Estimation Of Parameters For Log-logistic Distribution By Smooth Adaptive

Log-Logistic distribution has been paid more and more attention in practical application because of its good statistical properties,and also gradually attracted more and more scholars’ attention.As a very important content in statistical research,parameter estimation is also the focus of this paper.However,in real life,there are still some problems in parameter estimation of Log-Logistic distribution,and these problems need to be solved urgently.Therefore,this paper mainly studies parameter estimation of Log-Logistic distribution.This paper mainly studies the point estimation and interval estimation of shape parameters and scale parameters of Log-Logistic distribution.On the Log-logistic distribution parameter estimation,using the traditional maximum likelihood estimation(MLE)method to estimate parameters need to solve the simultaneous equations,namely no explicit form,and this estimation method under the condition of small sample is biased and no analytical solution,so using conventional parameter estimation methods for parameter estimation of the Log-logistic distribution effect is not very good.Therefore,we need to find more accurate estimators with explicit mathematical expressions in the case of small samples.In this paper,we proposed a smooth adaptive(SA),the Bootstrap deviation correction of SA estimation method(C-SA),the Bootstrap deviation correction of asymptotic maximum likelihood estimation method(C-AMLE)and function fitting correction method for estimating SA(F-SA)as well as the function fitting correction AMLE(F-AMLE)estimation method to solve these problems,and the results of these methods comparaed to the traditional maximum likelihood estimation(MLE)and the asymptotic maximum likelihood estimation(AMLE),We mainly use these parameter estimation methods and Monte Carlo simulation to evaluate the point estimation and interval estimation results of Log-Logistic distribution.According to the simulation results,the five parameter estimation methods proposed in this paper can obtain more accurate estimation results than the traditional maximum likelihood estimation and asymptotic maximum likelihood estimation,and do not need to solve simultaneous equations.In other words,the parameter estimation method proposed in this paper is more competitive and convincing than other methods.Point estimation is mainly through the deviation and mean square error estimators to measure parameter estimation effect,then the estimation of deviation and mean square error under different parameters and sample sizes is obtained by Monte Carlo simulation,through the simulation results we found that C-SA and C-AMLE method did very well in the small sample,the F-SA and F-AMLE method under the condition of large sample performance is very good.In addition,due to our research of the Log-logistic distribution is not symmetrical distribution,so this article is based on quantile Bootstrap method(PB)and deviation correction quantile Bootstrap method(BCPB)for all parameters mentioned in this paper to build the corresponding confidence interval estimation methods,after get the confidence interval with the Bootstrap method is used to estimate its coverage,to estimate the coverage and we set the confidence level of comparison,through its proximity to measure the accuracy of the estimation methods and their effects.We found from the simulation results that the coverage rate estimated by the Bootstrap deviation correction method using BCPB was closer to the confidence level we set,that is to say,under a certain probability guarantee degree,the interval estimation results obtained by the BCPB interval estimation method and the parameter estimation method we proposed were more accurate.The coverage rate obtained by the function fitting distribution correction method using PB estimation is closer to the confidence level we set.Finally,an example is using a data set to fit our research Log-logistic distribution,and use several estimation methods proposed in this paper to the instance data to carry on the point estimation and interval estimation is achieved in this paper,the proposed estimation method of empirical analysis,and use the Kolmogorov-Smirnov test to verify the effect of these parameters estimation method in practical application,this example also illustrates the application of the proposed parameter estimation method.