基于非参数逆向思维的参数推断设计与模拟实验

Parametric Statistical Inference Method Based on Nonparametric Inverse Thinking and Stochastic Simulation

为了求解参数估计中的统计推断问题,将不含参数的非参数核密度估计和经验分布函数作为"理论分布",含参数的密度函数和分布函数作为"拟合分布",逆向地将实际中的两类分布角色互换,然后构建拟合分布与理论分布偏差的目标函数。再基于优化理论得到使得目标函数达到最小的参数值作为未知参数的估计,由此构建了求解经典的参数推断问题的两种逆向求解法。通过柯西分布参数的随机模拟试验,说明了逆向法的可行性和普适性;再结合Bootstrap方法,对常见分布参数的区间估计和检验p值进行随机模拟。结果表明,逆向方法在上述参数统计推断问题上的可行性;基于分布函数的逆向法优于基于密度函数的逆向法。

In order to solve the statistical inference problem in parameter estimation,it treats the nonparametric kernel density estimation and empirical distribution function without parameters as the"theory distribution",and the density function and distribution function with parameters as the"fitted distribution",and then reverses the roles of the two functions and constructs the objective function of deviation between the fitted distribution and the theoretical distribution.It then searches for the parameter that minimizes the target function by optimization method.Through Monte-Carlo simulations for the Cauchy distribution,it shows the proposed reverse methods are feasible and general.And by combining the Bootstrap method,with the interval estimation and the p value of the common distribution parameters,the methods are feasible for dealing with the statistical inference problems.Simulation results also show that the inverse method based on the distribution function is better than the inverse method based on the density function.