Mean Homogeneity Test Of Heteroscedasticity Functional Data And Its Application In Aircraft Energy Analysis

The rapid development of information construction and the great improvement of data collection and storage capacity make functional data widely exist and accumulate in many fields such as economy,biology and medicine,and the relevant theories and methods of functional data analysis have been developed rapidly.An important fundamental problem in functional data analysis is to test whether two or more groups of data come from the same distribution,which is the homogeneity test of statistical functional data.The simplest case is whether the mean function of functional data is the same,which has been studied extensively.Cuevas et al.(2004),the global point-by-point F test proposed by Zhang?Liang(2013),and the function F test proposed by Shen?Faraway(2004).All the above methods assume that the covariance functions of different groups of data are the same,but this assumption is often not valid in reality due to the existence of various complex conditions.For example,in civil aviation safety flight research,civil aviation aircraft approach landing stage The energy function has different covariance function due to different landing airports.In this case,the existing test methods based on the same covariance function structure will have efficacy loss or lead to a high class I error rate,which urges us to develop a method of mean homogeneity test for functional data under heteroscedasticity structure.Considering that likelihood ratio test has good test efficacy under normal circumstances,we develop a likelihood ratio based test method for mean homogeneity test of multiple functional data with different covariance structures.We first introduce the likelihood ratio test under heteroscedasticity structure to examine the homogeneity test of multiple scalar data.Then we extend the likelihood ratio test method to the homogeneity test of functional data.Since functional data is composed of a series of single points,we construct two kinds of global likelihood ratio test(GLR)and maximum likelihood ratio test(LRmax)by integrating point by point likelihood ratio statistics.We determine that their limiting distributions are all functional of a normal process,and finally use bootstrap method to determine the critical value of the test.The simulation results show that GLR test performs as well as other methods in the case of homoscedasticity,and GLR test can not only control the first type error rate well in the case of heteroscedasticity,but also has good efficacy.To further illustrate,we apply our method to the analysis of civil aviation approach data.