**Sankhya: The Indian Journal of Statistics**

1996, Volume 58, Series B, Pt. 1, pp. 10--27

**SCORE TEST VERSUS BARTLETT TYPE MODIFIED TEST FOR TESTING HOMOGENEITY OF VARIANCES WITH AUTOCORRELATED ERRORS**

By

BRAJENDRA C. SUTRADHAR, *Memorial University of Newfoundland*

SUMMARY. Bartlett's (1937) test for the equality of several variances is a modification to the standard likelihood ratio test. The test assumes that the observations involved are independent and their distributions are normal. The violation of this assumption may have adverse effects on the statistical inference. For the case when observations are independent but their distributions are not normal, Box (1953) has shown that the distribution of Bartlett's test statistic involves a non-zero kurtosis which invalidates its chi-square approximation. The usual assumption for Bartlett's test is also violated when the distributions of the observations are correlated. Thus, the presence of autocorrelations invalidate the chi-square approximation to the distribution of the standard Bartlett's test statistic. In particular, the test would be highly misleading for large autocorrelations. When the observations are highly correlated, the standard likelihood ratio, Wald's and Rao's tests run into difficulties in testing the equality of variances of several groups. This is because the likelihood estimation method has convergence problems for large correlation parameter values.

This paper discusses Neyman's (1959) partial score for testing the homogeneity of variances of several independent time series. The test statistic has asymptotically optimal. We also provide a modification to Bartlett's test which accounts for the correlations of the data. The modified test statistic is simple to compute and it has asymptotically a chi-square distribution. The small and sample performance of these tests including standard Bartlett's test, are examined through a simulation study based on AR(1) and MA(1) data. The simulation study shows that the standard Bartlett's test highly over estimates the significance level. Thus, this test is extremely unreliable, in particular for large autocorrelations. The proposed modification to Bartlett's test as well as Nayman's partial score test perform well in controlling the size. When compared for the power, the score test is found to be uniformly more powerful than the modified Bartlett's test for all sample sizes, small or large. The modified test procedure is illustrated through an analysis of winter temperature data.

*AMS (1980) subject classification.* Primary 62G10; secondary 62H12, 62E25.

*Key words and phrases.* Independent groups; correlated errors within a group; partial score test; correlation for autocorrelations; Glacer's exact test; asymptotically chi-square tests; size and power of the tests.