The Selection Of The Kernel Function In The Test For Nonparametric Models

Nonparametric model is an important model in statistical analysis with a wide range of applications.There are many methods related to nonparametric models,but these methods either are usually computationally expensive or require the specification of the smoothness conditions of an unknown function in advance,which is much more involved.The kernel machine method has played an important role in various fields as a powerful learning technique for multi-dimensional data.Kernel machine testing is a kind of kernel machine method,which is widely used in hypothesis testing of nonparametric models or semiparametric models.Compared with the traditional multiple linear regression,the flexible framework of the kernel function can model complex nonlinear relationships without explicitly specifying the feature space and mapping function.However,there are many kinds of kernel functions,and most kernel functions depend on unknown parameters.Even for the same kernel function,the different parameters will greatly affect the performance of kernel machine methods,so the selection of kernel parameters has always been a problem in the research of kernel machine methods.Essentially,the kernel machine testing is to test whether the nonparametric explanatory variables are related to the explanatory variables under the nonparametric model or semiparametric model.This paper mainly studies how to choose the unknown parameter of kernel function for nonparametric hypothesis test.Power is an important concept in statistics,which is widely used in hypothesis testing.Neyman-Pearson lemma shows that the likelihood ratio test(LRT)is the uniformly most powerful(UMP)test.Therefore,we construct the LRT statistic in the hypothesis test of nonparametric function,and define a risk function based on the loss of power,and select the appropriate kernel parameters by comparing the loss.In this paper,the specific derivation process of LRT statistic for testing nonparametric function and the risk function based on the test are given.In addition,this paper also develops three efficient algorithms to calculate the kernel parameter.Through simulation experiments,we determined that the combination of greedy algorithm and bisection search is the best of the three algorithms.In addition,by simulating more complicated situations,the results show that the selection method of kernel parameters proposed in this paper performs well in kernel machine test method and has strong anti-interference.At the same time,it can be found that the type I error rate of the test statistics constructed in this paper is also well controlled.Comparing the power of the test statistic with the score test,it is found that the test statistic is robust to the signal strength and has higher power than the score test.We apply this method to the analysis of the superconducting critical temperature,and the results show the effectiveness of the values selected for .The main innovation of this paper lies in the selection method of kernel parameters.Referring to the method of selecting signal strength in MORST,the loss function is constructed by using power,which is a very useful index in hypothesis testing.At the same time,it incorporates the signal strength and significance level in the test,and three specific calculation methods of kernel parameters is given.This method does not need to get an estimator of the kernel parameters,and it is computationally more efficient to some extent.