Methods for Testing Normality of Hydrologic Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 47

sample sizes are available. For a random sample x 1 , x 2 , ... x n , assuming that it
follows normal distribution, the Geary's test-statistic ( U ) is defined as follows
(Walpole and Myers, 1985):

n

Ç

S 2 xxn

i

i 1

U = (8)

n

Ç

2

( xx n )

i

i 1

It should be noted that the denominator of Eqn. (8) is a reasonable estimator
of standard deviation whether the distribution is normal or non-normal. The
numerator is a good estimator of standard deviation if the distribution is
normal, but may overestimate or underestimate standard deviation when there
are departures from the normality. Thus, the values of U differing considerably
from 1.0 indicate that the hypothesis of normality should be rejected (Walpole
and Myers, 1985).

3.2.10 Jarque Bera Test

Jarque and Bera (1980) proposed a normality test using classical skewness
and kurtosis coefficients. The Jarque-Bera (JB) test is a goodness-of-fit measure
of departure from normality, based on the sample kurtosis and skewness. The
test-statistic JB is defined as (Jarque and Bera, 1987):

2

È Ø

n k 3

2

s

JB = (9)

É Ù

6 4

Ê Ú

where n = number of observations, s = sample skewness and k = sample
kurtosis.
The major disadvantage of the Jarque-Bera test is that asymptotic
convergence of the test-statistic is very slow. Therefore, decisions for testing
normality based on the quantile function of the chi-square distribution can
lead to serious errors (Bowman and Shenton, 1975; Jarque and Bera, 1987;
Lehmann, 1999).

3.2.11 D'Agostino Pearson Omnibus Test

The D'Agostino Pearson (DAP) Omnibus test first analyzes time series data
to determine skewness (to quantify the asymmetry of the data distribution)
and kurtosis (to quantify the shape of the data distribution). Thereafter, it
calculates how far each of the two values differs from the value expected with
a normal distribution, and computes a single P -value from the sum of the
squares of these discrepancies (D'Agostino, 1986). This test is a combination
of the D'Agostino skewness test and Anscombe-Glynn kurtosis test. The test-
statistic ( K 2 ) of the DAP Omnibus test is expressed as (D'Agostino et al.,
1990):