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Incomplete U -statistics of permanent design

A class of incomplete U -statistics based on what we call a "permanent design" is investigated. It is shown that the design is balanced and has minimal variance in a certain class of designs as well as is asymptotically efficient, i.e. , its asymptotic variance equals that of a complete U -statistic. Several examples of applications of our results to obtaining limiting distributions of certain statistics as well as to sub-sampling theory are discussed.