The Spectrum of a Model II Nested ANOVA and Its Applications

Parzen [4] and others have noted the similarity between probability densities, spectral densities, and their sample estimates. In the familiar x 2 goodness-of-fit test one seeks to determine whether an observed sample distribution could reasonably have resulted from sampling some specified probability distribution. One could ask a similar question concerning the spectral density of a process. That is, could the estimated spectral density ĝ(ω) of some sampled process have reasonably resulted from a process whose true spectrum is f(ω)? Durbin [l] considers a problem of this type in testing for randomness. The present paper considers a process {X(t), t = 1, 2, · · ·} where X(t) is an asymptotically stationary random process with spectrum g(ω). In Sections 1 and 2 it is shown that, if X(t) is generated by a Model II nested ANOVA, then X(t) is asymptotically stationary and possesses an asymptotic spectrum S(ω). This spectrum completely characterizes the process and is a function of the ANOVA variance component...