Summary Quenouille (1947), and Bartlett and Diananda (1950), have given large sample goodness-of-fit tests of the hypothesis that a time series can be represented by a linear autoregressive scheme with independent residuals. Asymptotic power functions of these tests are derived for alternative hypotheses specified by linear autoregressive schemes of higher order with independent residuals, and of the same or higher order with residuals among which there exists dependence of finite extent. On the null hypothesis, the constants of the scheme may either be given a priori, or be unspecified, having to be estimated from the data. Both these cases are considered in a detailed comparison of the asymptotic power functions for the two types of test. It is shown how, in the first case, the Quenouille tests may be conveniently modified to give greater power. A discussion is given of the application of the likelihood-ratio method for obtaining large sample tests, and of its relation to the Quenouille tests when the residuals are normally distributed.
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A. M. Walker (1952) studied this question.