Illustrations With Real Data

Our illustrations in this section uses real datasets in order to highlight overall performances of the newly proposed IIMs H21, H22, and H31 and to compare these with the widely used G and the CV. We consider a dataset on salaries of tenured and tenure-track (t/tt) professors in the mathematics department and

TABLE 1.13

Average Length (L) and Average Coverage Probability (p) Associated with the Confidence Interval from (1.25) for the IIMs for вы under the Lognormal Distribution from 2000 Replications: a = 0.10

statistics department from Texas A&M University, College Station, Texas. This dataset is widely available in the public domain (Texas Tribune) as shown:

Mathematics faculty: http://salaries.texastribune.org/texas-am- university / departments / mathematics / Statistics faculty: http://salaries.texastribune.org/texas-am-

niversity/departments/dept-f-statistics/

These websites are also included in the list of references at the end. Sections 1.5.1 and 1.5.2 respectively highlight a selection of both one-sample as well as two-sample comparisons among the inequality measures under consideration via approximate 95% confidence intervals.

One-Sample Problems

Tables 1.16-1.17 provide inequality measures including IIMs computed from salaries of full professors from the statistics department and mathematics department respectively. Tables 1.18-1.19 show the inequality measures including IIMs computed from salaries of all tenured and tenure-track (t/tt) professors from the statistics department and mathematics department respectively.

TABLE 1.14

Average Length (L) and Average Coverage Probability (p) Associated with the Confidence Interval from (1.25) for the IIMs for e_ki under the Lognormal Distribution from 2000 Replications: a = 0.05

TABLE 1.15

Average Length (L) and Average Coverage Probability (p) Associated with the Confidence Interval from (1.25) for the IIMs for e_ki under the Lognormal Distribution from 2000 Replications: a = 0.01

TABLE 1.16

Full Stat-prof: X = $177, 014.4, n = 20

TABLE 1.17

Full Math-prof: X = $132, 091.2, n = 43

The estimated variances, for H21, H22, and H31 were computed using the expression from (1.22). An expression for the estimated variance of G(= Нц) is routine in view of G being a U-statistic. The estimated variance of CV is also routine in view of Equation (10) from Curto and Pinto (2009). These five estimated variances (Est. var.) are reported in Tables 1.16-1.19.

From Tables 1.16-1.19 we find that the asymptotic variance for CV is the highest followed by that for the Gini inequality index. The asymptotic variance for the Gini inequality index exceeds that for H21 which exceeds that for H31. In the contexts of salaries of mathematics t/tt faculties and mathematics full professors separately from one another, we find that the asymptotic variance of H_3] is smaller than that of H₂2- But, for statistics faculties, the asymptotic variance for H22 is lower than that for H31.

Implementing Equation (10) from Curto and Pinto (2009), we note that for i.i.d. random variables, an approximate 100(l-rv)% confidence interval for the

TABLE 1.18

Stat-t/tt: X = $132,173.7, n= 39

TABLE 1.19

Math-t/tt: X = $177,014.4, n = 20

population CVf is given by:

One may refer to Bao (2009) to check on the expressions for the moments of the CV. A review of Nagar's (1959) results may throw some additional light in this direction.

In Tables 1.20-1.21, the intervals corresponding to G, H21, H22, H31, and CV are shown by implementing (1.25) and (1.26). The approximate 95% confidence intervals as shown in Tables 1.20-1.21 clearly indicate that considering all cases, the widths of the intervals corresponding to the newly proposed IIMs are overall smaller than the ones obtained via Gini index or CV.

We recall that the simulation studies presented in Section 1.4.2 showed clearly that the length of the confidence interval associated with CV was the highest, followed by that corresponding to G(= Ни), H21, H22, and H31. That is, the shortest confidence interval was observed in the context of H₃₁. Tables 1.20-1.21 validate a similar sentiment while comparing the performances of income inequality measures including the IIMs for income data from statistics faculty members. Income data from mathematics faculty members, however, show mixed signals. But, we should refrain from making too many judgment calls from such one-time analyses.

TABLE 1.20

95% Confidence Interval

TABLE 1.21

95% Confidence Interval

TABLE 1.22

95% Two-sample Confidence Interval

Two-Sample Problems

Next, we present some two-sample confidence intervals for the differences of the income inequality measures including the IIMs associated with professors and those associated with all tenured and tenure-track professors in both departments. Again, we find that the width of approximate 95% confidence intervals for the difference between the newly proposed IIMs corresponding to two departments is smaller than those corresponding to G and CV.

Again, Table 1.22 shows that our earlier sentiments expressed in the case of one-sample comparisons continue to hold in the context of two-sample comparisons between the IIMs and G, CV under consideration. We should make it clear that each inequality measure is compared between two populations. We recall that the simulation studies presented in Section 1.4.2 showed clearly that the length of the confidence interval associated with CV was the highest, followed by that corresponding to G(= H_n), H21, H22, and H31, Table 1.22 validates a similar sentiment.