A Method for Simulating Univariate and Multivariate ...

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examples are provided to show that -moment based Burr distributions are superior ... The parameters shown in Table 1 are the mean, standard deviation, skew,.
Section on Statistical Computing – JSM 2011

A Method for Simulating Univariate and Multivariate Burr Type III and Type XII Distributions with Specified -Moments and -Correlations

Mohan D. Pant1, Todd C. Headrick2 Illinois University Carbondale, 222-D Wham Building, Mail Code 4618, Carbondale, IL 62901; [email protected] 2Southern Illinois University Carbondale, 222-J Wham Building, Mail Code 4618, Carbondale, IL 62901; [email protected]

1Southern

Abstract Burr Type III and Type XII distributions are mainly used in statistical modeling and for simulating non-normal distributions with conventional moments (e.g., Skew and Kurtosis). However, conventional moment-based estimators can (a) be substantially biased, (b) have high variance, or (c) be influenced by outliers. In this context, a method for simulating univariate and multivariate Burr distributions through the method of -moments is introduced. Specifically, equations are derived for determining the shape parameters associated with user specified -moment ratios (e.g., -Skew and -Kurtosis). Further, a procedure is developed for generating multivariate Burr distributions with arbitrary -correlation matrices. Numerical examples are provided to show that -moment based Burr distributions are superior to their conventional moment based analogs in the context of estimation and distribution fitting. Monte Carlo simulation results are also provided to show that our procedure yields excellent agreement between specified -correlation matrices and empirical values of -correlation. Key Words: Monte Carlo simulation, -moments, -correlation, order statistics 1. Introduction Burr (1942) introduced twelve cumulative distribution functions ( s) with the primary purpose of distribution fitting. The Burr Type III and Type XII distributions (Burr, 1942, p. 217, equations 11 and 20) attract special attention because they include characteristics of distributions with varying degrees of skew and kurtosis (Burr, 1973; Rodriguez, 1977; Tadikamalla, 1980; Headrick, Pant, & Sheng, 2010). Some applications of the Burr Type III and Type XII distributions have been in statistical modeling in the fields such as forestry (Gove, Ducey, Leak, & Zhang, 2008; Lindsay, Wood, & Woollons, 1996), fracture roughness (Nadarajaha, & Kotz, 2006), life testing (Wingo, 1983; 1993), operational risk (Chernobai, Fabozzi, & Rachev, 2007), option market price distributions (Sherrick, Garcia, & Tirupattur, 1996), meteorology (Mielke, 1973), modeling crop prices (Tejeda, & Goodwin, 2008), and reliability (Mokhlis, 2005). For the purposes of this study, the quantile functions associated with Burr Type III and Type XII distributions can be given as (Headrick et al., 2010, eqs. 5-6) (1.1) (1.2)

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where and are strictly increasing monotonic functions of , that is, and for real-valued parameters and that define the shape of the distributions. The values of and can be determined by simultaneously solving the system of equations for specified values of skew and kurtosis (e.g., Headrick et al., 2010, eqs. 16-17). The p and associated with the quantile functions in either (1.1) or (1.2) can be given in parametric form ( ) as (Headrick et al., 2010, eqs. 9-10) (1.3) (1.4) where the derivative in (1.3). To demonstrate the procedure, the graphs of the p and associated with the non-normal Burr Type XII distribution with skew and kurtosis are shown in Figure 1. These values of skew and kurtosis were used in a number of studies (e.g., Berkovits, Hancock, and Nevitt, 2000; Enders, 2004; Headrick, 2011) for simulating non-normal distributions. The graphs in Figure 1 were obtained by using the solved values of and in equations (1.3), (1.4), and the Mathematica notebook package developed by Headrick, Sheng, and Hodis (2007). 1.0

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PDF

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CDF

Figure 1: PDF and CDF of the Burr Type XII distribution with skew and kurtosis . The solved values of and used in equations (1.3) and (1.4) are: and , which are also associated with the parameters in Tables 1 and 2. The parameters shown in Table 1 are the mean, standard deviation, skew, and kurtosis ( ) and are associated with the PDF in Figure 1. The conventional moment-based estimators can be (a) substantially biased, (b) highly dispersed, or (c) influenced by outliers (Hosking, 1990; Headrick, 2011) and thus may not be good representatives of the true parameters. Some of these limitations of conventional-moment based estimators are exemplified in Table 1. Inspection of Table 1 shows that the bootstrap estimates of skew and kurtosis are substantially attenuated below their corresponding parameters with greater bias and variance as the order of the estimate increases. In particular, for sample size = 50, the values of estimates are, respectively, 99.86%, 95.52%, 63.43%, and 22.71% of their corresponding parameters.

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According to Headrick (2011), the values of conventional moment-based estimators of skew and kurtosis based on sample data are algebraically bounded by the sample size such that and . This limitation implies that if a researcher wants to simulate non-normal data with kurtosis as in Table 1 by drawing samples of size , then the largest possible value of the computed estimator of kurtosis is only 10, which is 47.62% of the parameter (p. 3). The estimates in Table 1 were calculated from samples of size 50 using Fisher’s -statistics formulae given in Kendall and Stuart (1977, pp. 299-300). Table 1: Conventional moment-based parameters ( and their estimates for the p in Figure 1. Each estimate (Est.), associated 95% bootstrap confidence interval (95% C. I.), and the standard error (SE) were based on resampling 25000 statistics 10000 times. Each sample statistic was based on a sample size of Parameter

Est.

95% C. I. (0.2762, 0.2772) (0.2718, 0.2737) (1.8928, 1.9136) (4.7036, 4.8364)

SE 0.0003 0.0005 0.0053 0.0341

The method of -moments introduced by Hosking (1990) is an attractive alternative statistical technique that can be used in describing theoretical probability distributions, fitting distributions to real data, estimating parameters, and testing of hypotheses (Hosking, 1990; Headrick, 2011). The first four moments of any theoretical or empirical distribution are termed as -mean ( ), scale ( ), -skew ( ), and -kurtosis ( ) respectively (Headrick, 2011, p. 3). Some of the qualities of -moments that make them superior than the conventional moments are that: -moments (a) exist for any distribution with finite mean and finite variance, (b) are nearly unbiased for any sample size and less affected from sampling variability, (c) are more robust in the presence of outliers in the sample data, and (d) are not algebraically bounded by the sample size (Hosking, 1990; 1992; Hosking & Wallis, 1997; Headrick, 2011). Hosking (1990) also pointed out that the parameter estimation, under certain conditions, based on the -moment theory yields more accurate estimates than those obtained through the method of maximum likelihood and that the -momentbased methodology is capable of characterizing a wider range of distributions (p. 106). In this context, it is noted that the generalized lambda distributions (GLD) and the power method distributions have been characterized by the method of moments. (Asquith, 2007; Karvanen and Nuutinen, 2008; Headrick, 2011). -moments and -moment ratios are defined by using certain linear combinations of order statistics and are shown to have superior attributes than conventional moments. For example, the estimates in Table 2 are relatively closer to their respective parameters with much smaller variance compared to their conventional moment-based counterparts as in Table 1. Inspection of Table 2 shows that for the sample size , estimates are on average 100.18%, 100.07%, 97.62%, and 97.13% of their corresponding parameters.

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Table

2:

-moment based parameters and their estimates for the p in Figure 1. Each estimate (Est.), associated 95% bootstrap confidence interval (95% C. I.), and the standard error (SE) were based on resampling 25000 statistics 10000 times. Each sample statistic was based on a sample size of . Parameter

Est.

95% C. I. (0.2770, 0.2780) (0.1351, 0.1357) (0.3442, 0.3462) (0.1952, 0.1972)

SE 0.0003 0.0002 0.0005 0.0005

In view of the above concerns and the advantages of -moments over conventional moments, the main purpose of this study is to characterize Burr Type III and Type XII distributions through the method of -moments so that the problems associated with conventional moment based estimators can be obviated. Specifically, a system of equations is derived for determining the shape parameters associated with user specified -moment ratios (e.g., -Skew and -Kurtosis). Another purpose of this study is to develop the methodology for generating correlated non-normal Burr Type III and Type XII distributions using specified correlation matrix. The detailed methodology for computing -moments in the context of this study is provided in Chapter 2, below. 2. Methodology 2.1. Theoretical and Empirical Definitions of -Moments -moments can be expressed as certain linear combination of probability weighted moments ( s). Let be a random variable with the p , , and the quantile function . Then, the s associated with can be defined as (Headrick, 2011, Eq. 2.1) (2.1) where 2011, eq. 2.2)

. The first

-moments can be expressed as (Headrick, (2.2)

where are the coefficients from shifted orthogonal Legendre polynomials and these coefficients are determined as (Headrick, 2011, eq. 2.3) (2.3) In particular, the first four -moments based on (2.1) (2.3) are expressed in their simplified forms as (Headrick, 2011, eq. 2.4) (2.4) (2.5) (2.6) (2.7)

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The notations and denote the location and scale parameters. In the literature of -moments, is called the -location (which is equal to the arithmetic mean) and is called the -scale, which is one-hal o Gini’s oe i ient o mean difference (Kendall & Stuart, 1977, pp. 47-48). Dimensionless -moment ratios (i.e., -skew and -kurtosis) are defined as the ratios of higher-order moments (i.e., and ) to . Thus, and are, respectively, the indices of -skew and -kurtosis. In general, these indices of -skew and kurtosis are bounded such that and , and as in conventional moment theory, a symmetric distribution has -skew (Headrick, 2011). According to Jones (2004), for a continuous distribution the boundary region for skew and -kurtosis is given by the inequality (2.8) Empirical -moments for a sample of real data, are expressed as linear combinations of the sample order statistics . The unbiased estimates of the s are given as (Headrick, 2011, eq. 2.6): (2.9) where

. Here, is the sample mean. The first four sample -moments are obtained by substituting instead of in equations (2.4) (2.7). The symbols used for the sample -moment ratios (i.e., -skew and -kurtosis) are and , where and . 2.2. -Moments for Burr Type III Distributions Substituting from (1.1), , and (2.1), the -th for the Burr Type III distributions is given by

in (2.10)

After some manipulations, expression in (2.10) can be rewritten as: (2.11) Let . Then, rewritten as

Substituting in (2.11), the -th

can be (2.12)

The integral in (2.12) is a beta function, , where and such that . Integrating (2.12) for and substituting these s into (2.4) (2.7) and simplifying gives the following system of equations for the Burr Type III distribution (2.13) (2.14)

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(2.15)

(2.16)

2.3. -Moments for the Burr Type XII Distributions Substituting in (2.1), the -th

of (1.2), , and for the Burr Type XII distributions is given by (2.17)

After some manipulations, expression in (2.17) can be rewritten as: (2.18)

Let th

. Then, can be rewritten as

Substituting in (2.18), the (2.19)

Integrating (2.19) for and substituting these s into (2.4) (2.7) and simplifying gives the following system of equations for the Burr Type XII distribution (2.20) (2.21)

(2.22)

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(2.23)

For specified values of -skew and -kurtosis associated with the Burr Type III distributions, the system of equations (2.15) (2.16) can be simultaneously solved for real values of and . Similarly, for the Burr Type XII distributions with specified values of -skew and -kurtosis , the system of equations (2.22) (2.23) can be solved for real values of and . The solved values of and are then substituted in (1.1) and (1.2) for generating the quantile functions of the Burr Type III and Type XII distributions, respectively. Also, the solved values of and are substituted in equations (2.13) (2.14) and in equations (2.20) (2.21) for computing the values of -mean and -scale of the Burr Type III and Type XII distributions, respectively. To demonstrate the methodology, an example of parameter estimation based on the equations (2.20) (2.23) is given in Figure 2 and Table 3 in Chapter 3 for a Chi-square ( ) distribution. 2.4. -Correlation Let and be two random variables with s and respectively. The second -moments of and can be defined as (Serfling & Xiao, 2007) (2.24) (2.25) The second -comoment of

towards

and

towards

are given as (2.26) (2.27)

The -correlations of

towards

and

towards

are then defined as: (2.28)

(2.29) The -correlation given in equation (2.28) (or, equation 2.29) is bounded such that . A value of or implies that and have a strictly monotonically increasing (or, decreasing) relationship. See Serfling and Xiao (2007) for further details on the topics relevant with -correlation. 2.5. Multivariate Burr Type III and Type XII Distributions Extension of Burr Type III and Type XII distributions to multivariate level can be attained by specifying quantile functions as given in (1.1) and (1.2) with

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specified -Correlation. Specifically, let with s and the joint p associated with

and

denote standard normal variables given by following expressions: (2.30) (2.31) (2.32)

where in (2.32) is the intermediate correlation between and . Using the s in (2.30) and (2.31) as the uniform random variables, as , the quantile function defined in either (1.1) or (1.2) can be expressed as a function of or

eg

of towards denominator standardized to

or

. The formula for the -correlation can be expressed, by using (2.28) with , as (2.33)

The next step is to use equation (2.33) to solve for the values of the intermediate correlations such that specified Burr distributions also have the specified -correlations. The procedure for generating multivariate Burr distributions correlated with specified -correlations begins with collecting the solved intermediate correlations into a matrix and then factoring this matrix using Cholesky factorization. The elements of the matrix of Cholesky factorization are then used to generate standard normal variables that are correlated at the intermediate correlation levels as:

(2.34)

where are standard random variables and is the element in the -th row and -th column of the matrix yielded by Cholesky factorization. To generate the uniform deviates required for the quantile functions in (1.1) and (1.2), Headrick et al. (2010) suggested the use of Taylor series based expansion, as given in Marsaglia (2004), for the of , which is given as: (2.35) where is the p of . The advantage of using Taylor series based expansion in (2.35) is that its use obviates the problem of getting indefinite form in the quantile function either in (1.1) or (1.2) and that the absolute error associated with the value of this expression is less than

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3. Comparison of -Moments with Conventional Moments 3.1. Estimation An example to demonstrate the advantages of -moment-based estimation over conventional moment-based estimation is provided in Figure 2 and Table 3, where a Chi-square distribution is approximated by the Burr Type XII distribution in both systems of estimation. Specifically, Figure 2 shows the Burr Type XII parametric plot (dashed curve) based on the values of and parameters solved by using each system of equations (i.e., eqs. 16 17 from Headrick et al., 2010, and eqs. 2.22 2.23 from Chapter 2 of this study) superimposed on the p . The lower part of Figure 2 shows the parameters and the solved values of and , which are associated with Table 3. 0.10

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Distribution Transformed by the Burr Type XII Distribution

Conventional moment-based

-moment-based

Figure 2: Parametric plots (dashed curves) of the Burr Type XII approximations based on (a) conventional moment theory and (b) -moment theory superimposed on the PDF of the distribution. The quantile function associated with the Burr Type XII distribution was transformed as (a) and (b) respectively, where , and , are the values of mean, standard deviation whereas , and , are the values of -mean, -scale obtained from the distribution and Burr Type XII approximation, respectively. The advantage of -moment-based estimators being less biased than their conventional moment-based counterparts can be demonstrated in the context of Burr Type XII approximation to the p by considering the Monte Carlo simulation results associated with the indices for standard error (SE) and the percentage of relative bias (RBias%) reported in Table 3.

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Table 3: Conventional moment-based estimates and the -momentbased estimates for the p s in Figure 2. Each estimate (Est.), the associated 95% bootstrap confidence interval (95% C. I.), and the standard error (SE) were based on resampling 25000 statistics 10000 times.

: :

:

: : 3: 4:

: :

:

Est. (95% C. I.)

SE

RBias%

Est. (95% C. I.)

SE

.2738 (.2735, .2740) .1591 (.1588, .1593) .8530 (.8487, .8577) .7363 (.7186, .7539)

.0002

.0203

.3947 (.3944, .3951) .1230 (.1229, .1232) .1603 (.1596, .1610) .1303 (.1297, .1308)

.0002

.3949 (.3947, .3952) .1229 (.1228, .1230) .1628 (.1623, .1633) .1311 (.1307, .1315)

.0001

.3948 (.3947, .3949) .1230 (.1229, .1230) .1646 (.1645, .1648) .1312 (.1311, .1313)

.00005

.0001 .0027 .0107

.2735 (.2733, .2737) .1606 (.1605, .1608) .9191 (.9152, .9226) 1.059 (1.040, 1.075)

.0001

= 100 .0893

.0001

.9569

.0022

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.2737 (.2737, .2738) .1620 (.1620, .1621) .9905 (.9891, .9919) 1.443 (1.434, 1.452)

.00003 .00003 .00085 .00534

RBias%

.0001 .0004 .0003

.0377

.0001

.0845

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1.130

.0002

.1159

.0123

.00002 .00009 .00007

Specifically, a Fortran program was written to simulate 25000 independent samples of sizes 50, 100, and 1000, and the estimates and were computed for each of the samples based on the parameters and the values of and listed in the lower part of Figure 2. The estimates were ompute using Fisher’s -statistics formulae, whereas the estimates were computed using equation (2.9). Bootstrapped average estimates (Est.), the associated 95% confidence intervals (95% C. I.), and the standard errors (SE) were obtained for each type of estimates by resampling 25000 statistics 10000 times via the commercial software package Spotfire S+ (TIBCO, 2008). The percentage of relative bias (RBias%) was computed by using the formula: ias stimate parameter parameter The results in Table 3 illustrate that the -moment based estimators have relatively negligible bias and error compared with their conventional momentbased counterparts. These characteristics are most significant in the context of

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smaller sample sizes and higher-order moments. For example, for the sample size of 50, the conventional moment-based estimates were, on average, 14.7% and 50.91% less than their respective parameters . On the other hand, the estimates were, on average, 2.65% and 0.73% less than their respective parameters . Thus, the relative biases of estimators based on -moments are essentially negligible compared to those associated with the estimators based on conventional moments. Also, it can be verified that the standard errors associated with the estimates are relatively much smaller than the standard errors associated with the estimates . 3.2. Distribution Fitting Figure 3 shows the conventional moment- and the -moment-based Burr Type XII p s superimposed on the histogram of ankle circumference measured from adult males (http://lib.stat.cmu.edu/datasets/bodyfat) as cited in Headrick (2010, p. 48). 0.35

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(b) Fitting the -moment-based Burr Type XII p

Estimates

Estimates

Shape parameters

Shape parameters

Figure 3: Histograms of the Ankle circumference data superimposed by the (a) conventional moment-based and (b) -moment-based Burr Type XII distributions. The quantile function of the Burr Type XII distribution was transformed as (a) and (b) respectively, where , and , are the values of mean, standard deviation , whereas , and , are the values of -mean, -scale obtained from the sample data and the Burr Type XII distribution, respectively. The conventional moment-based estimates of skew and kurtosis and the -moment-based estimates of -skew and -kurtosis were computed for the sample ( ) of participants using Fisher’s -statistics formulae and equation (2.9), respectively. These sample estimates were then used to solve for the values of shape parameters an , which were subsequently used in equation (1.3) to plot the p s shown in Figure 3. Inspection of the two

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panels in Figure 3 illustrates that the -moment-based Burr Type XII p provides a better fit to the sample data. The Chi-square goodness of fit statistics along with their corresponding -values given in Table 4 provide evidence that the conventional moment-based Burr Type XII distribution does not provide a good fit to the actual data, whereas, the -moment-based Burr Type XII distribution fits very well. The degrees of freedom for the Chi-square goodness of fit tests were computed as = 5 = 10 (class intervals) – 4 (estimates of the parameters) – 1 (sample size). Table 4: Chi-square goodness of fit statistics for the conventional moment- and the -moment -based Burr Type XII approximations for the Ankle circumference data shown in Figure 3. Percent Expected Obs ( ) Obs ( ) Circumference ( ) Circumference ( ) 10 25.2 22 18 < 21.4866 < 21.3760 20 25.2 15 29 21.4866 – 21.7890 21.3760 – 21.8701 30 25.2 29 30 21.7890 – 22.0747 21.8701 – 22.2360 40 25.2 18 26 22.0747 – 22.3687 22.2360 – 22.5606 50 25.2 30 25 22.3687 – 22.6888 22.5606 – 22.8795 60 25.2 27 27 22.6888 – 23.0571 22.8795 – 23.2201 70 25.2 30 20 23.0571 – 23.5107 23.2201 – 23.6175 80 25.2 29 25 23.5107 – 24.1310 23.6175 – 24.1426 90 25.2 33 32 24.1310 – 25.1858 24.1426 – 25.0244 100 25.2 19 20 25.1858 or more 25.0244 or more

3.3. Correlated Non-normal Burr Distributions with Specified -Correlation Matrix. An example based on Monte Carlo simulation results is provided with the purpose of demonstrating that the methodology discussed in Chapter 2 for generating correlated Burr distributions works well with a mixture of Burr Type III and Type XII distributions. Specifically, presented in Table 5 are the -moment based parameters and the solved values of and for the two Burr Type III and two Burr Type XII distributions that approximate the Logistic (0, 1), the Chi-square ( ), the (3, 20), and the Extreme Value (0, 1) distributions, respectively. Table 6 shows a specified -correlation matrix. Table 7 gives the intermediate correlation matrix obtained by separately solving (2.33) for each of the six bivariate cases. Table 8 gives the results of the Cholesky factorization on the intermediate correlation matrix in Table 7. The elements in Table 8 are then used to generate that are correlated at the intermediate levels using formulae given in (2.34). The values of are then used in equation (2.35) to obtain the Taylor series based approximations of the s , , and , which are ultimately treated as and used in equations (1.1) and (1.2) to obtain the quantile functions of the Burr Type III and Type XII distributions. For the Monte Carlo simulation, a Fortran algorithm was written to generate 25000 independent estimates ( ) of each specified -correlation given in Table 6 based on samples of size . The estimates ( ) were computed using (2.28). Bootstrapped average estimates, confidence intervals, and standard errors

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were obtained for each of the estimates ( ) using 10000 resamples via the commercial software package Spotfire S+ (TIBCO, 2008). The results of the simulation are presented in Table 9. Inspection of Table 9 shows that all estimates ( ) are in very close agreement with their respective specified parameters ( ). Table 5: Burr Type III and Type XII distributions with -moment-based parameters and the corresponding values of shape parameters ( and ) for the results of Monte Carlo simulation. -mean -scale -skew -kurtosis Distribution

Table 6: Specified -correlations distributions in Table 5. 1.0 0.4 0.5 0.6

between the Burr Type III and Type XII

1.0 0.7 0.8

1.0 0.9

1.0

Table 7: Intermediate correlation matrix based on equation (2.33) for specified correlations of Table 6. 1.0 1.0 1.0 1.0 Table 8: Cholesky decomposition on the intermediate correlation matrix in Table 7.

Table 9: Bootstrap Estimates (Est.), 95% Bootstrap confidence intervals (95% C. I.), and Standard Errors (SE) for the specified -correlation matrix in Table 6. The entries are based on samples of size . -Correlation Est. 95% C. I. SE 0.400 (0.3990, 0.4006) 0.00040 0.500 (0.4992, 0.5008) 0.00041 0.600 (0.5991, 0.6006) 0.00039 0.700 (0.6987, 0.7015) 0.00071 0.800 (0.7986, 0.8015) 0.00073 0.900 (0.8991, 0.9011) 0.00051

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3.4. Statistical Computing and Software Packages FORTRAN PowerStation 4.0 (Microsoft, 1994) was used as the main computing software. The subroutines NORMB1, and UNI1 from Rangen (Blair, 1987) were used to generate pseudo-random normal and uniform deviates. Mathematica version 8.0.1 (Wolfram, 2008) was used for (a) solving the systems of equations for the values of shape parameters, (b) plotting the p s and s, and (c) solving for the intermediate correlations of the correlated multivariate Burr Type III and Type XII distributions with specified -correlations. Spotfire S+ (TIBCO, 2008) was used to obtain bootstrap estimates, the 95% confidence intervals, and the standard errors of the Fortran generated statistics. 4. Conclusion This study introduced -moment based approach for simulating the Burr Type III and Type XII distributions, which may be useful to the researchers in any discipline for generating non-normal distributions in their studies. An empirical comparison of the -moment based Burr distributions was made with conventional moment based Burr distributions in the context of estimation and distribution fitting. This study also introduced a methodology for generating correlated multivariate Burr Type III and Type XII distributions using specified -correlation matrix. A numerical example was given to demonstrate that the Monte Carlo simulation results of the empirical values of the estimates of -correlations among a mixture of Burr Type III and Type XII distributions are very close to their respective parameters (specified -correlations) with small standard errors. References Asquith, W. H. (2007). L-moments and TL-moments of the generalized lambda distribution. Computational Statistics & Data Analysis, 51, 4484-4496. Blair, R. C. (1987). Rangen. Boca Raton, FL: IBM. Burr, I. W. (1973). Parameters for a general system of distributions to match a grid of α3 an α4. Communications in Statistics - Theory and Methods, 2, 1-21. Burr, I. W. (1942). Cumulative frequency functions. The Annals of mathematical statistics,13, 215-232. Chernobai, A. S., Fabozzi, F. J., & Rachev, S. T. (2007). Operational Risk: A Guide toBasel II Capital Requirements, Models, and Analysis. New York: John Wiley & Sons. Gove, J. H., Ducey, M. J., Leak, W. B., & Zhang, L. (2008). Rotated sigmoid structures in managed uneven-aged northern hardwood stands: A look at the Burr Type III distribution, Forestry, 81, 21-36. Headrick, T. C. (2010). Statistical Simulation: Power Method Polynomials and Other Transformations. Boca Raton, FL: Chapman & Hall/CRC. Headrick, T. C. (2011). A characterization of power method transformations through -moments. Journal of Probability and Statistics, vol., 2011, 22 pages, doi:10.1155/2011/497463. Headrick, T. C., Pant, M. D., & Sheng, Y. (2010). On simulating univariate and multivariate Burr Type III and Type XII distributions. Applied Mathematical Sciences, 4, 2207–2240. Hosking, J. R. M. (1990). L-moments: analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society, Series B, 52, 105-124.

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