LQ-moments: Analogs of L-moments

Journal of Statistical Planning andInference 71 (1998) 191–208

LQ-moments: Analogs of L-moments

Govind S. Mudholkar a, Alan D. Hutson b; ∗a Department of Statistics, University of Rochester, Rochester, NY 14627, USA

b Division of Biostatistics, University of Florida, P.O. Box 100212, Gainesville, FL 32610-0212, USA

Received 4 April 1997; received in revised form 12 February 1998; accepted 13 February 1998

Abstract

The L-moments, certain linear functions of the expectations of order statistics, were intro-duced by Sillitto (1951) and comprehensively reviewed by Hosking (1990). L-moments havefound wide applications in such �elds of applied research as civil engineering, meteorology, andhydrology. We present a class of their analogs, labeled LQ-moments, obtained by replacing theexpectations by functionals inducing quick estimators such as the median, Gastwirth estimatorand the trimean. The LQ-moments always exist and are simpler to evaluate and estimate. Themeasures of skewness and kurtosis based on the LQ-moments can be used as more convenientand e�ective alternatives to the traditional beta coe�cients. Simple explicit schemes for their es-timation using readily obtainable sample quantiles are presented. The simplicity of their analysesis demonstrated in terms of the asymptotic distributions of the estimators of the LQ-moments,LQ-skewness and LQ-kurtosis. An application to the extreme value problem in ood data anal-ysis in hydrology is discussed, and several potential applications are outlined. c© 1998 ElsevierScience B.V. All rights reserved.

AMS classi�cation: 62G30; secondary 62G05; 62-07

Keywords: Moment problem; Order statistics; Quantile functions

1. Introduction and Summary

The term moments was acquired by Stieltjes (1894, 1895) from mechanics for nam-ing the now basic and familiar entities, and for discussing the ‘moment problem’ con-sidered earlier in a rudimentary form by Tchebyche� (1874). The moments, includingthe variations such as the central, factorial, and frequency moments, are now commonlyused in model speci�cation, as descriptors of probability distributions, and in inferenceproblems such as estimation and testing of hypotheses. The attention bestowed on themoments by a wide range of scholars from diverse disciplines is also because of theaura of the classical moment problem. The investigators of this problem in its classical

∗ Corresponding author.

0378-3758/98/$19.00 c© 1998 Elsevier Science B.V. All rights reserved.PII: S0378 -3758(98)00088 -3

192 G.S. Mudholkar, A.D. Hutson / Journal of Statistical Planning and Inference 71 (1998) 191–208

forms, as discussed in Shohat and Tamarkin (1943), constitute a virtual Who’s Whoin classical mathematics. Its modi�cations in new settings such as L-moments continueto sprout, ourish and attract new workers. For a general discussion see Kendall et al.(1987) and Huang (1989).Sillitto (1951) introduced a sequence of linear functions of the expectations of or-

der statistics, now known as the L-moments, as alternatives to the classical moments.Hosking (1990) uni�ed discussion of L-moments uses their ratios as new measuresof skewness and kurtosis to relate L-moments to the classical moments by classify-ing the well-known families, such as the Pearsonian system of distributions. Royston(1992) and Vogel and Fennessey (1993) discuss the advantages of L-skewness and L-kurtosis over their product-moment counterparts. Hosking (1990, 1992, 1994), Hoskingand Wallis (1995), Sillitto (1951, 1964, 1969) and Guttman (1994) consider varioustheoretical aspects and applications of L-moments. For their applications in engineer-ing, meteorology, and hydrology, see Angel and Hu� (1992), Gingras and Adamowski(1994), Guttman (1989, 1993), Guttman et al. (1994), Pearson (1993) and Pilon andAdamowski (1991, 1992).The purpose of this paper is to extend L-moments to new moment-like entities called

LQ-moments, which are linear functions of the medians, trimeans, or Gastwirth’s lo-cation estimators of the distributions of certain order statistics, and reduce to weightedaverages of certain population quantiles. LQ-moments always exist, are often eas-ier to evaluate and estimate than L-moments, and in general behave similarly to theL-moments when the latter exist.The class of LQ-moments based on the functionals de�ning arbitrary quick estimators

of location are de�ned, and the de�nition is simpli�ed, in Section 2. The measures ofskewness and kurtosis, respectively, labelled �1 and �2, based on the LQ-moments aredeveloped in Section 3. Section 4 is given to the (�1; �2)-plots for several familiesof distributions including those in the well-known Pearson system. It is seen that themeasures �1, �2 have properties very similar to those of the measures of skewness andkurtosis based on the L-moments. The estimation of the LQ-moments is discussed inSection 5, and the associated large sample theory is presented in Section 6. Section 7contains an application to the generalized extreme value distribution in the conext ofthe analysis of ood data. The conclusions and several related comments appear in the�nal section.

2. De�nition and properties

Let X1; X2; : : : ; Xn be a sample from a continuous distribution function FX (·) withquantile function QX (u)=F−1

X (u), and let X1 :n6X2 :n6 · · ·6Xn :n denote the orderstatistics. Then the rth L-moment �r is given by

�r = r−1r−1∑k=0(−1)k

(r − 1k

)E(Xr−k : r); r=1; 2; : : : : (2.1)

We de�ne the LQ-moments �r analogously

G.S. Mudholkar, A.D. Hutson / Journal of Statistical Planning and Inference 71 (1998) 191–208 193

De�nition 2.1. The rth LQ-moment �r of X is given by

�r = r−1r−1∑k=0(−1)k

(r − 1k

)�p; �(Xr−k : r); r=1; 2; : : : ; (2.2)

where 06�61=2, 06p61=2, and

�p; �(Xr−k : r)=pQXr−k : r (�) + (1− 2p)QXr−k : r (1=2) + pQXr−k : r (1− �): (2.3)

It is easily seen from Eqs. (2.1) and (2.2) that expectation E(·) in place of �p; �(·) inEq. (2.2) de�nes the L-moments. Other possible generalizations of L-moments includereplacing the expectation in Eq. (2.1) with its trimmed counterpart.The linear combination �p; � de�ned by Eq. (2.3) is a ‘quick’ measure of the location

of the sampling distribution of the order statistic Xr−k : r . See Mudholkar et al. (1996)and the references therein for a detailed study of the quick estimator �̂p; �(·). Thecandidates for �p; � include the functionals generating the common quick estimators,e.g. the

Median QXr−k : r (12 ); (2.4)

Trimean QXr−k : r (14 )=4 + QXr−k : r (

12 )=2 + QXr−k : r (

34 )=4; (2.5)

Gastwirth 0:3QXr−k : r (13 ) + 0:4QXr−k : r (

12 ) + 0:3QXr−k : r (

23 ): (2.6)

In Section 6 we show that when sampling from a normal distribution the Gastwirth-based estimators of the LQ-moments are the most e�cient among the choices givenin Eqs. (2.4)–(2.6).In most practical applications involving the LQ-moments, e.g. density classi�cation

and parameter estimation, one would generally use only the �rst four LQ-moments ofthe random variable X

�1 = �p; �(X ); (2.7)

�2 = 12 [�p; �(X2 : 2)− �p; �(X1 : 2)]; (2.8)

�3 = 13 [�p; �(X3 : 3)− 2�p; �(X2 : 3) + �p; �(X1 : 3)]; (2.9)

�4 = 14 [�p; �(X4 : 4)− 3�p; �(X3 : 4) + 3�p; �(X2 : 4)− �p;�(X1 : 4)]: (2.10)

It is obvious that the location measures �p; �(·) exist for any random variable X .Hence we have the following:

Theorem 2.2. The rth LQ-moment {�r : r=1; 2; : : :} always exists. It is unique if thec.d.f. F is continuous.

Furthermore, the evaluation of the LQ-moments of any continuous distribution issimpli�ed using the following lemma.


Lemma 2.3. If QX (·)=F−1X (·) is the quantile function of the random variable X then

the ‘quick’ location measure de�ned by Eq. (2.3) is equivalent to

�p; �(Xr−k : r) =pQX [B−1r−k : r(�)] + (1− 2p)QX [B−1r−k : r(1=2)]+pQX [B−1r−k : r(1− �)]; (2.11)

where B−1r−k : r(�) denotes the corresponding �th quantile of a beta random variablewith parameters r − k and k + 1.

Proof. The above re-expression of �p; �(Xr−k : r) used in the actual calculation of LQ-moments follows from the well-known fact that the uniform order statistic Ur−k : r isdistributed as a Beta random variable with parameters r − k and k + 1, and that theorder statistic Xr−k : r =QX (Ur−k : r) in distribution. Since the quantile function Q(·) ismonotone, the �th quantile Xr−k : r(�)=Q(�th quantile of Ur−k : r) =QX [B−1r−k : r(�)].Hence we have Eq. (2.11).Since the beta quantiles are included in most statistical software packages the LQ-

moments of any continuous distribution can be numerically evaluated using relation(2:11). This aspect is further clari�ed in Section 5.

3. LQ-skewness and LQ-kurtosis

The coe�cients of skewness√�1 and of kurtosis �2 have played an important

role in the classi�cation of statistical distributions, in model �tting, and in para-metric estimation. Because of their usefulness and because of some de�ciencies of√�1 and �2, e.g. their occasional nonexistence, alternative measures of skewness

and kurtosis have been de�ned. These include various classical measures discussed inKendall et al. (1987) and relatively recent measures such as u(F)= [F−1(1 − u) +F−1(u) − 2mF ]=[F−1(1 − u) + F−1(u)], which preserves van Zwet’s (1964) convexordering for all continuous distributions with �nite support; see MacGillivray (1986)for details. The quantile based measures of kurtosis for symmetric distributions include[Q(0:75 + u) + Q(0:75− u)− 2Q(0:75)]=[Q(0:75 + u)− Q(0:75− u)], 06u¡1=4 and[Q(0:5+u)−Q(0:5−u)]=[Q(0:75)−Q(0:25)], 06u¡ 1

2 . For an overview and discussionof the coe�cients of kurtosis see Balanda and MacGillivray (1988).Hosking (1990) de�nes the ratios �3 and �4, called L-skewness and L-kurtosis, re-

spectively, as

�3 = �3=�2 and �4 = �4=�2; (3.1)

and o�ers them as new alternatives to√�1 and �2, and describes their use in the

classi�cation of distributions. Hosking (1990) has shown that �3 satis�es convex or-dering and that �4 maintains van Zwet’s symmetric ordering. He also shows that fornon-degenerate distributions with �nite means that −1¡�3¡1 and (5�23−1)=4¿�4¡1;


see Hosking (1990). Furthermore, Hosking (1992) discusses the e�ectiveness of thesemeasures in explaining the power function of the Shapiro–Wilk W-test.

De�nition 3.1. The skewness and kurtosis measures �3 and �4 based upon the ratiosof LQ-moments to be called LQ-skewness and LQ-kurtosis are given by

�3 = �3=�2 and �4 = �4=�2; (3.2)

respectively.

It may be noted that LQ-skewness and LQ-kurtosis are location and scale invariant,and exist for all distributions. The LQ-skewness takes the value of zero for symmetricaldistributions. However, analogs of other properties of �3 and �4 mentioned above remainuninvestigated for �3 and �4.The behavior of the LQ-skewness �3 and the LQ-kurtosis �4 is now discussed by

comparing them with their established counterparts.

Symmetric distributions. Table 1 gives a comparison between the LQ-kurtosis �4 andthe L-kurtosis �4 and the classical measure �2 for a variety of symmetric distributions.Note that the magnitude of each measure of kurtosis increases for the distributionswith heavier tails. The notable exceptions are the Tukey-� distribution with �=5:2,and the cases where �2 and �4 do not exist. It is interesting to note that the normal,Tukey-� with �=5:2 and Tukey-� with �=0:1349, distributions agree in the �rst fourproduct moments, and to a lesser extent in terms of L-kurtosis, �4. On the other hand,LQ-kurtosis, �4, better distinguishes these three distributions. This distinction betweenthe three measures of kurtosis is noteworthy since the Tukey-� with �=0:1349 closelyresembles the normal distribution; but the Tukey-� with �=5:2 has high tails, alsoknown as truncated or no tails, and di�ers substantially in shape from it; see Freimeret al. (1988) for the relevant discussion. A graph of values of �4 as a function of �,for the Tukey-� distributions, −16�68, is given in Fig. 1.

Table 1A comparison of �4, �4, and �2 for some symmetric distributionsa

Distribution �4 �4 �2

N(0,1) 0.118 0.123 3.00Cauchy(0,1) 0.528 Does not exist Does not existt(10) 0.141 0.154 4.00Logistic(0,1) 0.156 0.167 4.20Laplace(0,1) 0.283 0.236 6.00Tukey-�(5.2) 0.376 0.178 3.00Tukey-�(0.1349) 0.121 0.124 3.00Tukey-�(−1:0) 0.602 Does not exist Does not existUniform(0,1) −0.004 0.000 1.80

a�3 and �4 are based upon the trimean functional �0:25; 0:25.


Fig. 1. Plot of �4 as a function of � for the Tukey-� distribution.

Table 2A comparison of some measures of skewness and kurtosis for selected asymmetric distributionsa

Distribution �3 �3√�1 �4 �4 �2

Beta(2,1) −0:152 −0:143 −0:57 0.050 0.048 2.40Beta(3,2) −0:061 −0:063 −0:29 0.073 0.070 2.36Chisq(4) 0.215 0.235 1.41 0.133 0.142 6.00Chisq(1) 0.503 0.464 2.83 0.230 0.227 15.0Weibull(3.6) −0:010 −0:003 0.00 0.107 0.108 2.72Weibull(4) −0:024 −0:019 −0:09 0.109 0.110 2.75Exponential(1) 0.323 0.333 2.00 0.156 0.167 9.00Gumbel(0,1) 0.151 0.170 1.14 0.141 0.150 5.40Lognormal(0,1)b 0.424 0.463 6.18 0.261 0.293 116.9Steiltjes(�=1=2)b 0.336 c 6.18 0.339 c 116.9a�3 and �4 are based upon the trimean functional �0:25;0:25.bThese distributions have identical moments of all orders; See Section 8 for a discussion.cThe L-moments for this distribution are computationally intricate.

Asymmetric distributions. Table 2 contains a comparison of some common skew dis-tributions with respect to various skewness and kurtosis measures. It can be seen thatthe LQ-ratios, �3 and �4 are similar in magnitude to the L-ratios, �3 and �4.

LQ-coe�cient of variation. Another ratio measure useful for comparing distributionswith common origin and scale is the analog of the coe�cient of variation �2 = �2=�1,where �1 and �2 are given by Eqs. (2.7) and (2.8), respectively. It is a common practicein survival modeling to plot

√b1 versus the sample coe�cient of variation ̂= s= �x in

the (√�1; ) plane in order to verify model selection. The use of �2 in place of will

serve a similar purpose.


Fig. 2. LQ-moment ratios for some common distributions.

4. (�3; �4)-Plots: Distributional classi�cation

The Pearson system is one of the earliest and the best-known family of distributions.Its members are classi�ed into types according to their placement in the regions ofthe plane of �1 and �2; e.g. chi-square is type III, and Student’s t is type VII andF is type VI; see Kendall et al. (1987). Because of the general familiarity with thisclassi�cation, other families of distributions are also classi�ed using �1 and �2; see forexample, Kendall et al. (1987) for the Burr system and the Johnson system, Freimeret al. (1988) for the generalized Tukey-� family, and Mudholkar and Kollia (1994)for the generalized Weibull family.

The (�3; �4)-Plots. Fig. 2 describes the loci of the LQ-kurtosis, LQ-skewness in (�3; �4)-plane for some common distributions and distributional families. Use of this graph isanalogous to that of the classical (�1; �2) graph for the Pearson family. Note thateven distributions such as Cauchy have representation in the (�3; �4) plane. From thediagram we see that LQ-moments behave in a reasonable manner, consistent with the(�1; �2)-plot. For example the Type VII line runs between the normal and Cauchydistributions, and the Type III line in the limit goes towards the normal point. We alsosee that the exponential distribution lies at the intersection of the Type III and Weibulldistributions. Furthermore, there is a natural ordering of the symmetric distributionswith respect to the ‘peakedness’ measure �4; e.g. uniform ≺ normal ≺ Laplace ≺Cauchy, where ‘≺’ denotes the peakedness ordering with respect to �4.


5. Estimation of LQ-moments

The LQ-moments can be estimated in a straightforward manner by estimating thequantiles of order statistics, in conjunction with Eq. (2.11), occurring in their de�ni-tions. The simplest quantile estimator suitable for the purpose is the linear interpolation-based quantile estimator available in common statistical packages such as MINITAB,SAS, IMSL and S-plus. However, more e�cient estimators such as some favorite ker-nel estimator or some quasi-quantile may be used in the estimation; see Mudholkarand Hutson (1996).For discussing the details of estimation we employ the linear interpolation-based

quantile estimator found in most statistical software packages. However, any alternativeestimator of the quantiles may also be used. Let X1:n6X2:n6 · · ·6Xn:n denote thesample order statistics then the quantile estimator of Q(u) is given by

Q̂X (u)= (1− �)X[n′u]:n + � X[n′u]+1:n; (5.1)

where �= n′u− [n′u] and n′= n+ 1. Thus, we have

De�nition 5.1. For samples of size n, the rth sample LQ-moment is given by

�̂r = r−1 r−1∑k = 0

(−1)k(r − 1k

)�̂p; �(Xr−k :r); (5.2)

where �̂p; �(Xr−k :r), the quick estimator of the location for the distribution of Xr−k :r ina random sample of size r.In particular, the �rst four sample LQ-moments from Eq. (5.2) are given by

�̂1 = �̂p; �(X ); (5.3)

�̂2 =12 [�̂p; �(X2:2)− �̂p; �(X1:2)]; (5.4)

�̂3 =13 [�̂p; �(X3:3)− 2�̂p; �(X2:3) + �̂p; �(X1:3)]; (5.5)

�̂4 =14 [�̂p; �(X4:4)− 3�̂p; �(X3:4) + 3�̂p; �(X2:4)− �̂p; �(X1:4)]; (5.6)

where the quick estimator �̂p; �(Xr−k:r) of the location of the order statistic Xr−k:r isgiven by

�̂p; �(Xr−k:r) =pQ̂Xr−k:r(�) + (1− 2p)Q̂Xr−k:r

(1=2) + pQ̂Xr−k:r(1− �);

=pQ̂X [B−1r−k:r(�)] + (1− 2p)Q̂X [B−1r−k:r(1=2)]

+ pQ̂X [B−1r−k:r(1− �)]; (5.7)

06�61=2, 06p61=2, B−1r−k:r(�) is the �th quantile of a beta random variable withparameters r− k and k +1, and Q̂X (·) denotes the linear interpolation estimator givenby Eq. (5.1). The computation of the sample LQ-moments �̂r is simpli�ed using thequantiles B−1r−k:r(�) which are easy to obtain from tables, packages such as SAS orIMSL or approximations such as those given in Mudholkar and Chaubey (1976).


Table 3A triangular representation for the estimates of the �rst four LQ-moments �r based upon the medianfunctionala

�̂4 �̂3 �̂2 �̂1ci ui ci ui ci ui ci ui

1=4 0.841 1=3 0.794 1=2 0.707 1 0.500−3=4 0.614 −2=3 0.500 −1=2 0.2933=4 0.386 1=3 0.206

−1=4 0.159

aThe LQ-moment �̂r =∑

iciQ̂X (ui), where Q̂X is given by Eq. (5.1).

Table 4A triangular representation for the estimates of the �rst four LQ-moments �r based upon the trimeanfunctionala

�̂4 �̂3 �̂2 �̂1ci ui ci ui ci ui ci ui

1=16 0.931 1/12 0.909 1/8 0.866 1/4 0.7501=8 0.841 1/6 0.794 1/4 0.707 1/2 0.500

−3=16 0.757 −1=6 0.674 −1=4 0.293 1/4 0.2501=16 0.707 1=12 0.630 −1=8 0.134

−3=8 0.614 −1=3 0.5003=16 0.544 1=12 0.370

−3=16 0.456 −1=16 0.3263=8 0.386 1=6 0.206

−1=16 0.293 1=2 0.0913=16 0.243

−1=8 0.159−1=16 0.069

aThe LQ-moment �̂r =∑

iciQ̂X (ui), where Q̂X is given by Eq. (5.1).

Now, we present explicit schemes for computing LQ-moments when the threebest known quick estimators, namely the median (p=0; �= ·), the trimean(p=1=4; �=1=4), and the Gastwirth estimator (p=0:3; �=1=3) are employed for�̂p; �(Xr−k:r) given by Eq. (5.7). The estimation of the �rst four sample LQ-momentsfrom Eq. (5.2) is simpli�ed using the pyramid schemes given in Tables 3–5 respec-tively.The sample LQ-skewness and sample LQ-kurtosis given by

�̂3 = �̂3=�̂2 and �̂4 = �̂4=�̂2; (5.8)

respectively, may be used for identifying the population in the (�3; �4)-plane and forestimating the parameters.

6. Large sample theory

The sample LQ-moments depend upon the choice of the quick estimator used and thequantile estimator used for estimating it. However, their asymptotic normality follows


Table 5A triangular representation for the estimates of the �rst four LQ-moments �r based upon the Gastwirthfunctionala

�̂4 �̂3 �̂2 �̂1ci ui ci ui ci ui ci ui3=40 0.904 1=10 0.874 3=20 0.816 3=10 0.3331=10 0.841 2=15 0.794 1=5 0.707 2=5 0.5003=40 0.760 1=10 0.693 3=20 0.577 3=10 0.667

−9=40 0.709 −1=5 0.613 −3=20 0.423−3=10 0.614 −4=15 0.500 −1=5 0.2939=40 0.514 −1=5 0.387 −3=20 0.184

−9=40 0.486 1=10 0.3073=10 0.386 2=15 0.2069=40 0.291 1=10 0.126

−3=40 0.240−1=10 0.159−3=40 0.096a The LQ-moment �̂r =

∑iciQ̂X (ui), where Q̂X is given by Eq. (5.1).

from the large sample theory of the linear functions of order statistics. In order todevelop large sample expressions for the mean and variance of the sample LQ-momentswe restrict attention to the class Q of quantile functions Q satisfying the followingconditions, as outlined in van Zwet (1964):1. The inverse function Q(u)=F−1(u) is uniquely de�ned for 0¡u¡1.2. Q(·) is twice di�erentiable on (0,1) with continuous second derivative Q′′(·) on(0,1).

3. Q′(u)¿0 for 0¡u¡1.

Lemma 6.1. Let 0¡u1¡ · · ·¡uk¡1 and suppose conditions (1)–(3) given aboveare satis�ed. Then [Q̂(u1); : : : ; Q̂(uk)] is asymptotically normal with mean vector[Q(u1); : : : ; Q(uk)] and covariances

�ij =Cov[Q̂(Ui)Q̂(Uj)]= ui(1− uj)Q′(ui)Q′(uj)=n for i6j; (6.1)

and �ij = �ji.

Proof. Follows from standard theory, e.g. see Ser ing (1980).In order to develop the asymptotic expressions for the covariances of the LQ-

moments, we �rst derive the Cov[�̂p; �(Xr−k:r); �̂p; �(Xs−l:s)] which is a function de-pendent upon six speci�c percentiles u1; : : : ; u6 used in the calculations of Eq. (6.1)assigned to the set of six percentiles B−1r−k:r(�); B

−1s−l:l(�); B

−1r−k:r(1=2); B

−1s−l:l(1=2); B

−1r−k:r

(1 − �); B−1s−l:l(1 − �); such that 0¡u1¡ · · ·¡u6¡1, where B−1r−k:r(�) denotes the�th quantile of a beta random variable with parameters r − k and k + 1. Then theCov[Q̂(ui)Q̂(uj)] can be obtained using Lemma 6.1, and the following result is easilyestablished:


Lemma 6.2. The covariance between the estimated quick estimators of the orderstatistics is

Cov[�̂p; �(Xr−k:r); �̂p; �(Xs−l:s)]

= p{pCov[Q̂(u1)Q̂(u2)] + (1− 2p)Cov[Q̂(u2)Q̂(u3)]+pCov[Q̂(u2)Q̂(u5)] + pCov[Q̂(u1)Q̂(u6)] + (1− 2p)Cov[Q̂(u3)Q̂(u6)]+pCov[Q̂(u5)Q̂(u6)]}+ (1− 2p){pCov[Q̂(u1)Q̂(u4)]+ (1− 2p)Cov[Q̂(u3)Q̂(u4)] + pCov[Q̂(u4)Q̂(u5)]}: (6.2)

Proof. Intricate but straightforward.

Theorem 6.3. The rth sample LQ-moment; �̂r ; r=1; 2; : : : ; is asymptotically normalwith mean �r; and for r6s the covariances of the LQ-moments are given by

Cov(�̂r ; �̂s) =1rs

r−1∑k=0

s−1∑l=0(−1)k+l

(r − 1k

)(s− 1l

)

×Cov[�̂p; �(Xr−k:r); �̂p; �(Xs−l:s)]; (6.3)

where Cov[�̂p; �(Xr−k:r); �̂p; �(Xs−l:s)] is given by Lemma 6.2 and u1; : : : ; u6 are givenabove. For r= s we get the variance of �̂r .

Proof. Follows from Lemmas 6.1 and 6.2.

Theorem 6.4. As n→∞ the sample measures of LQ-skewness; �̂3; and LQ-kurtosis�̂4 have a bivariate normal distribution with mean vector (�3; �4) and

Var(�̂3)=Var(�̂3)=�22; (6.4)

Cov(�̂3; �̂4)= [Cov(�̂3; �̂4)− �3Cov(�̂2; �̂4)− �4Cov(�̂2; �̂3) + �3�4Var(�̂2)]=�22;(6.5)

Var(�̂4)=Var(�̂4)=�22; (6.6)

where the Var(�̂r)=Cov(�̂r ; �̂r) and the variance and covariance terms on the rightare given by Eq. (6.3).

Proof. Follows from Theorem 6.3 in conjunction with Slutsky’s Theorem.

Application: The normal case. Consider sampling from a normal population and com-paring the use of the median, trimean, and Gastwirth estimators for estimating LQ-skewness and LQ-kurtosis. Then the estimators �̂3, and �̂4 given in Eq. (5.8), are jointlynormally distributed with the respective mean vectors (0; 0:116), (0; 0:118), (0; 0:117)


and covariance matrices

�MED =1n

(1:535 00 2:070

); �TRI=

1n

(0:824 00 0:381

);

and

�GAS =1n

(0:549 00 0:235

); (6.7)

respectively. We see from Eq. (6.7) that �̂3 and �̂4 are asymptotically uncorrelated foreach of the aforementioned quick estimators. The traditional moments based measuresof skewness and kurtosis

√b1 and b2 are similarly asymptotically uncorrelated; e.g.

see Kendall et al. (1987). Additionally, from the covariance matrices we see that, interms of the LQ-estimators of skewness and kurtosis, in case of large samples fromnear-normal populations the Gastwirth estimator is preferable to either the median ortrimean.

7. An application to the generalized extreme value distribution

As discussed in Section 1, the L-moments have been used in a wide variety of applied�elds. One such application involves some aspects of the extreme-value analysis of ood frequencies. It is well known that, if the limiting distribution of the appropriatelycentered and scaled maximum order statistic, anXn:n + bn, exists then it is one of thefollowing types:

F1(x)={0; x60; �¿0;exp(−x−�); x¿0;

F2(x)={exp[−(−x)�]; x60; �¿0;1; x¿0;

(7.1)

F3(x)= exp(−e−x) −∞¡x¡∞;e.g. see David (1981). So the generalized extreme value distribution which is an amal-gamation of the three-types given by Eq. (7.1) is commonly used in practice. Itsquantile function is given by

Q(u)= � + �Q0(u); (7.2)

where

Q0(u)= [1− (− log u)k ]=k; k 6=0: (7.3)

In this section we discuss the use of the LQ-moments to �t a generalized extremevalue model to observed ood frequencies. Hosking et al. (1985) give an approximatemethod based on probability-weighted moments, which in this instance


provides the same parameter estimates as those obtained by using L-moments, for�tting the model (7:2)–(7:3) to such data. Gingras and Adamowski (1994) have ap-plied the method to ood data using L-moments. They �rst estimate the shape pa-rameter as k̂ =7:8590z + 2:9554z2, where z=2=(3 + �̂3) − log 2= log 3. The estimatesof the location and parameters are then given by �̂= �̂1 + �̂[�(1 + k̂) − 1]=k̂ and�̂= �̂2k̂=[1− 2−k̂�(1+ k̂)]: The sample LQ-moments may be used directly for the pur-pose. So we propose �rst estimating the shape parameter k by numerically solving theequation

�̂3 = 2[Q0(0:091) + 2Q0(0:206)− 2Q0(0:326) + Q0(0:370)− 4Q0(0:5)+Q0(0:630)− 2Q0(0:674) + 2Q0(0:794) + Q0(0:909)]=3=[2Q0(0:707)− 2Q0(0:293) + Q0(0:866)− Q0(0:134)]; (7.4)

where Q0(·) is given by Eq. (7.3). The estimates of the parameters � and � are thengiven by

�̂ = �̂1 − �̂[Q̂0(1=4)=4 + Q̂0(1=2)=2 + Q̂0(3=4)=4]; (7.5)

�̂ = 8�̂2=[2Q̂0(0:707)− 2Q̂0(0:293) + Q̂0(0:866)− Q̂0(0:134)]; (7.6)

where Q̂0(u)= [1− (− log u)k̂ ]=k̂. The formulae used for the calculation of �̂1, �̂2 and�̂3 are given in Table 4. Note that for the LQ-moment estimation, unlike L-momentestimation, the expressions for � and � remain the same regardless of the form ofQ0(·).

The Flood Data. To illustrate the use of LQ-moments for �tting a generalized ex-treme value model we consider the data from the Feather River at Oroville, CA andthe Blackstone River at Woonsocket, RI, both given in Pericchi and Rodriguez-Iturbe(1985).The annual maximum ood data for the Feather River at Oroville, CA for a period

of 59 years are given in Table 6 and those for the Blackstone River at Woonsocket, RIfor a period of 37 years appear in Table 7. The estimates of the �rst three LQ-momentsfor these data are given in Table 8. In the computation we used the linear interpolation

Table 6The consecutive annual ood discharge rates of the Feather River at Oroville, CA

Years Flood discharge in (ft3=s)

1902–1909 42000, 102000, 118000, 81000, 128000, 230000, 16300,140000,1910–1917 31000, 75400, 16400, 16800, 122000, 81400, 42400, 80400,1918–1925 28200, 65900, 23400, 62300, 36400, 22400, 42400, 64300,1926–1933 55700, 94000, 185000, 14000, 80100, 11600, 22600, 8860,1934–1941 20300, 58600, 85400, 19200, 185000, 8080, 152000, 84200,1942–1949 110000, 108000, 24900, 60100, 54400, 45600, 36700, 16800,1950–1957 46400, 92100, 59200, 13000, 113000, 54800, 203000, 83100,1958–1960 102000, 34500, 135000.

Source: Pericchi and Rodriguez-Iturbe (1985).


Table 7The consecutive annual ood discharge rates of the Blackstone River at Woonsocket, RI

Years Flood discharge in (ft3=s)

1929–1938 4570, 1970, 8220, 4530, 5780, 6560, 7500, 15000, 6340, 15100,1939–1948 3840, 5860, 4480, 5330, 5310, 3830, 3410, 3830, 3150, 5810,1949–1958 2030, 3620, 4920, 4090, 5570, 9400, 32900, 8710, 3850, 4970,1959–1965 5398, 4780, 4020, 5790, 4510, 5520, 5300.

Source: Pericchi and Rodriguez-Iturbe (1985).

Table 8Feather River and Blackstone River moment estimates

Feather River data Blackstone River data

Sample Sample Sample SampleLQ-momentsa L-momentsb LQ-momentsa L-momentsb

�̂1 = 61325:00 �̂1 = 70265:08 �̂1 = 5158:75 �̂1 = 6372.92�̂2 = 27666:00 �̂2 = 28459:56 �̂2 = 1044:47 �̂2 = 2019:06�̂3 = 5934:83 �̂3 = 6594:45 �̂3 = 430:47 �̂3 = 1045:37�̂3 = 0:2145 �̂3 = 0:2317 �̂3 = 0:4123 �̂3 = 0:5178

a Estimates based upon the trimean, i.e. p= 14 and �=

14 .

b Estimated via U-statistics; see Hosking (1990).

Table 9Parameter estimates for the extreme value model at Eq. (7.2)Parameter Feather River Blackstone River

L-moment LQ-moment L-moment LQ-moment

� 44901.7 43548.3 4256.74 4496.21� 37355.7 40078.7 1441.13 1198.99k −0:0939 −0:1212 −0:4794 −0:484350 year ood 220,942 ft3=s 243,498 ft3=s 20,767.3 ft3=s 18,403.5 ft3=s1000 year ood 408,033 ft3=s 476,665 ft3=s 83,689.5 ft3=s 72,246.7 ft3=s

estimator (5:1). The value of the LQ-skewness 0.2145 for the Feather River, and 0.4123for the Blackstone River indicate that both distributions are right skewed. As can beexpected, the corresponding values for the L-skewness are of similar magnitude. The�rst three sample LQ-moments and sample L-moments were then used to estimatethe parameters of the extreme-value distribution with quantile function (7:2). Theseestimates for the generalized extreme value distribution �ts to the two datasets aregiven in Table 9. A substitution of the estimates into the quantile function (7:2) givepredictions for the 50 year maximum ood and the 1000 year maximum ood for eachset of parameter estimates. The results displayed Table 9, using the L-moments andthe LQ-moments, respectively, are comparable.


8. Conclusions and miscellaneous remarks

In this paper we have discussed some aspects of a class of analogs of L-moments,the LQ-moments which are obtained using some quick estimator � in place of theexpectations of the order statistics in the latter. It is shown that the LQ-moments,which always exist, amount to linear functions of some population quantiles and areeasy to evaluate and estimate. We have also described the use of their ratios called LQ-skewness and LQ-kurtosis in data modelling. Several theoretical and practical aspectsof the LQ-moments are under study or remain for future investigations. Some of theseare now indicated.1. Departures from normality. Royston (1992) argues that the sample L-skewness and

L-kurtosis statistics should be used in place of the sample estimators of the productmoment measures of skewness and kurtosis. In fact, Vogel and Fennessey (1993)argue for the replacement of the traditional moment diagrams, based on

√�1 and

�2, for the Pearson system with the corresponding diagrams based upon L-moments.Their arguments are based principally upon the ease of interpretation, the robustness tooutliers, approximate normality, ability to indicate the type of departure from normality,and ease of use in the case of censored data. We suggest that the sample LQ-momentsshare the same advantages. Moreover they always exist, are straightforward to compute,and their asymptotic distributions are easier to obtain. The values for the LQ-skewnessand LQ-kurtosis, based upon the trimean, for the normal distribution are �3 = 0 and�4 = 0:118.2. Approximating L-moments. It is obvious that analogs of the L-moments based on

any L-estimator, e.g. a trimmed mean, can be analyzed as in the analysis of the LQ-moments in this paper. Furthermore, it is well known that the population L-momentsare often analytically intractable and are obtained using Monte-Carlo methods. Alterna-tively, they may be approximated by averaging a su�ciently large number of populationquantiles and using the appropriate beta quantiles as in Lemma 2.3.3. Testing distributional assumptions. The classical sample measures

√b1 of skew-

ness and b2 of kurtosis are widely used to assess the validity of common assumptionssuch as normality. A study of the operating characteristics of the analogous tests basedupon �̂3 and �̂4 appears in order. Fig. 2 can be used to identify the population on thebasis of �3, �4 or their estimates.4. Shapiro–Wilk W-test. Hosking (1992) graphically illustrates that the measures

L-skewness and L-kurtosis given by Eq. (2.1) quantify deviations from the normaldistribution in a way that is more concordant with the powers of the Shapiro–Wilkgoodness-of-�t test of normality. Analogous graphs based upon LQ-skewness and LQ-kurtosis exhibit the same characteristics. This is to be expected because of the closenessbetween L-moments and LQ-moments.5. The moment problem. Rohatgi (1985) observes that ‘The problem of moments

has a long and glorious history and has led to the developments of several branchesof mathematics’. Its overview by Shohat and Tamarkin (1943) summarizes the contri-butions to the classical problem by a virtual who’s who of mathematicians in the late


Fig. 3. Two p.d.f.’s g(x; �) in the Stieltjes family∗ with identical moments. ∗ The p.d.f. g(x; �) as given byEq. (8.1). The distribution with �=0 is the standard lognormal distribution.

19th and early 20th century. Hosking’s (1990) review describes the characterizationof the distributions by L-moments; see also Huang (1989). Similar developments forLQ-moments are in progress.6. The lognormal distribution. An interesting counterexample to the set of all mo-

ments uniquely determining the distribution involves the lognormal distribution whichis widely used for modelling a variety of entities such as incomes in economic studiesand lifetimes in survival analyses. This example originally due to Stieltjes (1894) wasrediscovered independently by Heyde (1963). Speci�cally, they show that the momentsof all order for the p.d.f.

g(x; �)=1

x√2�exp[−(log x)2=2]× (1 + � sin[2� log(x)]); (8.1)

are the same for any choice of �, 0¡�¡1; �=0 being the standard lognormal distri-bution. On the other hand, since the moments of all order exist for the p.d.f. given inEq. (8.1) then the L-moments do uniquely determine the distribution for any choiceof �. We expect that the same holds true for the LQ-moments. Table 2 gives the �rstfour moments for �=0 and �= 1

2 , and Fig. 3 displays the p.d.f.’s of the distribution(8:1) for �=0 and �= 1

2 .

Acknowledgements

We are thankful to the referees for their detailed review and constructive commentswhich have helped in substantial improvements in the paper. Alan Hutson’s work byNIH General Clinical Center Grant RR00082.


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LQ-moments: Analogs of L-moments