INVARIANCE OF THE UNORDERED BEST ESTIMATOR IN THE T,–CLASS FOR MIDZUNO'S AND FOR IKEDA–SEN'S SAMPLING PROCEDURES

MIDZUNO'S sampling procedure is considered where the first (n – 1) draws are carried out with simple random sampling without replacement and the nth draw with varying probabilities. It is shown that for this scheme, the best estimator in the HORVITZ–THOMPSON (1952) T_t–class of linear estimators exists and rejects the last draw. When MURTHY'S technique of unordering of an ordered estimator is employed, the rejected draw is restored and the unordered estimator is obtained. Surprisingly, this unordered estimator is the same as the unordered best estimator in the T₁–class, derived for IKEDA–SEN'S sampling procedure.