Methods for Time Series Analysis - Hydrologic Time Series Analysis: Theory and Practice - page 57

4.1.8 Hartley Test

This is another parametric test to examine the significance of the differences
between the variances of ' K ' normally distributed populations. The size of the
' K ' samples should be approximately equal. The entire hydrologic time series
x t ( t = 1, 2, …, n ) is divided into ' K ' subseries with approximately equal size.
The sample variances of all the subseries are then calculated from Eqn. (20).
The test-statistic ( F ) is defined as (Kanji, 2001):

max
min

s
s

F max = (24)

where s 2 max is the largest of the ' K ' subseries variances, and s 2 min is the smallest
of the ' K ' subseries variances. The critical values of this test-statistic are given
in Kanji (2001). If the computed value of the test-statistic exceeds its critical
value, the null hypothesis of equal variances is rejected.

4.2 Methods for Checking Stationarity

A time series is said to be strictly stationary , if its statistical properties do not
vary with changes of time origin. That is, if two non-overlapping time intervals
are selected from a given time series, the two subseries will look almost the
same. In fact, both the subseries will differ from one another, but will be
scattered around the same mean value. Therefore, a stationary time series
cannot have any trend or periodic component. This is the reason that sometimes
trend and periodicity tests are used to check the stationarity of hydrologic time
series. There are two general approaches for checking stationarity: parametric
and nonparametric . The review of the literature reveals that the parametric
approach is usually used by the researchers working in the time domain, such
as economists, who make certain assumptions about the nature of their data.
On the other hand, the nonparametric approach is more commonly used by the
researchers working in the frequency domain, such as electrical engineers,
who often treat the system as a black box and cannot make any basic
assumptions about the nature of the system. However, in hydrology, both
parametric and nonparametric approaches are used. It should be noted that the
nonparametric tests are not based on the assumption that the population is
normally distributed (Bethea and Rhinehart, 1991). Hence, the nonparametric
tests are more widely applicable than the parametric tests which often require
normality in the data. Nevertheless, the nonparametric tests are reported to be
less powerful than the parametric tests. To arrive at the same conclusion with
the same confidence level, the nonparametric tests require 5 to 35% more data
than the parametric tests (Bethea and Rhinehart, 1991).
Only a couple of studies are reported wherein t -test has been used to
examine the stationarity of hydrologic time series (e.g., Jayawardena and Lai,