Bounded risk estimation of a finite population mean optimal strategies

Abstract

Bounded risk estimation of the mean of a finite population is considered under three simple random sampling strategies: (i) with replacement, mean per unit estimation, (ii) with replacement, mean per distinct unit estimation, and (iii) without replacement, mean per unit estimation. It is well known that in the fixed sample size scheme, (iii) fares better than (ii) and (ii) better than (i). However, in the current context, the sample sizes are dictated by (possibly, degenerate) stopping times, and visualizing the cost (due to measurements/recording, etc.) as a function of the number of distinct units in the sample, it is shown that the second strategy still fares better than the first, although the third strategy may not perform better than the second one. Actually, in the case of known population variance, it is shown that in the light of the number of distinct units, the difference of ASN for the second and third strategies can never be greater than two or less than minus one. A similar relationship also holds in the case of unknown population variance when we define the stopping rules in a coherent manner. The theoretical results are backed up by numerical examples too. Also, dominance of strategy (ii) over (i) in a general sequential setup constitutes an important task of the current study. Finally, to reconcile strategies (ii) and (iii) in a general sequential setup, the coherence of the associated stopping times has also been discussed thoroughly.

Keywords:

• Average cost function
• ASN
• bounded risk
• minimum risk
• sequential estimation
• simple random sampling
• SRSWR
• SRSWOR
• transitively sufficient