Efficacy of Time Series Tests: A Critical Assessment - Hydrologic Time Series Analysis: Theory and Practice - page 145

study for the entire period of collected data. Hence, it is believed that the
rainfall series could be homogeneous. Based on the results of the Cumulative
Deviations and the Bayesian tests, the rainfall series in this study are found
homogeneous. Thus, based on the available history of the raingauge station,
the Cumulative Deviations and the Bayesian tests seem to be superior to the
classical von Neumann test. Similar finding has also been reported by Buishand
(1982) based on the data generation method.

Table 7.2. Observed and critical test-statistics of the three homogeneity tests for
the annual and maximum rainfall time series of Kharagpur

Rainfall von Neumann Cumulative deviations Bayesian test

time series test (N) / Qn U A

/ Rn

1 2 3 4 5 6
Annual 1.908 0.990 0.990 0.249 1.471
(1.265) (1.540) (0.450) (2.460)
1-day max. 1.784 0.752 0.745 0.113 0.651
(1.265) (1.540) (0.450) (2.460)
2-day max. 1.873 0.637 0.631 0.067 0.407
(1.265) (1.540) (0.450) (2.460)
3-day max. 1.922 0.603 0.602 0.076 0.462
(1.265) (1.540) (0.450) (2.460)
4-day max. 2.008 0.670 0.657 0.087 0.504
(1.265) (1.540) (0.450) (2.460)
5-day max. 2.120 0.613 0.611 0.100 0.584
(1.265) (1.540) (0.450) (2.460)
6-day max. 2.022 0.575 0.571 0.099 0.601
(1.265) (1.540) (0.450) (2.460)
Note: max.: maximum; Bracketed figures in Columns 3 to 6 are critical values.

Moreover, four tests for multiple comparisons of homogeneity were also
applied to all the seven rainfall time series after fragmenting these in two
ways: (a) two fragmentations each comprising half of the entire series, and (b)
three fragmentations each consisting of one-third of the entire series. The
computed values of all the homogeneity test-statistics for multiple comparisons
were compared with their critical values (Table 7.3). It is clear from this table
that for the Tukey test, the calculated difference in the means of different
possible combinations of two subseries in two individual fragmentations are
always less than the critical limit. Therefore, the null hypotheses of homogeneity
cannot be rejected for annual, one-day, 2-, 3-, 4-, 5- and 6-day maximum
rainfall time series, which also have equal variances. Similarly, the observed
value of the test-statistic K L is found less than its critical value in case of Link-
Wallace test, which suggests the presence of homogeneity in the annual rainfall