international research status of Gumbel distribution's quantile,and the preparation knowledge needed in this paper is explained.In the second chapter,a new estimation method is adopted for the location parameter and scale parameter of the Gumbel distribution:the minimum risk equivariant estimators,at the same time,monte carlo method is used to compare the statistical properties of maximum likelihood estimation,It shows that it has more effective statistical significance in the case of small sample size.In the third chapter,which is based on the minimum risk equivariant estimators of Gumbel distribution's location-scale parameters to give the expressions of the generalized pivotal quanti-ty of location parameter,scale parameter and quantile,we figure out the quantile' s generalized confidence interval and obtain the coverage probability of the 0.95 confidence interval for quan-tile parameter by sampling.The simulation results show that the actual confidence level of the generalized confidence interval is very close to 0.95.It shows that the provided method performs very well,It also proves that the generalized confidence interval of the three interest parameters of Gumbel distribution is the actual confidence level in the frequency sense.Then we apply the method to practical problems,and use the Gumbel distribution to fit the highest sea level data of the southern Pirie from Australia and give the highest sea level' s confidence interval of the Pirie every T years.In the forth chapter,We will compare the generalized confidence intervals of the Gumbel distribution in this study with the generalized confidence interval of the studied Gumbel distribution quantiles,The first is the coverage of the confidence interval.The second aspect is the interval length,the data simulation shows that in the case of small sample the generalized confidence interval constructed in this paper has more effective statistical significance.In the fifth chapter,We discuss two inferences about the difference between two location pa-rameters in Gumbel distribution,including the generalized confidence interval and its test for the difference between two location parameters when the scale parameter is equal,and its test and scale parameter is not equal,The generalized confidence interval and the test of scale parameter ratio are also given.