Multivariate randomness 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 ni qi=1 are taken periodically for q periods. We shall assume each measurement is multivariate. A process is under control or on-target if all the observations are deemed to be independent and identically distributed. Let Fi represent the empirical distribution function of the ith sample. Let F¯ represent the empirical distribution function of all observations. Following Lehmann (1951) we propose statistics of the form i=1q -infinityinfinityFi s-F- s2d Fs. The asymptotics of nonparametric q-sample Cramer-Von Mises statistics were studied in Kiefer (1959). The emphasis there, however, is on the case where n(i) → infinity while q stayed fixed. Here we study the asymptotics of a family of randomness statistics, that includes the above. These asymptotics are in the quality control situation (i.e q → infinity while n( i) stay fixed). Such statistics can be used in many situations; in fact one can use randomness statistics in any situation where the problem amounts to a test of homoscedasticity or homogeneity of a collection of observations. We give two such applications. First we show how such statistics can be used in nonparametric regression. Second we illustrate the application to retrospective quality control.