Abstract
This paper introduces the Tukey family of symmetric and asymmetric -distributions in the contexts of univariate -moments and the -correlation. Included is the development of a procedure for specifying nonnormal distributions with controlled degrees of -skew, -kurtosis, and -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 -skew, -kurtosis, and -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 (or the -and- and Generalized Pareto) distributions (e.g., [1–3]) are often used in various applied mathematics contexts. Some examples include modeling events associated with operational risk [4], extreme oceanic wind speeds, [5], common stock returns [6], solar flare data [7], or in the context of Monte Carlo or simulation studies, for example, regression analysis [8].
The family of -distributions is based on the transformation where . . Equation (1.1) produces symmetric -distributions where the parameter 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 , ] where and are the parametric forms of the pdf and cdf with the mappings and with , , , and where and are the standard normal pdf and cdf, respectively. It is assumed that in (1.2) to ensure a valid pdf, that is, the transformation in (1.1) is strictly increasing, which requires . Further, if in (1.1) has a valid pdf where the -th order moment exists (for ), then must be bounded such that . That is, a distribution will not have a first moment (or mean) for [2, 3].
The variance () and kurtosis () of a distribution associated with (1.1) can be determined from [3, Equations (32), (36)] where the mean () and skew () of a distribution are both zero.
One of the extensions of (1.1)–(1.3) is the two parameter family of distributions introduced by Morgenthaler and Tukey [2]. More specifically, the family includes asymmetric distributions with heavy tails based on the transformation where 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 distributions have to be considered separately as they are weighted differently that is, in general, . For example, Figure 1 gives an example of an -distribution based on matching the values of and associated with a noncentral Student distribution. The values of and in Figure 1 were computed by simultaneously solving (A.3) and (A.4) in the appendix.
Conventional moment-based-estimators (e.g., , ) 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 and are only 78.83% and 42.81% of their associated population parameters. Note that each estimate of and in Figure 1 were calculated based on sample sizes of and the formulae currently used by most commercial software packages such as SAS, SPSS, and Minitab for computing skew and kurtosis.
However, -moment-based estimators such as -skew and -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 -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 and in Figure 1 are relatively much closer to their respective parameters with much smaller standard errors than their corresponding conventional moment based analogs (, ). More specifically, the estimates of and 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 -(or -and-) distributions with specified Pearson correlation structures [11, pages 140–148] [12]. This methodology is based on conventional product moments and the popular NORTA [13] approach, which begins with generating multivariate standard normal deviates. However, the NORTA approach is not without its limitations. Specifically, one limitation arises because the Pearson correlation is not invariant under nonlinear strictly increasing transformations such as (1.1). As such, the NORTA approach must begin with the computation of an intermediate correlation (IC) matrix, which is different than the specified correlation matrix between the nonnormal -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 -distributions—unlike the more popular power method [11] which has a straight-forward equation to solve for the ICs between distributions [11, page 30] and (b) may yield solutions to ICs that are not in the range of 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 -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 and families of distributions in the contexts of -moment and -correlation theory. Specifically, the purpose of this paper is to develop the methodology and a procedure for simulating nonnormal symmetric and asymmetric distributions with specified -moments and -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 as it is shown that the absolute value of an IC will be less than (or equal to) its associated specified -correlation.
The remainder of the paper is outlined as follows. In Section 2, a summary of univariate -moment theory is provided and the derivations of the systems of equations for the and distributions are provided for modeling or simulating nonnormal distributions with specified values of -skew and -kurtosis. In Section 3, the coefficient of -correlation is introduced and the equations are subsequently derived for determining ICs for specified -correlations between nonnormal or distributions. In Section 4, the steps for implementing the proposed -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 be random variables each with continuous pdf , cdf , order statistics denoted as , and -moments defined in terms of either linear combinations of (a) expectations of order statistics or (b) probability-weighted moments (). For the purposes considered herein, the first four -moments associated with are expressed as [10, pages 20–22] where the are determined from where . The coefficients associated with in (2.4) are obtained from shifted orthogonal Legendre polynomials and are computed as shown in [10, page 20] or in [15].
The -moments and in (2.1) are measures of location and scale and are the arithmetic mean and one-half the coefficient of mean difference (or Gini’s index of spread), respectively. Higher-order -moments are transformed to dimensionless quantities referred to as -moment ratios defined as for , and where and are the analogs to the conventional measures of skew and kurtosis. In general, -moment ratios are bounded in the interval as is the index of -skew () where a symmetric distribution implies that all -moment ratios with odd subscripts are zero. Other smaller boundaries can be found for more specific cases. For example, the index of -kurtosis () has the boundary condition for continuous distributions of [16]
2.2. -Moments for Symmetric -Distributions
The family of -distributions based on the method of -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 parameter on the interval for any distribution with finite -order -moments rather than for the th-order conventional moment to exist. This advantage is attributed to Hosking’s Theorem 1 [9] which states that if the mean () exists, then all other -moments will have finite expectations.
We begin the derivation for symmetric -distributions in the context of -moments by defining the probability-weighted moments based on (2.4) in terms of in (1.1) and the standard normal pdf and cdf as Integrating (2.6) for gives
The fourth -moment (and ) 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 as [17] where and are the pdf and cdf of the folded unit normal distribution at , respectively. The relevant expansions of the polynomial in (2.10) are where the expectations in (2.11) is linear combinations associated with the integrals denoted as and . The specific expressions for and are where it is convenient to use in (2.12) to standardize () to the unit normal distribution. Equation (2.13) may be integrated by parts based on , , and noting that and . As such, we have where the expression is Let us first consider the expression in (2.14), which can be expressed as Substituting (2.16) into (2.14) and, using Lichtenstein’s Theorem [18], and integrating first with respect to yield Using (2.17), the integral in (2.14) is expressed as Hence, using (2.3) and (2.11) and (2.12), -kurtosis can be expressed as where . Whence, it follows that we have a convenient closed formed solution for the parameter as Equation (2.19) has a lower limit of ) that is equivalent to the normal distribution and an upper limit (; ) that is equivalent to the Cauchy or distribution. Figure 2(d) gives an example of a symmetric -distribution with-kurtosis () of a logistic distribution.
(a)
(b)
(c)
(d)
2.3. -Moments for Asymmetric -Distributions
The derivation of the -moments for asymmetric -distributions associated with (1.5) begins with determining the probability-weighted moments in (2.6) by separately evaluating and summing two integrals as As such, using (2.1) and (2.21), it is straight-forward to obtain , , and the first two -moments as In terms of deriving and , it is convenient to consider in (2.2) as where can be obtained from (2.8). Thus, it is only necessary to determine as If we let and , where and are independent such that jointly follow the standard bivariate normal distribution, then the integral in the last part of (2.24) is where we are calculating the proportion of area between the -axis and the line 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 Hence, given (2.26) and (2.19), we can solve for , , and subsequently obtain the expressions for -skew () and -kurtosis () as Thus, given specified values of and , (2.27) can be numerically solved for the corresponding values of and . Figures 2(a), 2(b) and 2(c) provides some examples of various -distributions, which are used in the simulation portion of this study in Section 4.
3. -Correlations for the and -Distributions
The coefficient of -correlation (see [19]) is introduced by considering two random variables and with distribution functions and , respectively. The second -moments of and can alternatively be expressed as The second -comoments of toward and toward are As such, the -correlations of toward and toward are expressed as The -correlation in (3.5) or (3.6) is bounded such that , where a value of () indicates a strictly increasing (decreasing) monotone relationship between the two variables. In general, we would also note that .
In the context of -moment symmetric -distributions (), suppose it is desired to simulate distributions based on (1.1) with a specified -correlation matrix and where each distribution has its own specified value of . Define and as in (1.1), where and have Pearson correlation and standard normal bivariate density of Using (1.1), (1.3), and (3.5) with the denominator standardized to for the unit-normal distribution, and (3.7), the -correlation of toward can be expressed as where is the standardizing term in (2.12). Integrating (3.8) yields given that and . Analogously, the -correlation of toward is
From (3.9), the intermediate correlation (IC) can be determined by simply evaluating for a specified value of and a given value of from (2.20). Given from (3.11), the -correlation can be determined by evaluating (3.10) using the solved value of . Note the special case of where , in (3.9) and (3.10), then .
Remark 3.1. Inspection of (3.9) indicates that when either (a) is standard normal that is, , (b) , or (c) .
Remark 3.2. If the IC is such that and in (3.9), then we have the inequality
as from inspecting (3.9) it is evident that . Thus, solutions to ICs cannot exist outside the range of .
The extension of determining ICs for asymmetric -distributions is analogous to the method described above for -distributions, where (1.5) is standardized and subsequently integrated as in (3.8) to obtain
where it is straight-forward to see that if (or ), then (3.13) or (3.14) simplifies to (3.9) or (3.10) for the case of symmetric -distributions. The IC in (3.13) is determined by substituting the specified -correlation () and the solved values of the parameters and (from (2.27)) into the left- and right-hand sides of (3.13), respectively, and then numerically solving for . 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 - (or -) distributions with specified -moments and specified -correlations, we suggest the following five steps. (1)Specify the -moments for transformations of the form in (1.5), that is, and obtain the parameters of and by solving equations (2.27) using the specified values of -skew () and -kurtosis () for each distribution. Specify a matrix of -correlations () for toward , where . (2) Compute the (Pearson) intermediate correlations (ICs) by substituting the specified -correlation and the parameters of and from step into the left- and right-hand sides of (3.14), respectively, and then numerically solve for . Repeat this step separately for all pairwise combinations of correlations. (3) Assemble the ICs into a 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 standard normal variables correlated at the intermediate levels as follows: where are independent standard normal random variables and where represents the element in the -th row and the -th column of the matrix associated with the Cholesky factorization performed in step (3). (5) Substitute from step (4) into equations of the form in (1.5), as noted in step (1), to generate the nonnormal -distributions with the specified -moments and -correlations.
To demonstrate the steps above and evaluate the proposed procedure, a comparison between the new -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 -moment and conventional moment-based procedures, respectively. Note that the ICs for the conventional procedure were computed by adapting the Mathematica source code in [12, Table 1] for (1.5). Tables 4 and 5 give the results of the Cholesky decompositions on the IC matrices, which are then used to create with the specified ICs by making use of the formulae given in (4.1) of step 4 with . The values of are subsequently substituted into equations of the form in (1.5) to produce for both procedures.
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 (), kurtosis (), and Pearson correlation (); (b) -skew (), -kurtosis (), and -correlation () based on samples of sizes and . The estimates for were based on Fisher’s -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 for the standard normal distribution). The formulae used for computing estimates for were Headrick’s Equations (2.3) and (2.5) [15]. The estimate for was based on the usual formula for the Pearson product-moment of correlation statistic, and the estimate for was computed based on (3.13) using the empirical forms of the cdfs in (3.1) and (3.3). The estimates for and were both transformed using Fisher’s 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 () using 10,000 resamples via the commercial software package Spotfire S+ [20]. The bootstrap results for the estimates of and were transformed back to their original metrics. Further, if a parameter was outside its associated bootstrap C.I., then an index of relative bias (RB) was computed for the estimate as . The results of the simulation are reported in Tables 6, 7, 8, 9, 10, and 11 and are discussed in the next section.
5. Discussion and Conclusion
One of the advantages that -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 -moment ratios () have over their corresponding conventional moment-based counterparts () is obvious. For example, with samples of size , 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 -skew and -kurtosis were 87.9% and 96.10% of their respective parameters. It is also evident from comparing Tables 6 and 7 that -skew and -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 -correlations. Overall inspection of these tables indicates that the -correlation is superior to the Pearson correlation in terms of relative bias. For example, for moderate correlations (), 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 -correlation. It is also noted that the variability of the -correlation appears to be more stable than that of the Pearson correlation both within and across the different conditions.
In summary, the new -moment-based procedure is an attractive alternative to the traditional conventional-moment-based procedure. In particular, the -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 -moment-based procedures.
Appendix
System of Conventional Moment-Based Equations for Tukey -Distributions
The equations for the mean (), variance (), skew (), and kurtosis () for conventional moment-based -distributions are given below without the details of their derivation. We would note that we derived (A.1)–(A.4) based on the formulae given in Headrick et al. [3, equations –].