Asymptotic expansions for tests of goodness of fit for linear autoregressive schemes

Abstract

The general problem of discrimination in stationary time series analysis, especially when the length of the realized time series is short, is well known. A somewhat simpler problem is to test the hypothesis that the time series is a linear autoregressive scheme of order p where p is a specified positive integer. If the hypothesis is simple in the sense that a particular, well-defined autoregressive scheme is considered under the null hypothesis, one can use either Quenouille's test (1947) or Bartlett & Diananda's test (1950). If, on the other hand, the hypothesis is composite, i.e. if the order p is specified but the parameters of the process remain unknown, the only available admissible tests are those due to Quenouille (1947) and Whittle (1952). All these tests are large sample tests and have simple distributional properties when the sample size is large. In small samples, of course, the sampling properties will be different and considerable error would result if the large sample properties were assumed then. The extent of this error has been studied partially by Bartlett & Rajalakshman (1953) and more systematically by Chanda (1962). The method employed in each case, however, is somewhat inadequate for the purpose and it is believed that probably a more efficient approach would be to obtain asymptotic expansions of the distributions of the test-criteria considered in this connexion. Appropriate manipulation of these asymptotic forms would possibly provide more accurate approximations to the true small-sample distribution of the test-criteria than if we had simply used the corresponding large-sample distributions. The technique is rather well known and has been used extensively by mathematical statisticians in the past. Two different methods are available for this purpose, viz. (i) the method of Edgeworth expansion (see Cramer, 1946, 17.6) and (ii) the method of saddle-point approximation. The first is the classical method frequently used, whereas the second is of recent origin and has been investigated systematically by Daniels (1954). The second method has some advantage over the first, but computationally, it would possibly be more cumbersome. The purpose of this paper is to make a start in the direction of deriving asymptotic expansions for the distributions of the various test-criteria mentioned before by employing either or both of these methods. We have considered only the very simplest of these tests, namely that proposed by Bartlett & Diananda (1950). Thus let {Yt} (t = 1, 2, ...) be the sequence of autoregressive residuals, which we assume to have independent and identical distributions with zero means and finite higher moments and let Y, = Y'/or where o2 = V( Yt). Then the Bartlett-Diananda test criterion (without the end effect) can be written as