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Journal of Probability and Statistics
Volume 2011 (2011), Article ID 497463, 22 pages
http://dx.doi.org/10.1155/2011/497463
Research Article

A Characterization of Power Method Transformations through L-Moments

Section on Statistics and Measurement, Department EPSE, 222-J Wham Building, Mail Code 4618, Southern Illinois University Carbondale, Carbondale, IL 62901, USA

Received 4 July 2010; Revised 27 December 2010; Accepted 20 January 2011

Academic Editor: Rongling Wu

Copyright © 2011 Todd C. Headrick. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Power method polynomial transformations are commonly used for simulating continuous nonnormal distributions with specified moments. However, conventional moment-based estimators can (a) be substantially biased, (b) have high variance, or (c) be influenced by outliers. In view of these concerns, a characterization of power method transformations by L-moments is introduced. Specifically, systems of equations are derived for determining coefficients for specified L-moment ratios, which are associated with standard normal and standard logistic-based polynomials of order five and three. Boundaries for L-moment ratios are also derived, and closed-formed formulae are provided for determining if a power method distribution has a valid probability density function. It is demonstrated that L-moment estimators are nearly unbiased and have relatively small variance in the context of the power method. Examples of fitting power method distributions to theoretical and empirical distributions based on the method of L-moments are also provided.

1. Introduction

The power method (see Fleishman [1], Headrick [2], and [37]) is a traditional procedure used for simulating continuous nonnormal distributions. Some applications of the power method have included such topics as ANOVA [810], asset pricing theories [11], business-cycle features [12], cluster analysis [13], item parameter estimation [14], item response theory [15, 16], factor analysis [1719], price risk [20], structural equation models [2124], and toxicology [25].

In the context of univariate data generation and for the purposes considered herein, the power method can be generally summarized by the polynomial transformation as in Headrick [2, equation  (2.7)] 𝑝(𝑊)=𝑚𝑖=1𝑐𝑖𝑊𝑖1,(1.1) where 𝑊 can be either a standard normal (𝑍) or a standard logistic (L) random variable with probability density function (pdf) and cumulative distribution function (cdf) 𝑓𝑊𝑓(𝑤)=𝑊=𝑍(𝑤=𝑧)=𝜙(𝑧)=(2𝜋)(1/2)exp𝑧22,𝑓𝑊=𝐿(𝑤=𝑙)=𝜋/3exp𝜋/3𝑙1+exp𝜋/3𝑙2,𝐹𝑊𝐹(𝑤)=𝑊=𝑍(𝑤=𝑧)=Φ(𝑧)=𝑧𝐹𝜙(𝑢)𝑑𝑢,<𝑧<+,𝑊=𝐿1(𝑤=𝑙)=1+exp𝜋/3𝑙,<𝑙<+.(1.2) Setting 𝑚=4 (or 𝑚=6) in (1.1) gives the Fleishman [1] (or Headrick [3]) class of distributions associated with standard normal-based polynomials. The shape of 𝑝(𝑊) in (1.1) is contingent on the coefficients 𝑐𝑖, which are determined by moment-matching techniques. For example, see Headrick [2, equations  (2.18)–(2.21);  (2.22)–(2.25)] for determining the coefficients associated with standard normal orstandard logistic-based polynomials for 𝑚=4 in (1.1).

The pdf and cdf associated with 𝑝(𝑊) in (1.1) are given in parametric form (2) as in Headrick [2, equations  (2.12) and (2.13)] 𝑓𝑝(𝑊)(𝑝(𝑤))=𝑓𝑝(𝑊)(𝑝(𝑥,𝑦))=𝑓𝑝(𝑊)𝑓𝑝(𝑤),𝑊(𝑤)𝑝𝐹(𝑤),(1.3)𝑝(𝑊)(𝑝(𝑤))=𝐹𝑝(𝑊)(𝑝(𝑥,𝑦))=𝐹𝑝(𝑊)𝑝(𝑤),𝐹𝑊,(𝑤)(1.4) where the derivative 𝑝(𝑤)>0 in (1.3), that is, the polynomial 𝑝 is a strictly increasing monotonic function and where 𝑓𝑊(𝑤) and 𝐹𝑊(𝑤) are the pdf and cdf associated with the random variable 𝑊 in (1.2). To demonstrate, Figure 1 gives the graphs of a pdf and cdf associated with a standard normal-based power method distribution used in a Monte Carlo study by Berkovits et al. [8] (and, similarly, Enders [21] and Olsson et al. [24]).

fig1
Figure 1: Standard normal-based power method pdf and cdf. The coefficients used in (1.3) and (1.4) are 𝑐1=0.252299, 𝑐2=0.418610, 𝑐3=0.252299, and 𝑐4=0.147593 and are associated with the parameters in Tables 1 and 2.

The graphs in Figure 1 and numerical solutions for the coefficients 𝑐𝑖, which are based on the shape parameters in Table 1, were obtained using (1.3), (1.4), and the software package developed by Headrick et al. [26]. Note that the parameters in Table 1, 𝛾3 (skew), 𝛾4 (kurtosis), 𝛾5, and 𝛾6, are standardized cumulants and are scaled such that the normal distribution would have values of 𝛾3=𝛾4=𝛾5=𝛾6=0.

tab1
Table 1: Conventional moment-based parameters (𝛾) and their estimates (𝑔) for the pdf in Figure 1. Each estimate (Est.), and bootstrap confidence interval (C.I.) was based on resampling 25,000 statistics. Each sample statistic was based on a sample size of 𝑛=200.

Conventional moment-based estimators, such as 𝑔3,,𝑔6 in Table 1, have unfavorable attributes to the extent that they can be substantially biased, have high variance, or can be influenced by outliers and thus may not be representative of the true population parameters (e.g., see [27, 28]). Some of these attributes are exemplified in Table 1 as the estimates of 𝑔3,,𝑔6 and their respective bootstrap confidence intervals attenuate below the population parameters of 𝛾3,,𝛾6 with increased bias and variance as the order of the estimate increases. That is, on average, the estimates of 𝑔3,,𝑔6 are only 83.67%, 63.81%, 38.35%, and 21.38% of their associated population parameters, respectively.

The estimates of 𝑔3,,𝑔6 in Table 1 were calculated based on samples of size 𝑛=200 and Fisher’s 𝑘-statistics (see, e.g., [29, pages 299-300]), that is, the formulae currently used by most commercial software packages such as SAS, SPSS (PASW), Minitab, and so forth, for computing values of skew and kurtosis. Thus, it should also be pointed out that these estimates (𝑔3,,𝑔6) have another undesirable property of being algebraically bounded based on the sample size, that is, |𝑔3|𝑛1/2, 𝑔4𝑛, |𝑔5|𝑛3/2, 𝑔6𝑛2. As such, if a researcher was using a value of kurtosis in a Monte Carlo study, such as 𝛾4=21 in Table 1, and drawing samples of size 𝑛=15, then the largest sample estimate of kurtosis possible is 𝑔4=15 or 71.43% of the parameter.

The method of L-moments is an attractive alternative to conventional method of moments in terms of describing theoretical or empirical probability distributions, estimating parameters, and hypothesis testing (see [27, 28, 30]). More specifically, the first four L-moments are analogous to conventional moments as they describe the location, scale, skew, and kurtosis of a data set. However, L-moments have demonstrated to be superior to conventional moments to the extent that they (a) exist whenever the mean of the distribution exists, (b) only require that a distribution have finite variance for their standard errors to be finite, (c) are nearly unbiased for all sample sizes and distributions, (d) do not suffer from the deleterious effects of sampling variability, (e) are more robust in the presence of outliers, and (f) are not algebraically bounded based on sample size (see [27, 28, 3033]). Further, it has been demonstrated that there are conditions where the method of L-moments can also yield more accurate and efficient parameter estimates than the method of maximum likelihood when sample sizes are small to moderate (see [27, 3437]). Other advances have also been made. For example, Elamir and Seheult (see [38, 39]) introduced trimmed L-moments and derived expressions for the exact variances and covariances of sample L-moments. Further, Necir and Meraghni [40] demonstrated that L-moments and trimmed L-moments are useful for estimating L-functionals in the context of heavy-tailed distributions.

Estimates of L-moments and L-moment ratios are based on linear combinations of order statistics unlike conventional moments that are based on raising the data to successive powers which in turn gives greater weight to data points located farther away from the mean and thus may result with estimates (𝑔𝑗) having substantial bias and (or) variance. For example, the L-moment ratio estimates (𝑡𝑗) in Table 2 are relatively much closer to their respective population parameters (𝜏𝑗) with smaller variance than their corresponding conventional moment-based analogs (𝑔𝑗) in Table 1. Specifically, the ratios (𝑡𝑗) are, on average, 99.03%, 99.55%, 98.35%, and 99.29% of their respective parameters.

tab2
Table 2: L-moment ratios (𝜏) and their estimates (𝑡) for the pdf in Figure 1. Each estimate and bootstrap confidence interval (C.I.) was based on resampling 25,000 statistics. Each sample statistic was based on a sample size of 𝑛=200.

In view of the above, the present aim is to obviate the problems associated with conventional moments in the context of power method transformations of the form in (1.1) by characterizing these transformations through L-moments. The focus is on standard normal and standard logistic-based polynomials. In Section 2, the essential requisite information associated with L-moments for theoretical and empirical distributions is provided as well as the derivations for the systems of equations for computing polynomial coefficients. Closed-formed formulae are provided for determining if any particular polynomial has a valid pdf. Further, the boundary regions for valid pdfs are derived and graphed for polynomials of order three and for the symmetric case associated with polynomials of order five. In Section 3, conventional moments and L-moments are compared in terms of estimation and distribution fitting to demonstrate the superior characteristics that L-moments have in the context of the power method. In Section 4, third- and fifth-order standard normal and logistic-based polynomials are compared in terms of their upper and lower L-moment ratio boundary points.

2. Methodology

2.1. Preliminaries

L-moments are defined as linear combinations of probability weighted moments (PWMs). For a continuous theoretical distribution with a cdf denoted as 𝐹(𝑥), the PWMs can be generally defined as in Hosking [27] 𝛽𝑟=𝑥{𝐹(𝑥)}𝑟𝑑𝐹(𝑥),(2.1) where 𝑟=0,1,2,. The L-moments can be determined by summing the PWMs as 𝜆𝑟+1=𝑟𝑖=0𝑝(𝑟,𝑖)𝛽𝑖,(2.2) where 𝑝(𝑟,𝑖)=(1)𝑟𝑖(𝑟+𝑖)!(𝑖!)2(𝑟𝑖)!(2.3) are coefficients from shifted orthogonal Legendre polynomials. Specifically, the first six L-moments based on (2.1)–(2.3) are expressed as 𝜆1=𝛽0,𝜆2=2𝛽1𝛽0,𝜆3=6𝛽26𝛽1+𝛽0,𝜆4=20𝛽330𝛽2+12𝛽1𝛽0,𝜆5=70𝛽4140𝛽3+90𝛽220𝛽1+𝛽0,𝜆6=252𝛽5630𝛽4+560𝛽3210𝛽2+30𝛽1𝛽0.(2.4)

Analogous to conventional moment theory, the values of 𝜆1 and 𝜆2 are parameters associated with the location and scale of the distribution. More specifically, the L-mean (𝜆1) is the usual arithmetic mean, and L-scale (𝜆20) is one-half of Gini’s coefficient of mean difference (see, e.g., [29, pages 47-48]). Higher-order L-moments are transformed to dimensionless indices referred to as L-moment ratios defined as 𝜏𝑟+1=𝜆𝑟+1/𝜆2 for 𝑟2. In general, L-moment ratios are bounded such that |𝜏𝑟+1|<1 as is the index of L-skew (𝜏3) where a symmetric distribution implies that 𝜏2𝑟1=0. Smaller boundaries can be found for specific cases. For example, in the context of continuous distributions, L-kurtosis (𝜏4) and 𝜏6 have boundaries of (see Jones [41]) 5𝜏2314<𝜏41<1,2542𝜏2414𝜏43<𝜏6<1,(2.5) which indicate that 𝜏4 and 𝜏6 have lower bounds of 1/4 and 1/6, respectively. An example of a set of computed L-moments (𝜆1,𝜆2) and L-moment ratios (𝜏3,,𝜏6) based on (2.1) through (2.4) is provided in the first column of Table 3 for a Beta distribution.

tab3
Table 3: L-moment parameters (𝜆,𝜏) and percentiles for the Beta (𝛼=5,𝛽=4) distribution and power method approximations in Figure 3.

Empirical L-moments for a set of data of size 𝑛 are linear combinations of the sample order statistics 𝑋1𝑛𝑋2𝑛𝑋𝑛𝑛. The unbiased estimates of the PWMs are 𝑏𝑟=1𝑛𝑛𝑖=𝑟+1(𝑖1)(𝑖2)(𝑖𝑟)𝑋(𝑛1)(𝑛2)(𝑛𝑟)𝑖𝑛(2.6) for 𝑟=0,1,2, and where 𝑏0 is the sample mean. The sample L-moments are obtained by substituting 𝑏𝑟 in place of 𝛽𝑟 in (2.4). The notations used for sample L-moments and L-moment ratios are 𝑡𝑟+1=𝑟+1/2 for 𝑟2.

2.2. L-Moments for Standard Normal Polynomial Transformations

Using (1.1)-(1.2) and (2.1), the PWMs for power method polynomials based on the standard normal distribution are 𝛽𝑟=𝑚𝑖=1𝑐𝑖𝑧𝑖1{Φ(𝑧)}𝑟𝜙(𝑧)𝑑𝑧.(2.7)

Integrating (2.7) for 𝛽𝑟=0,,5 and subsequently substituting these PWMs into (2.4) (and after several tedious manipulations) and simplifying yields the following system of equations for fifth-order polynomials (𝑚=6): 𝜆1=𝑐1+𝑐3+3𝑐5,𝜆(2.8)2=Δ2=4𝑐2+10𝑐4+43𝑐6/2𝜋2𝜏,(2.9)3=6𝑐3+26𝑐5𝜏3Δ𝜋,(2.10)4=202𝛿1𝑐2+𝛿2𝑐4+𝛿3𝑐6Δ𝜋3/232𝜏,(2.11)5=845𝛿4𝑐3+𝛿5𝑐55𝜋33𝑐3+13𝑐53Δ𝜋2𝜏,(2.12)6=15Δ𝜋5/2560𝜋2𝛿1𝑐2+𝛿2𝑐4+𝛿3𝑐6+13443𝛿6𝑐2+𝛿7𝑐4+𝛿8𝑐68Δ𝜋5/2,(2.13) where Δ=(4𝑐2+10𝑐4+43𝑐6)/(2𝜋) is the coefficient of mean difference. The analytical expressions for the constants 𝛿𝑖=1,,8 in (2.11)–(2.13) are given in Appendix A.

The derivations above yield a system of six equations (2.8)–(2.13) expressed in terms of six real variables 𝑐1,,𝑐6. The first two equations of this system can be standardized by setting (2.8) to zero and (2.9) to 1/𝜋. The next four equations, (2.10)–(2.13), are set to the desired values of 𝜏3,,𝜏6. Simultaneously solving this system of equations yields the solutions of 𝑐1,,𝑐6. The coefficients are then substituted into (1.1) to generate 𝑝(𝑊) which has zero mean (𝜆1=0), one-half of the coefficient of mean difference for the unit-normal distribution (𝜆2=1/𝜋 ), and the specified values of 𝜏3,,𝜏6. If the negatives of 𝜏3 and 𝜏5 are desired, then inspection of (2.8)–(2.13) indicates that all that is needed are simultaneous sign reversals between 𝑐1,𝑐3, and 𝑐5. These sign reversals will have no effect on 𝜏2,𝜏4, or 𝜏6. It is worthy to point out that it is not necessary to numerically solve the system of equations in (2.8)–(2.13) as the coefficients 𝑐1,,𝑐6 have unique solutions which can be determined by evaluating equations (A.9)–(A.14) in Appendix A. See Figures 2(a) and 2(b) for examples of standard normal-based fifth-order power method pdfs.

fig2
Figure 2: Standard normal and standard logistic-based power method pdfs.

Setting 𝑐5=𝑐6=0 in (2.8)–(2.11) and simplifying yields the system of equations for the smaller class of distributions associated with third-order polynomials (𝑚=4) in (1.1) as 𝜆1=𝑐1+𝑐3,𝜆(2.14)2=Δ2=4𝑐2+10𝑐4/2𝜋2𝜏,(2.15)3=23𝑐3𝜏Δ𝜋,(2.16)4=202𝛿1𝑐2+𝛿2𝑐4Δ𝜋3/232,(2.17) where the solutions for the coefficients in (2.14)–(2.17) are 𝑐1=𝑐3=𝜏3𝜋3,𝑐2=16𝛿2+23+2𝜏4𝜋85𝛿12𝛿2,𝑐4=40𝛿123+2𝜏4𝜋205𝛿12𝛿2.(2.18)

Examples of a fifth-order (𝑐1,,𝑐6) and a third-order (𝑐1,,𝑐4) power method distribution are given in Figure 3, both of which provide an approximation of the Beta (𝛼=5,𝛽=4) distribution. The coefficients associated with the two polynomial approximations in Figure 3 were used in (1.4) to compute the percentiles in Table 3 for the power method pdfs. Inspection of the graphs in Figure 3 and the percentiles in Table 3 indicate that the fifth-order polynomial provides a much more accurate approximation of the Beta distribution than the third-order polynomial. The reason for this is because the fifth-order system could produce a valid pdf that is based on an exact match with the Beta distribution’s L-moment ratios 𝜏3,,𝜏6, whereas the third-order system was unable to produce a valid pdf that has an exact match with 𝜏4,𝜏5, and 𝜏6. Thus, it is important to consider the boundary conditions for valid power method pdfs.

fig3
Figure 3: Third- and fifth-order standard normal-based polynomial approximations (dashed lines) of a Beta (𝛼=5, 𝛽=4) distribution. The L-moment ratios for this Beta distribution are given in Table 3. The fifth-order polynomial coefficients are 𝑐1=.02697, 𝑐2=1.071, 𝑐3=.0297, 𝑐4=.0312, 𝑐5=.0009, and 𝑐6=.00065. The third-order polynomial coefficients are 𝑐1=.02576, 𝑐2=0.9993, 𝑐3=.02576, and 𝑐4=.00027.
2.3. L-Moment Boundaries for Standard Normal Polynomial Transformations

The restriction that 𝑝(𝑤)>0 in (1.3) implies that a set of solved coefficients may not necessarily produce a valid pdf. To determine if a third-order polynomial produces a valid pdf, we first set the quadratic equation 𝑝(𝑤)=0 and subsequently solve for 𝑤 as 𝑤=𝑐3±𝑐233𝑐2𝑐41/23𝑐4.(2.19)

A set of solved coefficients will produce a valid pdf if the discriminant 𝑐233𝑐2𝑐4 in (2.19) is negative. That is, the complex solutions for 𝑤 must have nonzero imaginary parts. As such, setting 𝑐23=3𝑐2𝑐4 will yield the point where the discriminant vanishes and thus real-valued solutions exist to 𝑝(𝑤)=0.

Standardizing (2.15), by setting the coefficient of mean difference to 1/𝜋, and solving for 𝑐4 give 𝑐4=252𝑐25.(2.20)

Substituting the right-hand side of (2.20) into (2.16) and (2.17) and setting 𝑐3=(3𝑐2𝑐4)1/2, because we only need to consider positive values of L-skew, gives 𝜏3=325𝜋𝑐21𝑐2,𝜏(2.21)4=225𝛿1𝑐2+2𝛿22𝛿2𝑐2𝜋32.(2.22)

Inspection of (2.21) indicates that for real values of 𝜏3 to exist then we must have 𝑐2[0,1] and thus from (2.20) 𝑐4[0,0.4]. Using (2.21) and (2.22), the graph of the region for valid third-order power method pdfs is given in Figure 4 along with the minimum and maximum values of 𝜏3 and 𝜏4. In summary, a valid standardized nonnormal third-order pdf will have the properties of (a) 0<𝑐2<1, (b) 0<𝑐4<2/5, and (c) 𝑐233𝑐2𝑐4<0.

497463.fig.004
Figure 4: Boundary region and values of L-skew |𝜏3| and L-kurtosis 𝜏4 for third-order standard normal power method pdfs. For asymmetric distributions, valid pdfs exist in the region inside the graphed boundary. For symmetric distributions, valid pdfs exist for 𝜏4 between 𝑎4<𝜏4<𝑏4. The lower boundary point (𝑎3,4) denotes the normal distribution. Note that 𝑎3 and 𝑎4 are lower limits and 𝑏3 and 𝑏4 are upper limits.

In terms of fifth-order polynomials, the formulae that solve the (quartic) equation 𝑝(𝑤)=0 must also have nonzero imaginary parts to ensure a valid pdf. Specifically, the closed-formed expressions for evaluating pdfs associated with symmetric distributions are ±𝑤=±9𝑐2420𝑐2𝑐61/23𝑐410𝑐61/2,±𝑤=9𝑐2420𝑐2𝑐61/23𝑐410𝑐61/2,(2.23) and for asymmetric distributions the expressions are 1𝑤=±2𝐴1/2±12(𝐵±𝐶)1/2𝑐55𝑐6,1𝑤=±2𝐴1/212(𝐵±𝐶)1/2𝑐55𝑐6.(2.24) The expressions for 𝐴, 𝐵, and 𝐶 in (2.24) are given in Appendix B.

The boundary for the larger class of symmetric fifth-order pdfs (i.e., 𝜏3=𝜏5=0;𝑐1=𝑐3=𝑐5=0) can be viewed by making use of (2.23). Specifically, the graph of the region for valid symmetric pdfs is given in Figure 5 along with the minimum and maximum values of 𝜏4 and 𝜏6. The elliptical graph in Figure 5 consists of four separate segments where 𝜏4 in (2.11) and 𝜏6 in (2.13) are expressed solely as functions of 𝑐2. See Appendix C for further details.

497463.fig.005
Figure 5: Boundary region and values of L-kurtosis 𝜏4 and 𝜏6 for symmetric fifth-order standard normal-based power method pdfs. Valid pdfs exist inside the elliptical graph. The point at the origin represents the regular uniform distribution (𝜏4=𝜏6=0). Note that 𝑎4 and 𝑎6 are lower limits and 𝑏4 and 𝑏6 are upper limits.
2.4. L-Moments and Boundaries for Standard Logistic Polynomial Transformations

The method for deriving the system of equations for power method transformations based on the standard logistic distribution is similar to the derivation of the system associated with (2.8)–(2.13). However, the derivation is more straight forward because the cdf for the logistic distribution is available in closed form. As such, the system of equations for fifth-order polynomials in (1.1) is 𝜆1=𝑐1+𝑐3+21𝑐55,𝜆(2.25)2=Δ2=23𝑐2+3𝑐4+21𝑐6/𝜋2𝜏,(2.26)3=12𝑐3+72𝑐5Δ𝜋2𝜏,(2.27)4=2390𝑐4+900𝑐6+𝜋2𝑐2+3𝑐4+21𝑐66Δ𝜋3𝜏,(2.28)5=5𝜋2𝑐3+252𝑐5+30𝜋2𝑐5Δ𝜋4𝜏,(2.29)6=2Δ𝜋5+23315𝜋2𝑐4+11340𝑐6+3150𝜋2𝑐630Δ𝜋5,(2.30) where Δ=23(𝑐2+3𝑐4+21𝑐6)/𝜋 is the coefficient of mean difference and for standardized distributions the solutions for the coefficients 𝑐1,,𝑐6 are given in Appendix D. The system of equations for the smaller family of distributions based on third-order polynomials can be obtained by setting 𝑐5=𝑐6=0 in (2.25)–(2.28), and for standardized distributions, the solutions for the coefficients are 𝑐1=𝑐3𝜏=3𝜋23,𝑐2=30+𝜋216𝜏4,𝑐304=𝜋26𝜏41.90(2.31)

The derivation of the L-skew (𝜏3) and L-kurtosis (𝜏4) boundary for third-order polynomials is also analogous to the steps taken in Section 2.3. As such, the graph of the boundary for valid third-order power method pdfs is given in Figure 6 along with the minimum and maximum values of 𝜏3 and 𝜏4. Similarly, a valid standardized nonlogistic third-order pdf will have the properties of (a) 0<𝑐2<1, (b) 0<𝑐4<1/3, and (c) 𝑐233𝑐2𝑐4<0. In terms of fifth-order polynomials, the boundary region for symmetric pdfs and minimum and maximum values of 𝜏4 and 𝜏6 are given in Figure 7. The elliptical graph in Figure 7 also consists of four segments. See Appendix E for further details. Note that the expressions in (2.23) and (2.24) are also used to determine whether fifth-order pdfs are valid or not. An example of a logistic-based power method pdf is given in Figure 2(c).

497463.fig.006
Figure 6: Boundary region and values of L-skew |𝜏3| and L-kurtosis 𝜏4 for third-order standard logistic power method pdfs. For asymmetric distributions, valid pdfs exist in the region inside the graphed boundary. For symmetric distributions, valid pdfs exist for 𝜏4 between 𝑎4<𝜏4<𝑏4. The lower boundary point (𝑎3,4) represents the standard logistic distribution. Note that 𝑎3 and 𝑎4 are lower limits and 𝑏3 and 𝑏4 are upper limits.
497463.fig.007
Figure 7: Boundary region and values of L-kurtosis 𝜏4 and 𝜏6 for symmetric fifth-order standard logistic-based power method pdfs. Valid pdfs exist inside the elliptical graph. Note that 𝑎4 and 𝑎6 are lower limits and 𝑏4 and 𝑏6 are upper limits.

3. A Comparison of Conventional Moments and L-Moments

3.1. Estimation

One of the advantages that sample L-moment ratios (𝑡3,,6) have overconventional moment-based estimates, such as skew (𝑔3) and kurtosis (𝑔4), is that 𝑡3,,6 are less biased (e.g., [28]). This advantage can be demonstrated in the context of the power method by considering the simulation results associated with the indices for the percentage of relative bias (RBias%) and standard error (SE) reported in Table 4.

tab4
Table 4: Overall average estimates (Est.), bootstrap confidence intervals (C.I.), standard errors (SE), and percentage of relative bias (RBias%) for the conventional moment-based statistics (𝑔) and L-moment ratios (𝑡).

Specifically, a Fortran algorithm was coded to generate twenty-five thousand independent samples of size 𝑛=50,100,1000, and the estimates of 𝑔3,,6 and 𝑡3,,6 were computed for each of the (3 × 25000) samples based on the parameters and coefficients listed in Table 4. The estimates of 𝑔3,,6 were computed based on Fisher’s 𝑘-statistics, and the estimates of 𝑡3,,6 were based on (2.6). Bootstrapped average estimates, confidence intervals, and SEs were obtained for 𝑔3,,6 and 𝑡3,,6 using twenty-five thousand resamples via the commercial software package Spotfire S+ [42]. The percentage of relative bias for each estimate was computed as RBias%,(𝑔𝑗)=100×(𝑔𝑗𝛾𝑗)/𝛾𝑗 and RBias%,(𝑡𝑗)=100×(𝑡𝑗𝜏𝑗)/𝜏𝑗.

The results in Table 4 demonstrate the substantial advantage that L-moment ratios have overconventional estimates in terms of both bias and error. These advantages are most pronounced in the context of smaller sample sizes and higher-order moments. For example, given a sample size of 𝑛=50, the conventional estimates of 𝑔3 (𝑔5) and 𝑔4 (𝑔6) generated in the simulation were, on average, 11.40% (56.50%) and 31.93% (78.36%) less than their respective parameters. On the other hand, the amounts of relative bias associated with the L-moment ratios were essentially negligible. Further, the SEs associated with 𝑡3,,6 are relatively much smaller than the corresponding SEs for 𝑔3,,6.

3.2. Distribution Fitting

Presented in Figure 8 are conventional moment and L-moment-based power method pdfs superimposed on a histogram of data from the Project Match Research Group studies (see [43, 44]). The data are associated with participants who were assigned to a Twelve-Step Facilitation (TSF) treatment condition for alcoholism. Specifically, these data are the total number of drinks per ninety-day period which were determined by each participant’s reports of daily standard ethanol drinks (one drink was equal to one-half of an ounce). Drinking was assessed approximately every three months, beginning at pretreatment (baseline), immediately following treatment, and at three-month intervals over a twelve-month posttreatment follow-up period (i.e., six, nine, twelve, and fifteen months posttreatment).

fig8
Figure 8: Histograms and fifth-order power method approximations for the TSF group. aThe value of 𝑔6 had to be increased to −0.28898 to ensure a valid pdf.

The sample estimates (𝑔3,6; 𝑡3,6) associated with Figure 8 were based on a sample size of 𝑛=311 participants. The estimates were also used to solve for the two sets of coefficients, which produced the power method pdfs based on (1.3). Note that the two polynomials were linearly transformed using the location and scale estimates (𝑚,𝑠;1,2) from the data. Visual inspection of the approximations in Figure 8 and the goodness of fit statistics given in Table 5 indicate that the L-moment pdf provides a more accurate fit to the actual data. The reason for this is partially attributed to the fact that the conventional moment-based power method pdf does not have an exact match with 𝑔6, whereas the L-moment pdf is based on an exact match with all of the sample estimates. Note also that the asymptotic P values in Table 5 are based on a chi-square distribution with degrees of freedom: 𝑑𝑓 = 13(classes) − 6(estimates) − 1(sample size) = 6.

tab5
Table 5: Chi-square goodness of fit statistics for the conventional (𝐶) moment and L-moment approximations for the TSF group data (𝑛=311) in Figure 8.

4. Discussion and Conclusion

This paper introduced L-moments in the context of standard normal and standard logistic-based power method polynomials. The boundaries were analytically derived for polynomials of order three and for symmetric pdfs associated with polynomials of order five. The lower boundary point associated with the standard normal third-order system is the standard normal distribution (i.e., 𝜏40.1226 and 𝜏60.0437). As such, it is worthy to point out that the larger family of pdfs for the standard normal fifth-order system includes the L-moment ratios associated with the regular uniform or Beta (𝛼=1,𝛽=1) distribution (i.e., 𝜏4=0 and 𝜏6=0). Further, the upper boundary of L-kurtosis for the third-order system of 𝜏4.5728 extends up to a boundary of 𝜏4.8233 for the fifth-order system which is a substantial increase. That is, in terms of conventional moments, this is equivalent to increasing kurtosis from 𝛾443 to 𝛾4833.

In terms of the third-order logistic-based family of pdfs, it is noted that maximum L-skew is 𝜏3.5513 whereas the maximum value of L-skew for the standard normal-based system is 𝜏3.5352. Again, this is actually a considerable difference—as these values correspond to values of conventional skew of 𝛾38.4 and 𝛾34, respectively. Similarly, the maximum values of L-kurtosis for the standard logistic-based and normal-based systems are 𝜏4.6733 and 𝜏4.5728 and are equivalent to values of conventional kurtosis of 𝛾4472 and 𝛾443, respectively.

Finally, we note that Mathematica ([45], Version  7.0) software for computing polynomial coefficients, cumulative probabilities, and fitting (graphing) power method pdfs to data are available on request.

Appendices

A. Analytical Expressions and Polynomial Coefficients (Normal)

The expressions for 𝛿𝑖=1,,8 in (2.11)–(2.13) and the coefficients 𝑐𝑖=1,,6 in (2.8)–(2.13) are 𝛿1=3tan1223𝜋42𝛿=0.360451474758732546420622,(A.1)2=15tan122215𝜋82+14𝛿=1.151128686896831366051556,(A.2)3=129tan1242129𝜋162+77𝛿48=5.479020020323041540688357,(A.3)4=15tan153𝜋154𝛿=0.489313210040355820119964,(A.4)5=15tan153853tan1+1519𝜋453+14𝛿=2.370357243508208553853177,(A.5)6=5𝜋tan1243+583sin11325𝜋sin1[]1/3835𝜋2163+532𝜀1𝛿=0.261016893769534900409394,(A.6)7=25𝜋tan1283+2543sin113225𝜋sin11/34325𝜋2643+5𝜋86546+2534𝜀2𝛿=0.960807335426376575971986,(A.7)8=215𝜋tan12163+21583sin1132215𝜋sin11/383215𝜋21283+385𝜋966385486+21538𝜀3𝑐=4.908966006122127514487203,(A.8)2=403202𝛿3𝛿7𝛿2𝛿850443𝛿710𝛿83+2𝜏4𝜋+5643𝛿210𝛿3𝜋227+28𝜏4+8𝜏610080243𝛿2𝛿610𝛿3𝛿643𝛿1𝛿7+4𝛿3𝛿7+10𝛿1𝛿84𝛿2𝛿8,𝑐(A.9)3=140𝜏3𝛿53𝜋+13𝜋3/252𝜏5+5𝜏314013𝛿43𝛿5,𝑐(A.10)4=403202𝛿3𝛿6𝛿1𝛿8+50443𝛿64𝛿83+2𝜏4𝜋+5643𝛿1+4𝛿3𝜋227+28𝜏4+8𝜏610080243𝛿2𝛿610𝛿3𝛿643𝛿1𝛿7+4𝛿3𝛿7+10𝛿1𝛿84𝛿2𝛿8,𝑐(A.11)5=28𝜏3𝛿43𝜋3𝜋3/22𝜏5+5𝜏3/5364𝛿484𝛿5,𝑐(A.12)6=𝛿201602𝛿6𝛿1𝛿725225𝛿62𝛿73+2𝜏4𝜋+535𝛿12𝛿2𝜋227+28𝜏4+8𝜏6504043𝛿2𝛿610𝛿3𝛿643𝛿1𝛿7+4𝛿3𝛿7+10𝛿1𝛿84𝛿2𝛿8,𝑐(A.13)1=𝑐33𝑐5.(A.14) Equations (A.6)–(A.8) were derived by adapting the techniques used by Renner [46] and where 𝜀1, 𝜀2, and 𝜀3 are based on the standard recurrence relation for sec𝑝𝑢𝑑𝑢. The coefficients in (A.9)–(A.14) are for standardized distributions (i.e., 𝜆1=0,𝜆2=1/𝜋).

B. Expressions for (2.24)

The expressions for 𝐴, 𝐵, and 𝐶 in (2.24) are 𝐴=4𝑐2525𝑐26+𝐸1/2+𝐷1/315𝑐621/3+21/3𝐹5𝑐6𝐸1/2+𝐷1/32𝑐45𝑐6,𝐵=8𝑐2525𝑐26𝐸1/2+𝐷1/315𝑐621/321/3𝐹5𝑐6𝐸1/2+𝐷1/34𝑐45𝑐6,𝐶=𝐴(1/2)48𝑐4𝑐5100𝑐2616𝑐320𝑐664𝑐35500𝑐36,𝐷=54𝑐34216𝑐3𝑐4𝑐5+432𝑐2𝑐25+540𝑐23𝑐61080𝑐2𝑐4𝑐6,𝐸=𝐷249𝑐2424𝑐3𝑐5+60𝑐2𝑐63,𝐹=3𝑐248𝑐3𝑐5+20𝑐2𝑐6.(B.1)

C. Symmetric Distributions Boundary (Normal)

The elliptical graph in Figure 5 is based on (2.11) and (2.13) and setting 𝑐4=(44𝑐243𝑐6)/10 from (2.9) and then setting 𝑐6 as follows: Segment1:0𝑐21,𝑐64=137𝑐2387+16641405387𝑐2262𝑐22,16641Segment2:1𝑐2387262,𝑐64=137𝑐2387+16641405387𝑐2262𝑐22,16641Segment3:0𝑐21,𝑐64=137𝑐238716641405387𝑐2262𝑐22,16641Segment4:1𝑐2387262,𝑐64=137𝑐238716641405387𝑐2262𝑐22.16641(C.1)

D. Polynomial Coefficients (Logistic)

The coefficients for the standard logistic-based system in (2.25)–(2.30) are 𝑐2=840+28𝜋216𝜏4+𝜋4114𝜏4+20𝜏6,𝑐8403=42𝜏3𝜋+𝜋35𝜏312𝜏5843,𝑐4=𝜋2426𝜏41+5𝜋214𝜏4120𝜏6,𝑐37805=𝜋312𝜏55𝜏35043,𝑐6=𝜋4114𝜏4+20𝜏6,𝑐75601=𝑐321𝑐55.(D.1)

E. Symmetric Distributions Boundary (Logistic)

The elliptical graph in Figure 7 is based on (2.28) and (2.30) and setting 𝑐4=(1𝑐221𝑐6)/3 from (2.26) and then setting 𝑐6 as follows: Segment1:0𝑐21,𝑐6=2111𝑐2+2441521𝑐216𝑐22,441Segment2:1𝑐22116,𝑐6=2111𝑐2+2441521𝑐216𝑐22,441Segment3:0𝑐21,𝑐6=2111𝑐22441521𝑐216𝑐22,441Segment4:1𝑐22116,𝑐6=2111𝑐22441521𝑐216𝑐22.441(E.1)

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