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International Journal of Business and Statistical Analysis ISSN (2384-4663) Int. J. Bus. Stat. Ana. 2, No. 2 (July-2015) Linear Moments: An Overview A.F. Kandeel1 1 Department of Statistics and Mathematics, Faculty of Commerece, Benha University, Benha, Egypt Received 5 February 2015, Revised 1 April,. 2015, Accepted 5 April 2015, Published 1st July 2015 Abstract: Many statistical techniques are based on the use of linear combinations of order statistics that called linear moments. L-moments are a sequence of statistics used to summarize the shape of a probability distribution. They are linear combinations of order statistics analogous to conventional moments, and can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis respectively. In this paper an overview for recent works in L- moments is presented. Keywords: Order statistics, L-moments, TL-moments, LQ-moments, Quantile function. to the standard errors of L-moments to be finite, only the 1. INTRODUCTION distribution is required to have finite variance and no Many statistical techniques are based on the use of higher-order moments need be finite. Also, though linear combinations of order statistics (or quantile moment ratios can be arbitrarily large, sample moment function), but there has not been developed a unified ratios have algebraic bounds but sample L-moment ratios theory of estimation covering the characterization of can take any values that the corresponding population probability distribution, until Hosking (1990) introduced quantities can. L-moments, as an alternative to the classical moments. In addition, L-moments have properties that hold in a So, L-moments are summary statistics for probability wide range of practical situations. L-moments also give distributions and data samples. They provide measures of asymptotic approximations to sampling distributions location, dispersion, skewness, kurtosis, and other aspects better than classical moments and provide better of the shape of probability distributions or data samples. identification of the parent distribution which generated a However, The L in L-moments emphasizes the particular data sample (see Hosking (1990)). construction of L-moments from linear combinations of Furthermore, L-moments are less sensitive to outlying order statistics. data values and yield, sometimes, more efficient Sillitto (1969) has derived an approximation to the parameter estimates than the maximum likelihood inverse distribution function (the quantile function) in estimates (Vogel and Fennessey (1993)). terms of population L-moments without referring to L- But, the main shortage in L-moments is that they do moments. Also, he gave the sample version of the not exist for the distribution which has infinite mean and quantile function in terms of sample L-moments without less efficient for heavy tail distributions. studying its properties. A formal and comprehensive Mudholkar and Hutson (1998), introduced, as a treatment of L-moments was developed by Hosking result of the previous shortage, the LQ-moments, as a (1990), who established foundational results supporting a robust version of L-moments in which expectation of the new methodology in data analysis and inference based on conceptual order statistics (in the definition of population L-moments. L-moments) is replaced by a class of robust location Hosking (1990) concluded that the L-moments, as a measures defined in terms of simple linear combinations function of the quantile function, have various theoretical of symmetric quantiles of the distributions of the order advantages over the classical moments. For example, for statistics, such as median and trimean. L-moments of a probability distribution to be meaningful, Elamir and Seheult (2003) introduced Trimmed L- we require only that the distribution has a finite mean; no moments (TL-moments) as an alternative to LQ-moments higher-order moments need be finite. Similarly, in order and natural generalization of L-moments that do not E-mail address: [email protected] http://journals.uob.edu.bh 86 A.F. Kandeel: Linear Moments: An Overview require the mean of underlying distribution to exist. TL- characterizing life distributions and other applications. moments depend on giving zero weight to extreme The role of certain quantile functions and quantile-based observations. Therefore, they are defined for heavy tailed concepts in reliability analysis are also investigated. distributions where they do not involve some values at Several studies had been done using and applying the the extreme ends of the distribution. method of TL-moments as natural generalization of L- As interested in statistical modeling using heavy- moments: tailed distributions is increasing, so is the important of(a) Some of these studies concerned to introduce the potential by the L-moment and TL-moment theoretical results for TL-moments method are as approaches. The need of linear moments has been follows: developed in support of regional frequency analysis in Karvanen (2006) proposed parametric families from environmental science, which treats the quantile of quantile functions of distributions. This class of distributions of variables such as annual maximum parametric families is called quantile mixtures, which precipitation, stream flow, wind speed observed at each contain a wide range of different distributions that have site in a given network. Hosking and Wallis (1997) practical importance. He introduced two parametric provide an excellent exposition. Also linear moments families: the normal-polynomial quantile mixture and the approach has special utility in applications where Cauchy-polynomial quantile mixture. And he used the descriptive estimates (location, spread, skewness, and methods of L-moments and TL-moments to estimate the kurtosis) more stable than the usual central moments are parameters of the two quantile mixtures respectively. The critically needed. Such concern arise, for example, in proposed quantile mixtures are applied to model monthly, volatility estimation in financial risk management weekly and daily returns of some major stock indexes. involving market variables such as stock indices, interest Abdul-Moniem (2007) applied the method of L- rates; see, Serfling (1980) , Embrechts et al. (1997), moment and TL-moment estimators to estimate the Leonowicz et al. (2005) and Willinger et al. (1998). parameters of exponential distribution. TL-moments have various theoretical advantages. Asquith (2007) derived analytical solutions for the For example, TL-moments give more robust estimators first five L-moments and TL-moments (in case of than L-moments in the presence of outliers. Moreover, symmetrical trimming (1,1) in terms of a four-parameters population TL-moments may be well defined where the generalized Lambda distribution (GLD) in asymmetric corresponding population L-moments (or central case. And, because that asymmetric GLD needs moments ) do not exist , for example, the first population numerical methods to compute the parameters from the TL-moment is well defined for a Cauchy distribution, but L-moments or TL-moments, algorithms are suggested for the first population L-moment, the population mean, does parameter estimation. Application of the GLD using both not exist (see, Elamir and Seheult (2003)). L-moments and TL-moment parameter estimates from Also, their sample variance and covariance can be example data is demonstrated, a small simulation study obtained in closed form (see, Elamir and Seheult (2003)). of the 98th percentile (far-right tail) is conducted for a In addition, TL-moments ratios are bounded for any heavy-tail GLD with high-outlier contamination. The trimmed value (see, Hosking (2007 and Elamir et al simulations showed, with respect to estimation of the (2010)). 98th-percent quantile, that TL-moments are less biased Hosking (2007) has derived of approximation to the (more robust) in the presence of high-outlier quantile function in terms of population TL-moments. contamination. Elamir (2009) introduced properties of this approximate Hosking (2007) derived some further theoretical function by minimizing the weighted mean square error results for TL-moments. He defined the TL-moments in between the population quantile function and its TL- terms of shifted jacobi polynomials, and he introduced a moments representation. Also, he studied properties of representation for the quantile function as a weighted the corresponding sample estimator. He concluded that function of orthogonal polynomials in which the the estimators have a good approximation to population coefficients are related to TL-moments. quantile for a broad class of probability distribution Abdul-Moniem and Selim (2009) derived the TL- functions. moments and L-moments of the generalized Pareto Also, Elamir (2010) derived an optimal choice for distribution (GPD), and they used it to obtain the first the amount of trimming from known distributions, based four TL-moments and L-moments. They introduced the on the minimum sum of the absolute value of the errors TL-skewness, L-skewness, TL-kurtosis and L-kurtosis between the quantile probability function and its TL- for the GPD. They used the TL-moments and L-moments moments representation. Nair and Vineshkumar (2010) to estimate the parameters for the GPD. Also, they study the properties of L-moments of residual life in the introduced a numerical illustrate for the new results. context of modeling lifetime data. They introduced
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