Probability Density Estimation Of Two-dimensions Continuous Random Variable

As the basis of statistics,probability density estimation has been the focus ofmost scholars.C.F Gaussproposed a parameter estimation method ofprobability density estimation.With the rise of non-parametric estimation,Parzen and other mathe-maticians have provided a series of non-parametric density estimation methods such as histogram estimation,kernel density estimation and nearest-neighbor estimation for probability density estimation.These methods not only enrich the solving methods of probability density estimation,but also solve the probability density estimation better.With the development of computer technology,probability density estimation of one-dimensional random variables is powerless while describing the complex process in the new fields of machine learning,target tracking and so on.Therefore,the research of probability density estimation about multiple random variables is significant and indispensable.However,the existingmethods of parametric density estimation and non-parametric density estimationcan't solve the multiple probability density estimation effectively.In this paper,the method of integral operator is applied to the probability density estimation,which can solve theprobability density estimation of two-dimensional random variables effectively,and provide reference for the further research of the probability density estimation problem of multiple dimensional random variables.According to definition of probability density estimation,probability density estimation can come down to the derivation of probability distribution function.When solving the problem of probability density estimation innumerical method,there are differences between the empirical distribution function and the true distribution functionbecause of the limitation of sample size.These tiny differences may cause a huge deviation of probability density estimation.Therefore,the probability density estimation problem based on numerical method is ill-posed.Through comprehensive analysis ofthe solving methods of the numerical differential,it can bediscoveredthat the integral operator method is used to transform the differential into integral equation problem.The integral operator method which is not restricted by the problem of the dimension and can be applied to the differential problems of one-dimensional and multiple dimensional flexibly.The probability density estimation based on integral operator method using integral operator method into one-dimensional probability density estimation could be presented under the help of Taylor expanding.Considering the non-negativity and the regularity of the probability density function,integral operator method is proved tobe rational.In thecase of two-dimensional random variable,the integral operator methodfinds out the second order mixed partial derivative throughcalculatingpartial derivativeof the two variables in turn,namely,the two-dimensional probability density estimation is the approximation of the true probability density function.The verification of the non-negativity and regularity proves the rationality of the integral method in the two-dimensional probability density estimation problemsufficiently.In this thesis,the reliability of the integral operator method in the probability density estimationis demonstrated by numerical simulation.It is concluded that the integral operator method asequal to the kernel density estimation in both computational accuracy and computational speed.