**5. Application**

The data that we will use were collected by the Australian Institute of Sport and reported by Cook and Weisberg (1994) [10]. The data set consists of values of several variables measured on *n* = 202 Australian athletes. Specifically, we shall consider the pair of variables (Ht,Wt) which are the height (cm) and the weight (Kg) measured for each athlete.

We fitted the bivariate Mudholkar-Hutson distributions for five different cases. In addition to the general case which is given by the pdf *f* (*<sup>x</sup>*, *y*; *αi*, *β<sup>i</sup>*, *γi*, *ν*, *μ*, **<sup>Σ</sup>**), based on (9), where *i* = 1, 2, 3, and *μ* = (*μ*1, *μ*2) is the location parameter and

$$
\Sigma = \begin{pmatrix}
\Sigma\_{11} & \Sigma\_{12} \\
\Sigma\_{21} & \Sigma\_{22}
\end{pmatrix},
$$

is the symmetric positive definite scale matrix, we also consider four special cases:


$$f\left(\mathbf{x}, y; \varepsilon\_i, \mu\_j, \Sigma\_{11}, \Sigma\_{22}, \Sigma\_{12}\right) = f\left(\mathbf{x}, y; 1 + \varepsilon\_i, 1 - \varepsilon\_i, \frac{1 + \varepsilon\_i}{2}, \mu, \Sigma\right), f$$

which is the bivariate MH distribution specified using the special case (1.), where |*<sup>ε</sup>i*| < 1 for all *i* = 1, 2, 3.

3. When the pdf is given by

$$f\left(\mathbf{x}, \mathbf{y}; \varepsilon, \mu\_j, \Sigma\_{11}, \Sigma\_{22}, \Sigma\_{12}\right) = f\left(\mathbf{x}, \mathbf{y}; 1 + \varepsilon, 1 - \varepsilon, \frac{1 + \varepsilon}{2}, \mu, \Sigma\right),$$

which is the bivariate MH distribution specified using the special case (2.), where |*ε*| < 1. 4. When the pdf is given by

$$f\left(\mathbf{x}, y; \mu\_{j\prime}\Sigma\_{11\prime}\Sigma\_{22\prime}\Sigma\_{12}\right) = f\left(\mathbf{x}, y; 1, 1, \frac{1}{2}, \mu\_{\prime}\Sigma\right),$$

which is the bivariate Cauchy distribution (as in (3)).

All fits were done by maximizing the likelihood through numerical methods which combine algorithms based on the Hessian matrix and the Simulated Annealing algorithm. Standard errors of the estimations were computed based on 1000 bootstrap data samples.

To compare model fits, we used the Akaike criterion (see Akaike, 1974 [11]), namely

$$AIC = -2\ln\left[L\left(\hat{\theta}\right)\right] + 2k\_r$$

where *k* is the dimension of *θ* which is the vector of parameters of the model being considered.

Table 1 displays the results of the fits for 100 women. In Table 1, we see the results of fitting the five competing models. They are the general bivariate skew−*t* MH model with pdf given in Equation (9), General bivariate MH with pdf given in Equation (4), Bivariate MH (1.) with pdf given in Equation (5), Bivariate MH (2.) with pdf given in Equation (6) and Bivariate Cauchy. It shows the maximum likelihood estimates (mle's) of the five models. The last column shows the estimated standard errors (se) of the estimates. Non identifiability of the bivariate skew−*t* MH model is evidenced by the huge value of the estimated standard error of the estimate of *ν*. However, the AIC criterion indicates that data are better fitted by the general bivariate skew−*t* MH (9) model. Figure 2 shows the contour lines of the fitted pdf.


**Table 1.** Bivariate Mudholkar-Hutson fits for women.

Table 2 displays the results of the fits for 102 men. In Table 2 we again compare five competing models. The maximum likelihood estimates (mle's) of the five models, the corresponding Akaike criterion values and estimated standard errors of the estimates are displayed in the table. Again, the AIC indicates that the data are better fitted by the general bivariate skew−*t* MH (9) model. Figure 3 shows the contour lines of the fitted pdf. Non identifiability of the general bivariate skew−*t* MH (9) and General bivariate MH (4) models is shown by the large values of the estimated standard errors for some estimates.


**Table 2.** Bivariate Mudholkar-Hutson fits for men.

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**Figure 2.** Female Australian athletes data: scatter plot (Ht, Wt) and fitted General bivariate MH.

**Figure 3.** Male Australian athletes data: scatter plot (Ht, Wt) and fitted General bivariate MH.

Table 3 displays the results of the fits for the full set of *n* = 202 Australian athletes, regardless of gender. Table 3 shows the maximum likelihood estimates (mle's) of the five models together with the corresponding Akaike criterion values and estimated standard errors of the estimates. The AIC here also indicates that the data are better fitted by the general bivariate skew−*t* MH model (9). Figure 4 shows the contour lines of the fitted pdf. Thus, in all three cases, males, females and combined, the best fitting model was the general bivariate skew−*t* MH.


**Table 3.** Bivariate Mudholkar-Hutson fits.

**Figure 4.** Australian athletes data: scatter plot (Ht, Wt) and fitted General bivariate MH. Red point for Men and sign + for women.

For the three real cases analyzed the general bivariate skew-t MH model (9) was indicated as the best fitted. That means that it seems worth considering the more general model to explain the variability of these data sets.
