# Characterizing Tukeyh and hh-Distributions through L-Moments and the L-Correlation

Authors: Todd C. Headrick; Mohan D.  Pant
Journal: ISRN Applied Mathematics
 (2012)

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## Abstract

This paper introduces the Tukey family of symmetrich and asymmetric hh-distributions in the contexts of univariate L-moments and the L-correlation. Included is the development of a procedure for specifying nonnormal distributions with controlled degrees of L-skew, L-kurtosis, and L-correlations. The procedure can be applied in a variety of settings such as modeling events (e.g., risk analysis, extreme events) and Monte Carlo or simulation studies. Further, it is demonstrated that estimates of L-skew, L-kurtosis, and L-correlation are substantially superior to conventional product-moment estimates of skew, kurtosis, and Pearson correlation in terms of both relative bias and efficiency when heavy-tailed distributions are of concern.


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## Body

## 1. Introduction

The conventional moment-based Tukey families ofh (or the g-and-h and Generalized Pareto) distributions (e.g., [1–3]) are often used in various applied mathematics contexts. Some examples include modeling events associated with operational risk [4], extreme oceanic wind speeds, [5], common stock returns [6], solar flare data [7], or in the context of Monte Carlo or simulation studies, for example, regression analysis [8].The family ofh-distributions is based on the transformation(1.1)q(Z)=Zexp⁡(hZ22),
where Z~i.i.d.  N(0,1). Equation (1.1) produces symmetric h-distributions where the parameter h controls the tail weight or elongation of any particular distribution and is positively related with kurtosis. The pdf and cdf associated with (1.1) are expressed as in [3, equations (12), (13)] (1.2)fq(Z)(q(z))=f̅(z)=(q(z),ϕ(z)q′(z)),(1.3)Fq(Z)(q(z))=F̅(z)=(q(z),Φ(z)),
where f̅:ℜ↦ℜ2 and F̅:ℜ↦ℜ2 are the parametric forms of  the pdf and cdf with the mappings z↦(x,y) and z↦(x,v) with x=q(z), y=ϕ(z)/q′(z), v=Φ(z), and where ϕ(z) and Φ(z) are the standard normal pdf and cdf, respectively. It is assumed that q′(z)>0 in (1.2) to ensure a valid pdf, that is, the transformation in (1.1) is strictly increasing, which requires h≥0. Further, if q(Z) in (1.1) has a valid pdf where the k-th order moment exists (for k=1,2,…), then h must be bounded such that 0≤h<1/k. That is, a distribution will not have a first moment (or mean) for h≥1 [2, 3].The variance (α22) and kurtosis (α4) of a distribution associated with (1.1) can be determined from [3, Equations (32), (36)] (1.4)α22=1(1-2h)3/2,α4=3(1-2h)3(1(1-4h)5/2+1(2h-1)3),
where the mean (α1) and skew (α3) of a distribution are both zero.One of the extensions of (1.1)–(1.3) is the two parameter hh family of distributions introduced by Morgenthaler and Tukey [2]. More specifically, the hh family includes asymmetric distributions with heavy tails based on the transformation
(1.5)q(Z)={Zexp⁡(hLZ22),forZ≤0,Zexp⁡(hRZ22),forZ≥0,
where hℒ(hℛ) is the parameter for the left (right) tail of a distribution. The properties of the transformation in (1.5) are the same as those associated with (1.1). However, the tails of hh distributions have to be considered separately as they are weighted differently that is, in general, hℒ≠hℛ. For example, Figure 1 gives an example of an hh-distribution based on matching the values of α3 and α4 associated with a noncentral Student t(df=5,δ=1) distribution. The values of hℒ and hℛ in Figure 1 were computed by simultaneously solving (A.3) and (A.4) in the appendix.Figure 1
Graph of anhh-distribution based on matching the conventional moments of a noncentral Student t(df=5,δ=1) distribution. The values of hℒ and hℛ were determined by solving equations (A.3) and (A.4) in the appendix. The estimates (α̂3,4; τ̂3,4) and bootstrap confidence intervals (C.I.s) were based on resampling 25,000 statistics. Each sample statistic was based on a sample size of n=250.Conventional moment-based-estimators (e.g.,α̂3, α̂4) have unfavorable attributes insofar as they can be substantially biased, have high variance, or can be influenced by outliers. For example, inspection of Figure 1 indicates, on average, that the estimates of α̂3 and α̂4 are only 78.83% and 42.81% of their associated population parameters. Note that each estimate of α̂3 and α̂4 in Figure 1 were calculated based on sample sizes of n=250 and the formulae currently used by most commercial software packages such as SAS, SPSS, and Minitab for computing skew and kurtosis.However,L-moment-based estimators such as L-skew and L-kurtosis have been introduced to address some of the limitations associated with conventional estimates of skew and kurtosis [9, 10]. Specifically, some of the advantages that L-moments (or their estimators) have over conventional moments are that they (a) exist whenever the mean of the distribution exists, (b) are nearly unbiased for all sample sizes and distributions, and (c) are more robust in the presence of outliers. For example, the estimates τ̂3 and τ̂4 in Figure 1 are relatively much closer to their respective parameters with much smaller standard errors than their corresponding conventional moment based analogs (α̂3, α̂4). More specifically, the estimates of τ̂3 and τ̂4 that were simulated are, on average, 98.74% and 99.64% of their parameters.In the context of multivariate data generation, the methodology has been developed for simulatingh-(or g-and-h) distributions with specified Pearson correlation structures [11, pages 140–148] [12]. This methodology is based on conventional product moments and the popular NORTA [13] approach, which begins with generating multivariate standard normal deviates. However, the NORTA approach is not without its limitations. Specifically, one limitation arises because the Pearson correlation is not invariant under nonlinear strictly increasing transformations such as (1.1). As such, the NORTA approach must begin with the computation of an intermediate  correlation (IC) matrix, which is different than the specified correlation matrix between the nonnormal h-distributions. The purpose of the IC matrix is to adjust for the nonnormalization effect of the transformation in (1.1) such that the resulting nonnormal distributions have their specified skew, kurtosis, and specified correlation matrix.Some additional consequences associated with NORTA in this context are that it (a) requires numerical integration to compute solutions to ICs betweenh-distributions—unlike the more popular power method [11] which has a straight-forward equation to solve for the ICs between distributions [11, page 30] and (b) may yield solutions to ICs that are not in the range of [-1,+1] as the absolute values of ICs must be greater than (or equal to) their specified Pearson correlations [14]. Further, these two problems which can be exacerbated when h-distributions with heavy tails are used as functions performing numerical integration will more frequently either fail to converge or yield incorrect solutions to ICs.In view of the above, the present aim is to derive theh and hh families of distributions in the contexts of L-moment and L-correlation theory. Specifically, the purpose of this paper is to develop the methodology and a procedure for simulating nonnormal symmetric h and asymmetric hh distributions with specified L-moments and L-correlations. Some of the advantages of the proposed procedure are that ICs (a) can be solved directly with a single equation, that is, numerical integration is not required and (b) cannot exist outside the range of [-1,+1] as it is shown that the absolute value of an IC will be less than (or equal to) its associated specified L-correlation.The remainder of the paper is outlined as follows. In Section2, a summary of univariate L-moment theory is provided and the derivations of the systems of equations for the h and hh distributions are provided for modeling or simulating nonnormal distributions with specified values of L-skew and L-kurtosis. In Section 3, the coefficient of L-correlation is introduced and the equations are subsequently derived for determining ICs for specified L-correlations between nonnormal h or hh distributions. In Section 4, the steps for implementing the proposed L-moment procedure are described. A numerical example and results of a simulation are also provided to confirm the derivations and compare the new procedure with the traditional or conventional moment-based procedure. In Section 5, the results of the simulation are discussed.


## 2. Methodology

### 2.1. Preliminaries

LetX1,…,Xj,…,Xn be iid random variables each with continuous pdf f(x), cdf F(x), order statistics denoted as X1:n≤⋯≤Xj:n≤⋯≤Xn:n, and L-moments defined in terms of either linear combinations of (a) expectations of order statistics or (b) probability-weighted moments (βi). For the purposes considered herein, the first four L-moments associated with Xj:n are expressed as [10, pages 20–22] (2.1)λ1=E[X1:1]=β0,λ2=12E[X2:2-X1:2]=2β1-β0,(2.2)λ3=13E[X3:3-2X2:3+X1:3]=6β2-6β1+β0,(2.3)λ4=14E[X4:4-3X3:4+3X2:4-X1:4]=20β3-30β2+12β1-β0,
where the βi are determined from (2.4)βi=∫x{F(x)}if(x)dx,
where i=0,…,3. The coefficients associated with βi in (2.4) are obtained from shifted orthogonal Legendre polynomials and are computed as shown in [10, page 20] or in [15].TheL-moments λ1 and λ2 in (2.1) are measures of location and scale and are the arithmetic mean and one-half the coefficient of mean difference (or Gini’s index of spread), respectively. Higher-order L-moments are transformed to dimensionless quantities referred to as L-moment ratios defined as τr=λr/λ2 for r≥3, and where τ3 and τ4 are the analogs to the conventional measures of skew and kurtosis. In general, L-moment ratios are bounded in the interval -1<τr<1 as is the index of L-skew (τ3) where a symmetric distribution implies that all L-moment ratios with odd subscripts are zero. Other smaller boundaries can be found for more specific cases. For example, the index of L-kurtosis (τ4) has the boundary condition for continuous distributions of [16] (2.5)(5τ32-1)4<τ4<1.


### 2.2.L-Moments for Symmetric h-Distributions

The family ofh-distributions based on the method of L-moments is less restrictive than the family based on conventional method of moments as described in the previous section to the extent that we may consider the h parameter on the interval 0≤h<1 for any distribution with finite k-order L-moments rather than 0≤h<1/k for the kth-order conventional moment to exist. This advantage is attributed to Hosking’s Theorem 1 [9] which states that if the mean (λ1) exists, then all other L-moments will have finite expectations.We begin the derivation for symmetrich-distributions in the context of L-moments by defining the probability-weighted moments based on (2.4) in terms of q(z) in (1.1) and the standard normal pdf and cdf as
(2.6)βi=∫-∞+∞q(z){Φ(z)}iϕ(z)dz.
Integrating (2.6) for i=0,1,2 gives (2.7)λ1=0,(2.8)λ2=-2π(h-1)2-h,(2.9)λ3=τ3=0.The fourthL-moment λ4 (and τ4) is subsequently derived in terms of the expectations of order statistics as in (2.3) by making use of the following expression for standard normal-based expectations and for n=4 as [17](2.10)E[q(Z)j:4]=14(3j-1)∫0∞(q(z)φ(z)[1+Ψ(z)]j-1[1-Ψ(z)]4-j-[1-Ψ(z)]j-1[1+Ψ(z)]4-j)dz,
where φ(z)=2ϕ(z) and Ψ(z)=2Φ(z)-1 are the pdf and cdf of the folded unit normal distribution at z=0, respectively. The relevant expansions of the polynomial in (2.10) are (2.11)E[q(Z)3:4]=-E[q(Z)2:4]=(32)(I1-I3),E[q(Z)4:4]=-E[q(Z)1:4]=(12)(3I1+I3),
where the expectations in (2.11) is linear combinations associated with the integrals denoted as I1 and I3. The specific expressions for I1 and I3 are (2.12)I1=δ⋅λ2=(-(h-1)1-h2)⋅(-2π(h-1)2-h)=1π,(2.13)I3=∫0∞q(z)φ(z)[Ψ(z)]3dz,
where it is convenient to use δ in (2.12) to standardize I1 (λ2) to the unit normal distribution. Equation (2.13) may be integrated by parts based on Ψ′(z)=φ(z), φ′(z)=-φ(z)z, and noting that Ψ(0)=0 and limz→+∞φ(z)=0. As such, we have (2.14)I3=3∫0∞ξ(z)φ(z)[Ψ(z)]2dz,
where the expression ξ(z) is (2.15)ξ(z)=-∫δq(z)φ(z)dz=exp⁡{(1/2)(h-1)z2}2-hπ.
Let us first consider the expression [Ψ(z)]2 in (2.14), which can be expressed as (2.16)[Ψ(z)]2=2π[∫0zexp⁡{-12u2}du]2=2π∬0zexp⁡{-12(z12+z22)}dz1dz2=1-4π∫0π/4exp⁡{-12z2sec2θ1}dθ1.
Substituting (2.16) into (2.14) and, using Lichtenstein’s Theorem [18], and integrating first with respect to z yield (2.17)π∫0∞ξ(z)φ(z)2exp⁡{-12z2sec2θ1}dz=4-2h22-h+sec2θ1.
Using (2.17), the integral in (2.14) is expressed as (2.18)I3=3π{1-4π∫0π/44-2h22-h+sec2θ1}dθ1=3π-12tan-1[(h-4/h-2)-1/2]π3/2.
Hence, using (2.3) and (2.11) and (2.12), L-kurtosis can be expressed as (2.19)τ4=6-30tan-1[(h-4/h-2)-1/2]π,
where 0≤h<1. Whence, it follows that we have a convenient closed formed solution for the parameter h as (2.20)h=3-sec⁡[π15(τ4-6)].
Equation (2.19) has a lower limit of τ4≈0.1226(h=0) that is equivalent to the normal distribution and an upper limit (τ4→1; h→1) that is equivalent to the Cauchy or t(df=1) distribution. Figure 2(d) gives an example of a symmetric h-distribution withL-kurtosis (τ4) of a logistic distribution.Three asymmetrichh-distributions (a)–(c) and one symmetric h distribution (d) with their conventional and L-moment parameters of skew (α3) and L-skew (τ3), kurtosis (α4) and L-kurtosis (τ4), and corresponding shape parameters for (1.5) and (1.1).
(a)(b)(c)(d)


### 2.3.L-Moments for Asymmetric hh-Distributions

The derivation of theL-moments for asymmetric hh-distributions associated with (1.5) begins with determining the probability-weighted moments βi in (2.6) by separately evaluating and summing two integrals as (2.21)βi=ILi(hL)+IRi(hR)=∫-∞0q(z,hL){Φ(z)}iϕ(z)dz+∫0+∞q(z,hR){Φ(z)}iϕ(z)dz.
As such, using (2.1) and (2.21), it is straight-forward to obtain β0, β1, and the first two L-moments as (2.22)λ1=12π(hL-1)+12π(1-hR),λ2=2-hL+2-hR-hL2-hL-hR2-hR2π(hL-1)(hR-1)(hL-2)(hR-2).
In terms of deriving λ3 and λ4, it is convenient to consider β2 in (2.2) as (2.23)β2=IL2(hL)+IR2(hR)=IL2(hL)+(-IL2(hR)+λ2(hR)2),
where λ2(hℛ) can be obtained from (2.8). Thus, it is only necessary to determine Iℒ2(hℒ) as
(2.24)IL2(hL)=∫-∞0q(z,hL){Φ(z)}2ϕ(z)dz=∫-∞0z2πexp⁡{(hL-1)z22}{Φ(z)}2dz=12π(hL-1)∫-∞0{Φ(z)}2dexp⁡{(hL-1)z22}=12π(hL-1)[{Φ(z)}2exp⁡{(hL-1)z22}|-∞0-∫-∞012πexp⁡{(hL-2)z22}2Φ(z)dz]=-142π(1-hL)+22π2-hL(1-hL)∫-∞02-hL2πexp⁡{-(2-hL)z22}Φ(z)dz.
If we let X~N(0,1/(2-hℒ)) and ~N(0,1), where X and Y are independent such that (X/2-hℒ,Y) jointly follow the standard bivariate normal distribution, then the integral in the last part of (2.24) is (2.25)∫-∞02-hL2πexp⁡{-(2-hL)z22}Φ(z)dz=Pr{X<0,Y<X}=Pr{X2-hL<0,Y<2-hLX2-hL}=14(1-2cot-1⁡(2-hL)π),
where we are calculating the proportion of area between the y-axis and the line y=x(2-hℒ) as a sector because the independent standard normal bivariate density has rotational symmetry about the origin. Combining terms from (2.24) and (2.25) yields (2.26)IL2(hL)=14(1-hL)2π[22-hL(1-2cot-1⁡(2-hL)π)-1].
Hence, given (2.26) and (2.19), we can solve for β2, β3, and subsequently obtain the expressions for L-skew (τ3) and L-kurtosis (τ4) as (2.27)τ3={122-hL(hR-2)(hR-1)cot-1(2-hL)-π(hL-hR)(hL-2)(hR-2)-122-hR(hL-2)(hL-1)cot-1⁡(2-hR)}/{2π(hL-2)(hR-2)(hL2-hL-2-hL-2-hR+hR2-hR)},τ4={(1+2hR-4)6π(hR(hL-4)(hL-2)(hL-1)(hR-2)-2(hL-4)(hL-1)-hL(hL-3)(hL-4)(hL-1)-(hL-4)(hL-2)(hL-1)(hR-2))+30(hL-1)(hL-4)(hL-2)(hR-2)hL-1(hR-1)tan-1⁡(1+2hL-4)+30(hL-2)(hL-4)(hL-1)(hL-1)tan-1⁡(1+2hR-4)}/{π(4-hL)(2-hL)(1-hL)(hL2-hL-2-hL-2-hR+hR2-hR)}.
Thus, given specified values of τ3 and τ4, (2.27) can be numerically solved for the corresponding values of hℒ and hℛ. Figures 2(a), 2(b) and 2(c) provides some examples of various hh-distributions, which are used in the simulation portion of this study in Section 4.


## 2.1. Preliminaries

LetX1,…,Xj,…,Xn be iid random variables each with continuous pdf f(x), cdf F(x), order statistics denoted as X1:n≤⋯≤Xj:n≤⋯≤Xn:n, and L-moments defined in terms of either linear combinations of (a) expectations of order statistics or (b) probability-weighted moments (βi). For the purposes considered herein, the first four L-moments associated with Xj:n are expressed as [10, pages 20–22] (2.1)λ1=E[X1:1]=β0,λ2=12E[X2:2-X1:2]=2β1-β0,(2.2)λ3=13E[X3:3-2X2:3+X1:3]=6β2-6β1+β0,(2.3)λ4=14E[X4:4-3X3:4+3X2:4-X1:4]=20β3-30β2+12β1-β0,
where the βi are determined from (2.4)βi=∫x{F(x)}if(x)dx,
where i=0,…,3. The coefficients associated with βi in (2.4) are obtained from shifted orthogonal Legendre polynomials and are computed as shown in [10, page 20] or in [15].TheL-moments λ1 and λ2 in (2.1) are measures of location and scale and are the arithmetic mean and one-half the coefficient of mean difference (or Gini’s index of spread), respectively. Higher-order L-moments are transformed to dimensionless quantities referred to as L-moment ratios defined as τr=λr/λ2 for r≥3, and where τ3 and τ4 are the analogs to the conventional measures of skew and kurtosis. In general, L-moment ratios are bounded in the interval -1<τr<1 as is the index of L-skew (τ3) where a symmetric distribution implies that all L-moment ratios with odd subscripts are zero. Other smaller boundaries can be found for more specific cases. For example, the index of L-kurtosis (τ4) has the boundary condition for continuous distributions of [16] (2.5)(5τ32-1)4<τ4<1.


## 2.2.L-Moments for Symmetric h-Distributions

The family ofh-distributions based on the method of L-moments is less restrictive than the family based on conventional method of moments as described in the previous section to the extent that we may consider the h parameter on the interval 0≤h<1 for any distribution with finite k-order L-moments rather than 0≤h<1/k for the kth-order conventional moment to exist. This advantage is attributed to Hosking’s Theorem 1 [9] which states that if the mean (λ1) exists, then all other L-moments will have finite expectations.We begin the derivation for symmetrich-distributions in the context of L-moments by defining the probability-weighted moments based on (2.4) in terms of q(z) in (1.1) and the standard normal pdf and cdf as
(2.6)βi=∫-∞+∞q(z){Φ(z)}iϕ(z)dz.
Integrating (2.6) for i=0,1,2 gives (2.7)λ1=0,(2.8)λ2=-2π(h-1)2-h,(2.9)λ3=τ3=0.The fourthL-moment λ4 (and τ4) is subsequently derived in terms of the expectations of order statistics as in (2.3) by making use of the following expression for standard normal-based expectations and for n=4 as [17](2.10)E[q(Z)j:4]=14(3j-1)∫0∞(q(z)φ(z)[1+Ψ(z)]j-1[1-Ψ(z)]4-j-[1-Ψ(z)]j-1[1+Ψ(z)]4-j)dz,
where φ(z)=2ϕ(z) and Ψ(z)=2Φ(z)-1 are the pdf and cdf of the folded unit normal distribution at z=0, respectively. The relevant expansions of the polynomial in (2.10) are (2.11)E[q(Z)3:4]=-E[q(Z)2:4]=(32)(I1-I3),E[q(Z)4:4]=-E[q(Z)1:4]=(12)(3I1+I3),
where the expectations in (2.11) is linear combinations associated with the integrals denoted as I1 and I3. The specific expressions for I1 and I3 are (2.12)I1=δ⋅λ2=(-(h-1)1-h2)⋅(-2π(h-1)2-h)=1π,(2.13)I3=∫0∞q(z)φ(z)[Ψ(z)]3dz,
where it is convenient to use δ in (2.12) to standardize I1 (λ2) to the unit normal distribution. Equation (2.13) may be integrated by parts based on Ψ′(z)=φ(z), φ′(z)=-φ(z)z, and noting that Ψ(0)=0 and limz→+∞φ(z)=0. As such, we have (2.14)I3=3∫0∞ξ(z)φ(z)[Ψ(z)]2dz,
where the expression ξ(z) is (2.15)ξ(z)=-∫δq(z)φ(z)dz=exp⁡{(1/2)(h-1)z2}2-hπ.
Let us first consider the expression [Ψ(z)]2 in (2.14), which can be expressed as (2.16)[Ψ(z)]2=2π[∫0zexp⁡{-12u2}du]2=2π∬0zexp⁡{-12(z12+z22)}dz1dz2=1-4π∫0π/4exp⁡{-12z2sec2θ1}dθ1.
Substituting (2.16) into (2.14) and, using Lichtenstein’s Theorem [18], and integrating first with respect to z yield (2.17)π∫0∞ξ(z)φ(z)2exp⁡{-12z2sec2θ1}dz=4-2h22-h+sec2θ1.
Using (2.17), the integral in (2.14) is expressed as (2.18)I3=3π{1-4π∫0π/44-2h22-h+sec2θ1}dθ1=3π-12tan-1[(h-4/h-2)-1/2]π3/2.
Hence, using (2.3) and (2.11) and (2.12), L-kurtosis can be expressed as (2.19)τ4=6-30tan-1[(h-4/h-2)-1/2]π,
where 0≤h<1. Whence, it follows that we have a convenient closed formed solution for the parameter h as (2.20)h=3-sec⁡[π15(τ4-6)].
Equation (2.19) has a lower limit of τ4≈0.1226(h=0) that is equivalent to the normal distribution and an upper limit (τ4→1; h→1) that is equivalent to the Cauchy or t(df=1) distribution. Figure 2(d) gives an example of a symmetric h-distribution withL-kurtosis (τ4) of a logistic distribution.Three asymmetrichh-distributions (a)–(c) and one symmetric h distribution (d) with their conventional and L-moment parameters of skew (α3) and L-skew (τ3), kurtosis (α4) and L-kurtosis (τ4), and corresponding shape parameters for (1.5) and (1.1).
(a)(b)(c)(d)


## 2.3.L-Moments for Asymmetric hh-Distributions

The derivation of theL-moments for asymmetric hh-distributions associated with (1.5) begins with determining the probability-weighted moments βi in (2.6) by separately evaluating and summing two integrals as (2.21)βi=ILi(hL)+IRi(hR)=∫-∞0q(z,hL){Φ(z)}iϕ(z)dz+∫0+∞q(z,hR){Φ(z)}iϕ(z)dz.
As such, using (2.1) and (2.21), it is straight-forward to obtain β0, β1, and the first two L-moments as (2.22)λ1=12π(hL-1)+12π(1-hR),λ2=2-hL+2-hR-hL2-hL-hR2-hR2π(hL-1)(hR-1)(hL-2)(hR-2).
In terms of deriving λ3 and λ4, it is convenient to consider β2 in (2.2) as (2.23)β2=IL2(hL)+IR2(hR)=IL2(hL)+(-IL2(hR)+λ2(hR)2),
where λ2(hℛ) can be obtained from (2.8). Thus, it is only necessary to determine Iℒ2(hℒ) as
(2.24)IL2(hL)=∫-∞0q(z,hL){Φ(z)}2ϕ(z)dz=∫-∞0z2πexp⁡{(hL-1)z22}{Φ(z)}2dz=12π(hL-1)∫-∞0{Φ(z)}2dexp⁡{(hL-1)z22}=12π(hL-1)[{Φ(z)}2exp⁡{(hL-1)z22}|-∞0-∫-∞012πexp⁡{(hL-2)z22}2Φ(z)dz]=-142π(1-hL)+22π2-hL(1-hL)∫-∞02-hL2πexp⁡{-(2-hL)z22}Φ(z)dz.
If we let X~N(0,1/(2-hℒ)) and ~N(0,1), where X and Y are independent such that (X/2-hℒ,Y) jointly follow the standard bivariate normal distribution, then the integral in the last part of (2.24) is (2.25)∫-∞02-hL2πexp⁡{-(2-hL)z22}Φ(z)dz=Pr{X<0,Y<X}=Pr{X2-hL<0,Y<2-hLX2-hL}=14(1-2cot-1⁡(2-hL)π),
where we are calculating the proportion of area between the y-axis and the line y=x(2-hℒ) as a sector because the independent standard normal bivariate density has rotational symmetry about the origin. Combining terms from (2.24) and (2.25) yields (2.26)IL2(hL)=14(1-hL)2π[22-hL(1-2cot-1⁡(2-hL)π)-1].
Hence, given (2.26) and (2.19), we can solve for β2, β3, and subsequently obtain the expressions for L-skew (τ3) and L-kurtosis (τ4) as (2.27)τ3={122-hL(hR-2)(hR-1)cot-1(2-hL)-π(hL-hR)(hL-2)(hR-2)-122-hR(hL-2)(hL-1)cot-1⁡(2-hR)}/{2π(hL-2)(hR-2)(hL2-hL-2-hL-2-hR+hR2-hR)},τ4={(1+2hR-4)6π(hR(hL-4)(hL-2)(hL-1)(hR-2)-2(hL-4)(hL-1)-hL(hL-3)(hL-4)(hL-1)-(hL-4)(hL-2)(hL-1)(hR-2))+30(hL-1)(hL-4)(hL-2)(hR-2)hL-1(hR-1)tan-1⁡(1+2hL-4)+30(hL-2)(hL-4)(hL-1)(hL-1)tan-1⁡(1+2hR-4)}/{π(4-hL)(2-hL)(1-hL)(hL2-hL-2-hL-2-hR+hR2-hR)}.
Thus, given specified values of τ3 and τ4, (2.27) can be numerically solved for the corresponding values of hℒ and hℛ. Figures 2(a), 2(b) and 2(c) provides some examples of various hh-distributions, which are used in the simulation portion of this study in Section 4.


## 3.L-Correlations for the h and hh-Distributions

The coefficient ofL-correlation (see [19]) is introduced by considering two random variables Yj and Yk with distribution functions F(Yj) and F(Yk), respectively. The second L-moments of Yj and Yk can alternatively be expressed as (3.1)λ2(Yj)=2Cov(Yj,F(Yj)),(3.2)λ2(Yk)=2Cov(Yk,F(Yk)).
The second L-comoments of Yj toward Yk and Yk toward Yj are (3.3)λ2(Yj,Yk)=2Cov(Yj,F(Yk)),(3.4)λ2(Yk,Yj)=2Cov(Yk,F(Yj)).
As such, the L-correlations of Yj toward Yk and Yk toward Yj are expressed as (3.5)ηjk=λ2(Yj,Yk)λ2(Yj),(3.6)ηkj=λ2(Yk,Yj)λ2(Yk).
The L-correlation in (3.5) or (3.6) is bounded such that -1≤ηjk≤1, where a value of ηjk=1 (ηjk=-1) indicates a strictly increasing (decreasing) monotone relationship between the two variables. In general, we would also note that ηjk≠ηkj.In the context ofL-moment symmetric h-distributions (0≤h<1), suppose it is desired to simulate T distributions based on (1.1) with a specified L-correlation matrix and where each distribution has its own specified value of τ4. Define q(Zj) and q(Zk) as in (1.1), where Zj and Zk have Pearson correlation ρjk and standard normal bivariate density of (3.7)fjk=(2π(1-ρjk2)1/2)-1exp⁡{-(2(1-ρjk2))-1(zj2+zk2-2ρjkzjzk)}.
Using (1.1), (1.3), and (3.5) with the denominator standardized to λ2=1/π for the unit-normal distribution, and (3.7), the L-correlation of q(Zj) toward q(Zk) can be expressed as (3.8)ηjk=2πCov(q(zj),Fq(Zk)(q(zk)))=2πCov(q(zj),Φ(zk))=2πE[q(zj)Φ(zk)]-2πE[q(zj)]E[Φ(zk)]=2π∬-∞+∞δq(zj)Φ(zk)fjkdzjdzk-2πE[q(zj)]E[Φ(zk)],
where δ is the standardizing term in (2.12). Integrating (3.8) yields (3.9)ηjk=ρjk2-hj2+hj(ρjk2-2)
given that E[q(zj)]=0 and E[Φ(zk)]=1/2. Analogously, the L-correlation of q(Zk) toward q(Zj) is (3.10)ηkj=ρjk2-hk2+hk(ρjk2-2).From (3.9), the intermediate correlation (IC) ρjk can be determined by simply evaluating (3.11)ρjk=±2hjηjk2-ηjk2hjηjk2+hj-2
for a specified value of ηjk and a given value of hj from (2.20). Given ρjk from (3.11), the L-correlation ηkj can be determined by evaluating (3.10) using the solved value of hk. Note the special case of where hj=hk, in (3.9) and (3.10), then ηjk=ηkj.Remark 3.1.
Inspection of (3.9) indicates that ηjk=ρjk when either (a) q(Zj) is standard normal that is, hj=0, (b) ρjk=0, or (c) ρjk=1.Remark 3.2.
If the IC is such that0<|ρjk|<1 and 0<hj<1 in (3.9), then we have the inequality
(3.12)0<|ρjk|<|ηjk|<1,
as from inspecting (3.9) it is evident that [(2-hj)/(2+hj(ρjk2-2))]1/2>1. Thus, solutions to ICs cannot exist outside the range of [-1,+1].
The extension of determining ICs for asymmetrichh-distributions is analogous to the method described above for h-distributions, where (1.5) is standardized and subsequently integrated as in (3.8) to obtain
(3.13)ηjk=12ρjk2-hLj2+hLj(ρjk2-2)+12ρjk2-hRj2+hRj(ρjk2-2),(3.14)ηkj=12ρjk2-hLk2+hLk(ρjk2-2)+12ρjk2-hRk2+hRk(ρjk2-2),
where it is straight-forward to see that if hℒj=hℛj (or hℒk=hℛk), then (3.13) or (3.14) simplifies to (3.9) or (3.10) for the case of symmetric h-distributions. The IC in (3.13) is determined by substituting the specified L-correlation (ηjk) and the solved values of the parameters hℒj and hℛj (from (2.27)) into the left- and right-hand sides of (3.13), respectively, and then numerically solving for ρjk. We would also note that Remarks 3.1 and 3.2 also apply to (3.13) and (3.14).


## 4. The Procedure and Simulation Study

To implement the procedure for simulatinghh- (or h-) distributions with specified L-moments and specified L-correlations, we suggest the following five steps. (1)
Specify theL-moments for T transformations of the form in (1.5), that is, q(Z1),…,q(ZT)and obtain the parameters of hℒj and hℛjby solving equations (2.27) using the specified values of L-skew (τ3) and L-kurtosis (τ4) for each distribution. Specify a T×T matrix of L-correlations (ηjk) for q(Zj) toward q(Zk), where j<k∈{1,2,…,T}.(2)
Compute the (Pearson) intermediate correlations (ICs)ρjk by substituting the specified L-correlation ηjk and the parameters of hℒj and hℛj from step (1) into the left- and right-hand sides of (3.14), respectively, and then numerically solve for ρjk. Repeat this step separately for all T(T-1)/2 pairwise combinations of correlations.(3)
Assemble the ICs into aT×T matrix and decompose this matrix using a Cholesky factorization. Note that this step requires the IC matrix to be positive definite.(4)
Use the results of the Cholesky factorization from step (3) to generateT standard normal variables (Z1,…,ZT) correlated at the intermediate levels as follows:
(4.1)Z1=a11V1,Z2=a12V1+a22V2,⋮Zj=a1jV1+a2jV2+⋯+aijVi+⋯+ajjVj,⋮ZT=a1TV1+a2TV2+⋯+aiTVi+⋯+ajTVj+⋯+aTTVT,
where V1,…,VT are independent standard normal random variables and where aij represents the element in the i-th row and the j-th column of the matrix associated with the Cholesky factorization performed in step (3).(5)
SubstituteZ1,…,ZT from step (4) into T equations of the form in (1.5), as noted in step (1), to generate the nonnormal hh-distributions with the specified L-moments and L-correlations.To demonstrate the steps above and evaluate the proposed procedure, a comparison between the newL-moment and conventional moment-based procedures is subsequently described. Specifically, the distributions in Figure 2 are used as a basis for a comparison using the specified correlation matrices in Table 1 where both strong and moderate levels of correlation are considered. Tables 2 and 3 give the solved IC matrices for the L-moment and conventional moment-based procedures, respectively. Note that the ICs for the conventional procedure were computed by adapting the Mathematica source code in [12, Table  1] for (1.5). Tables 4 and 5 give the results of the Cholesky decompositions on the IC matrices, which are then used to create Z1,…,Z4 with the specified ICs by making use of the formulae given in (4.1) of step 4 with T=4. The values of Z1,…,Z4 are subsequently substituted into equations of the form in (1.5) to produce q(Z1),…,q(Z4) for both procedures.Table 1
Specified correlation matrices for the distributions in Figure2.
(a)12341120.70130.700.70140.850.700.701(b)12341120.40130.500.40140.600.500.401Table 2
Intermediate correlations for the conventional moment procedure.
(a)12341120.726130.7220.715140.8810.7170.7071(b)12341120.424130.5210.414140.6260.5150.4051Table 3
Intermediate correlations for theL-moment procedure.
(a)12341120.689130.6890.685140.8430.6850.6911(b)12341120.389130.4880.386140.5880.4850.3921Table 4
Cholesky decompositions for the Conventional moment procedure.
(a)a11=1a12=0.726a13=0.722a14=0.8810a22=0.687a23=0.277a24=0.11200a33=0.633a34=0.062000a44=0.455(b)a11=1a12=0.424a13=0.521a14=0.6260a22=0.906a23=0.213a24=0.27500a33=0.827a34=0.025000a44=0.729Table 5
Cholesky decompositions for theL-moment procedure.
(a)a11=1a12=0.689a13=0.689a14=0.8430a22=0.725a23=0.291a24=0.14500a33=0.664a34=0.104000a44=0.508(b)a11=1a12=0.389a13=0.488a14=0.5880a22=0.921a23=0.213a24=0.27800a33=0.846a34=0.0543000a44=0.758In terms of the simulation, a Fortran algorithm was written for both procedures to generate 25,000 independent sample estimates for the specified parameters of (a) conventional skew (α3), kurtosis (α4), and Pearson correlation (ρjk*); (b) L-skew (τ3), L-kurtosis (τ4), and L-correlation (ηjk) based on samples of sizes n=25 and n=1000. The estimates for α3,4 were based on Fisher’s k-statistics that is, the formulae currently used by most commercial software packages such as SAS, SPSS, and Minitab for computing indices of skew and kurtosis (where α3,4=0 for the standard normal distribution). The formulae used for computing estimates for τ3,4 were Headrick’s Equations (2.3) and (2.5) [15]. The estimate for ρjk* was based on the usual formula for the Pearson product-moment of correlation statistic, and the estimate for ηjk was computed based on (3.13) using the empirical forms of the cdfs in (3.1) and (3.3). The estimates for ρjk* and ηjk were both transformed using Fisher’s z′ transformation. Bias-corrected accelerated bootstrapped average estimates, confidence intervals (C.I.s), and standard errors were subsequently obtained for the estimates associated with the parameters (α3,4,τ3,4,zρjk*′,zηjk′) using 10,000 resamples via the commercial software package Spotfire S+ [20]. The bootstrap results for the estimates of zρjk*′ and zηjk′ were transformed back to their original metrics. Further, if a parameter (P) was outside its associated bootstrap C.I., then an index of relative bias (RB) was computed for the estimate (E) as RB=((E-P)/P)×100. The results of the simulation are reported in Tables 6, 7, 8, 9, 10, and 11 and are discussed in the next section.Table 6
Skew (α3) and Kurtosis (α4) results for the conventional moment procedure.
(a)
n=25
DistParameterEstimate95% Bootstrap C.I.Standard errorRelative bias %1α3=20.6490.638, 0.6590.0056−67.6α4=401.6771.64, 1.720.0200−95.82α3=10.3050.293, 0.3170.0060−69.5α4=151.731.69, 1.760.0175−88.53α3=0.50.2480.239, 0.2570.0048−50.4α4=30.970.95, 1.000.0128−67.74α3=00.0056−0.0024, 0.01400.0042—α4=1.20.6240.606, 0.6430.0095−48.0(b)
n=1000
DistParameterEstimate95% Bootstrap C.I.Standard errorRelative bias %1α3=21.6121.597, 1.6270.0077−19.4α4=4011.4111.11, 11.720.1542−71.52α3=10.8340.823, 0.8450.0057−16.6α4=157.687.50, 7.870.0962−48.83α3=0.50.4770.472, 0.4810.0021−4.6α4=32.652.61, 2.700.0223−11.74α3=0−0.0003−0.0023, 0.00180.0011—α4=1.21.1661.158, 1.1750.0046−2.83Table 7
L-skew (τ3) and L-kurtosis (τ4) results.
(a)
n=25
DistParameterEstimate95% Bootstrap C.I.Stand. errorRelative bias %1τ3=0.1106100.097240.09574, 0.098970.00082−12.1τ4=0.1957580.18820.1869, 0.18930.00061−3.92τ3=0.0500400. 043610.04193, 0.045340.00086−12.8τ4=0.2108390.20380.2026,  0.20500.00060−3.33τ3=0.0410250.037260.03592,  0.038830.00074−9.2τ4=0.1750840.17230.1713,  0.17340.00054−1.64τ3=00.0008−0.0007,  0.00210.00070—τ4=0.1601730.15910.1582,  0.16020.00050—(b)
n=1000
DistParameterEstimate95% Bootstrap C.I.Stand. errorRelative bias %1τ3=0.1106100.11010.1098,  0.11040.00014−0.46τ4=0.1957580. 19550. 1953,  0.19580.00010—2τ3=0.0500400.04980.04946,  0.0500410.00015—τ4=0.2108390.21060.2104,  0.21090.00010—3τ3=0.0410250.040860.04062,  0.041090.00012—τ4=0.1750840.17500.1748,  0.17510.00010—4τ3=0−0.00003−0.00027,  0.000150.00011—τ4=0.1601730.16020.1600,  0.16030.00010—Table 8
Correlation (strong) results for the conventional moment procedure.
(a)
n=25
ParameterEstimate95% Bootstrap C.I.Standard errorRelative bias%ρ12*=0.700.7250.724, 0.7270.00153.57ρ13*=0.700.7230.722, 0.7240.00143.29ρ14*=0.850.8730.873, 0.8740.00132.71ρ23*=0.700.7180.716, 0.7190.00142.57ρ24*=0.700.7180.717, 0.7190.00142.57ρ34*=0.700.7120.710, 0.7130.00141.71(b)
n=1000
ParameterEstimate95% Bootstrap C.I.Standard errorRelative bias%ρ12*=0.700.70240.7021, 0.70270.000300.34ρ13*=0.700.70250.7023, 0.70280.000270.36ρ14*=0.850.85350.8532, 0.85360.000370.41ρ23*=0.700.70100.7008, 0.70130.000240.14ρ24*=0.700.70130.7011, 0.70150.000240.19ρ34*=0.700.70050.7003, 0.70070.000210.07Table 9
Correlation (strong) results for theL-moment procedure.
(a)
n=25
ParameterEstimate95% Bootstrap C.I.Standard errorRelative bias%η12=0.700.7100.708, 0.7110.00151.40η13=0.700.7090.708, 0.7110.00151.34η14=0.850.8560.855, 0.8570.00150.71η23=0.700.7080.706, 0.7090.00151.10η24=0.700.7070.706, 0.7090.00151.01η34=0.700.7080.707, 0.7100.00141.20(b)
n=1000
ParameterEstimate95% Bootstrap C.I.Standard errorRelative bias%η12=0.700.70140.7012, 0.70160.000220.20η13=0.700.70160.7014, 0.70180.000220.23η14=0.850.85110.8509, 0.85120.000220.13η23=0.700.70020.6999, 0.70040.00022—η24=0.700.70030.7001, 0.70050.000220.04η34=0.700.70040.7002, 0.70060.000210.06Table 10
Correlation (moderate) results for the conventional moment procedure.
(a)
n=25
ParameterEstimate95% Bootstrap C.I.Standard errorRelative bias%ρ12*=0.400.4240.422, 0.4270.00146.00ρ13*=0.500.5220.520, 0.5240.00144.40ρ14*=0.600.6240.622, 0.6260.00134.00ρ23*=0.400.4160.414, 0.4180.00134.00ρ24*=0.500.5160.514, 0.5180.00133.20ρ34*=0.400.4100.408, 0.4120.00132.50(b)
n=1000
ParameterEstimate95% Bootstrap C.I.Standard ErrorRelative Bias%ρ12*=0.400.40200.4016, 0.40230.000240.50ρ13*=0.500.50220.5018, 0.50250.000240.44ρ14*=0.600.60230.6020, 0.60260.000240.38ρ23*=0.400.40050.4001, 0.40080.000210.12ρ24*=0.500.50100.5007, 0.50130.000210.20ρ34*=0.400.40030.4000, 0.40060.000200.08Table 11
Correlation (moderate) results for theL-moment procedure.
(a)
n=25
ParameterEstimate95% Bootstrap C.I.Standard errorRelative bias%η12=0.400.4100.408, 0.4120.00152.50η13=0.500.5090.507, 0.5110.00151.80η14=0.600.6100.608, 0.6110.00151.67η23=0.400.4070.404, 0.4090.00151.75η24=0.500.5070.505, 0.5090.00151.40η34=0.400.4080.406, 0.4100.00142.00(b)
n=1000
ParameterEstimate95% Bootstrap C.I.Standard errorRelative bias%η12=0.400.40130.4010, 0.40170.000220.32η13=0.500.50170.5014, 0.50200.000220.34η14=0.600.60160.6014, 0.60190.000220.27η23=0.400.39990.3995, 0.40020.00022—η24=0.500.50040.5001, 0.50070.000220.08η34=0.400.40030.3999, 0.40060.00021—


## 5. Discussion and Conclusion

One of the advantages thatL-moment ratios have over conventional moment-based estimators is that they can be far less biased when sampling is from distributions with more severe departures from normality [10, 19]. And inspection of the simulation results in Tables 6 and 7 clearly indicates that this is the case. That is, the superiority that estimates of L-moment ratios (τ3,τ4) have over their corresponding conventional moment-based counterparts (α3,α4) is obvious. For example, with samples of size n=25, the estimates of skew and kurtosis for Distribution 1 were, on average, only 32.4% and 4.2% of their associated population parameters, whereas the estimates of L-skew and L-kurtosis were 87.9% and 96.10% of their respective parameters. It is also evident from comparing Tables 6 and 7 that L-skew and L-kurtosis are more efficient estimators as their standard errors are significantly smaller than the conventional-moment-based estimators of skew and kurtosis.Presented in Tables8, 9, 10, and 11 are the results associated with the conventional Pearson and L-correlations. Overall inspection of these tables indicates that the L-correlation is superior to the Pearson correlation in terms of relative bias. For example, for moderate correlations (n=25), the relative bias for the two heavy-tailed distributions (i.e., distributions 1 and 2) was 6% for the Pearson correlation compared with only 2.5% for the L-correlation. It is also noted that the variability of the L-correlation appears to be more stable than that of the Pearson correlation both within and across the different conditions.In summary, the newL-moment-based procedure is an attractive alternative to the traditional conventional-moment-based procedure. In particular, the L-moment-based procedure has distinct advantages when distributions with large departures from normality are used. Finally, we note that Mathematica Version 8.0.1 [21] source code is available from the authors for implementing both the conventional and new L-moment-based procedures.



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*Source: 980153-2012-02-12.xml*