Since hydrological processes are random in nature, statistical parameters are
useful in projecting the central tendency and variability of a probability
distribution (Hong, 2009). Therefore, aside from the emphasis on which type of
distribution function should be adopted in fitting related hydrological data,
methods of parameters estimation have been studied as well. However, the
competency of the parameter estimation methods is subjected to the choice of
distribution functions and sample size (Martins & Stedinger, 2000).
2.1.3.1 Method of moments
The method of moments is a relatively old and perhaps the simplest method for
37 of taking a linear functional equation and representing it by a linear matrix
equation, a technique that was developed almost a century ago (Harrington, 1990;
Wooldridge, 2001). Through the voluminous studies that have been carried out in
the field of statistics, this method is becoming less and less relevant. Wilks (2006)
stated that the method of moments does not fully utilise the information in the
data set and it causes the value of the estimated parameters to become unreliable.
Moreover, the traditional moment-based measure of skewness, γ, is difficult to
estimate if the distribution is distinctly skewed; it is too sensitive to the extreme
tails compared to the L-moments method according to Hosking (1990). Hence,
the method of moments is less suitable to estimate parameters for distribution
with more than two parameters.
2.1.3.2 Method of maximum likelihood
The method of maximum likelihood estimates the parameters by maximising the
likelihood function in which the probability of the observed data gains the highest
probability (Rice, 2007).
According to In (2003), maximum likelihood estimation does not require
or only needs very few distribution assumptions to summarise observed data by
its moments and it also gives smaller variance (Suhaila & Jemain, 2007b).
Therefore, the maximum likelihood estimation is still widely adopted in practice
(Suhaila & Jemain, 2007a; Suhaila & Jemain, 2007b; Park & Jung, 2002;
38 suitable for five-parameter Wakeby distribution because the probability
distribution function of Wakeby is not clearly defined (Park et al., 2001).
2.1.3.3 L-Moments method
The L-moments method was introduced by Hosking (1990) for hydrological data
analysis. According to Hosking (1990), the L-moments method is more robust to
the existence of outliers in the data, is sturdier in its adaptability to a wider range
of distributions and is more accurate for data with small sample size. Furthermore,
the L-moments method would not exaggerate the value because this method does
not raise the number to power (Koutsoyiannis & Baloutsos, 2000).
Due to the above mentioned advantages, the L-moments has been applied
to different regions, for example India (Parida, 1999), Korea (Park & Jung, 2002),
China (Su et al., 2008) using precipitation data, and using flood data in Iran
(Borujeni & Sulaiman, 2009), Canada (Yue & Wang, 2004). In a local context,
several studies were also carried out using L-moments in Peninsular Malaysia
(Zalina et al., 2002) as well as east Malaysia (Lim & Lye, 2003).
L-moments is found to be an effective approach in hydrological statistics
studies carried out internationally using different types of data, such as stream
flow data (Borujeni & Sulaiman, 2009), rainfall data (Koutsoyiannis & Baloutsos,
2000; Parida, 1999) and number of days without rainfall - dry-spell data (Nasri &
Moradi, 2011). Moreover, the estimation of parameters for generalised extreme
value distribution using the L-moments method has a lower root-mean-square
39 2.1.3.4 L-Moments related methods
Ever since the L-moments method was introduced by Hosking (1990), several
extensions of L-moments have been developed along the way, including LQ-
moments (Mudholkar & Hutson, 1998) and Trimmed L-moments (TL-moments)
(Elamir & Seheult, 2003).
LQ-moments is developed for the estimation of Kappa parameters, and
according to Shabri & Jemain (2010), LQ-moments is able to give an estimation
for the Kappa distribution whereas sometimes L-moments fails to give a reliable
estimation. Another study has been carried out to compare the robustness of
conventional L-moments with LQ-moments in finding the most suitable
distribution to fit the annual maximum daily rainfall in Peninsular Malaysia (Wan
Zawiah et al., 2009).
The TL-moments method is quite beneficial in parameter estimation for
data with outliers and for distributions that do not have a second-order moment
(mean) such as the Cauchy distribution (Elamir & Seheult, 2003). However,
Shabri & Mohd Ariff (2010) found that in the study of identifying the most
suitable distribution for annual maximum rainfall by L-moments and TL-
moments, L-moments method still be able to give a more precise result. Yet, the
result of parameters estimation by these two methods does not differ significantly.
Therefore, it is acceptable to use either TL-moments or LQ-moments to
replace L-moments as the parameter estimation method (Shabri & Mohd Ariff,
40 In short, the advantages and disadvantages of the previously mentioned
parameter estimation methods are as shown in Table 2.4.
Table 2.4: Advantages and disadvantages of parameter estimation methods
Function Advantages Disadvantages
Method of Moments
Easy to compute Easy to give accurate
estimation if the distribution is distinctly skewed
Less appropriate for
distribution with more than two parameters Maximum Likelihood Required minimum distribution assumptions Smaller variance
Less appropriate for Wakeby distribution
L-Moments More robust to the existence of outliers
The performance is not consistent with Kappa distribution
L-Moments Related Methods
Robust with the existence of outliers
Suitable for distributions that do not have second- order moments
Lesser studies on these methods