Abstract:
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To characterize the spread of a multivariate data set, Liu, Parelius and Singh (1999) introduced the depth-based scale curve but left its properties open. Here we study the asymptotic behavior of generalized depth-based scale processes, which include the scale curve of Liu, Parelius and Singh as a special case. The uniform strong and weak convergences of the generalized depth-based scale processes are established. As applications, we obtain asymptotic distributions of some nonparametric multivariate kurtosis measures. Applications to assess multivariate normality and compare two multivariate data sets with respect to kurtosis will also be discussed.
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