Sankhya: The Indian Journal of Statistics

1997, Volume 59, Series B, Pt. 1, 44-65

ON CONDITIONAL INFERENCE FROM SAMPLE SURVEY DATA ABOUT AN EXISTING UNIVERSE

By

J. C. KOOP, *3201 Clark Avenue, Rayleigh*

SUMMARY. This paper is both a continuation and a generalization of the author's previous paper on the fundamental principles of ancillarity and conditionality as applied to the data of independent interpenetrating samples. The application of these two principles to boot-strap, the subsampling from a simple random sample, stratified sampling (both single stage and multi stage), is now considered. By way of example, only the conditional probability distribution and the corresponding conditional probability limits, which are the analogues of the classical conditional limits, are derived for the bootstrap method. The steps leading to conditional probability limits are similar for other methods. The general theory for conditional inference with more than two interpenetrating samples is given in Sec 6; all these theories are considered in the context of Fisher's levels of uncertainty of Rank A, Rank B and Rank C. With respect to Rank A, Rank B is a higher level of uncertainty. And with respect to Rank B, Rank C is yet another higher level of uncertainty. An uncertainty of Rank C takes into account all nonsampling errors. The probabilistic nature of these levels of uncertainty is explained in Sec2. The ancillary statistics produced are not always unique. An example for the general theory is given in Subsection 6.5, with data from the National Sample Survey of India on agricultural labour for rural areas during the period August 1956 to August 1957. An ancillary statistics in the entire body of sampling theory from a finite universe have their source in the fundamental principles of randomization, replication and control, they are classified in Sec.7.

*AMS (1980) subject classification.* 62D05

*Key words and phrases*. Assumption free methods, estimand, strata, sampling procedure, interpenetrating samples, conditional probability uncertainty of Rank C.