Control Charts Based on Weighted Sums

Abstract

In a continuous production process, samples of fixed size are taken at regular intervals of time and a statistic $X_n$ is computed from the $n$th sample, $n = 1,2, \cdots$. In this paper, we study process inspection schemes which stop the production and take corrective action with $N = \text{first} n \geqq 1$ such that $\sum^n_{i=1} c_{n-i} X_i \geqq h$, where $h$ is a preassigned constant and $c_0 \geqq c_1 \geqq \cdots \geqq c_{k-1} > 0 = c_k = c_{k+1} = \cdots$ is a suitably chosen sequence of weights. The average run length of such procedures is examined, and in the normal case, numerical comparisons with the average run length of the usual Shewhart Chart are given. In connection with the normal case, the first passage times of more general Gaussian sequences are studied and an asymptotic theorem is obtained. The first passage time $N$ for more general weighted sums, where the sequence $(c_n)$ is not assumed to be eventually zero but is assumed to be at least square summable, is also considered.