Estimators for particulate sampling derived from a multinomial distribution

B. Geelhoed and H. J. Glass

Statistica Neerlandica, 2004, vol. 58, issue 1, 57-74

Abstract: Counting the number of units is not always practical during the sampling of particulate materials: it is often much easier to sample a fixed volume or fixed mass of particles. Hence, a class of sampling designs is proposed which leads to samples that have approximately a constant mass or a constant volume. For these sampling designs, estimators were derived which are a ratio of arbitrary sample totals. A Taylor expansion was used to obtain a first‐order approximation for the expected value and variance in the limit of a large batch‐to‐sample size ratio. Furthermore, a π‐estimator for a ratio of batch totals was found by deriving expressions for the first‐ and second‐order inclusion probabilities. Practical application of the π‐estimator is limited because it requires inaccessible batch information. However, when the denominator of the estimated batch ratio is the batch size, the π‐estimator becomes equal to a sample total divided by the sample size in the limit of a large sample‐to‐particle size ratio. As a consequence, the obtained sample ratio becomes an unbiased estimator for the corresponding batch ratio. Retaining unbiasedness, the Horvitz–Thompson estimator for the variance, which also contains inaccessible batch information, is replaced by an estimator containing sample information only. Practical application of this estimator is illustrated for the sampling of slag, produced during the production of steel.

Date: 2004
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