On the Decomposition of Empirical Processes in Sampling with Replacement into Sufficient and Ancillary Components

Abstract

The randomness in SRSWR sampling scheme has two components: the choice of distinct units and their repetitions. Basu (1958) invoked the notion of sufficiency in this framework showing that the distinct units behave like a sufficient statistic in usual sense. Going one step ahead we can call the repetitions of the distinct units ‘ancillary’ since they do not contain any information about population characteristics. It is s.bown that the above fact is reflected in an interesting way in the behaviour of empirical processes in large samples. In large samples such a break-up of variability is similar to the celebrated theorem of Basu (1955). AMS ( 1980) Subject Classification: Primary 60007; Secondary 62DOS, 62CIS