LQ-Moments : Application to the Log-Normal distribution 1

Mudolkar and Hutson (1998) extended L-moments to new moment like entitles called LQmoments (LQMOM). The LQMOM are constructed by using functional defining the quick estimators, where the parameters of quick estimator take the values = = α p for the median, = = α p for the trimean and = = α p for the Gastwirth, in places of expectations in L-moments (LMOM). The objective of this paper is to develop improved LQMOM that do not impose restrictions on the value of p and α such as the median, trimean or the Gastwirth but we explore an extended class of LQMOM with consideration combinations of p and α values in the range 0 and 0.5. The popular quantile estimator namely the weighted kernel quantile (WKQ) estimator will be proposed to estimate the quantile function. Monte Carlo simulations are conducted to illustrate the performance of the proposed estimators of the log-normal 3 (LN3) distribution were compared with the estimators based on conventional LMOM and MOM (method of moments) for various sample sizes and return periods.


INTRODUCTION
The problem of estimating the parameter of a probability distribution from a sample is crucial to many fields of science and engineering, particularly for predicting future behavior of phenomenon from previously observed behavior. A wide variety of methods have been developed to perform parameter estimation. Despite the efforts of many researchers, there is on-going need to create a method that is easily used, has the flexibility to accommodate many different distributions and can procedure accurate and robust parameter estimates from sets of data.
The oldest and most widely method for fitting frequency distributions to observed data is known as the method of moments (MOM) [1] . In this method the parameters of a probability distribution are estimated by equating the sample moments to those of the theoretical moments of the distribution. The MOM has been one of the simplest and conventional parameter estimation techniques used in statistical hydrology [2] .
The L-moments (LMOM) are linear combinations of order statistics [3] . Analogously to the conventional moments, the LMOM of order one to four characterize location, scale, skewness and kurtosis, respectively. The main advantages of using the method of LMOM are that the parameter estimates are more reliable (i.e., smaller mean-squared error of estimation) and are more robust against outliers than MOM and are usually computationally more tractable than ML (maximum likelihood) method [4] . The LMOM have found wide applications in such fields of applied research as civil engineering, meteorology and hydrology. The method of LMOM has become a standard procedure in hydrology for estimating the parameters of certain statistical distributions. The LMOM are defined as  (1) Mudolkar and Hutson [5] extended LMOM to new moment like entitles called LQ moments (LQMOM). They found the LQMOM always exists, are often easier to compute than LMOM and in general behave similarly to the LMOM. The method of LQMOM was originally introduced as an extension of the LMOM method for parameter estimation. The LQMOM are defined as is the linear combination and defined as a quick measures of the location of the sampling distribution of the order statistic r k r X − and u Q X is the quantile function. Mudholkar and Hutson [5] discussed a robust modification in which the mean of the distribution of assuming that the underlying distribution is correctly specified. More specifically, we develop the method of LQMOM for the three-parameter lognormal (LN3) distribution, which is often employed in statistical analyses of hydrological data. The popular quantile estimator namely the weighted kernel quantile (WKQ) estimator will be proposed to estimate the quantile function. The performance of the proposed estimators of the LN3 distribution was compared with the estimators based on conventional LMOM and MOM (method of moments) for various sample sizes and return periods. One goal of LQMOM method was to enhance the accuracy of the LMOM method by more fully utilizing the mathematical properties of the underlying probability distribution.
The three-parameter log-normal distribution: The three-parameter lognormal distribution (LN3) may be one of the most versatile distributions. It has been seen to have applications in many fields, such as agriculture, entomology, economics, geology, industry, quality control and hydrology [6] . The LN3 has the probability density function (pdf) is the distribution function [7] . The popular class of L quantile estimators is called kernel quantile estimators has been widely applied [8] . The L quantile estimators is given by where K is a density function symmetric about 0 and In our study, the approximation of the L quantile estimator is called as the weighted kernel quantile estimator (WKQ) is used [9] . The WKQ is given by is an optimal bandwith [8] .

Definition of L-moments:
Let n X X X be a random sample from a continuous distribution function denote the corresponding order statistics. Then the r L-moments r λ is given by [6] ( ) The L in 'L-moments' emphasizes that r λ is a linear function of the expected order statistic. The expectation of an order statistic may be written as The first four L moments of the random variable X are defined as The LQ-skewness and LQ-kurtosis of the random variable X, respectively are defined as λ λ τ = and λ λ λ = (17) Estimation of L-moments: Given a ranked sample, , then the first four sample L moments are given by [10]  denote the corresponding order statistics. Then the r LQ-moments r ξ is given by [6] ( ) ( ) The LQ-skewness and LQ-kurtosis, respectively are defined as Estimation of LQ-moments: For samples of size n, the rth sample LQ-moment is given by is the quick estimator of location and is the quantile of a beta random variable with parameter k r − and + k and Q denotes the weighted kernel quantile estimator (WKQ) given by Eq. (9). The first four sample LQ-moments of the random variable X are defined as The sample LQ-skewness and LQ-kurtosis, respectively are defined as The relationship between η and y σ from Eq.
(39) (for example p = 0.2 and = α ) is shown in Fig. 1. Hoshi et al. [11] compared the ML, MOM and two quantiles-lower bound estimators in combination with two moments in real or in log space. Monte Carlo simulations are conducted to illustrate the performance of using the method of ML and MOM with the other methods. The result shows the method of maximum likelihood and Stedinger's quantile-lower bound method provide more accurate estimators of the lower return period of the distributions. No method was found is the best for estimating the large return period. In addition the maximum likelihood is often highly computational and does not always work well in small samples. For these reasons, the maximum likelihood is not considered here.
The LMOM method becomes a standard procedure in hydrology for estimating the parameters of certain statistical distributions. For regional frequency analysis the LN3 distribution has received special attention based on LMOM [2,12] .

Method of L-moments:
The LMOM estimators for the LN3 distributions are given by [3] " !!! " Stedinger [13] and Hoshi et al. [11] and the location τ was fixed to the value τ = 0. The chosen values of           The RMSE increases as the Cv increases for all methods. The LQMOM has the smallest RMSE for n = 30 and ≥ T 100-year. However for n = 100, the LMOM has the smallest RMSE for T = 100-year and perform as well as LQMOM and MOM for T = 1000year return period.
The RMSE and BIAS of quantile estimators for the LN3 distribution for different sample size and T = 100year return period are compared and shown in Table 2.
The results are quite similar. For sample size, n < , the LQMOM estimator has the smallest RMSE in comparison to the other estimators. The MOM performs next followed by LMOM. However for sample size, ≥ n , the LQMOM method was comparable to the LMOM and MOM method in terms of RMSE. The LMOM method consistently shows the lowest BIAS in comparison to the other estimators. The LQMOM performs next followed by MOM method.

Annual flood data from Malaysia stations:
To compare the performance of LQMOM, LMOM and MOM methods in a more realistic setting actual annual flood data collected at various stations in Malaysia were analyzed. Here, numerical results are presented for two stations, namely, the river Linggui in Johor, with 15 annual maximum floods covering 1978-1992 and the river Pari in Perak, with 42 annual maximum floods covering 1961-2002. These stations are chosen for no particular reason other than the fact that they represent typical short (15 years) and long (42 years), term data, respectively, in the available data-base of annual flood data.
Observed and computed frequency curves for the two data sets are plotted in Fig. 5 and 6. The observed data values are plotted against the corresponding EV1 reduced variates -log(-logF i ), i=1,K,n, where Fi=(i-0.44)/( +0.12) is the Gringorton (1963) plotting position for the i th smallest of n observations. For the river Linggui, the curves fitted by matching LQMOM better capture the trends shown by the larger flows. The LQMOM estimates of large return periods events are less influenced by the small annual maximum flows.
For the river Pari, the frequency curves obtained by the LQMOM and MOM methods are in close agreement and lie much closer to the data than LMOM.
This suggests that from the LN3 distribution may reasonably be fitted to the annual maximum flood series, by the LQMOM than the LMOM or MOM methods.

CONCLUSION
The LQ-moments are constructed by using functional defining the quick estimators, such as the median, trimean or Gastwirth, in places of expectations in L-moments are re-examined. The quick estimators using weighted kernel estimators are introduced for fitting the data for characterizing the upper part of distributions in a sample.
This study has demonstrated that the choice α and p of quick estimator for LQ-moments based on the median, trimeans or Gaswirth is not optimal for the estimation of LN3 quantiles. For all combination of return period and sample size, the optimal values of p is mostly 0.2 and α is 0.04 for T = 10-year and 0.02 for T = 100-year quantile.
The LQMOM method always performs better than the LMOM and MOM methods with respect to RMSE in estimating high quantiles for small samples. It also was seen comparable with the other two methods in estimating high quantiles with large samples.
This study has demonstrated that the conventional LMOM is not optimal for the estimation of the LN3 distribution. The new method of estimation, denoted the LQMOM method, in many cases represents higher efficiency in the quantile estimation compared the LMOM and MOM method. The simplicity and generally good performance of this method make it an attractive option for estimating quantiles in the LN3 distribution.