On the asymptotic distribution of the mean of distinct units in sampling from a finite population

Pathak, P. K. and Sethuraman, J. (1964) On the asymptotic distribution of the mean of distinct units in sampling from a finite population. A MAGYAR TUDOMÁNYOS AKADÉMIA MATEMATIKAI KUTATÓ INTÉZETÉNEK KÖZLEMÉNYEI, 9 (1-2). pp. 113-116.

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Abstract

For each integer k, SNₖ = {1, 2, . . ., Nₖ} is a finite population of Nₖ units with characteristics {Yₖ,₁ , Yₖ,₂, . . . , Yₖ,Nₖ}. A simple random sample, Sₙₖ, of size nₖ is drawn with replacement from Sₙₖ. The set of distinct units in Sₙₖ is denoted by sₘₖ and it contains mₖ units. Let Ῡₖ = 1/mₖ ∑ᵢ∈ₛₘₖ Yₖ,ᵢ . This note is concerned with the asymptotic distribution of Ῡₖ. In section 3, we show this to be normal by the following device. The conditional distribution of sₘₖ when mₖ is fixed is that of a simple random sample without replacement from SNₖ. The mean of a simple random sample without replacement and mₖ are known to be asymptotically normal. These facts and the theorem Sethuraman [3] quoted in Section 2 (Lemma 3), establish the result.

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