Estimation of a standard measuring error by repeated measurements and sortings

A sorting-and-measuring machine (SMM) measures and sorts (classifies) on-line produced items into several groups according to their size. The measuring devices of the SMM perceive the actual item size with a random error ε and classify the item as being smaller than b iff z+ε<b. Here ε is a normal zero-mean r.v. with unknown standard deviation σ which is the main parameter characterizing the precision and technical condition of an SMM. The paper gives the following method of estimating σ. N₀ items are measured and N₁ of them are recognized by the SMM as belonging to the group a<z≤b. These N₁ items are sorted again and N₂ of them return to this group, these are sorted again, and so on. The estimation of σ is based on the statistics N_m/N_n. Moments of the ratio statistics N_m/N_n and their distributional properties are investigated. It turns out that the expected value of N_m/N_n depends almost linearly on σ which allows us to construct 'almost' unbiased estimators of type σ ̌_mn=A{radical dot}N_m/N_n+B with good propert including robustness with respect to the distribution of item size. Convex combinations of σ ̌_mn statistics are considered to obtain an estimator with minimal variance.