Table of Contents
ISRN Applied Mathematics
Volumeย 2012, Article IDย 980153, 20 pages
Research Article

Characterizing Tukey โ„Ž and โ„Žโ„Ž-Distributions through ๐ฟ-Moments and the ๐ฟ-Correlation

Section on Statistics and Measurement, Department EPSE, Southern Illinois University Carbondale, P.O. Box 4618, 222-J Wham Building, Carbondale, IL 62901-4618, USA

Received 10 October 2011; Accepted 31 October 2011

Academic Editors: M.ย Cho and K.ย Karamanos

Copyright ยฉ 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.


This paper introduces the Tukey family of symmetric โ„Ž and asymmetric โ„Žโ„Ž-distributions in the contexts of univariate ๐ฟ-moments and the ๐ฟ-correlation. Included is the development of a procedure for specifying nonnormal distributions with controlled degrees of ๐ฟ-skew, ๐ฟ-kurtosis, and ๐ฟ-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 ๐ฟ-skew, ๐ฟ-kurtosis, and ๐ฟ-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 Tukey families of โ„Ž (or the ๐‘”-and-โ„Ž and Generalized Pareto) distributions (e.g., [1โ€“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 of โ„Ž-distributions is based on the transformation๎‚ต๐‘ž(๐‘)=๐‘expโ„Ž๐‘22๎‚ถ,(1.1) where ๐‘โˆผ๐‘–.๐‘–.๐‘‘. โ€‰๐‘(0,1). Equation (1.1) produces symmetric โ„Ž-distributions where the parameter โ„Ž 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)] ๐‘“๐‘ž(๐‘)(๐‘ž(๐‘ง))=๎‚ต๐‘“(๐‘ง)=๐‘ž(๐‘ง),๐œ™(๐‘ง)๐‘ž๎…ž๎‚ถ,(๐‘ง)(1.2)๐น๐‘ž(๐‘)(๐‘ž(๐‘ง))=๐น(๐‘ง)=(๐‘ž(๐‘ง),ฮฆ(๐‘ง)),(1.3) where ๐‘“โˆถโ„œโ†ฆโ„œ2 and ๐นโˆถโ„œโ†ฆโ„œ2 are the parametric forms of the pdf and cdf with the mappings ๐‘งโ†ฆ(๐‘ฅ,๐‘ฆ) and ๐‘งโ†ฆ(๐‘ฅ,๐‘ฃ) with ๐‘ฅ=๐‘ž(๐‘ง), ๐‘ฆ=๐œ™(๐‘ง)/๐‘ž๎…ž(๐‘ง), ๐‘ฃ=ฮฆ(๐‘ง), and where ๐œ™(๐‘ง) and ฮฆ(๐‘ง) are the standard normal pdf and cdf, respectively. It is assumed that ๐‘ž๎…ž(๐‘ง)>0 in (1.2) to ensure a valid pdf, that is, the transformation in (1.1) is strictly increasing, which requires โ„Žโ‰ฅ0. Further, if ๐‘ž(๐‘) in (1.1) has a valid pdf where the ๐‘˜-th order moment exists (for ๐‘˜=1,2,โ€ฆ), then โ„Ž must be bounded such that 0โ‰คโ„Ž<1/๐‘˜. That is, a distribution will not have a first moment (or mean) for โ„Žโ‰ฅ1 [2, 3].

The variance (๐›ผ22) and kurtosis (๐›ผ4) of a distribution associated with (1.1) can be determined from [3, Equations (32), (36)] ๐›ผ22=1(1โˆ’2โ„Ž)3/2,๐›ผ4=3(1โˆ’2โ„Ž)3๎‚ต1(1โˆ’4โ„Ž)5/2+1(2โ„Žโˆ’1)3๎‚ถ,(1.4) where the mean (๐›ผ1) and skew (๐›ผ3) of a distribution are both zero.

One of the extensions of (1.1)โ€“(1.3) is the two parameter โ„Žโ„Ž family of distributions introduced by Morgenthaler and Tukey [2]. More specifically, the โ„Žโ„Ž family includes asymmetric distributions with heavy tails based on the transformation โŽงโŽชโŽจโŽชโŽฉ๎‚ตโ„Ž๐‘ž(๐‘)=๐‘expโ„’๐‘22๎‚ถ,for๎‚ตโ„Ž๐‘โ‰ค0,๐‘expโ„›๐‘22๎‚ถ,for๐‘โ‰ฅ0,(1.5) where โ„Žโ„’(โ„Žโ„›) 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 โ„Žโ„Ž distributions have to be considered separately as they are weighted differently that is, in general, โ„Žโ„’โ‰ โ„Žโ„›. For example, Figure 1 gives an example of an โ„Žโ„Ž-distribution based on matching the values of ๐›ผ3 and ๐›ผ4 associated with a noncentral Student ๐‘ก(df=5,๐›ฟ=1) distribution. The values of โ„Žโ„’ and โ„Žโ„› in Figure 1 were computed by simultaneously solving (A.3) and (A.4) in the appendix.

Figure 1: Graph of an โ„Žโ„Ž-distribution based on matching the conventional moments of a noncentral Student ๐‘ก(df=5,๐›ฟ=1) distribution. The values of โ„Žโ„’ and โ„Žโ„› were determined by solving equations (A.3) and (A.4) in the appendix. The estimates (๎๐›ผ3,4; ฬ‚๐œ3,4) and bootstrap confidence intervals (C.I.s) were based on resampling 25,000 statistics. Each sample statistic was based on a sample size of ๐‘›=250.

Conventional moment-based-estimators (e.g., ๎๐›ผ3, ๎๐›ผ4) 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 ๎๐›ผ3 and ๎๐›ผ4 are only 78.83% and 42.81% of their associated population parameters. Note that each estimate of ๎๐›ผ3 and ๎๐›ผ4 in Figure 1 were calculated based on sample sizes of ๐‘›=250 and the formulae currently used by most commercial software packages such as SAS, SPSS, and Minitab for computing skew and kurtosis.

However, ๐ฟ-moment-based estimators such as ๐ฟ-skew and ๐ฟ-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 ๐ฟ-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 ฬ‚๐œ3 and ฬ‚๐œ4 in Figure 1 are relatively much closer to their respective parameters with much smaller standard errors than their corresponding conventional moment based analogs (๎๐›ผ3, ๎๐›ผ4). More specifically, the estimates of ฬ‚๐œ3 and ฬ‚๐œ4 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 simulating โ„Ž-(or ๐‘”-and-โ„Ž) distributions with specified Pearson correlation structures [11, pages 140โ€“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 โ„Ž-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 between โ„Ž-distributionsโ€”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 [โˆ’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 โ„Ž-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 the โ„Ž and โ„Žโ„Ž families of distributions 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 nonnormal symmetric โ„Ž and asymmetric โ„Žโ„Ž distributions with specified ๐ฟ-moments and ๐ฟ-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 [โˆ’1,+1] as it is shown that the absolute value of an IC will be less than (or equal to) its associated specified ๐ฟ-correlation.

The remainder of the paper is outlined as follows. In Section 2, a summary of univariate ๐ฟ-moment theory is provided and the derivations of the systems of equations for the โ„Ž and โ„Žโ„Ž distributions are provided for modeling or simulating nonnormal distributions with specified values of ๐ฟ-skew and ๐ฟ-kurtosis. In Section 3, the coefficient of ๐ฟ-correlation is introduced and the equations are subsequently derived for determining ICs for specified ๐ฟ-correlations between nonnormal โ„Ž or โ„Žโ„Ž distributions. In Section 4, the steps for implementing the proposed ๐ฟ-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

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 [10, pages 20โ€“22] ๐œ†1๎€บ๐‘‹=๐ธ1โˆถ1๎€ป=๐›ฝ0,๐œ†2=12๐ธ๎€บ๐‘‹2โˆถ2โˆ’๐‘‹1โˆถ2๎€ป=2๐›ฝ1โˆ’๐›ฝ0,(2.1)๐œ†3=13๐ธ๎€บ๐‘‹3โˆถ3โˆ’2๐‘‹2โˆถ3+๐‘‹1โˆถ3๎€ป=6๐›ฝ2โˆ’6๐›ฝ1+๐›ฝ0,(2.2)๐œ†4=14๐ธ๎€บ๐‘‹4โˆถ4โˆ’3๐‘‹3โˆถ4+3๐‘‹2โˆถ4โˆ’๐‘‹1โˆถ4๎€ป=20๐›ฝ3โˆ’30๐›ฝ2+12๐›ฝ1โˆ’๐›ฝ0,(2.3) where the ๐›ฝ๐‘– are determined from ๐›ฝ๐‘–=๎€œ๐‘ฅ{๐น(๐‘ฅ)}๐‘–๐‘“(๐‘ฅ)๐‘‘๐‘ฅ,(2.4) where ๐‘–=0,โ€ฆ,3. The coefficients associated with ๐›ฝ๐‘– in (2.4) are obtained from shifted orthogonal Legendre polynomials and are computed as shown in [10, page 20] or in [15].

The ๐ฟ-moments ๐œ†1 and ๐œ†2 in (2.1) are measures of location and scale and are the arithmetic mean and one-half the coefficient of mean difference (or Giniโ€™s index of spread), respectively. Higher-order ๐ฟ-moments are transformed to dimensionless quantities referred to as ๐ฟ-moment ratios defined as ๐œ๐‘Ÿ=๐œ†๐‘Ÿ/๐œ†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 [16] ๎€ท5๐œ23๎€ธโˆ’14<๐œ4<1.(2.5)

2.2. ๐ฟ-Moments for Symmetric โ„Ž-Distributions

The family of โ„Ž-distributions based on the method of ๐ฟ-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 โ„Ž parameter on the interval 0โ‰คโ„Ž<1 for any distribution with finite ๐‘˜-order ๐ฟ-moments rather than 0โ‰คโ„Ž<1/๐‘˜ for the ๐‘˜th-order conventional moment to exist. This advantage is attributed to Hoskingโ€™s Theorem 1 [9] which states that if the mean (๐œ†1) exists, then all other ๐ฟ-moments will have finite expectations.

We begin the derivation for symmetric โ„Ž-distributions in the context of ๐ฟ-moments by defining the probability-weighted moments based on (2.4) in terms of ๐‘ž(๐‘ง) in (1.1) and the standard normal pdf and cdf as ๐›ฝ๐‘–=๎€œ+โˆžโˆ’โˆž๐‘ž(๐‘ง){ฮฆ(๐‘ง)}๐‘–๐œ™(๐‘ง)๐‘‘๐‘ง.(2.6) Integrating (2.6) for ๐‘–=0,1,2 gives ๐œ†1,=0(2.7)๐œ†2โˆš=โˆ’2โˆšโˆš๐œ‹(โ„Žโˆ’1),2โˆ’โ„Ž(2.8)๐œ†3=๐œ3=0.(2.9)

The fourth ๐ฟ-moment ๐œ†4 (and ๐œ4) 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 ๐‘›=4 as [17]๐ธ๎€บ๐‘ž(๐‘)๐‘—โˆถ4๎€ป=14โŽ›โŽœโŽœโŽ3โŽžโŽŸโŽŸโŽ ๎€œ๐‘—โˆ’1โˆž0๎€ท[]๐‘ž(๐‘ง)๐œ‘(๐‘ง)1+ฮจ(๐‘ง)๐‘—โˆ’1[]1โˆ’ฮจ(๐‘ง)4โˆ’๐‘—โˆ’[]1โˆ’ฮจ(๐‘ง)๐‘—โˆ’1[]1+ฮจ(๐‘ง)4โˆ’๐‘—๎€ธ๐‘‘๐‘ง,(2.10) where ๐œ‘(๐‘ง)=2๐œ™(๐‘ง) and ฮจ(๐‘ง)=2ฮฆ(๐‘ง)โˆ’1 are the pdf and cdf of the folded unit normal distribution at ๐‘ง=0, respectively. The relevant expansions of the polynomial in (2.10) are ๐ธ๎€บ๐‘ž(๐‘)3โˆถ4๎€ป๎€บ=โˆ’๐ธ๐‘ž(๐‘)2โˆถ4๎€ป=๎‚€32๎‚๎€ท๐ผ1โˆ’๐ผ3๎€ธ,๐ธ๎€บ๐‘ž(๐‘)4โˆถ4๎€ป๎€บ=โˆ’๐ธ๐‘ž(๐‘)1โˆถ4๎€ป=๎‚€12๎‚๎€ท3๐ผ1+๐ผ3๎€ธ,(2.11) where the expectations in (2.11) is linear combinations associated with the integrals denoted as ๐ผ1 and ๐ผ3. The specific expressions for ๐ผ1 and ๐ผ3 are ๐ผ1=๐›ฟโ‹…๐œ†2=๎ƒฉ๎‚™โˆ’(โ„Žโˆ’1)1โˆ’โ„Ž2๎ƒชโ‹…๎ƒฉโˆ’โˆš2โˆšโˆš๐œ‹(โ„Žโˆ’1)๎ƒช=12โˆ’โ„Žโˆš๐œ‹,(2.12)๐ผ3=๎€œโˆž0[]๐‘ž(๐‘ง)๐œ‘(๐‘ง)ฮจ(๐‘ง)3๐‘‘๐‘ง,(2.13) where it is convenient to use ๐›ฟ in (2.12) to standardize ๐ผ1 (๐œ†2) to the unit normal distribution. Equation (2.13) may be integrated by parts based on ฮจโ€ฒ(๐‘ง)=๐œ‘(๐‘ง), ๐œ‘๎…ž(๐‘ง)=โˆ’๐œ‘(๐‘ง)๐‘ง, and noting that ฮจ(0)=0 and lim๐‘งโ†’+โˆž๐œ‘(๐‘ง)=0. As such, we have ๐ผ3๎€œ=3โˆž0[]๐œ‰(๐‘ง)๐œ‘(๐‘ง)ฮจ(๐‘ง)2๐‘‘๐‘ง,(2.14) where the expression ๐œ‰(๐‘ง) is ๎€œ๎€ฝ๐œ‰(๐‘ง)=โˆ’๐›ฟ๐‘ž(๐‘ง)๐œ‘(๐‘ง)๐‘‘๐‘ง=exp(1/2)(โ„Žโˆ’1)๐‘ง2๎€พโˆš2โˆ’โ„Žโˆš๐œ‹.(2.15) Let us first consider the expression [ฮจ(๐‘ง)]2 in (2.14), which can be expressed as []ฮจ(๐‘ง)2=2๐œ‹๎‚ธ๎€œ๐‘ง0๎‚†โˆ’1exp2๐‘ข2๎‚‡๎‚น๐‘‘๐‘ข2=2๐œ‹๎€๐‘ง0๎‚†โˆ’1exp2๎€ท๐‘ง21+๐‘ง22๎€ธ๎‚‡๐‘‘๐‘ง1๐‘‘๐‘ง24=1โˆ’๐œ‹๎€œ0๐œ‹/4๎‚†โˆ’1exp2๐‘ง2sec2๐œƒ1๎‚‡๐‘‘๐œƒ1.(2.16) Substituting (2.16) into (2.14) and, using Lichtensteinโ€™s Theorem [18], and integrating first with respect to ๐‘ง yield โˆš๐œ‹๎€œโˆž0๐œ‰(๐‘ง)๐œ‘(๐‘ง)2๎‚†โˆ’1exp2๐‘ง2sec2๐œƒ1๎‚‡โˆš๐‘‘๐‘ง=4โˆ’2โ„Žโˆš2โˆš2โˆ’โ„Ž+sec2๐œƒ1.(2.17) Using (2.17), the integral in (2.14) is expressed as ๐ผ3=3โˆš๐œ‹๎ƒฏ41โˆ’๐œ‹๎€œ0๐œ‹/4โˆš4โˆ’2โ„Žโˆš2โˆš2โˆ’โ„Ž+sec2๐œƒ1๎ƒฐ๐‘‘๐œƒ1=3โˆš๐œ‹โˆ’12tanโˆ’1๎€บ(โ„Žโˆ’4/โ„Žโˆ’2)โˆ’1/2๎€ป๐œ‹3/2.(2.18) Hence, using (2.3) and (2.11) and (2.12), ๐ฟ-kurtosis can be expressed as ๐œ4=6โˆ’30tanโˆ’1๎€บ(โ„Žโˆ’4/โ„Žโˆ’2)โˆ’1/2๎€ป๐œ‹,(2.19) where 0โ‰คโ„Ž<1. Whence, it follows that we have a convenient closed formed solution for the parameter โ„Ž as โ„Ž=3โˆ’sec๎‚ƒ๐œ‹๎€ท๐œ154๎€ธ๎‚„.โˆ’6(2.20) Equation (2.19) has a lower limit of ๐œ4โ‰ˆ0.1226(โ„Ž=0) that is equivalent to the normal distribution and an upper limit (๐œ4โ†’1; โ„Žโ†’1) that is equivalent to the Cauchy or ๐‘ก(df=1) distribution. Figure 2(d) gives an example of a symmetric โ„Ž-distribution with๐ฟ-kurtosis (๐œ4) of a logistic distribution.

Figure 2: Three asymmetric โ„Žโ„Ž-distributions (a)โ€“(c) and one symmetric โ„Ž distribution (d) with their conventional and ๐ฟ-moment parameters of skew (๐›ผ3) and ๐ฟ-skew (๐œ3), kurtosis (๐›ผ4) and ๐ฟ-kurtosis (๐œ4), and corresponding shape parameters for (1.5) and (1.1).
2.3. ๐ฟ-Moments for Asymmetric โ„Žโ„Ž-Distributions

The derivation of the ๐ฟ-moments for asymmetric โ„Žโ„Ž-distributions associated with (1.5) begins with determining the probability-weighted moments ๐›ฝ๐‘– in (2.6) by separately evaluating and summing two integrals as ๐›ฝ๐‘–=๐ผโ„’๐‘–๎€ทโ„Žโ„’๎€ธ+๐ผโ„›๐‘–๎€ทโ„Žโ„›๎€ธ=๎€œ0โˆ’โˆž๐‘ž๎€ท๐‘ง,โ„Žโ„’๎€ธ{ฮฆ(๐‘ง)}๐‘–๎€œ๐œ™(๐‘ง)๐‘‘๐‘ง+0+โˆž๐‘ž๎€ท๐‘ง,โ„Žโ„›๎€ธ{ฮฆ(๐‘ง)}๐‘–๐œ™(๐‘ง)๐‘‘๐‘ง.(2.21) As such, using (2.1) and (2.21), it is straight-forward to obtain ๐›ฝ0, ๐›ฝ1, and the first two ๐ฟ-moments as ๐œ†1=1โˆš๎€ทโ„Ž2๐œ‹โ„’๎€ธ+1โˆ’1โˆš๎€ท2๐œ‹1โˆ’โ„Žโ„›๎€ธ,๐œ†2=โˆš2โˆ’โ„Žโ„’+โˆš2โˆ’โ„Žโ„›โˆ’โ„Žโ„’โˆš2โˆ’โ„Žโ„’โˆ’โ„Žโ„›โˆš2โˆ’โ„Žโ„›โˆš๎€ทโ„Ž2๐œ‹โ„’โ„Žโˆ’1๎€ธ๎€ทโ„›๎€ธ๎”โˆ’1๎€ทโ„Žโ„’โ„Žโˆ’2๎€ธ๎€ทโ„›๎€ธ.โˆ’2(2.22) In terms of deriving ๐œ†3 and ๐œ†4, it is convenient to consider ๐›ฝ2 in (2.2) as ๐›ฝ2=๐ผโ„’2๎€ทโ„Žโ„’๎€ธ+๐ผโ„›2๎€ทโ„Žโ„›๎€ธ=๐ผโ„’2๎€ทโ„Žโ„’๎€ธ+๎ƒฉโˆ’๐ผโ„’2๎€ทโ„Žโ„›๎€ธ+๐œ†2๎€ทโ„Žโ„›๎€ธ2๎ƒช,(2.23) where ๐œ†2(โ„Žโ„›) can be obtained from (2.8). Thus, it is only necessary to determine ๐ผโ„’2(โ„Žโ„’) as ๐ผโ„’2๎€ทโ„Žโ„’๎€ธ=๎€œ0โˆ’โˆž๐‘ž๎€ท๐‘ง,โ„Žโ„’๎€ธ{ฮฆ(๐‘ง)}2๎€œ๐œ™(๐‘ง)๐‘‘๐‘ง=0โˆ’โˆž๐‘งโˆš๎ƒฏ๎€ทโ„Ž2๐œ‹expโ„’๎€ธ๐‘งโˆ’122๎ƒฐ{ฮฆ(๐‘ง)}2=1๐‘‘๐‘งโˆš๎€ทโ„Ž2๐œ‹โ„’๎€ธ๎€œโˆ’10โˆ’โˆž{ฮฆ(๐‘ง)}2๎ƒฏ๎€ทโ„Ž๐‘‘expโ„’๎€ธ๐‘งโˆ’122๎ƒฐ=1โˆš๎€ทโ„Ž2๐œ‹โ„’๎€ธโŽกโŽขโŽขโŽฃโˆ’1{ฮฆ(๐‘ง)}2๎ƒฏ๎€ทโ„Žexpโ„’๎€ธ๐‘งโˆ’122๎ƒฐ|||||0โˆ’โˆžโˆ’๎€œ0โˆ’โˆž1โˆš๎ƒฏ๎€ทโ„Ž2๐œ‹expโ„’๎€ธ๐‘งโˆ’222๎ƒฐโŽคโŽฅโŽฅโŽฆ=2ฮฆ(๐‘ง)๐‘‘๐‘งโˆ’14โˆš๎€ท2๐œ‹1โˆ’โ„Žโ„’๎€ธ+2โˆšโˆš2๐œ‹2โˆ’โ„Žโ„’๎€ท1โˆ’โ„Žโ„’๎€ธ๎€œ0โˆ’โˆžโˆš2โˆ’โ„Žโ„’โˆš๎ƒฏโˆ’๎€ท2๐œ‹exp2โˆ’โ„Žโ„’๎€ธ๐‘ง22๎ƒฐฮฆ(๐‘ง)๐‘‘๐‘ง.(2.24) If we let ๐‘‹โˆผ๐‘(0,1/(2โˆ’โ„Žโ„’)) and โˆผ๐‘(0,1), where ๐‘‹ and ๐‘Œ are independent such that โˆš(๐‘‹/2โˆ’โ„Žโ„’,๐‘Œ) jointly follow the standard bivariate normal distribution, then the integral in the last part of (2.24) is ๎€œ0โˆ’โˆžโˆš2โˆ’โ„Žโ„’โˆš๎ƒฏโˆ’๎€ท2๐œ‹exp2โˆ’โ„Žโ„’๎€ธ๐‘ง22๎ƒฐ=ฮฆ(๐‘ง)๐‘‘๐‘งPr={๐‘‹<0,๐‘Œ<๐‘‹}Pr๎ƒฏ๐‘‹โˆš2โˆ’โ„Žโ„’โˆš<0,๐‘Œ<2โˆ’โ„Žโ„’๐‘‹โˆš2โˆ’โ„Žโ„’๎ƒฐ=14โŽ›โŽœโŽœโŽ21โˆ’cotโˆ’1๎‚€โˆš2โˆ’โ„Žโ„’๎‚๐œ‹โŽžโŽŸโŽŸโŽ ,(2.25) where we are calculating the proportion of area between the ๐‘ฆ-axis and the line โˆš๐‘ฆ=๐‘ฅ(2โˆ’โ„Žโ„’) 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 ๐ผโ„’2๎€ทโ„Žโ„’๎€ธ=14๎€ท1โˆ’โ„Žโ„’๎€ธโˆšโŽกโŽขโŽขโŽฃ22๐œ‹โˆš2โˆ’โ„Žโ„’โŽ›โŽœโŽœโŽ21โˆ’cotโˆ’1๎‚€โˆš2โˆ’โ„Žโ„’๎‚๐œ‹โŽžโŽŸโŽŸโŽ โŽคโŽฅโŽฅโŽฆ.โˆ’1(2.26) Hence, given (2.26) and (2.19), we can solve for ๐›ฝ2, ๐›ฝ3, and subsequently obtain the expressions for ๐ฟ-skew (๐œ3) and ๐ฟ-kurtosis (๐œ4) as ๐œ3=๎‚†โˆš122โˆ’โ„Žโ„’๎€ทโ„Žโ„›โ„Žโˆ’2๎€ธ๎€ทโ„›๎€ธโˆ’1cotโˆ’1๎‚€โˆš2โˆ’โ„Žโ„’๎‚๎€ทโ„Žโˆ’๐œ‹โ„’โˆ’โ„Žโ„›โ„Ž๎€ธ๎€ทโ„’โ„Žโˆ’2๎€ธ๎€ทโ„›๎€ธโˆšโˆ’2โˆ’122โˆ’โ„Žโ„›๎€ทโ„Žโ„’โ„Žโˆ’2๎€ธ๎€ทโ„’๎€ธโˆ’1cotโˆ’1๎‚€โˆš2โˆ’โ„Žโ„›๎‚ป๎”๎‚๎‚‡๎‚‚2๐œ‹๎€ทโ„Žโ„’โ„Žโˆ’2๎€ธ๎€ทโ„›๎€ธ๎‚€โ„Žโˆ’2โ„’โˆš2โˆ’โ„Žโ„’โˆ’โˆš2โˆ’โ„Žโ„’โˆ’โˆš2โˆ’โ„Žโ„›+โ„Žโ„›โˆš2โˆ’โ„Žโ„›๎‚๎‚ผ,๐œ4=๎ƒฏ๎‚ตโ„Ž6๐œ‹โ„›๎”๎€ทโ„Žโ„’โ„Žโˆ’4๎€ธ๎€ทโ„’โ„Žโˆ’2๎€ธ๎€ทโ„’โ„Žโˆ’1๎€ธ๎€ทโ„›๎€ธ๎”โˆ’2โˆ’2๎€ทโ„Žโ„’โ„Žโˆ’4๎€ธ๎€ทโ„’๎€ธโˆ’1โˆ’โ„Žโ„’๎€ทโ„Žโ„’๎€ธ๎”โˆ’3๎€ทโ„Žโ„’โ„Žโˆ’4๎€ธ๎€ทโ„’๎€ธโˆ’๎”โˆ’1๎€ทโ„Žโ„’โ„Žโˆ’4๎€ธ๎€ทโ„’โ„Žโˆ’2๎€ธ๎€ทโ„’โ„Žโˆ’1๎€ธ๎€ทโ„›๎€ธ๎‚ถ๎€ทโ„Žโˆ’2+30โ„’๎€ธ๎ƒŽโˆ’1๎€ทโ„Žโ„’โ„Žโˆ’4๎€ธ๎€ทโ„’โ„Žโˆ’2๎€ธ๎€ทโ„›๎€ธโˆ’2โ„Žโ„’๎€ทโ„Žโˆ’1โ„›๎€ธโˆ’1tanโˆ’1๎ƒฉ๎ƒŽ21+โ„Žโ„’๎ƒช๎€ทโ„Žโˆ’4+30โ„’๎€ธ๎”โˆ’2๎€ทโ„Žโ„’โ„Žโˆ’4๎€ธ๎€ทโ„’โ„Žโˆ’1๎€ธ๎€ทโ„’๎€ธโˆ’1tanโˆ’1๎ƒฉ๎ƒŽ21+โ„Žโ„›๎‚‚๎‚ป๐œ‹๎”โˆ’4๎ƒช๎ƒฐ๎€ท4โˆ’โ„Žโ„’๎€ธ๎€ท2โˆ’โ„Žโ„’๎€ธ๎€ท1โˆ’โ„Žโ„’๎€ธ๎‚€โ„Žโ„’โˆš2โˆ’โ„Žโ„’โˆ’โˆš2โˆ’โ„Žโ„’โˆ’โˆš2โˆ’โ„Žโ„›+โ„Žโ„›โˆš2โˆ’โ„Žโ„›๎‚๎‚ผ.(2.27) Thus, given specified values of ๐œ3 and ๐œ4, (2.27) can be numerically solved for the corresponding values of โ„Žโ„’ and โ„Žโ„›. Figures 2(a), 2(b) and 2(c) provides some examples of various โ„Žโ„Ž-distributions, which are used in the simulation portion of this study in Section 4.

3. ๐ฟ-Correlations for the โ„Ž and โ„Žโ„Ž-Distributions

The coefficient of ๐ฟ-correlation (see [19]) is introduced by considering two random variables ๐‘Œ๐‘— and ๐‘Œ๐‘˜ with 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 as ๐œ†2๎€ท๐‘Œ๐‘—๎€ธ=2Cov๎€ท๐‘Œ๐‘—๎€ท๐‘Œ,๐น๐‘—,๎€ธ๎€ธ(3.1)๐œ†2๎€ท๐‘Œ๐‘˜๎€ธ=2Cov๎€ท๐‘Œ๐‘˜๎€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‘— are ๐œ†2๎€ท๐‘Œ๐‘—,๐‘Œ๐‘˜๎€ธ=2Cov๎€ท๐‘Œ๐‘—๎€ท๐‘Œ,๐น๐‘˜,๎€ธ๎€ธ(3.3)๐œ†2๎€ท๐‘Œ๐‘˜,๐‘Œ๐‘—๎€ธ=2Cov๎€ท๐‘Œ๐‘˜๎€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 as ๐œ‚๐‘—๐‘˜=๐œ†2๎€ท๐‘Œ๐‘—,๐‘Œ๐‘˜๎€ธ๐œ†2๎€ท๐‘Œ๐‘—๎€ธ,(3.5)๐œ‚๐‘˜๐‘—=๐œ†2๎€ท๐‘Œ๐‘˜,๐‘Œ๐‘—๎€ธ๐œ†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 ๐ฟ-moment symmetric โ„Ž-distributions (0โ‰คโ„Ž<1), suppose it is desired to simulate ๐‘‡ distributions based on (1.1) with a specified ๐ฟ-correlation matrix and where each distribution has its own specified value of ๐œ4. Define ๐‘ž(๐‘๐‘—) and ๐‘ž(๐‘๐‘˜) as in (1.1), where ๐‘๐‘— and ๐‘๐‘˜ have Pearson correlation ๐œŒ๐‘—๐‘˜ and standard normal bivariate density of 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.(3.7) Using (1.1), (1.3), and (3.5) with the denominator standardized to ๐œ†2โˆš=1/๐œ‹ for the unit-normal distribution, and (3.7), the ๐ฟ-correlation of ๐‘ž(๐‘๐‘—) toward ๐‘ž(๐‘๐‘˜) can be expressed as ๐œ‚๐‘—๐‘˜โˆš=2๐œ‹Cov๎€ท๐‘ž๎€ท๐‘ง๐‘—๎€ธ,๐น๐‘ž(๐‘๐‘˜)๎€ท๐‘ž๎€ท๐‘ง๐‘˜โˆš๎€ธ๎€ธ๎€ธ=2๐œ‹Cov๎€ท๐‘ž๎€ท๐‘ง๐‘—๎€ธ๎€ท๐‘ง,ฮฆ๐‘˜โˆš๎€ธ๎€ธ=2๎€บ๐‘ž๎€ท๐‘ง๐œ‹๐ธ๐‘—๎€ธฮฆ๎€ท๐‘ง๐‘˜โˆš๎€ธ๎€ปโˆ’2๎€บ๐‘ž๎€ท๐‘ง๐œ‹๐ธ๐‘—๐ธ๎€บฮฆ๎€ท๐‘ง๎€ธ๎€ป๐‘˜โˆš๎€ธ๎€ป=2๐œ‹๎€+โˆžโˆ’โˆž๎€ท๐‘ง๐›ฟ๐‘ž๐‘—๎€ธฮฆ๎€ท๐‘ง๐‘˜๎€ธ๐‘“๐‘—๐‘˜๐‘‘๐‘ง๐‘—๐‘‘๐‘ง๐‘˜โˆšโˆ’2๎€บ๐‘ž๎€ท๐‘ง๐œ‹๐ธ๐‘—๐ธ๎€บฮฆ๎€ท๐‘ง๎€ธ๎€ป๐‘˜,๎€ธ๎€ป(3.8) where ๐›ฟ is the standardizing term in (2.12). Integrating (3.8) yields ๐œ‚๐‘—๐‘˜=๐œŒ๐‘—๐‘˜๎„ถ๎„ต๎„ต๎„ตโŽท2โˆ’โ„Ž๐‘—2+โ„Ž๐‘—๎‚€๐œŒ2๐‘—๐‘˜๎‚โˆ’2(3.9) given that ๐ธ[๐‘ž(๐‘ง๐‘—)]=0 and ๐ธ[ฮฆ(๐‘ง๐‘˜)]=1/2. Analogously, the ๐ฟ-correlation of ๐‘ž(๐‘๐‘˜) toward ๐‘ž(๐‘๐‘—) is ๐œ‚๐‘˜๐‘—=๐œŒ๐‘—๐‘˜๎„ถ๎„ต๎„ตโŽท2โˆ’โ„Ž๐‘˜2+โ„Ž๐‘˜๎‚€๐œŒ2๐‘—๐‘˜๎‚.โˆ’2(3.10)

From (3.9), the intermediate correlation (IC) ๐œŒ๐‘—๐‘˜ can be determined by simply evaluating ๐œŒ๐‘—๐‘˜โˆš=ยฑ2๎”โ„Ž๐‘—๐œ‚2๐‘—๐‘˜โˆ’๐œ‚2๐‘—๐‘˜๎”โ„Ž๐‘—๐œ‚2๐‘—๐‘˜+โ„Ž๐‘—โˆ’2(3.11) for a specified value of ๐œ‚๐‘—๐‘˜ and a given value of โ„Ž๐‘— from (2.20). Given ๐œŒ๐‘—๐‘˜ from (3.11), the ๐ฟ-correlation ๐œ‚๐‘˜๐‘— can be determined by evaluating (3.10) using the solved value of โ„Ž๐‘˜. Note the special case of where โ„Ž๐‘—=โ„Ž๐‘˜, in (3.9) and (3.10), then ๐œ‚๐‘—๐‘˜=๐œ‚๐‘˜๐‘—.

Remark 3.1. Inspection of (3.9) indicates that ๐œ‚๐‘—๐‘˜=๐œŒ๐‘—๐‘˜ when either (a) ๐‘ž(๐‘๐‘—) is standard normal that is, โ„Ž๐‘—=0, (b) ๐œŒ๐‘—๐‘˜=0, or (c) ๐œŒ๐‘—๐‘˜=1.

Remark 3.2. If the IC is such that 0<|๐œŒ๐‘—๐‘˜|<1 and 0<โ„Ž๐‘—<1 in (3.9), then we have the inequality ||๐œŒ0<๐‘—๐‘˜||<||๐œ‚๐‘—๐‘˜||<1,(3.12) as from inspecting (3.9) it is evident that [(2โˆ’โ„Ž๐‘—)/(2+โ„Ž๐‘—(๐œŒ2๐‘—๐‘˜โˆ’2))]1/2>1. Thus, solutions to ICs cannot exist outside the range of [โˆ’1,+1].
The extension of determining ICs for asymmetric โ„Žโ„Ž-distributions is analogous to the method described above for โ„Ž-distributions, where (1.5) is standardized and subsequently integrated as in (3.8) to obtain ๐œ‚๐‘—๐‘˜=12๐œŒ๐‘—๐‘˜๎„ถ๎„ต๎„ต๎„ตโŽท2โˆ’โ„Žโ„’๐‘—2+โ„Žโ„’๐‘—๎‚€๐œŒ2๐‘—๐‘˜๎‚+1โˆ’22๐œŒ๐‘—๐‘˜๎„ถ๎„ต๎„ต๎„ตโŽท2โˆ’โ„Žโ„›๐‘—2+โ„Žโ„›๐‘—๎‚€๐œŒ2๐‘—๐‘˜๎‚,โˆ’2(3.13)๐œ‚๐‘˜๐‘—=12๐œŒ๐‘—๐‘˜๎„ถ๎„ต๎„ตโŽท2โˆ’โ„Žโ„’๐‘˜2+โ„Žโ„’๐‘˜๎‚€๐œŒ2๐‘—๐‘˜๎‚+1โˆ’22๐œŒ๐‘—๐‘˜๎„ถ๎„ต๎„ตโŽท2โˆ’โ„Žโ„›๐‘˜2+โ„Žโ„›๐‘˜๎‚€๐œŒ2๐‘—๐‘˜๎‚,โˆ’2(3.14) where it is straight-forward to see that if โ„Žโ„’๐‘—=โ„Žโ„›๐‘— (or โ„Žโ„’๐‘˜=โ„Žโ„›๐‘˜), then (3.13) or (3.14) simplifies to (3.9) or (3.10) for the case of symmetric โ„Ž-distributions. The IC in (3.13) is determined by substituting the specified ๐ฟ-correlation (๐œ‚๐‘—๐‘˜) and the solved values of the parameters โ„Žโ„’๐‘— and โ„Žโ„›๐‘— (from (2.27)) into the left- and right-hand sides of (3.13), respectively, and then numerically solving for ๐œŒ๐‘—๐‘˜. 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 simulating โ„Žโ„Ž- (or โ„Ž-)โ€‰distributions with specified ๐ฟ-moments and specified ๐ฟ-correlations, we suggest the following five steps. (1)Specify the ๐ฟ-moments for ๐‘‡ transformations of the form in (1.5), that is, ๐‘ž(๐‘1),โ€ฆ,๐‘ž(๐‘๐‘‡)and obtain the parameters of โ„Žโ„’๐‘— and โ„Žโ„›๐‘—by solving equations (2.27) 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‘ž(๐‘๐‘—) toward ๐‘ž(๐‘๐‘˜), where ๐‘—<๐‘˜โˆˆ{1,2,โ€ฆ,๐‘‡}. (2) Compute the (Pearson) intermediate correlations (ICs) ๐œŒ๐‘—๐‘˜ by substituting the specified ๐ฟ-correlation ๐œ‚๐‘—๐‘˜ and the parameters of โ„Žโ„’๐‘— and โ„Žโ„›๐‘— from step (1) into the left- and right-hand sides of (3.14), respectively, and then numerically solve for ๐œŒ๐‘—๐‘˜. 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‘‰1,๐‘2=๐‘Ž12๐‘‰1+๐‘Ž22๐‘‰2,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‹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 ๐‘‡ equations of the form in (1.5), as noted in step (1), to generate the nonnormal โ„Žโ„Ž-distributions 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 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 ๐ฟ-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, Tableโ€‰โ€‰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 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 substituted into equations of the form in (1.5) to produce ๐‘ž(๐‘1),โ€ฆ,๐‘ž(๐‘4) for both procedures.

Table 1: Specified correlation matrices for the distributions in Figure 2.
Table 2: Intermediate correlations for the conventional moment procedure.
Table 3: Intermediate correlations for the ๐ฟ-moment procedure.
Table 4: Cholesky decompositions for the Conventional moment procedure.
Table 5: Cholesky decompositions for the ๐ฟ-moment procedure.

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‘˜) based on samples of sizes 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 (2.3) and (2.5) [15]. 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 (3.13) 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‘งโ€ฒ 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+ [20]. 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, and 11 and are discussed in the next section.

Table 6: Skew (๐›ผ3) and Kurtosis (๐›ผ4) results for the conventional moment procedure.
Table 7: ๐ฟ-skew (๐œ3) and ๐ฟ-kurtosis (๐œ4) results.
Table 8: Correlation (strong) results for the conventional moment procedure.
Table 9: Correlation (strong) results for the ๐ฟ-moment procedure.
Table 10: Correlation (moderate) results for the conventional moment procedure.
Table 11: Correlation (moderate) results for the ๐ฟ-moment procedure.

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 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 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 32.4% and 4.2% of their associated population parameters, whereas the estimates of ๐ฟ-skew and ๐ฟ-kurtosis were 87.9% and 96.10% 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 significantly smaller than the conventional-moment-based estimators of skew and kurtosis.

Presented in Tables 8, 9, 10, and 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 moderate correlations (๐‘›=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 ๐ฟ-correlation. 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 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 ๐ฟ-moment-based procedures.


System of Conventional Moment-Based Equations for Tukey โ„Žโ„Ž-Distributions

The equations for the mean (๐›ผ1), variance (๐›ผ22), skew (๐›ผ3), and kurtosis (๐›ผ4) for conventional moment-based โ„Žโ„Ž-distributions are given below without the details of their derivation. We would note that we derived (A.1)โ€“(A.4) based on the formulae given in Headrick et al. [3, equations (16)โ€“(18)]. ๐›ผ1=1โˆš๎€ทโ„Ž2๐œ‹โ„’๎€ธ+1โˆ’1โˆš๎€ทโ„Ž2๐œ‹โ„›๎€ธ,โˆ’1(A.1)๐›ผ22=๐ถ2=๎€ท1/1โˆ’2โ„Žโ„›๎€ธ3/2๎€ท+1/1โˆ’2โ„Žโ„’๎€ธ3/2โˆ’๎€ทโ„Žโ„’โˆ’โ„Žโ„›๎€ธ2๎€ทโ„Ž/๐œ‹โ„’๎€ธโˆ’12๎€ทโ„Žโ„›๎€ธโˆ’122,(A.2)๐›ผ3=โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ4๐œ‹๎€ท1โˆ’3โ„Žโ„›๎€ธ2โˆ’4๐œ‹๎€ท1โˆ’3โ„Žโ„’๎€ธ2โˆ’2๎€ทโ„Žโ„’โˆ’โ„Žโ„›๎€ธ3๎€ทโ„Žโ„’๎€ธโˆ’13๎€ทโ„Žโ„›๎€ธโˆ’13โˆ’๎ƒฌ๎€ทโ„Ž3๐œ‹โ„›โˆ’โ„Žโ„’๎€ธ๎ƒฉ1๎€ท1โˆ’2โ„Žโ„’๎€ธ3/2+1๎€ท1โˆ’2โ„Žโ„›๎€ธ3/2๎‚‚โ„Ž๎ƒช๎ƒญ๎€บ๎€ทโ„’โ„Žโˆ’1๎€ธ๎€ทโ„›โŽซโŽชโŽชโŽฌโŽชโŽชโŽญ๎‚‚๎€ท๐œ‹โˆ’1๎€ธ๎€ป3/2๐ถ3/2๎€ธ,(A.3)๐›ผ4โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ6โŽ›โŽœโŽœโŽœโŽœโŽ1=โˆ’3+๎€ท1โˆ’4โ„Žโ„›๎€ธ5/2+1๎€ท1โˆ’4โ„Žโ„’๎€ธ5/2โˆ’๎€ทโ„Žโ„’โˆ’โ„Žโ„›๎€ธ4๐œ‹2๎€ทโ„Žโ„’๎€ธโˆ’14๎€ทโ„Žโ„›๎€ธโˆ’14โˆ’๎€ทโ„Žโ„’โˆ’โ„Žโ„›๎€ธ2๐œ‹๎€ทโ„Žโ„’๎€ธโˆ’12๎€ทโ„Žโ„›๎€ธโˆ’12+๎ƒฉ2๎€ทโ„Žโ„’โˆ’โ„Žโ„›๎€ธ2๎ƒฉ1๎€ท1โˆ’2โ„Žโ„’๎€ธ3/2+1๎€ท1โˆ’2โ„Žโ„›๎€ธ3/2๐œ‹๎€ทโ„Ž๎ƒช๎ƒช๎‚‚๎‚€โ„’๎€ธโˆ’12๎€ทโ„Žโ„›๎€ธโˆ’12๎‚+8๎€ทโ„Žโ„’โˆ’โ„Žโ„›๎€ธ2๎€ท3โ„Žโ„’+3โ„Žโ„›๎€ธโˆ’2๎‚€๐œ‹๎€ท1โˆ’3โ„Žโ„’๎€ธ2๎€ท1โˆ’3โ„Žโ„›๎€ธ2๎€ทโ„Žโ„’โ„Žโˆ’1๎€ธ๎€ทโ„›๎€ธ๎‚โŽžโŽŸโŽŸโŽŸโŽŸโŽ โŽซโŽชโŽชโŽฌโŽชโŽชโŽญ๎‚‚๐ถโˆ’12.(A.4)


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