Sampling Inspection by Variables Under Weibull Distribution and Type I Censoring

Abstract

The lifetime (time to failure) of a product is modeled as Weibull distributed (with unknown parameters); in this case the logarithms of the lifetimes are Gumbel distributed. Lots of items shall be accepted if their fraction p of nonconforming items (items the lifetime of which is smaller than a lower specification limit t _L ) is not larger than a specified acceptable quality limit. The acceptance decision is based on the r ≤ n observed lifetimes of a sample of size n which is put under test until a defined censoring time t _C is reached (Type I censoring). A lot is accepted if r = 0 or if the test statistic $y = \hat {\mu } - k\hat {\sigma }$ is not smaller than the logarithm of the specification limit, $x_L = \log (t_L)$, where k is an acceptance factor and $\hat {\mu }$ and $\hat {\sigma }$ are the Maximum Likelihood estimates of the parameters of the Gumbel distribution. The parameters of the sampling plan (acceptance factor k, sample size n and censoring time t _C ) are derived so that lots with p ≤ p ₁ shall be accepted with probability not smaller than 1 − α. On the other hand, lots with fractions nonconforming larger than a specified value p ₂ shall be accepted with probability not larger than β. n and t _C are not obtained separately but as a function that relates the sample size n to the censoring time t _C . Of course, n decreases if the censoring time t _C is increased. For t _C  →∞ the smallest sample size, i.e. that of the uncensored sample, is obtained. Unfortunately, the parameters of the sampling plan do not only depend on the two specified points of the OC, P ₁(p ₁, 1 − α) and P ₂(p ₂, β), but directly on the parameters τ and δ of the underlying Weibull distribution or equivalently, on the parameters $\mu = \log (\tau )$ and σ = 1/δ of the corresponding Gumbel distribution. Since these parameters are unknown we assume that the hazard rate of the underlying Weibull distribution is nondecreasing (δ ≥ 1). For the design of the sampling plan we use the limiting case δ = 1 or σ = 1/δ = 1. A simulation study shows that the OC of the sampling plan is almost independent of σ if the censoring time t _C is not smaller than the specification limit t _L .