# Probability and Statistics Seminar

### An ARL-unbiased $np-$chart

We usually assume that counts of nonconforming items have a binomial distribution with parameters $(n,p)$, where $n$ and $p$ represent the sample size and the fraction nonconforming, respectively.

The non-negative, discrete and usually skewed character and the target mean $(np_0)$ of this distribution may prevent the quality control engineer to deal with a chart to monitor $p$ with: a pre-specified in-control average run length (ARL), say $1/\alpha$; a positive lower control limit; the ability to control not only increases but also decreases in $p$ in a expedient fashion. Furthermore, as far as we have investigated, the $np-$ and $p-$charts proposed in the Statistical Process Control literature are ARL-biased, in the sense that they take longer, in average, to detect some shifts in the fraction nonconforming than to trigger a false alarm.

Having all this in mind, this paper explores the notions of uniformly most powerful unbiased tests with randomization probabilities to eliminate the bias of the ARL function of the $np-$chart and to bring its in-control ARL exactly to $1/\alpha$.