The U-statistics Testing Methods About Coefficient Of Variation, Coefficient Of Skewness And Coefficient Of Kurtosis

Coefficient of variation, coefficient of skewness and coefficient of kurtosis are importantparameters in fields of reliability engineering, finance and insurance, biotechnology andnatural science. Among them, coefficient of variation is a digital feature of reflect the relativedispersion degree. Coefficient of skewness is a digital feature of test the random variableswhich have symmetry. Coefficient of kurtosis is a digital feature of measure the randomvariables which have fat-tailed distributions. Therefore, hypothesis testing problems aboutcoefficient of variation, coefficient of skewness and coefficient of kurtosis have importantpractical significances. Although the hypothesis testing problems about the three parametershave discussed in previous literatures,which were confined to the situation which the overallmust be obeyed to certain special distribution. They haven’t discussed the general overalldistributions. The paper gave the U-statistics testing methods about the three parameters underthe general overall distributions and discussed the excellent properties.The paper was divided into four sections.Firstly, the paper introduced the research background, research significance and researchproperties at home and abroad about coefficient of variation, coefficient of skewness,coefficient of kurtosis and U-statistics testing methods.Secondly, the paper gave the U-statistics testing method about coefficient of variations’differences by using the method as discussed below. First, it constructed U-statistics offunction2of coefficient of variation’s numerator, then the result wass2. Second, itconstructed U-statistics of coefficient of variation’s denominator,then the result was x.Third,by using the multiple central limit theorem obtained the joint asymptotic distribution of x, s2and worked out the asymptotic normality ofsx. Four, the paper gave out the U-statisticstesting method about coefficient of variation by using the conclusion of literature27(Asymptotic relative efficiency of two tests that is imit of the required sample size’s ratio, theessence is limit of asymptotic variance’s ration of estimate.), and worked out the asymptoticrelative efficiency of this method in relation to the other method. Last, through the concreteexample introduced practical applications of U-statistics testing method.The last two parts gave the U-statistics testing methods about coefficient of skewness and coefficient of kurtosis’ differences separately.It gave the asymptotic relative efficiency byadopting the method of compared to required sample size’s of two tests and Noether theorem.By using the U-statistics the paper constructed hypothesis testing methods aboutcoefficient of variation, coefficient of skewness and coefficient of kurtosis. Those methodsbroke traditional model of the past, not only applicable to the parametric statistical structureof the overall distribution types are known but the unknown parameters, also applicable to thenon-parametric statistical structure of the overall distribution completely unknown. Therefore,the research has important theory significance and comprehensive application value.