Multivariate non-parametric quality control statistics.

Abstract

During the startup phase of a production process while statistics on the product quality are being collected it is useful to establish that the process is under control. Small samples $\{ n\sb{i}\} \sbsp{i=1}{q}$ are taken periodically for $q$ periods. We shall assume each measurement is bivariate. A process is under control or on-target if all the observations are deemed to be independent and identically distributed and moreover the distribution of each observation is a product distribution. This would be the case if each coordinate of an observation is a nominal value plus noise. Let $F\sp{i}$ represent the empirical distribution function of the $i\sp{-th}$ sample. Let $\overline {F}$ represent the empirical distribution function of all observations. Following Lehman (1951) we propose statistics of the form$${\sum\limits\sbsp{i = 1}{q}}\int\sbsp{-\infty}{\infty}\int\sbsp{-\infty}{\infty}(F\sp{i}(s,t) - \overline{F}(s)\overline{F}(t))\sp2 d\overline{F}(s,t)\eqno(1)$$The emphasis there, however, is on the case where $n\sb{i}\ \to\ \infty$ while $q$ stayed fixed. Here we study the following family of statistics$$S\sb{q}={\sum\limits\sbsp{i = 1}{q}}\int\sbsp{-\infty}{\infty}\int\sbsp{-\infty}{\infty}k\sb{q}(n, i, F\sp{i}(s,t),\overline{F}(s)\overline{F}(t))n\sb{i}dF\sp{i}(s,t)\eqno(2)$$in the above quality control situation, where $q\to\infty$ while $n\sb{i}$ stays fixed. (Abstract shortened by UMI.)