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This paper introduces and studies multivariate spacings. The spacings aredeveloped using the order statistics derived from data depth. Specifically, thespacing between two consecutive order statistics is the region which bridgesthe two order statistics, in the sense that the region contains all the pointswhose depth values fall between the depth values of the two consecutive orderstatistics. These multivariate spacings can be viewed as a data-drivenrealization of the so-called ``statistically equivalent blocks.'' Thesespacings assume a form of center-outward layers of ``shells'' (``rings'' in thetwo-dimensional case), where the shapes of the shells follow closely theunderlying probabilistic geometry. The properties and applications of thesespacings are studied. In particular, the spacings are used to constructtolerance regions. The construction of tolerance regions is nonparametric andcompletely data driven, and the resulting tolerance region reflects the truegeometry of the underlying distribution. This is different from most existingapproaches which require that the shape of the tolerance region be specified inadvance. The proposed tolerance regions are shown to meet the prescribedspecifications, in terms of $\beta$-content and $\beta$-expectation. They arealso asymptotically minimal under elliptical distributions. Finally, asimulation and comparison study on the proposed tolerance regions is presented.

Multivariate spacings based on data depth: I. Construction of nonparametric multivariate tolerance regions