Some Research On Parameter Estimation

The estimation of probability density is one of the important contents of sta-tistical inference,which is widely used in machine learning,data mining,finance and communication.In a sense,the probability density provides all the infor-mation about the variables to be studied.With the probability density,you can answer any question about the subset of variables.In traditional statistics,we usually assume that data are independent and identically distributed.However,in practice,the assumption of independent identically distributed is difficult to hold.The nonlinear expectation theory proposed by Professor Shige Peng is an effective tool to solve the problem that.the independent identically distributed hypothesis is not tenable in reality.It is found that most random variables in reality satisfy the assumption of independent and identical distribution under the nonlinear expectation framework.The classical probability theory is mainly applicable to situations where the probability measure can be determined or approximately determined,while the nonlinear expectation theory is applicable to situations where the probability measure can not be determined,and is suitable for quantitative analysis in a very wide range of situations.Both the study of probability density estimation under traditional statistics and the study of parameter estimation under nonlinear expectation have very important theoretical and practical significance.This paper is divided into two parts.The first part is the study of probabil-ity density estimation under classical statistics.When the form of probability density is known,parameter estimation methods such as maximum likelihood es-timation can be used to estimate the parameters of the known distribution,so as to obtain the probability density obeyed by a group of samples.When the form of probability density is unknown,nonparametric estimation methods such as kernel density estimation can be used to estimate the probability density obeyed by a group of samples.In this paper,a new method is proposed to estimate the probability density obeyed by a group of samples,that is,using the convolutional neural network(CNN)in deep learning to estimate the probability density obeyed by the samples.Our estimation is divided into two steps:first,we predict which distribution the samples obey,and then estimate the parameters of the known distribution.We can understand these two estimat.ion steps as an estimation sys-tem.When the estimation model is established,we can encapsulate it.As long as the client inputs a group of samples,it can automatically output the probability density of the group of samples.Whether in predicting which distribution the samples obey,or in estimating the parameters of a known distribution,our new method has achieved good estimation results.The second part is about parameter estimation under nonlinear expectations.First,we studied the relationship between m(the number of groups)and n(the number of samples in each group)in the ?-max-mean algorithm.The results show that m(the number of groups)and n(the number of samples in each group)should satisfy an inequality,that is,m(the number of groups)should be greater than or equal to the nth power of k(the number of distributions that generated the samples).Then,we propose a new method to estimate the unknown parameters of the G-normal distribution.