Two-stage procedure for the control of reliability by one control level (Q1269943)

A control of reliability characteristics (RC) of products to confirm that they satisfy the required level is usually reduced to the assignment of necessary test sizes, that is, to the design of experiments. In this connection, these tests are called determinable if their aim is the establishment of the actually achieved reliability level of a product. To this end, the design of experiments involves the assignment of a test size (that is, the number of observations and the duration of testing) such that the reliability characteristic \(R\), in which the consumer is interested, is estimated with given accuracy and confidence. We shall use the relative error of an estimator of the RC \(R\) as a measure of accuracy. By confidence we mean a given probability level \(\beta (\beta=0.8 -0.95)\) with which the relative error does not exceed an admissible bound. Some procedures of designing determinable experiments proposed in the literature are not exactly correct because they use a one-stage process for the determination of the required sample size at the expense of assuming that the a priori information concerning the variation coefficient \(V\) (the so-called ``expected'' variation coefficient \(V_{ex}\) is used) in the case of two-parameter lifetime distributions is known. However, since the value \(V_{ex}\) is often merely theoretical, we propose a correction of the decision in the case where the estimator \(\widehat V_1\) of the variation coefficient \(V\) based on the designed tests differs from \(V_{ex}\) in the ``bad direction'' (that is, \(\widehat V_1> V_{ex})\). This actually leads to a two-stage procedure of the experiment when in addition to the sample size \(n_1\) at the first stage we assign a new sample size \(n_2\) according to the same rules as those at the first stage, but instead of \(V_{ex}\) we substitute \(\widehat V_1\), the estimator of the variation coefficient based on the sample with size \(n_1\). Since the estimator \(\widehat V_1\) is a random variable, \(n_2\) is also random and does not provide the required accuracy of the estimator of the RC \(R\), and its confidence. We restrict ourselves to the case important in practice where the lifetime \(\tau \sim N(\mu, \sigma)\), that is, \(\tau\) is a normally distributed random variable (we neglect the fact that \(\tau\) has a truncated distribution when \(\mu>3 \sigma)\), the RC \(R= \mu\), where \(\mu\) is the mean lifetime, and the sample is complete, that is, we test \(n\) products up to the failures of \(n\) products.