The Mean Deviation from the Median of the Dagum Distribution

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1. Introduction

Camilo Dagum in 1977 introduced a new distribution model particularly suited to describe the personal distribution of income. This model, in fact, thanks to the presence of a greater number of parameters (4 in the more general version) than other models, proves to be particularly flexible and adaptable to describe even deeply dissimilar income distributions (Dagum, 1977) [1]. For this model Dagum had obtained the main characteristic values (mean, mode, median, variance, moments, Lorenz curve and concentration ratio). The Dagum distribution has been studied by several authors that have proposed several variations to increase the flexibility of the Dagum distribution in modeling lifetime data. Some recent modifications concern log-Dagum distribution (Domma and Perri, 2009) [2], Mc-Dagum distribution (Oluyede and Rajasoorya, 2013) [3], beta-Dagum distribution (Domma and Condino, 2013) [4], gamma-Dagum distribution (Oluyede et al., 2014) [5], weighted Dagum distribution (Oluyede and Ye, 2014) [6], exponentiated Kumaraswamy-Dagum distribution (Huang and Oloyede, 2014) [7], transmuted Dagum distribution (Elbatal and Aryal, 2015) [8], extended Dagum distribution (Silva et al., 2015) [9] and Dagum-Poisson distribution (Oluyede et al., 2016) [10], exponentiated generalized exponential Dagum distribution (Nasiru et al., 2019) [11], moreover, regarding properties and methods of estimation of the parameters of the Dagum distribution. Domma et al. (2011a [12], 2011b [13] ) determined the observed information matrix in right censored samples and debated aspects of the maximum likelihood estimation for censored data. In the 2013 Shahzad and Asghar [14] obtained the L-moments and TL-moments in closed form to estimate the parameters of the Dagum distribution. Al-Zahrani (2016) [15] proposed a reliability test plan under the assumption that the life of a product follows a Dagum distribution. Dey et al. (2017) [16] studied the properties and different methods of estimating the parameters of the Dagum distribution.

2. Dagum Distribution

Girone and Viola (2009) [17] and Girone (2010) [18] obtained the expression of the mean difference and the mean deviation. It is very important to underline that the mean deviation from the median is invariant with respect to translations and it is homogeneous to the variable. Therefore, without losing generality, we can consider the density function with only one shape parameters

$f\left(x\right)=\beta \delta {x}^{-\left(\delta +1\right)}{\left(1+{x}^{-\delta}\right)}^{-\left(\beta +1\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}0<x<\infty ,$

and the distribution function

$F\left(x\right)={\left(1+{x}^{-\delta}\right)}^{-\beta}.$

The mean value and the median of this distribution are:

$\mu =\beta B\left(\beta +1/\delta ,1-1/\delta \right),$

$Me={\left({2}^{1/\beta}-1\right)}^{-1/\delta}.$

3. The Mean Deviation from the Median

The formula of the mean deviation from the median is:

${S}_{Me}={\displaystyle {\int}_{-\infty}^{\infty}\left|x-Me\right|f}\left(x\right)\text{d}x.$

A formula that avoids the absolute value and that splits the calculation into two parts is:

${S}_{Me}={\displaystyle {\int}_{-\infty}^{Me}\left(Me-x\right)f}\left(x\right)\text{d}x+{\displaystyle {\int}_{Me}^{\infty}\left(x-Me\right)f}\left(x\right)\text{d}x;$

the above formula represents the first attempt in simplifying the calculations. After simple steps, the formula becomes

${S}_{Me}=Me{\displaystyle {\int}_{-\infty}^{Me}f}\left(x\right)\text{d}x-{\displaystyle {\int}_{-\infty}^{Me}xf}\left(x\right)\text{d}x+{\displaystyle {\int}_{Me}^{\infty}xf}\left(x\right)\text{d}x-Me{\displaystyle {\int}_{Me}^{\infty}f}\left(x\right)\text{d}x,$

considering that the first and last terms offset each other, we arrive at the formula

${S}_{Me}={\displaystyle {\int}_{Me}^{\infty}xf}\left(x\right)\text{d}x-{\displaystyle {\int}_{-\infty}^{Me}xf}\left(x\right)\text{d}x,$

formula that can be simplified taking into account that

${\int}_{Me}^{\infty}xf}\left(x\right)\text{d}x-{\displaystyle {\int}_{-\infty}^{Me}xf}\left(x\right)\text{d}x={\displaystyle {\int}_{-\infty}^{\infty}xf\left(x\right)\text{d}x}=\mu ,$

and that allows to obtain

${S}_{Me}=\mu -2{\displaystyle {\int}_{-\infty}^{Me}xf}\left(x\right)\text{d}x.$

Then we have to calculate the only integral present in the formula of the mean deviation from the median considering that, in our case, the Dagum density function starts from 0. With the aid of the Mathematica software we obtain a very heavy expression of the integral that, however, after a few steps can be simplified into the following formula:

$\int}_{0}^{Me}xf}\left(x\right)\text{d}x=\frac{\beta {\delta}_{2}{F}_{1}\left[\beta +1/\delta ,1+\beta ,1+\beta ,1+\beta +1/\delta ,-{\left({2}^{1/\beta}-1\right)}^{-1}\right]}{\left(1+\beta \delta \right){\left({2}^{1/\beta}-1\right)}^{\beta +1/\delta$

and then the formula of the mean deviation from the median in the Dagum model results:

${S}_{Me}=\beta B\left(\beta +1/\delta ,1-1/\delta \right)-{2}^{\frac{\beta {\delta}_{2}{F}_{1}\left[\beta +1/\delta ,1+\beta ,1+\beta ,1+\beta +1/\delta ,-{\left({2}^{1/\beta}-1\right)}^{-1}\right]}{\left(1+\beta \delta \right){\left({2}^{1/\beta}-1\right)}^{\beta +1/\delta}}}$

an expression that cannot be simplified but that, for some values of β and δ, gives more compact results.

4. Expressions of the Mean Deviation from the Median for Some Values of δ and β

In this paragraph the expressions of the mean deviation from the median are given for some values of δ and β.

For $\delta =\text{2}$ and $\beta =\text{1}$ ${S}_{Me}=1$,

for $\delta =\text{2}$ and $\beta =\text{2}$ ${S}_{Me}=\sqrt{\frac{1}{2}+\frac{5}{\sqrt{2}}}+\frac{3\pi}{4}-3\text{arccot}\sqrt{-1+\sqrt{2}}$,

for $\delta =\text{3}$ and $\beta =\text{1}$ ${S}_{Me}=1-\frac{2\mathrm{log}2}{3}$,

for $\delta =\text{4}$ and $\beta =\text{1}$ ${S}_{Me}=\frac{1}{4}\left(4+\sqrt{2}\mathrm{log}\left[2-\sqrt{2}\right]-\sqrt{2}\mathrm{log}\left[2+\sqrt{2}\right]\right)$.

Table 1 shows the mean deviation from the median values for some values of δ and β. The same and other values are shown in Figure 1.

The values shown in Table 1 represent the single values assumed by the mean deviation from the median, obtained by crossing some values assumed by the δ parameter and the β parameter.

With δ is equal to 1.5, the mean deviation from the median is 1.21 when β is 0.5 and, as β increases until it reaches 5.0, the mean deviation from the median

Table 1. Values of the mean deviation from the median for some values of δ and β.

Figure 1. Graphical representation of the mean deviation from the median.

increases until 5.51. With δ is equal to 2.0, the mean deviation from the median is 0.73 when β is 0.5 and, as β increases until it reaches 5.0, the mean deviation from the median increases until 2.11. With δ is equal to 2.5, the mean deviation from the median is 0.56 when β is 0.5 and, as β increases until it reaches 5.0, the mean deviation from the median increases until 1.21. With δ is equal to 3.0, the mean deviation from the median is 0.47 when β is 0.5 and, as β increases until it reaches 5.0, the mean deviation from the median increases until 0.82. With δ is equal to 3.5, the mean deviation from the median is 0.41 when β is 0.5 and, as β increases until it reaches 5.0, the mean deviation from the median increases until 0.61. With δ is equal to 4.0, the mean deviation from the median is 0.36 when β is 0.5 and, as β increases until it reaches 5.0, the mean deviation from the median increases until 0.49. With δ is equal to 4.5, the mean deviation from the median is 0.33 when β is 0.5 and, as β increases until it reaches 5.0, the mean deviation from the median increases until 0.40. Finally with δ is equal to 5.0, the mean deviation from the median is 0.30 when β is 0.5 and, as β increases until it reaches 5.0, the mean deviation from the median increases until 0.34.

So we can state that the mean deviation from the median decreases as δ increases, but increases as β increases.

The values shown in Table 1 are displayed in Figure 1. And this graph allows us to have a visual perception of the trend of the mean deviation from the median at the values of δ and β.

As it can be seen, the mean deviation from the median seems to increase as β increases and decrease as δ increases; moreover, it increases as the scale parameter λ increases.

5. Conclusion

In this paper an explicit and compact expression of the mean deviation from the median for the distribution of Dagum was obtained. This expression allows us to examine, with great evidence, the behavior of the scale and shape parameters in terms of absolute variability and in terms of relative variability.

References

[1] Dagum, C. (1977) A New Model of Personal Income Distribution: Specification and Estimation. Economie Appliquée, 30, 413-436.

[2] Domma, F. and Perri, P.F. (2009) Some Developments on the Log-Dagum Distribution. Statistical Methods and Applications, 18, 205-209.

https://doi.org/10.1007/s10260-007-0091-3

[3] Oluyede, B.O. and Rajasooriya, S. (2013) The Mc-Dagum Distribution and Its Statistical Properties with Applications. Asian Journal of Mathematics and Applications, 2013, ama0085.

[4] Domma, F. and Condino, F. (2013) The Beta-Dagum Distribution: Definition and Properties. Communication in Statistics—Theory and Methods, 42, 4070-4090.

https://doi.org/10.1080/03610926.2011.647219

[5] Oluyede, B.O., Huang, S. and Pararai, M. (2014) A New Generalized Dagum Distribution with Applications to Income and Lifetime Data. Journal of Statistical and Econometric Methods, 3, 125-151.

[6] Oluyede, B.O. and Ye, Y. (2014) Weighted Dagum and Related Distributions. Afrika Matematika, 25, 1125-1141.

https://doi.org/10.1007/s13370-013-0176-0

[7] Huang, S. and Oluyede, B.O. (2014) Exponentiated Kumaraswamy-Dagum Distribution with Applications to Income and Lifetime Data. Journal of Statistical Distribution and Application, 1, 1-20.

https://doi.org/10.1186/2195-5832-1-8

[8] Elbatal, I. and Aryal, G. (2015) Transmuted Dagum Distribution with Applications. Chilean Journal of Statistics, 6, 31-45.

[9] Silva, A., da Silva, L.C.M. and Cordeiro, G.M. (2015) The Extended Dagum Distribution: Properties and Application. Journal of Data Science, 13, 53-72.

[10] Oluyede, B.O., Motsewabagale, G., Huang, S., Warahena-Liyanage, G. and Pararai, M. (2016) The Dagum-Poisson Distribution: Model, Properties and Application. Electronic Journal of Applied Statistical Analysis, 9, 169-197.

[11] Nasiru, S., Mwita, P.N. and Ngesa, O. (2019) Exponentiated Generalized Exponential Dagum Distribution. Journal of King Saud University—Science, 31, 362-371.

https://doi.org/10.1016/j.jksus.2017.09.009

[12] Domma, F., Giordano, S. and Zenga, M. (2011) The Fisher Information Matrix in Right Censored Samples from the Dagum Distribution. Working Paper No. 8, Department of Economics and Statistics, University of Calabria, Calabria.

[13] Domma, F., Giordano, S. and Zenga, M. (2011) Maximum Likelihood Estimation in Dagum Distribution with Censored Samples. Journal of Applied Statistics, 38, 2971-2985.

https://doi.org/10.1080/02664763.2011.578613

[14] Shahzad, M.N. and Ashgar, Z. (2013) Comparing TL-Moments, L-Moments and Conventional Moments of Dagum Distribution by Simulated Data. Revista Colombiana de Estadìstica, 36, 79-93.

[15] Al-Zahrani, B. (2016) Reliability Test Plan Based on Dagum Distribution. International Journal of Advanced Statistics and Probability, 4, 75-78.

https://doi.org/10.14419/ijasp.v4i1.6165

[16] Dey, S., Al-Zahrani, B. and Basloom, S. (2017) Dagum Distribution: Properties and Different Methods of Estimation. International Journal of Advanced Statistics and Probability, 6, 74-92.

https://doi.org/10.5539/ijsp.v6n2p74

[17] Girone, G. and Viola, D. (2009) La differenza media della distribuzione di Dagum. Annali del Dipartimento di Scienze statistiche dell’Università di Bari, Vol. VIII.

[18] Girone, G. (2010) Lo scarto semplice medio della distribuzione di Dagum, in Studi in onore del Prof. Umberto Belviso.