AISM 52, 239-254

## Optimal allocation for symmetric distributions in ranked set sampling

### Amarjot Kaur, G. P. Patil and C. Taillie

Center for Statistical Ecology and Environmental
Statistics, Department of Statistics, Pennsylvania State University,
University Park, PA 16802, U.S.A.

(Received August 5, 1996; revised October 22, 1998)

Abstract.
Ranked set sampling (RSS) is a cost efficient method of
sampling that provides a more precise estimator of population mean
than simple random sampling. The benefits due to ranked set sampling further
increase when appropriate allocation of sampling units is made. For highly skew
distributions, allocation based on the Neyman criterion achieves a substantial
precision gain over equal allocation. But the same is not true for symmetric distributions; in fact, the gains due to using the Neyman allocation are
typically very marginal for symmetric distributions. This paper, determines
optimal RSS allocations for two classes of symmetric distributions.
Depending upon the class, the optimal allocation assigns all measurements
either to the extreme ranks or to the middle rank(s). This allocation
outperforms both equal and Neyman allocations in terms of the precision of the
estimator which remains unbiased. The two classes of distributions are
distinguished by different growth patterns in the variance of their order
statistics regarded as a function of the rank order. For one class,
the variance peaks for middle rank orders and tapers off in the tails;
for the other class, the variance peaks for the two extreme rank orders
and tapers off toward the middle. Kurtosis appears to effectively
discriminate between the two classes of symmetic distributions. The Neyman
allocation is required to quantify all rank orders at least once (to ensure general unbiasedness) but then quantifies most frequently the more variable rank orders.
Under symmetry, unbiasedness can be obtained without quantifying all rank orders
and the optimal allocation quantifies the least variable rank
order(s), resulting in a high precision estimator.

Key words and phrases:
Equal allocation, kurtosis, Neyman allocation,
order statistics, relative precision, skewness, symmetry.

**Source**
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