The seasonal Kendall slope estimator is a generalization of Sen's estimator of slope .
The individual \( N_i \) slope estimates for the \( i^{th} \) season are computed as:
$$ \Large Q_i = \frac{x_{il} - x_{ik}}{l-k} $$Where \( x_{il} \) is the datum for the \( i^{th} \) season for the \( l^{th} \) year and \( x_{ik} \) is the datum for the \( i^{th} \) season of the \( k^{th} \) year, where \( l > k \) ; . This is repeated for each of the \( m \) seasons. All \( N' \) individual slope estimates are ranked and the median is found. This median is the Seasonal Kendall slope estimator.
$$ VAR( S_i)= \frac{1}{18} [n_i ( n_i - 1)(2 n_i + 5) - \displaystyle\sum_{p=1}^{g_i} t_{ip} (t_{ip} -1)(2 t_{ip} + 5) - \displaystyle\sum_{q=1}^{h_i} u_{ik} (u_{ik} -1)(2 u_{ik} + 5)] $$
$$ \Large +\frac{ \displaystyle\sum_{p=1}^{g_i} t_{ip} (t_{ip} -1)( t_{ip} - 1) \displaystyle\sum_{q=1}^{h_i} u_{ik} (u_{ik} -1)( u_{ik} - 2)}{9 n_i ( n_i - 1)( n_i - 2)} $$ $$ \Large +\frac{ \displaystyle\sum_{p=1}^{g_i} t_{ip} (t_{ip} -1)\displaystyle\sum_{q=1}^{h_i} u_{ik} (u_{ik} -1)}{ 2 n_i ( n_i - 1)} $$
\( g_i \) is the number of tied groups for the \( i^{th} \) season and \( t_{ip} \) is the number of data in the \( p^{th} \) group for the \( i^{th} \) season. For example, if the sequence of measurements over 9 years for \( i^{th} \) season is {23, 24, 29, 6, 29, 24, 24, 29, 23} we have \( g_i = 3 \) tied groups for that season: \( t_{i1} = 2 \) for the tied value 23, \( t_{i2} = 3 \) for the tied value 24, and \( t_{i3} \) for the tied value 29.
\( h_i \) is the number of sampling times (or time periods) in season \( i \) than contain multiple data, and \( u_{iq} \) is the number of multiple data in the \( q^{th} \) time period in season \( i \).
The overall variance of \( S_i \) is computed as:
$$ \Large VAR(S') = \displaystyle\sum_{i=1}^m VAR( S_i ) $$A 100(1- \( \alpha \) )% confidence interval about the true slope is obtained by:
First, the value
\( C_a = Z_{1- \alpha /2} \sqrt{VAR(S')} \)
is computed, where
\( Z_{1- \alpha /2} \quad \mbox{is the} \quad 100(1- \alpha /2) \)th percentile of the standard normal distribution.
Next, the values
\( M_1 = (N' - C_a )/2 \)
\( M_2 = (N' + C_a )/2 \)
are computed.
The lower confidence limit is the \( M_1 \)th largest of the \( N' \) ordered slope estimates.
The upper confidence limit is the \( M_2 + 1\)th largest of the \( N' \) ordered slope estimates.
Gilbert, R.O. 1987. Statistical Methods for Environmental Pollution Monitoring, Wiley, New York. p. 225-228.