Estimation of a standard measuring error by repeated measurements and sortings (Q1067727)

The following estimation problem arising for instance in the inspection of automatic sorting and measuring machines in the ball-bearing industry is investigated: Let \(X_ i\) be i.i.d. random variables, the problem is to sort the population into three parts: \(X_ i\leq a\), \(a<X_ i\leq b\) and \(b<X_ i\). There are two kinds of errors to be considered, a systematic shift in the adjustment of the measurement devices and pure random errors. As a result an item having the dimension X is classified into the (a,b]-group if X satisfies: \[ a+\delta_ a+\epsilon '<X\leq b+\delta_ b+\epsilon '' \] where \(\epsilon\) ' and \(\epsilon\) '' represent the pure random errors. Assuming that \(\epsilon\) ' and \(\epsilon\) '' are normally distributed with zero mean and variance \(\sigma^ 2\), a ratio type estimator for \(\sigma\) is proposed which is based on repeated measurements and sortings.