Goodness-of-fit Tests For Multivariate Distribution And Its Application

Goodness-of-fit test is an important field in statistics and application. As isknown, there are rigorous systems of theory in statistic inference under theassumption of normal distribution; the common Bayes discrimination isdiscussed under the assumption of the normal distribution. Failure of thepopulation distribution means failure of the model, the conclusions according tothe model are then invalid. Consequently, the importance of model checking,which is the essence of goodness-of-fit test is self-evident.The uniform distribution on the surface of a unit sphere is a foundationaldistribution in multivariate distribution, the elliptically symmetric distribution(including the multivariate normal distribution) can be constructed by uniformdistribution on the surface of a unit sphere. The multivariate normal distributionis the basic hypothesis of classic multivariate analysis, elliptically symmetricdistribution is the basic hypothesis of general multivariate analysis. Therefore,the goodness-of-fit test for uniform distribution on the surface of a unit sphere isof great significance.In this paper, we introduce two statistics for testing the uniformity on thesurface of a unit sphere: smooth test statistic based on spherical harmonicfunctions, we define the highest order of spherical harmonic functions as theorder of smooth test statistic; characteristic test statistic based on the generalizedinverse, and simulate the power performance of these two statistics. We study thecharacteristic test statistics for elliptically symmetric distribution and conduct thepower simulation. The main conclusions are as follows:(1) The conclusions of smooth test statistic for uniform distribution on thesurface of a unit sphere: a、simulation shows that for various alternatives thesmooth test statistic shows a superiority when the order of spherical harmonicfunction is no more than2. b、The components of smooth test statistic providepowerful directional tests, it can show the way that the alternatives depart fromuniform distribution on the surface of a unit sphere: first order smooth teststatistic shows good power against the first moment(the departure of center ofmass)、second order smooth test statistic shows good power against the secondmoment(the difference of moment of inertia).(2) As for characteristic test statistic of uniform distribution on the surface of a unit sphere, simulation shows that the statistic has a nice property indetecting large departures from the moment of inertia of uniform distribution onthe surface of a unit sphere. These two statistics both have their own strengthsand supplement each other.(3) We suggest the characteristic test statistic of elliptically symmetricdistribution and obtain the asymptotic distribution of the transformed samples ofelliptically symmetric distribution is univariate t-distribution, therefore, thegoodness-of-fit test can be transformed into the goodness-of-fit test for theunvaried t-distribution.