International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 725754 | https://doi.org/10.5402/2012/725754

Todd C. Headrick, Mohan D. Pant, "A Doubling Method for the Generalized Lambda Distribution", International Scholarly Research Notices, vol. 2012, Article ID 725754, 19 pages, 2012. https://doi.org/10.5402/2012/725754

A Doubling Method for the Generalized Lambda Distribution

Academic Editor: H. C. So
Received14 Jan 2012
Accepted06 Feb 2012
Published07 May 2012

Abstract

This paper introduces a new family of generalized lambda distributions (GLDs) based on a method of doubling symmetric GLDs. The focus of the development is in the context of L-moments and L-correlation theory. As such, included is the development of a procedure for specifying double GLDs 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 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.

1. Introduction

The conventional moment-based family of generalized lambda distributions (GLDs) is often used in various applied mathematics contexts to model and describe data by a single functional form [1, page 5]. Some examples include modeling non-log-normal security price distributions [2], biological and physical phenomena [3], and solar radiation data [4]. The GLD is also a popular tool for generating random variables for Monte Carlo or simulation studies. Some examples include studies in such areas as operations research [5], microarray research [6], and structural equation modeling [7].

The family of GLDs is based on the transformation [1, page 21], [9, 11], [10, page 127]๐‘ž(๐‘ข)=๐œ†1+๐‘ข๐œ†3โˆ’(1โˆ’๐‘ข)๐œ†4๐œ†2,(1.1) where ๐‘ข is uniformly distributed on the interval (0,1). The parameters ๐œ†1 and ๐œ†2 are location and scale parameters, while ๐œ†3 and ๐œ†4 are shape parameters that determine the skew and kurtosis of a GLD. The pdf and cdf associated with (1.1) can be expressed as [10, page 127]๐‘“๐‘ž(๐‘ข)(๐‘ž(๐‘ข))=๎‚ต1๐‘“(๐‘ข)=๐‘ž(๐‘ข),๐‘ž๎…ž๎‚ถ๐น(๐‘ข),(1.2)๐‘ž(๐‘ข)(๐‘ž(๐‘ข))=๐น(๐‘ข)=(๐‘ž(๐‘ข),๐‘ข),(1.3) where ๐‘“โˆถโ„œโ†ฆโ„œ2 and ๐นโˆถโ„œโ†ฆโ„œ2 are parametric forms of the pdf and cdf with the mappings ๐‘ขโ†ฆ(๐‘ฅ,๐‘ฆ) and ๐‘ขโ†ฆ(๐‘ฅ,๐‘ฃ) with ๐‘ฅ=๐‘ž(๐‘ข), ๐‘ฆ=1/๐‘žโ€ฒ(๐‘ข), ๐‘ฃ=๐‘ข, and where 1 and ๐‘ข are the regular uniform pdf and cdf, respectively. It is assumed that ๐‘ž๎…ž(๐‘ข)>0 in (1.2) to ensure a valid pdf that is, the transformation in (1.1) is strictly increasing. An essential requirement for a valid pdf is that ๐œ†2, ๐œ†3, ๐œ†4 in (1.1) all have the same sign [1, page 24]. For more specific details on the parameter space and conditions related to valid GLDs, see Karian and Dudewicz [1, pages 21โ€“47]. Provided in Figure 1 is an example of a valid GLD pdf based on (1.1) and (1.2).

Symmetric GLDs are produced for the case where ๐œ†3=๐œ†4 in (1.1) and where the mean (๐›ผ1), variance (๐›ผ22), skew (๐›ผ3), and kurtosis (๐›ผ4) can be determined from [8]๐›ผ1=๐œ†1,๐›ผ(1.4)22=๎€ท2/1+2๐œ†3๎€ธ๎€บโˆ’2๐›ฝ1+๐œ†3,1+๐œ†3๎€ป๐œ†22๐›ผ,(1.5)3๐›ผ=0,(1.6)4=๎€บ6๐›ฝ1+2๐œ†3,1+2๐œ†3๎€ป๎€บโˆ’4๐›ฝ1+๐œ†3,1+3๐œ†3๎€ป๎€บโˆ’4๐›ฝ1+3๐œ†3,1+๐œ†3๎€ป๎€ท+2/1+4๐œ†3๎€ธ๐œ†42.(1.7) Numerical solutions for ๐œ†2 and ๐œ†3 in (1.5) and (1.7) can be found in [1, Appendix B], which are associated with standardized GLDs (i.e., ๐›ผ1=๐œ†1=0, ๐›ผ22=1). Note that the term ๐›ฝ in (1.5) and (1.7) represents the complete beta function where the arguments cannot be negative. As such, for the ๐‘˜th moment to exist then we must have ๐œ†3>โˆ’1/๐‘˜ [8, 9, 11]. Thus, the condition ๐œ†3>โˆ’1/4 ensures that the first four moments exist.

We propose a new family of asymmetric GLDs based on a technique referred to herein as doubling symmetric GLDs. More specifically, a family of double GLDs can be created by selecting a pair of constants (๐œ†โ„’, ๐œ†โ„›) and transforming separately for ๐‘ขโ‰ค1/2 and ๐‘ขโ‰ฅ1/2 as follows: โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๎€ท๐‘ข๐‘ž(๐‘ข)=๐œ†โ„’โˆ’(1โˆ’๐‘ข)๐œ†โ„’๎€ธ๎€ท๐›ฟ๐œ†โ„’22โˆ’๐œ†โ„’๎€ธ1,for0<๐‘ขโ‰ค2,๎€ท๐‘ข๐œ†โ„›โˆ’(1โˆ’๐‘ข)๐œ†โ„›๎€ธ๎€ท๐›ฟ๐œ†โ„›22โˆ’๐œ†โ„›๎€ธ1,for2โ‰ค๐‘ข<1,(1.8) where ๐œ†โ„’ (๐œ†โ„›) is the nonzero shape parameter for the left (right) side of a distribution and ๐›ฟ is the (positive) parameter that determines the height of the double GLD at ๐‘ข=1/2. Provided in Figure 2 is an example of a double GLD pdf based on (1.2) and (1.8). Inspection of Figures 1 and 2 clearly indicates that these two GLDs are markedly different even though both distributions have the same values of skew (๐›ผ3=3) and kurtosis (๐›ผ4=65). Note that the values of ๐œ†โ„’ and ๐œ†โ„› in Figure 2 were determined based on the equations for ๐›ผ3 and ๐›ผ4 given in the appendix.

Conventional moment-based estimators (e.g., ๎๐›ผ3,4) have unfavorable attributes insofar as they can be substantially biased, have high variance, or can be influenced by outliers. For example, inspection of Figure 2 indicates, on average, that the estimates of ๎๐›ผ3,4 are only 67.70% and 14.70% of their associated population parameters. Note that each estimate of ๎๐›ผ3,4 in Figure 2 was calculated based on samples of size ๐‘›=250 and the formulae currently used by most commercial software packages such as SAS, SPSS, and Minitab, for computing skew and kurtosis.

๐ฟ-moment-based estimators, such as ๐ฟ-skew (๐œ3) and ๐ฟ-kurtosis (๐œ4), have been introduced to address the limitations associated with conventional moment-based estimators [12, 13]. Specifically, some of the advantages that ๐ฟ-moments 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 of ฬ‚๐œ3,4 in Figure 2 are relatively much closer to their respective parameters ๐œ3,4 with much smaller standard errors than their corresponding conventional moment-based analogs ๎๐›ผ3,4. More specifically, the estimates of ฬ‚๐œ3,4 that were simulated are, on average, 98.89% and 99.23% of their parameters.

In the context of multivariate data generation, the methodology has been developed for simulating symmetric (or asymmetric) GLDs based on (1.1) with specified Pearson correlation structures [2, 14]. This methodology is based on conventional product moments and the popular NORTA (NORmal To Anything, [15]) 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 GLDs. The purpose of the IC matrix is to adjust for the effect of the transformation in (1.1) such that the resulting GLDs have their specified skew, kurtosis, and Pearson correlation matrix.

Two additional limitations associated with the NORTA approach in this context are that solutions to an IC matrix may neither (a) exist in the range of [โ€“1, +1] as the absolute values of the ICs must be greater than (or equal to) their specified Pearson correlations nor (b) yield a positive definite IC matrix albeit the specified Pearson correlation matrix is positive definite [16]. Further, these two problems can be exacerbated when heavy tailed distributions are involved in the computation of ICs as functions performing numerical integration can more frequently either fail to converge or yield incorrect solutions. In contradistinction, it has been demonstrated in the context of the ๐ฟ-correlation that the limitations associated with the NORTA approach are less pronounced because the solution values of an IC matrix are in closer proximity to their specified (positive definite) ๐ฟ-correlation matrix [17].

In view of the above, the present aim is to derive the double GLD family of distributions based on (1.8) in the contexts of ๐ฟ-moment and ๐ฟ-correlation theory. Specifically, the purpose of this paper is to develop the methodology and a procedure for simulating double GLDs with specified ๐ฟ-moments and ๐ฟ-correlations. The primary advantages of the proposed procedure are that estimates of ๐ฟ-skew, ๐ฟ-kurtosis, and ๐ฟ-correlation are less biased and more efficient.

The remainder of this paper is organized into four sections. The next section provides a summary of univariate ๐ฟ-moment theory and the derivations of the system of equations and boundary conditions for generating double GLDs with specified values of ๐ฟ-skew and ๐ฟ-kurtosis. The section thereafter introduces the coefficient of ๐ฟ-correlation and the equation is subsequently derived for determining ICs for specified ๐ฟ-correlations between double GLDs. The steps for implementing the proposed ๐ฟ-moment procedure are subsequently described. A numerical example and results of a simulation are also provided to confirm the derivations and compare the new ๐ฟ-moment-based procedure with the conventional product-moment-based procedure. In the last section, the results of the simulation are discussed.

2. Methodology

2.1. Preliminaries

Let ๐‘‹1,โ€ฆ,๐‘‹๐‘—,โ€ฆ,๐‘‹๐‘› be ๐‘–๐‘–๐‘‘ random variables each with continuous pdf ๐‘“(๐‘ฅ), cdf ๐น(๐‘ฅ), order statistics denoted as ๐‘‹1โˆถ๐‘›โ‰คโ‹ฏโ‰ค๐‘‹๐‘—โˆถ๐‘›โ‰คโ‹ฏโ‰ค๐‘‹๐‘›โˆถ๐‘›, and ๐ฟ-moments defined in terms of either linear combinations of (a) expectations of order statistics or (b) probability weighted moments (๐›ฝ๐‘–). For the purposes considered herein, the first four ๐ฟ-moments associated with ๐‘‹๐‘—โˆถ๐‘› are expressed as [13, pages 20โ€“22]ฮ›1๎€บ๐‘‹=๐ธ1โˆถ1๎€ป=๐›ฝ0ฮ›,(2.1)2=12๐ธ๎€บ๐‘‹2โˆถ2โˆ’๐‘‹1โˆถ2๎€ป=2๐›ฝ1โˆ’๐›ฝ0ฮ›,(2.2)3=13๐ธ๎€บ๐‘‹3โˆถ3โˆ’2๐‘‹2โˆถ3+๐‘‹1โˆถ3๎€ป=6๐›ฝ2โˆ’6๐›ฝ1+๐›ฝ0ฮ›,(2.3)4=14๐ธ๎€บ๐‘‹4โˆถ4โˆ’3๐‘‹3โˆถ4+3๐‘‹2โˆถ4โˆ’๐‘‹1โˆถ4๎€ป=20๐›ฝ3โˆ’30๐›ฝ2+12๐›ฝ1โˆ’๐›ฝ0,(2.4) where the ๐›ฝ๐‘– are determined from๐›ฝ๐‘–=๎€œ๐‘ฅ{๐น(๐‘ฅ)}๐‘–๐‘“(๐‘ฅ)๐‘‘๐‘ฅ,(2.5) where ๐‘–=0,โ€ฆ,3. The coefficients associated with ๐›ฝ๐‘– in (2.5) are obtained from shifted orthogonal Legendre polynomials and are computed as shown in [13, page 20].

The ๐ฟ-moments ฮ›1 and ฮ›2 in (2.1) and (2.2) are measures of location and scale and are the arithmetic mean and one-half the coefficient of mean difference, respectively. Higher-order ๐ฟ-moments are transformed to dimensionless quantities referred to as ๐ฟ-moment ratios defined as ๐œ๐‘Ÿ=ฮ›๐‘Ÿ/ฮ›2 for ๐‘Ÿโ‰ฅ3, and where ๐œ3 and ๐œ4 are the analogs to the conventional measures of skew and kurtosis. In general, ๐ฟ-moment ratios are bounded in the interval โˆ’1<๐œ๐‘Ÿ<1 as is the index of ๐ฟ-skew (๐œ3) where a symmetric distribution implies that all ๐ฟ-moment ratios with odd subscripts are zero. Other smaller boundaries can be found for more specific cases. For example, the index of ๐ฟ-kurtosis (๐œ4) has the boundary condition for continuous distributions of [18]5๐œ23โˆ’14<๐œ4<1.(2.6)

2.2. ๐ฟ-Moments for Double GLDs

The family of double GLDs, associated with (1.8), based on ๐ฟ-moments is less restrictive than the family based on conventional moments insofar as we may consider the nonzero parameters of ๐œ†โ„’,๐œ†โ„›>โˆ’1 for any distribution with finite ๐‘˜ order ๐ฟ-moments rather than ๐œ†โ„’,๐œ†โ„›>โˆ’1/๐‘˜ for the ๐‘˜th-order conventional moment to exist. This advantage is attributed to Hoskingโ€™s Theorem 1 [12] which states that if the mean (ฮ›1) exists, then all other ๐ฟ-moments will have finite expectations.

As such, the family of double GLDs can be derived in the context of ๐ฟ-moments by defining the probability weighted moments based on (2.5) in terms of ๐‘ž(๐‘ข) in (1.1) and the regular uniform pdf and cdf as๐›ฝ๐‘–=๎€œ01/2๐‘ž๎€ท๐‘ข,๐œ†โ„’๎€ธ๐‘ข,๐›ฟ๐‘–๎€œ๐‘‘๐‘ข+11/2๐‘ž๎€ท๐‘ข,๐œ†โ„›๎€ธ๐‘ข,๐›ฟ๐‘–๐‘‘๐‘ข.(2.7) Integrating (2.7) for ๐‘–=0,1,2,3 and using (2.1)โ€“(2.4) yields ฮ›1=1โˆ’2๐œ†โ„’๐›ฟ๎€ท1+๐œ†โ„’๎€ธ4๐œ†โ„’+2๐œ†โ„›โˆ’1๐›ฟ๎€ท1+๐œ†โ„›๎€ธ4๐œ†โ„›ฮ›,(2.8)2=2๐œ†โ„’โˆ’2๐›ฟ๎€ท2+3๐œ†โ„’+๐œ†2โ„’๎€ธ+2๐œ†โ„›โˆ’2๐›ฟ๎€ท2+3๐œ†โ„›+๐œ†2โ„›๎€ธ๐œ,(2.9)3=๎€ฝ๎€ท212๐œ†โ„’โˆ’2๐œ†โ„›๎€ธ+6๐œ†2โ„’๎€ท5+2๐œ†โ„›+1๎€ท๐œ†โ„›๎€ธโˆ’1+๐œ†โ„›๎€ธ+๐œ†3โ„’๎€ท5+2๐œ†โ„›+1๎€ท๐œ†โ„›๎€ธโˆ’1+๐œ†โ„›๎€ธ+๐œ†โ„’๎€ท49โˆ’(11)2๐œ†โ„›+1+(11)2๐œ†โ„›+1๐œ†โ„›โˆ’6๐œ†2โ„›โˆ’๐œ†3โ„›โˆ’2๐œ†โ„’+1๎€ท1+๐œ†โ„›๎€ธ๎€ท2+๐œ†โ„›๎€ธ๎€ท3+๐œ†โ„›๎€ธ๎€ธ+๐œ†โ„›๎€ท(11)2๐œ†โ„’+1โˆ’49+(3)2๐œ†โ„›+2+๎€ท2๐œ†โ„’+1๎€ธ๐œ†โˆ’5โ„›๎€ท6+๐œ†โ„›๎€ฝ2๎€ท๎€ธ๎€ธ๎€พ๎€น3+๐œ†โ„’๎€ธ๎€ท3+๐œ†โ„›2๎€ธ๎€ท๐œ†โ„›๎€ท1+๐œ†โ„’๎€ธ๎€ท2+๐œ†โ„’๎€ธ+2๐œ†โ„’๎€ท1+๐œ†โ„›๎€ธ๎€ท2+๐œ†โ„›,๐œ๎€ธ๎€ธ๎€พ(2.10)4=๎ƒฏ๎€ท1+๐œ†โ„’๎€ธ๎€ท2+๐œ†โ„’๎€ธ๎€ท1+๐œ†โ„›๎€ธ๎€ท2+๐œ†โ„›๎€ธร—๎ƒฉ2๐œ†โ„’๎€ท๐œ†โ„’๐œ†โˆ’2๎€ธ๎€ทโ„’๎€ธโˆ’1๎€ท1+๐œ†โ„’๎€ธ๎€ท2+๐œ†โ„’๎€ธ๎€ท3+๐œ†โ„’๎€ธ๎€ท4+๐œ†โ„’๎€ธ+2๐œ†โ„›๎€ท๐œ†โ„›๐œ†โˆ’2๎€ธ๎€ทโ„›๎€ธโˆ’1๎€ท1+๐œ†โ„›๎€ธ๎€ท2+๐œ†โ„›๎€ธ๎€ท3+๐œ†โ„›๎€ธ๎€ท4+๐œ†โ„›๎€ธ๎€น๎€ฝ2๎ƒช๎ƒฐ๐œ†โ„›๎€ท1+๐œ†โ„’๎€ธ๎€ท2+๐œ†โ„’๎€ธ+2๐œ†โ„’๎€ท1+๐œ†โ„›๎€ธ๎€ท2+๐œ†โ„›.๎€ธ๎€พ(2.11) Thus, given specified values of ๐œ3 and ๐œ4, (2.10) and (2.11) can be numerically solved for the corresponding values of ๐œ†โ„’ and ๐œ†โ„›. Note that the values of ๐ฟ-skew (๐œ3) and ๐ฟ-kurtosis (๐œ4) in (2.10) and (2.11) are independent of the value of ๐›ฟ selected in (1.8). Further, inspection of (2.10) and (2.11) indicates that interchanging values for the parameters ๐œ†โ„’ and ๐œ†โ„› reverses the direction of ๐œ3 and has no effect on ๐œ4.

For the special case of ๐œ†โ„’=๐œ†โ„› the double GLD is symmetric where ๐œ3=0 in (2.10) and ๐œ4 in (2.11) will simplify to the expression๐œ4=๎€ท๐œ†โ„’๐œ†โˆ’2๎€ธ๎€ทโ„’๎€ธโˆ’1๎€ท๐œ†โ„’๐œ†+3๎€ธ๎€ทโ„’๎€ธ+4.(2.12) Differentiating (2.12) with respect to ๐œ†โ„’ and equating the resulting expression to zero and solving for ๐œ†โ„’ yields a minimum value of ๐ฟ-kurtosis of๎€ท๐œmin4๎€ธ=โˆš12โˆ’56โˆš12+56=โˆ’0.010205โ€ฆ,(2.13) where ๐œ†โ„’=๐œ†โ„›=โˆš6โˆ’1. As such, using (2.10) and (2.11) with ๐œ†โ„’=โˆš6โˆ’1 and ๐œ†โ„›โˆšโˆˆ(โˆ’1,6โˆ’1), provided in Figure 3 is a graph of the region for feasible combinations of ๐œ3 and ๐œ4 for double GLDs. Feasible combinations of ๐œ3 and ๐œ4 for (2.10) and (2.11) will lie in the region above the curve graphed in the |๐œ3|,๐œ4 plane of Figure 3.

Provided in Figure 4 are some examples of various double GLDs. These distributions are used in the simulation portion of this study in Section 4. The next section begins with an introduction to the ๐ฟ-correlation.

3. The ๐ฟ-Correlation for Double GLDs

The coefficient of ๐ฟ-correlation (see [19]) is introduced by considering two random variables ๐‘Œ๐‘— and ๐‘Œ๐‘˜ with continuous distribution functions ๐น(๐‘Œ๐‘—) and ๐น(๐‘Œ๐‘˜), respectively. The second ๐ฟ-moments of ๐‘Œ๐‘— and ๐‘Œ๐‘˜ can alternatively be expressed asฮ›2๎€ท๐‘Œ๐‘—๎€ธ๎€ท๐‘Œ=2Cov๐‘—๎€ท๐‘Œ,๐น๐‘—,ฮ›๎€ธ๎€ธ(3.1)2๎€ท๐‘Œ๐‘˜๎€ธ๎€ท๐‘Œ=2Cov๐‘˜๎€ท๐‘Œ,๐น๐‘˜๎€ธ๎€ธ.(3.2) The second ๐ฟ-comoments of ๐‘Œ๐‘— toward ๐‘Œ๐‘˜ and ๐‘Œ๐‘˜ toward ๐‘Œ๐‘— areฮ›2๎€ท๐‘Œ๐‘—,๐‘Œ๐‘˜๎€ธ๎€ท๐‘Œ=2Cov๐‘—๎€ท๐‘Œ,๐น๐‘˜,ฮ›๎€ธ๎€ธ(3.3)2๎€ท๐‘Œ๐‘˜,๐‘Œ๐‘—๎€ธ๎€ท๐‘Œ=2Cov๐‘˜๎€ท๐‘Œ,๐น๐‘—๎€ธ๎€ธ.(3.4) As such, the ๐ฟ-correlations of ๐‘Œ๐‘— toward ๐‘Œ๐‘˜ and ๐‘Œ๐‘˜ toward ๐‘Œ๐‘— are expressed as๐œ‚๐‘—๐‘˜=ฮ›2๎€ท๐‘Œ๐‘—,๐‘Œ๐‘˜๎€ธฮ›2๎€ท๐‘Œ๐‘—๎€ธ๐œ‚,(3.5)๐‘˜๐‘—=ฮ›2๎€ท๐‘Œ๐‘˜,๐‘Œ๐‘—๎€ธฮ›2๎€ท๐‘Œ๐‘˜๎€ธ.(3.6) The ๐ฟ-correlation in (3.5) or (3.6) is bounded such that โˆ’1โ‰ค๐œ‚๐‘—๐‘˜โ‰ค1 where a value of ๐œ‚๐‘—๐‘˜=1 (๐œ‚๐‘—๐‘˜=โˆ’1) indicates a strictly increasing (decreasing) monotone relationship between the two variables. In general, we would also note that ๐œ‚๐‘—๐‘˜โ‰ ๐œ‚๐‘˜๐‘—.

In the context of the ๐ฟ-moment-based double GLD, suppose it is desired to simulate ๐‘‡ distributions of the form in (1.8) with a specified ๐ฟ-correlation matrix and where each distribution has its own specified values of ๐œ3 and ๐œ4. Let ๐‘1,โ€ฆ,๐‘๐‘‡ denote standard normal variables where the distribution functions and bivariate density function associated with ๐‘๐‘— and ๐‘๐‘˜ are expressed as ฮฆ๎€ท๐‘ง๐‘—๎€ธ๎€ฝ๐‘=Pr๐‘—โ‰ค๐‘ง๐‘—๎€พ=๎€œ๐‘ง๐‘—โˆ’โˆž(2๐œ‹)โˆ’1/2๎ƒฏexpโˆ’๐‘ค2๐‘—2๎ƒฐ๐‘‘๐‘ค๐‘—,(3.7)ฮฆ๎€ท๐‘ง๐‘˜๎€ธ๎€ฝ๐‘=Pr๐‘˜โ‰ค๐‘ง๐‘˜๎€พ=๎€œ๐‘ง๐‘˜โˆ’โˆž(2๐œ‹)โˆ’1/2๎ƒฏexpโˆ’๐‘ค2๐‘˜2๎ƒฐ๐‘‘๐‘ค๐‘˜,(3.8)๐‘“๐‘—๐‘˜=๎‚ต๎‚€2๐œ‹1โˆ’๐œŒ2๐‘—๐‘˜๎‚1/2๎‚ถโˆ’1๎‚ปโˆ’๎‚€2๎‚€exp1โˆ’๐œŒ2๐‘—๐‘˜๎‚๎‚โˆ’1๎€ท๐‘ง2๐‘—+๐‘ง2๐‘˜โˆ’2๐œŒ๐‘—๐‘˜๐‘ง๐‘—๐‘ง๐‘˜๎€ธ๎‚ผ.(3.9) Using (3.7), it follows that the ๐‘—th double GLD associated with (1.8) can be expressed as ๐‘ž๐‘—(ฮฆ(๐‘ง๐‘—)) because ฮฆ(๐‘ง๐‘—)โˆผ๐‘ˆ(0,1). As such, using (3.5), the ๐ฟ-correlation of ๐‘ž๐‘—(ฮฆ(๐‘ง๐‘—)) toward ๐‘ž๐‘˜(ฮฆ(๐‘ง๐‘˜)) can be evaluated using solved values of ๐œ†โ„’๐‘— and ๐œ†โ„›๐‘— for ๐‘ž๐‘—(ฮฆ(๐‘ง๐‘—)), a specified intermediate correlation (IC) ๐œŒ๐‘—๐‘˜ in (3.9), and the following integral expressed as๐œ‚๐‘—๐‘˜โˆš=2๐œ‹๎€+โˆžโˆ’โˆž๐‘ฅ๐‘—๎‚€๐‘ž๐‘—๎‚€ฮฆ๎€ท๐‘ง๐‘—๎€ธ,๐œ†โ„’๐‘—,๐œ†โ„›๐‘—ฮฆ๎€ท๐‘ง๎‚๎‚๐‘˜๎€ธ๐‘“๐‘—๐‘˜๐‘‘๐‘ง๐‘—๐‘‘๐‘ง๐‘˜.(3.10) The double GLD in (3.10) is standardized by a linear transformation such that it has a mean of zero and one-half the coefficient of mean difference equal to that of the unit-normal distribution as๐‘ฅ๐‘—๎‚€๐‘ž๐‘—๎‚€ฮฆ๎€ท๐‘ง๐‘—๎€ธ,๐œ†โ„’๐‘—,๐œ†โ„›๐‘—๎‚€๐‘ž๎‚๎‚=๐œ‰๐‘—๎‚€ฮฆ๎€ท๐‘ง๐‘—๎€ธ,๐œ†โ„’๐‘—,๐œ†โ„›๐‘—๎‚โˆ’ฮ›1๎‚,(3.11) where ฮ›1 is the mean from (2.8) and ๐œ‰ is a constant that scales ฮ›2 in (2.9) and in the denominator of (3.5) to โˆš1/๐œ‹ as๎‚€๐œ‰=๐›ฟ41+๐œ†โ„’๐‘—๎‚๎‚€2+๐œ†โ„’๐‘—๎‚๎‚€1+๐œ†โ„›๐‘—๎‚๎‚€1+๐œ†โ„›๐‘—๎‚2๐œ†โ„›๐‘—๎‚€1+๐œ†โ„’๐‘—๎‚๎‚€2+๐œ†โ„’๐‘—๎‚โˆš๐œ‹+2๐œ†โ„’๐‘—๎‚€1+๐œ†โ„›๐‘—๎‚๎‚€1+๐œ†โ„›๐‘—๎‚โˆš๐œ‹.(3.12) Analogously, the ๐ฟ-correlation of ๐‘ž๐‘˜(ฮฆ(๐‘ง๐‘˜)) toward ๐‘ž๐‘—(ฮฆ(๐‘ง๐‘—)) is expressed as๐œ‚๐‘˜๐‘—โˆš=2๐œ‹๎€+โˆžโˆ’โˆž๐‘ฅ๐‘˜๎€ท๐‘ž๐‘˜๎€ทฮฆ๎€ท๐‘ง๐‘˜๎€ธ,๐œ†โ„’๐‘˜,๐œ†โ„›๐‘˜ฮฆ๎€ท๐‘ง๎€ธ๎€ธ๐‘—๎€ธ๐‘“๐‘—๐‘˜๐‘‘๐‘ง๐‘˜๐‘‘๐‘ง๐‘—.(3.13) Note for the special case that if ๐‘ž๐‘—(ฮฆ(๐‘ง๐‘—)) in (3.10) and ๐‘ž๐‘˜(ฮฆ(๐‘ง๐‘˜)) in (3.13) have the same parameters that is, ๐œ†โ„’๐‘—=๐œ†โ„’๐‘˜ and ๐œ†โ„›๐‘—=๐œ†โ„›๐‘˜, then ๐œ‚๐‘—๐‘˜=๐œ‚๐‘˜๐‘—. Provided in Algorithm 1 is source code written in Mathematica [20] that implements the computation of an IC (๐œŒ๐‘—๐‘˜) based on (3.10). The details for simulating double GLDs with specified๐ฟ-correlations are described in the next section.

ฮฆ ๐‘— = C D F [ N o r m a l D i s t r i b u t i o n [ 0 , 1 ] , ๐‘ ๐‘— ] ;
ฮฆ ๐‘˜ = C DF [ NormalDistribution [ 0,1 ] , ๐‘ ๐‘˜ ] ;
๐œ† โ„’ = 0 . 2 7 7 9 7 3 3 5 9 8 2 8 3 2 3 2 ;
๐œ† โ„› = โˆ’ 0 . 0 8 3 2 9 4 7 9 1 4 3 3 1 2 7 6 2 ;
๐‘ž โ„’ = ( ฮฆ ๐œ† โ„’ ๐‘— โˆ’ ( 1 โˆ’ ฮฆ ๐‘— ) ๐œ† โ„’ ) / ( ๐œ† โ„’ ร— 2 ( 3 / 2 ) โˆ’ ๐œ† โ„’ / โˆš ๐œ‹ ) ;
๐‘ž โ„› = ( ฮฆ ๐œ† โ„› ๐‘— โˆ’ ( 1 โˆ’ ฮฆ ๐‘— ) ๐œ† โ„› ) / ( ๐œ† โ„› ร— 2 ( 3 / 2 ) โˆ’ ๐œ† โ„› / โˆš ๐œ‹ ) ;
(* Standardizing constants ฮ› 1 from Eq. ( 2 . 8 ) and ๐œ‰ from Eq. ( 3 . 1 2 ) *)
๐‘‹ โ„’ = ๐œ‰ ( ๐‘ž โ„’ โˆ’ ฮ› 1 ) ;
๐‘‹ โ„› = ๐œ‰ ( ๐‘ž โ„› โˆ’ ฮ› 1 ) ;
(* Intermediate Correlation *)
๐œŒ ๐‘— ๐‘˜ = 0.395685;
Needs [ โ€œMultivariateStatistics`โ€ ]
๐‘“ ๐‘— ๐‘˜ = P DF [ MultinormalDistribution [ { 0,0}, { { 1, ๐œŒ ๐‘— ๐‘˜ } , { ๐œŒ ๐‘— ๐‘˜ ,1 } } ] , { ๐‘ ๐‘— , ๐‘ ๐‘˜ } ] ;
(* Compute the specified ๐ฟ -correlation *)
๐œ‚ ๐‘— ๐‘˜ โˆš = 2 ๐œ‹ โˆ— NIntegrate [ Piecewise [ { { ๐‘‹ โ„’ , ฮฆ ๐‘— โ‰ค 0 . 5 } , { ๐‘‹ โ„› , ฮฆ ๐‘— > 0 . 5 } } ] ร— ฮฆ ๐‘˜ ร— ๐‘“ ๐‘— ๐‘˜ ,
{ ๐‘ ๐‘— , โˆ’ 10, 10}, { ๐‘ ๐‘˜ , โˆ’10, 10}, Method โ†’ MultiDimensional ]
0.40

4. The Procedure and Simulation Study

To implement the procedure for simulating double GLDs with specified ๐ฟ-moments and ๐ฟ-correlations we suggest the following six steps. (1)Specify the ๐ฟ-moments for ๐‘‡ transformations of the form in (1.8), that is, ๐‘ž1(ฮฆ(๐‘ง1)),โ€ฆ,๐‘ž๐‘‡(ฮฆ(๐‘ง๐‘‡))and obtain the solutions for the parameters of ๐œ†โ„’๐‘— and ๐œ†โ„›๐‘—by solving (2.10) and (2.11) using the specified values of ๐ฟ-skew (๐œ3) and ๐ฟ-kurtosis (๐œ4) for each distribution. Specify a ๐‘‡ร—๐‘‡ matrix of ๐ฟ-correlations (๐œ‚๐‘—๐‘˜) for ๐‘ž๐‘—(ฮฆ(๐‘ง๐‘—)) toward ๐‘ž๐‘˜(ฮฆ(๐‘ง๐‘˜)), where ๐‘—<๐‘˜โˆˆ{1,2,โ€ฆ,๐‘‡}. (2)Compute the (Pearson) intermediate correlations (ICs) ๐œŒ๐‘—๐‘˜ by substituting the solutions of ๐œ†โ„’๐‘— and ๐œ†โ„›๐‘— from Step (1) into (3.10) and then numerically integrate to solve for ๐œŒ๐‘—๐‘˜ (see Algorithm 1 for an example). Repeat this step separately for all ๐‘‡(๐‘‡โˆ’1)/2 pairwise combinations of correlations. (3)Assemble the ICs into a ๐‘‡ร—๐‘‡ 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 generate ๐‘‡ standard normal variables (๐‘1,โ€ฆ,๐‘๐‘‡) correlated at the intermediate levels as follows: ๐‘1=๐‘Ž11๐‘‰1๐‘2=๐‘Ž12๐‘‰1+๐‘Ž22๐‘‰2โ‹ฎ๐‘๐‘—=๐‘Ž1๐‘—๐‘‰1+๐‘Ž2๐‘—๐‘‰2+โ‹ฏ+๐‘Ž๐‘–๐‘—๐‘‰๐‘–+โ‹ฏ+๐‘Ž๐‘—๐‘—๐‘‰๐‘—โ‹ฎ๐‘๐‘‡=๐‘Ž1๐‘‡๐‘‰1+๐‘Ž2๐‘‡๐‘‰2+โ‹ฏ+๐‘Ž๐‘–๐‘‡๐‘‰๐‘–+โ‹ฏ+๐‘Ž๐‘—๐‘‡๐‘‰๐‘—+โ‹ฏ+๐‘Ž๐‘‡๐‘‡๐‘‰๐‘‡,(4.1) where ๐‘‰1,โ€ฆ,๐‘‰๐‘‡ are independent standard normal random variables and where ๐‘Ž๐‘–๐‘— represents the element in the ๐‘–th row and the ๐‘—th column of the matrix associated with the Cholesky factorization performed in Step 3. (5)Substitute ๐‘1,โ€ฆ,๐‘๐‘‡ from Step (4) into the following Taylor series-based expansion for the standard normal cdf [21]: ฮฆ๎€ท๐‘๐‘—๎€ธ=๎‚€12๎‚๎€ท๐‘+๐œ™๐‘—๎€ธ๎ƒฏ๐‘๐‘—+๐‘3๐‘—3+๐‘5๐‘—+๐‘(3โ‹…5)7๐‘—๎ƒฐ(3โ‹…5โ‹…7)+โ‹ฏ,(4.2) where ๐œ™(๐‘๐‘—) denotes the standard normal pdf and where the absolute error associated with (4.2) is less than 8ร—10โˆ’16. (6)Substitute the regular uniform deviates, ฮฆ(๐‘๐‘—), generated from Step (5) into the ๐‘‡ equations of the form in (1.8), as noted in Step (1), to generate the double GLDs with the specified ๐ฟ-moments and ๐ฟ-correlations.

To demonstrate the steps above and evaluate the proposed procedure, a comparison between the new ๐ฟ-moment and conventional product moment-based procedures is subsequently described. Specifically, the distributions in Figure 4 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 conventional moment- and ๐ฟ-moment-based procedures, respectively. See Algorithm 2 for the algorithm and an example for computing ICs for the conventional procedure. Tables 4 and 5 give the results of the Cholesky decompositions on the IC matrices, which are then used to create ๐‘1,โ€ฆ,๐‘4 with the specified ICs by making use of the formulae given in (4.1) of Step (4) with ๐‘‡=4. The values of ๐‘1,โ€ฆ,๐‘4 are subsequently transformed to ฮฆ(๐‘1),โ€ฆ,ฮฆ(๐‘4) using (4.2) and then substituted into equations of the form in (1.8) to produce ๐‘ž1(ฮฆ(๐‘1)),โ€ฆ,๐‘ž4(ฮฆ(๐‘4)) for both procedures.

(a)

1234

11
20.701
30.700.701
40.850.700.701

(b)

1234

11
20.401
30.500.401
40.600.500.401

(a)

1234

11
2 0.815469 1
3 0.728537 0.762929 1
4 0.899099 0.741935 0.709597 1

(b)

1234

11
2 0.441594 1
3 0.527660 0.429990 1
4 0.636315 0.527047 0.406786 1

(a)

1234

11
2 0.686456 1
3 0.686456 0.694980 1
4 0.841013 0.694980 0.695427 1

(b)

1234

11
2 0.387335 1
3 0.485840 0.395135 1
4 0.585468 0.494609 0.395685 1

(a)

๐‘Ž 1 1 = 1 ๐‘Ž 1 2 = 0 . 8 1 5 4 6 9 ๐‘Ž 1 3 = 0 . 7 2 8 5 3 7 ๐‘Ž 1 4 = 0 . 8 9 9 0 9 9
0 ๐‘Ž 2 2 = 0 . 5 7 8 8 0 1 ๐‘Ž 2 3 = 0 . 2 9 1 6 8 9 ๐‘Ž 2 4 = 0 . 0 1 5 1 3 3
00 ๐‘Ž 3 3 = 0 . 6 1 9 8 0 0 ๐‘Ž 3 4 = 0 . 0 8 0 9 3 2
000 ๐‘Ž 4 4 = 0 . 4 2 9 9 3 3

(b)

๐‘Ž 1 1 = 1 ๐‘Ž 1 2 = 0 . 4 4 1 5 9 4 ๐‘Ž 1 3 = 0 . 5 2 7 6 6 0 ๐‘Ž 1 4 = 0 . 6 3 6 3 1 5
0 ๐‘Ž 2 2 = 0 . 8 9 7 2 1 5 ๐‘Ž 2 3 = 0 . 2 1 9 5 4 4 ๐‘Ž 2 4 = 0 . 2 7 4 2 4 2
00 ๐‘Ž 3 3 = 0 . 8 2 0 5 9 4 ๐‘Ž 3 4 = 0 . 0 1 3 1 8 5
000 ๐‘Ž 4 4 = 0 . 7 2 0 9 1 7

(a)

๐‘Ž 1 1 = 1 ๐‘Ž 1 2 = 0 . 6 8 6 4 5 6 ๐‘Ž 1 3 = 0 . 6 8 6 4 5 6 ๐‘Ž 1 4 = 0 . 8 4 1 0 1 3
0 ๐‘Ž 2 2 = 0 . 7 2 7 1 7 1 ๐‘Ž 2 3 = 0 . 3 0 7 7 1 0 ๐‘Ž 2 4 = 0 . 1 6 1 8 0 7
00 ๐‘Ž 3 3 = 0 . 6 5 8 8 5 7 ๐‘Ž 3 4 = 0 . 1 0 3 6 9 3
000 ๐‘Ž 4 4 = 0 . 5 0 5 7 3 0

(b)

๐‘Ž 1 1 = 1 ๐‘Ž 1 2 = 0 . 3 8 7 3 3 5 ๐‘Ž 1 3 = 0 . 4 8 5 8 4 0 ๐‘Ž 1 4 = 0 . 5 8 5 4 6 8
0 ๐‘Ž 2 2 = 0 . 9 2 1 9 3 9 ๐‘Ž 2 3 = 0 . 2 2 4 4 7 5 ๐‘Ž 2 4 = 0 . 2 9 0 5 1 5
00 ๐‘Ž 3 3 = 0 . 8 4 4 7 3 1 ๐‘Ž 3 4 = 0 . 0 5 4 4 8 8
000 ๐‘Ž 4 4 = 0 . 7 5 4 8 9 0

ฮฆ ๐‘— = C DF[NormalDistribution [ 0,1], ๐‘ ๐‘— ];
ฮฆ ๐‘˜ = C DF[NormalDistribution [ 0,1], ๐‘ ๐‘˜ ];
๐œ† โ„’ ๐‘— = 0 . 2 7 7 9 7 3 3 5 9 8 2 8 3 2 3 2 ;
๐œ† โ„› ๐‘— = โˆ’ 0 . 0 8 3 2 9 4 7 9 1 4 3 3 1 2 7 6 2 ;
๐œ† โ„’ ๐‘˜ = 0 . 3 6 6 2 5 9 9 0 6 9 2 9 8 9 9 4 ;
๐œ† โ„› ๐‘˜ = 0 . 1 2 9 7 7 7 1 2 6 3 0 8 0 0 0 4 ;
๐‘ž โ„’ ๐‘— = ( ฮฆ ๐œ† โ„’ ๐‘— ๐‘— โˆ’ ( 1 โˆ’ ฮฆ ๐‘— ) ๐œ† โ„’ ๐‘— ) / ( ๐œ† โ„’ ๐‘— ร— 2 ( 3 / 2 ) โˆ’ ๐œ† โ„’ ๐‘— / โˆš ๐œ‹ ) ;
๐‘ž โ„› ๐‘— = ( ฮฆ ๐œ† โ„› ๐‘— ๐‘— โˆ’ ( 1 โˆ’ ฮฆ ๐‘— ) ๐œ† โ„› ๐‘— ) / ( ๐œ† โ„› ๐‘— ร— 2 ( 3 / 2 ) โˆ’ ๐œ† โ„› ๐‘— / โˆš ๐œ‹ ) ;
๐‘ž โ„’ ๐‘˜ = ( ฮฆ ๐œ† โ„’ ๐‘˜ ๐‘˜ โˆ’ ( 1 โˆ’ ฮฆ ๐‘˜ ) ๐œ† โ„’ ๐‘˜ ) / ( ๐œ† โ„’ ๐‘˜ ร— 2 ( 3 / 2 ) โˆ’ ๐œ† โ„’ ๐‘˜ / โˆš ๐œ‹ ) ;
๐‘ž โ„› ๐‘˜ = ( ฮฆ ๐œ† โ„› ๐‘˜ ๐‘˜ โˆ’ ( 1 โˆ’ ฮฆ ๐‘˜ ) ๐œ† โ„› ๐‘˜ ) / ( ๐œ† โ„› ๐‘˜ ร— 2 ( 3 / 2 ) โˆ’ ๐œ† โ„› ๐‘˜ / โˆš ๐œ‹ ) ;
(* Standardizing constants ๐›ผ 1 and ๐›ผ 2 from Equation ( A . 2 ) in the Appendix *)
๐‘‹ โ„’ ๐‘— = ( ๐‘ž โ„’ ๐‘— โˆ’ ๐›ผ 1 ๐‘— ) / ๐›ผ 2 ๐‘— ;
๐‘‹ โ„› ๐‘— = ( ๐‘ž โ„› ๐‘— โˆ’ ๐›ผ 1 ๐‘— ) / ๐›ผ 2 ๐‘— ;
๐‘‹ โ„’ ๐‘˜ = ( ๐‘ž โ„’ ๐‘˜ โˆ’ ๐›ผ 1 ๐‘˜ ) / ๐›ผ 2 ๐‘˜ ;
๐‘‹ โ„› ๐‘˜ = ( ๐‘ž โ„› ๐‘˜ โˆ’ ๐›ผ 1 ๐‘˜ ) / ๐›ผ 2 ๐‘˜ ;
(* Intermediate Correlation *)
๐œŒ ๐‘— ๐‘˜ = 0 . 4 0 6 7 8 6 ;
Needs[โ€œMultivariateStatistics`โ€]
๐‘“ ๐‘— ๐‘˜ = P DF [ MultinormalDistribution [ { 0,0}, { { 1, ๐œŒ ๐‘— ๐‘˜ } , { ๐œŒ ๐‘— ๐‘˜ ,1 } } ] , { ๐‘ ๐‘— , ๐‘ ๐‘˜ } ] ;
(* Compute the specified conventional Pearson correlation *)
๐œŒ โˆ— ๐‘— ๐‘˜ = NIntegrate [ Piecewise [ { { ๐‘‹ โ„’ ๐‘— , ฮฆ ๐‘— โ‰ค 0 . 5 } , { ๐‘‹ โ„› ๐‘— , ฮฆ ๐‘— > 0 . 5 } } ] ร— Piecewise [ { { ๐‘‹ โ„’ ๐‘˜ , ฮฆ ๐‘˜ โ‰ค
0 . 5 } , { ๐‘‹ โ„› ๐‘˜ , ฮฆ ๐‘˜ > 0 . 5 } } ] ร— ๐‘“ ๐‘— ๐‘˜ , { ๐‘ ๐‘— , โˆ’10, 10}, { ๐‘ ๐‘˜ , โˆ’10, 10}, Method โ†’ MultiDimensional ]
0.40

In 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 (๐›ผ3), kurtosis (๐›ผ4), and Pearson correlation (๐œŒโˆ—๐‘—๐‘˜); (b) ๐ฟ-skew (๐œ3), ๐ฟ-kurtosis (๐œ4), and ๐ฟ-correlation (๐œ‚๐‘—๐‘˜). The estimates of ๐œ3,4 were based on samples of size ๐‘›=250 and the estimates of ๐œ‚๐‘—๐‘˜ were based on sample sizes of ๐‘›=25 and ๐‘›=1000. The estimates for ๐›ผ3,4 were based on Fisherโ€™s ๐‘˜-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 ๐›ผ3,4=0 for the standard normal distribution). The formulae used for computing estimates for ๐œ3,4 were Headrickโ€™s equations (2.4) and (2.6) [22]. The estimate for ๐œŒโˆ—๐‘—๐‘˜ was based on the usual formula for the Pearson product-moment of correlation statistic and the estimate for ๐œ‚๐‘—๐‘˜ was computed based on (3.5) using the empirical forms of the cdfs in (3.1) and (3.3). The estimates for ๐œŒโˆ—๐‘—๐‘˜ and ๐œ‚๐‘—๐‘˜ were both transformed using Fisherโ€™s ๐‘ง๎…ž 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 (๐›ผ3,4, ๐œ3,4, ๐‘ง๎…ž๐œŒโˆ—๐‘—๐‘˜, ๐‘ง๎…ž๐œ‚๐‘—๐‘˜) using 10,000 resamples via the commercial software package Spotfire S+ [23]. The bootstrap results for the estimates of ๐‘ง๎…ž๐œŒโˆ—๐‘—๐‘˜ and ๐‘ง๎…ž๐œ‚๐‘—๐‘˜ were transformed back to their original metrics. Further, if a parameter (๐‘ƒ) was outside its associated bootstrap C.I., then an index of relative bias (RB) was computed for the estimate (๐ธ) as RB=((๐ธโˆ’๐‘ƒ)/๐‘ƒ)ร—100. The results of the simulation are reported in Tables 6, 7, 8, 9, 10, 11 and are discussed in the next section.


DistParameterEstimate95% Bootstrap C.I.Standard errorRelative bias %

1 ๐›ผ 3 = 2 . 7 0 1.731.72, 1.750.0078 โˆ’35.92
๐›ผ 4 = 8 7 . 4 8.938.75, 9.120.0939 โˆ’89.78
2 ๐›ผ 3 = โˆ’ 1 . 3 8 โˆ’1.23 โˆ’1.24, โˆ’1.220.0039 โˆ’10.87
๐›ผ 4 = 5 . 6 7 3.783.71, 3.850.0360 โˆ’33.33
3 ๐›ผ 3 = 0 . 8 8 3 0.8240.819, 0.8290.0026 โˆ’6.68
๐›ผ 4 = 2 . 5 5 2.042.01, 2.080.0185 โˆ’20.00
4 ๐›ผ 3 = 0 . 2 9 5 0.2920.291, 0.2940.0008 โˆ’1.02
๐›ผ 4 = โˆ’ 0 . 2 3 2 โˆ’0.227 โˆ’0.231, โˆ’0.2240.0018 โˆ’2.16


DistParameterEstimate95% Bootstrap C.I.Standard errorRelative bias %

1 ๐œ 3 = 0 . 1 5 0.14790.1473, 0.14840.0003 โˆ’1.40
๐œ 4 = 0 . 2 0 0.19870.1982, 0.19910.0002 โˆ’0.65
2 ๐œ 3 = โˆ’ 0 . 1 5 โˆ’0.1487 โˆ’0.1492, โˆ’0.14830.0002 โˆ’0.87
๐œ 4 = 0 . 1 5 0.14940.1490, 0.14970.0002 โˆ’0.40
3 ๐œ 3 = 0 . 1 0 0.09920.0988, 0.09960.0002 โˆ’0.80
๐œ 4 = 0 . 1 5 0.14970.1494, 0.15010.0002โ€”
4 ๐œ 3 = 0 . 0 5 0.04980.0495, 0.05010.0002โ€”
๐œ 4 = 0 . 1 0 0.10020.0999, 0.10040.0001โ€”

(a) ๐‘› = 2 5

ParameterEstimate95% Bootstrap C.I.Standard errorRelative bias %

๐œŒ โˆ— 1 2 = 0 . 7 0 0.76060.7597, 0.76150.001118.66
๐œŒ โˆ— 1 3 = 0 . 7 0 0.72770.7263, 0.72910.001463.96
๐œŒ โˆ— 1 4 = 0 . 8 5 0.88540.8847, 0.88580.001304.16
๐œŒ โˆ— 2 3 = 0 . 7 0 0.72900.7279, 0.73000.001134.14
๐œŒ โˆ— 2 4 = 0 . 7 0 0.72090.7198, 0.72200.001162.99
๐œŒ โˆ— 3 4 = 0 . 7 0 0.71200.7108, 0.71340.001361.71

(b) ๐‘› = 1 0 0 0

ParameterEstimate95% Bootstrap C.I.Standard errorRelative bias %

๐œŒ โˆ— 1 2 = 0 . 7 0 0.70670.7063, 0.70710.000380.96
๐œŒ โˆ— 1 3 = 0 . 7 0 0.70360.7033, 0.70390.000300.51
๐œŒ โˆ— 1 4 = 0 . 8 5 0.85620.8560, 0.85650.000540.73
๐œŒ โˆ— 2 3 = 0 . 7 0 0.70140.7012, 0.70160.000200.20
๐œŒ โˆ— 2 4 = 0 . 7 0 0.70100.7008, 0.70120.000200.14
๐œŒ โˆ— 3 4 = 0 . 7 0 0.70050.7002, 0.70060.000200.07

(a) ๐‘› = 2 5

ParameterEstimate95% Bootstrap C.I.Standard errorRelative bias %

๐œ‚ 1 2 = 0 . 7 0 0.70820.7068, 0.70970.001481.17
๐œ‚ 1 3 = 0 . 7 0 0.70780.7064, 0.70930.001491.11
๐œ‚ 1 4 = 0 . 8 5 0.85540.8546, 0.85620.001510.64
๐œ‚ 2 3 = 0 . 7 0 0.70910.7077, 0.71050.001461.30
๐œ‚ 2 4 = 0 . 7 0 0.70900.7076, 0.71040.001471.29
๐œ‚ 3 4 = 0 . 7 0 0.70970.7083, 0.71110.001481.39

(b) ๐‘› = 1 0 0 0

ParameterEstimate95% Bootstrap C.I.Standard errorRelative bias %

๐œ‚ 1 2 = 0 . 7 0 0.70000.6998, 0.70020.00023โ€”
๐œ‚ 1 3 = 0 . 7 0 0.70020.7000, 0.70040.00023โ€”
๐œ‚ 1 4 = 0 . 8 5 0.85020.8500, 0.85030.00022โ€”
๐œ‚ 2 3 = 0 . 7 0 0.69990.6997, 0.70020.00022โ€”
๐œ‚ 2 4 = 0 . 7 0 0.70050.6998, 0.70030.00022โ€”
๐œ‚ 3 4 = 0 . 7 0 0.70010.6999, 0.70030.00021โ€”

(a) ๐‘› = 2 5

ParameterEstimate95% Bootstrap C.I.Standard errorRelative bias %

๐œŒ โˆ— 1 2 = 0 . 4 0 0.42070.4187, 0.42270.001235.18
๐œŒ โˆ— 1 3 = 0 . 5 0 0.52650.5244, 0.52850.001425.30
๐œŒ โˆ— 1 4 = 0 . 6 0 0.63080.6292, 0.63240.001355.13
๐œŒ โˆ— 2 3 = 0 . 4 0 0.41780.4156, 0.41970.001264.45
๐œŒ โˆ— 2 4 = 0 . 5 0 0.51650.5146, 0.51830.001273.30
๐œŒ โˆ— 3 4 = 0 . 4 0 0.41030.4081, 0.41250.001342.58

(b) ๐‘› = 1 0 0 0

ParameterEstimate95% Bootstrap C.I.Standard errorRelative bias %

๐œŒ โˆ— 1 2 = 0 . 4 0 0.39060.3903, 0.39100.00020โˆ’2.35
๐œŒ โˆ— 1 3 = 0 . 5 0 0.50290.5026, 0.50330.000250.58
๐œŒ โˆ— 1 4 = 0 . 6 0 0.60350.6032, 0.60390.000260.58
๐œŒ โˆ— 2 3 = 0 . 4 0 0.40050.4001, 0.40070.000180.13
๐œŒ โˆ— 2 4 = 0 . 5 0 0.50060.5003, 0.50090.000190.12
๐œŒ โˆ— 3 4 = 0 . 4 0 0.40030.3999, 0.40060.000200.08

(a) ๐‘› = 2 5

ParameterEstimate95% Bootstrap C.I.Standard errorRelative bias %

๐œ‚ 1 2 = 0 . 4 0 0.40840.4061, 0.41090.001482.10
๐œ‚ 1 3 = 0 . 5 0 0.50730.5051, 0.50950.001491.46
๐œ‚ 1 4 = 0 . 6 0 0.60800.6061, 0.60980.001501.33
๐œ‚ 2 3 = 0 . 4 0 0.40830.4060, 0.41070.001452.08
๐œ‚ 2 4 = 0 . 5 0 0.50940.5073, 0.51160.001471.88
๐œ‚ 3 4 = 0 . 4 0 0.40890.4066, 0.41120.001422.23

(b) ๐‘› = 1 0 0 0

ParameterEstimate95% Bootstrap C.I.Standard errorRelative bias %

๐œ‚ 1 2 = 0 . 4 0 0.40000.3996, 0.40040.00023โ€”
๐œ‚ 1 3 = 0 . 5 0 0.50020.4999, 0.50050.00023โ€”
๐œ‚ 1 4 = 0 . 6 0 0.60020.5999, 0.60050.00022โ€”
๐œ‚ 2 3 = 0 . 4 0 0.39970.3994, 0.40010.00022โ€”
๐œ‚ 2 4 = 0 . 5 0 0.50010.4998, 0.50040.00022โ€”
๐œ‚ 3 4 = 0 . 4 0 0.40000.3996, 0.40030.00021โ€”

5. Discussion and Conclusion

One of the advantages that ๐ฟ-moment ratios have over conventional moment-based estimators is that they can be far less biased when sampling is from distributions with heavy tails [13, 19]. Inspection of the simulation results in Tables 6 and 7 clearly indicates that this is the case. Specifically, the superiority that estimates of ๐ฟ-moment ratios (๐œ3,๐œ4) have over their corresponding conventional moment-based counterparts (๐›ผ3,๐›ผ4) is obvious. For example, with samples of size ๐‘›=25 the estimates of skew and kurtosis for Distribution 1 were, on average, only 64.07% and 10.22% of their associated population parameters, whereas the estimates of ๐ฟ-skew and ๐ฟ-kurtosis were 98.6% and 99.35% of their respective parameters. It is also evident from comparing Tables 6 and 7 that ๐ฟ-skew and ๐ฟ-kurtosis are more efficient estimators as their standard errors are substantially smaller and more stable than the conventional moment-based estimators of skew and kurtosis.

Presented in Tables 8, 9, 10, 11 are the results associated with the conventional Pearson and ๐ฟ-correlations. Overall inspection of these tables indicates that the ๐ฟ-correlation is superior to the Pearson correlation in terms of relative bias. For example, for strong correlations (๐‘›=25) the relative bias for the two heavy tailed distributions (i.e., distributions 1 and 2) was 8.66% for the Pearson correlation compared to only 1.17% for the ๐ฟ-correlation. Further, for large sample sizes (๐‘›=1000), the ๐ฟ-correlation bootstrap C.I.s contained all of the population parameters, whereas the Pearson correlation C.I.s contained none of the parameters. It is also noted that the variability of the ๐ฟ-correlation appears to be more stable than that of the Pearson correlation both within and across the different conditions.

In summary, the new ๐ฟ-moment-based procedure is an attractive alternative to the traditional conventional moment-based procedure. In particular, the ๐ฟ-moment-based double GLD procedure has distinct advantages when distributions with heavy tails are used. Finally, we note that Mathematica Version 8.0.1 [20] source code is available from the authors for implementing both the conventional and new ๐ฟ-moment-based procedures.

Appendix

System of Conventional Moment-Based Equations for Double GLDs

The moments (๐œ‡๐‘Ÿ=1,โ€ฆ,4) based on (1.8) can be determined from๐œ‡๐‘Ÿ=๎€œ01/2๐‘ž๎€ท๐‘ข,๐œ†โ„’๎€ธ,๐›ฟ๐‘Ÿ๎€œ๐‘‘๐‘ข+11/2๐‘ž๎€ท๐‘ข,๐œ†โ„›๎€ธ,๐›ฟ๐‘Ÿ๐‘‘๐‘ข.(A.1) The mean, variance, skew, and kurtosis are in general (e.g., [24])๐›ผ1=๐œ‡1,๐›ผ22=๐œ‡2โˆ’๐œ‡21,๐›ผ3=๎€ท๐œ‡3โˆ’3๐œ‡2๐œ‡1+2๐œ‡31๎€ธ๐œ‡23/2,๐›ผ4=๎€ท๐œ‡4โˆ’4๐œ‡3๐œ‡1โˆ’3๐œ‡22+12๐œ‡2๐œ‡21โˆ’6๐œ‡41๎€ธ๐œ‡22.(A.2) In terms of the double GLD in Figure 2, setting โˆš๐›ฟ=1/2๐œ‹ in (A.1) for the unit normal distribution, the moments associated with the location and scale parameters in (A.2) are ๐›ผ1=๐œ‡1=12๎‚™๐œ‹2๎ƒฉ1โˆ’2๐œ†โ„’๐œ†โ„’+๐œ†2โ„’+2๐œ†โ„›โˆ’1๐œ†โ„›+๐œ†2โ„›๎ƒช,๐œ‡2=๐œ‹4๐œ†โ„’๎€ท1/1+2๐œ†โ„’๎€ทโˆ’2๐›ฝ1/2,1+๐œ†โ„’,1+๐œ†โ„’๎€ธ๎€ธ8๐œ†2โ„’+๐œ‹4๐œ†โ„›๎€ท1/1+2๐œ†โ„›๎€ท+2๐›ฝ1/2,1+๐œ†โ„›,1+๐œ†โ„›๎€ธ๎€ธ8๐œ†2โ„›โˆ’๐œ‹3/2ฮ“๎€ท๐œ†โ„›๎€ธ8๐œ†โ„›ฮ“๎€ท3/2+๐œ†โ„›๎€ธ(A.3) and the moments related to skew and kurtosis are as follows: ๐œ‡3=(3)8๐œ†โ„’๐›ฝ๎€ท1/2,1+๐œ†โ„’,1+2๐œ†โ„’๎€ธโˆš162๐œ†3โ„’๐œ‹3/2โˆ’(3)2๐œ†โ„’๐œ†โ„’ฮ“๎€ท2๐œ†โ„’๎€ธ๎€ทH2F1Regโˆ’๐œ†โ„’,1+2๐œ†โ„’,2+2๐œ†โ„’๎€ธ,1/2โˆš162๐œ†3โ„’๐œ‹3/2+โˆ’๎€ท8๐œ†โ„’โˆ’1๎€ธ๐œ†3โ„›/๐œ†3โ„’๎€ท1+3๐œ†โ„’๎€ธโˆš162๐œ†3โ„›๐œ‹3/2+๎€ท8๐œ†โ„›โˆ’1๎€ธ/๎€ท1+3๐œ†โ„›๎€ธโˆš162๐œ†3โ„›๐œ‹3/2โˆ’(3)8๐œ†โ„›๐›ฝ๎€ท1/2,1+๐œ†โ„›,1+2๐œ†โ„›๎€ธโˆš162๐œ†3โ„›๐œ‹3/2+(3)2๐œ†โ„›๐œ†โ„›ฮ“๎€ท2๐œ†โ„›๎€ธ๎€ทH2F1Regโˆ’๐œ†โ„›,1+2๐œ†โ„›,2+2๐œ†โ„›๎€ธ,1/2โˆš162๐œ†3โ„›๐œ‹3/2,(A.4)๐œ‡4=24๐œ†โ„’โˆ’6๐œ‹2๐œ†4โ„’๎€ท1+4๐œ†โ„’๎€ธ+24๐œ†โ„›โˆ’6๐œ‹2๐œ†4โ„›๎€ท1+4๐œ†โ„›๎€ธโˆ’24๐œ†โ„’โˆ’4๐œ‹2๐›ฝ๎€ท1/2,1+๐œ†โ„’,1+3๐œ†โ„’๎€ธ๐œ†4โ„’โˆ’24๐œ†โ„’โˆ’4๐œ‹2๐›ฝ๎€ท1/2,1+3๐œ†โ„’,1+๐œ†โ„’๎€ธ๐œ†4โ„’+24๐œ†โ„›โˆ’4๐œ‹2๐›ฝ๎€ท1/2,1+๐œ†โ„›,1+3๐œ†โ„›๎€ธ๐œ†4โ„›โˆ’(3)24๐œ†โ„›โˆ’5๐œ‹2๐›ฝ๎€ท1/2,1+2๐œ†โ„›,1+2๐œ†โ„›๎€ธ๐œ†4โ„›+24๐œ†โ„›โˆ’4๐œ‹2๐›ฝ๎€ท1/2,1+3๐œ†โ„›,1+๐œ†โ„›๎€ธ๐œ†4โ„›+3๐œ‹5/2ฮ“๎€ท1+2๐œ†โ„›๎€ธ64๐œ†4โ„›ฮ“๎€ท3/2+2๐œ†โ„›๎€ธโˆ’24๐œ†โ„›โˆ’3๐œ‹2ฮ“๎€ท๐œ†โ„›๎€ธฮ“๎€ท1+3๐œ†โ„›๎€ธ๐œ†3โ„›ฮ“๎€ท2+4๐œ†โ„›๎€ธ+(3)22๐œ†โ„’โˆ’6๐œ‹2ฮ“๎€ท1+2๐œ†โ„’๎€ธ๎€ทH2F1Regโˆ’2๐œ†โ„’,1+2๐œ†โ„’,2+2๐œ†โ„’๎€ธ,1/2๐œ†4โ„’,(A.5) where H2F1Reg(),ฮ“(), and ๐›ฝ() are the regularized hypergeometric, gamma, and incomplete beta functions, respectively.

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Copyright © 2012 Todd C. Headrick and Mohan D. Pant. 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.

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