Preliminaries, Illustrations, and Economic Persuasions Behind the New IIMs
An interpretation of a population Gini index в from (1.1) is that it quantifies how any two randomly chosen individuals' incomes differ on an average after scaling. In other words, in coming up with the sample G from (1.2), every person's income is compared with every other person's income whatever may be the income distribution. We propose new IIMs by capitalizing on this fundamental premise. We demonstrate with the help of illustrations and economic persuasions that such proposed new IIMs are more apt to detect certain intricate features in an income distribution which are perhaps missed by G.
Some Preliminaries
We may begin by comparing (X, + X,) with X* for each triplet X„X,, Xj... That is, an income inequality index could be constructed by comparing any one individual's income with the average income of any two other individuals. This fundamental idea germinates from Mukhopadhyay and Chat topadhyay's (2011) useful constructions of new estimators for the standard deviation in a normal population.
IIMs of Degree 3
Let us begin by introducing one new ИМ which compares average income of every pair of individuals with another individual's income. We begin with a basic kernel i(X, + X,)  X_{k} and symmetrize it which gives the kernel:
This kernel leads to a new IIM estimator of degree 3:
Corresponding to the population IIM:
where X, Xj, X3 are the i.i.d. copies of X.
IIMs of Degree 4
Along lines similar to those in Section 1.2.1.1, we additionally propose the following two IIM estimators based on Ustatistics of degree 4.
We first define:
with an associated kernel:
Corresponding to the population IIM:
where X, X2, X3 and X4 are i.i.d. copies of X.
We may also define:
With an associated kernel
Corresponding to the population IIM:
where X, X2, X3 and X4 are i.i.d. copies of X.
Both measures H22 and H31 have degree 4. H22 quantifies on an average, how the average income of any two individuals is different from that of any two other individuals, all individuals chosen randomly from a population. Similarly, H31 quantifies on an average, how the average income of any three individuals is different from the income of any other individual, all chosen randomly from a population. Analogously defined H13 will obviously coincide with H31.
Remark 1.1. In Section 1.3, we have shown general constructs of IIMs in a similar spirit based on Ustatistics of degree m. But before we move in that direction, in what follows, we first explain a number of economic persuasions and motivations behind these new IIMs through illustrations.
Economic Persuasions and Motivations via Illustrations
Now, we illustrate the performance of G from (1.2) and the CV(= S/X) along with the new IIMs H21, H22, and H31 with the help of realistic datasets (populations). We note that S is the sample standard deviation. We present a number of situations, and in each situation we first argue which population values are more (or less) unequal based entirely upon our basic understanding of economic persuasions and feelings.
Then, we examine whether some of the new IIMs, H21, H22, or H31, may point in the right direction with regard to intrinsic features felt within data whereas G may not provide immediate guidance.
The realistic datasets under the microscope are necessarily small in size. That way, any economic persuasion or perception of income inequality may be genuinely felt before any specific measure is considered. We emphasize that the fact that a newly proposed IIM may (or may not) coincide with G or CV is not the important issue. The important issue is that some finer economic features guiding a population's dynamics may be incorrectly missed by G thus misleading policy makers if they focus on G alone. We find that our newly proposed IIMs are more apt to draw attention to some delicate features than G in a number of scenarios.
Illustration 2.1: Different Income DistributionsWith Same Misleading G
We show two sets (Situations A and B) of income data in Table 1.1, along the lines of Fernando (2007), with the same total income ($200,000).
Economic Arguments: Situations A and В provide two different income distributions. Lorenz curves of both cross each other. Indeed, the utility of an additional $11,000 for the poorest person in Situation A is much more than the utility of an additional $2000 for the second richest person in Situation A and the utility of an additional $5000 for the richest person in Situation A. From first principles, it appears to us that a decision maker should treat Situation A as having less income inequality than Situation B.
Inequality Measures: Table 1.2 shows the associated G, CV, and the newly proposed IIMs H21, H22, and H31. We find that H21, H22, and H31 in Situation В exceed the corresponding entries in Situation A. In other words, the new IIMs validate our strong empirical feelings argued via elementary economic principles. We note that the CV points in that same direction too, but the Gini index remains same in both Situations A and B.
Appraisal: In this illustration, we contend that the same G value misleads users by creating a perception that the extent of income disparity in Situations A and В remain the same. But, the reality is different and our new IIMs are
TABLE 1.1
Income Distribution (in S1000)
Serial No 
Total 

1 
2 
3 
4 
5 

Situation A 
20 
30 
40 
50 
60 
200 
Situation В 
9 
40 
48 
48 
55 
200 
TABLE 1.2
Absolute Differences of Inequality Measures
G(=H„) 
H21 
H22 
H31 
CV 

Situation A 
0.2500 
0.2042 
0.1583 
0.2000 
0.3953 
Situation В 
0.2500 
0.2254 
0.2067 
0.2067 
0.4531 
right on the mark. In other words, reporting the CV and our newly proposed IIMs H21, H22, and H31 along with G should make a good sense in arriving at inequality comparisons.
Illustration 2.2: Same Income Distribution with Different G
Household data may be collected annually to estimate G for an intermediate period between two censuses. For instance, the National Sample Survey under the Ministry of Statistics and Program Implementation in India conducts household surveys to collect data periodically on a number of economic and social factors from selected households to assess economic conditions in the country. In order to estimate the Gini index in an intermediate period between two censuses, one may exploit data on monthly household expenditure obtained from the household survey.
Deininger and Squire (1996) found that a Gini index, G, based on individual expenditures behaved differently from that based on household expenditures. In order to examine the performance of our newly proposed IIMs, we may simply think of a scenario having five households with two individuals in each family. Monthly consumption expenditures for these five households are recorded in two different ways. In Situation A, income disparity was assessed from the monthly expenditure of each individual from five households. For example, the data values "2", "3" in Situation A show the monthly expenditures of two individuals in the same household. In Situation B, income disparity was assessed from total monthly expenditures recorded from each household. Suppose Ia(Ib) denotes an inequality measure corresponding to Situation A(B).
Economic Arguments: From economic considerations, the two situations should appear almost the same. That is, we would expect the absolute difference for each inequality measure for the two datasets should be negligible. No inequality measure should ideally show widely different results when applied to individuals' expenditures instead of households' expenditures within the same economy.
Inequality Measures: Compared with the newly proposed IIMs and CV, from Table 1.3 we especially note the highest value for the absolute difference, IaIb, when using the Gini index, G. In this case, the inequality index H31 is associated with the smallest value among all IaIb  
Appraisal: The proposed IIMs H21, H22, and H31 appear more consistent showing smaller values of the absolute difference, IaIb, when compared with that under G. In this illustration, we contend that the G value may mislead users by creating a perception that the extent of income disparity in Situations A and В are rather greatly different. But, the reality is on the opposite side of that and our new IIMs are right on the mark. In other words, reporting the CV and our newly proposed IIMs H21, H22, and H31 along with G should make a good sense in arriving at inequality comparisons.