# A General Class of New IIMs

Now, we proceed to broaden the core idea which we had laid out in Section 1.2.1. Suppose that we have income data from n individuals. First, we consider a subset of random variables X,i, Хд,..., X,„, of fixed size m obtained from Xj, X_{2},.. •, X„ assuming that *n > m.* We emphasize that *m* is held fixed throughout and *m* does not functionally depend upon *n.*

Then we compare two arbitrary linear function i X/p and j *Xty*

based on *Хц,* Хд,..., X* and X*+i, X*+2,..., X*_{+}f, respectively, where *к, l *are fixed positive integers such that *k + l = m.* That is, we begin with the idea of any two subsets of *к* individuals and *l* individuals, compare their sample average incomes, and then take the absolute value of the difference. Next, we symmetrize the difference and take their average over all possi- *n n*

ble combinations ( ^ )( ^ ) with *к, l* held fixed, *к + I = m.* This construction

leads us to propose a general ИМ, denoted by *Oki(F),* which has the following form:

With

where *X*,..., *X _{m}* are i.i.d. copies of X.

Thus, an estimator of *в _{м}* based on U-statistics of order

*m(= к*+ /) would be given by:

where *hki* is the corresponding symmetric kernel of degree *m* defined via

- (1.14) . This way, we come up with a general class of IIMs based on U-statistics with kernel of degree higher than 2. Section 1.2.1 included illustrations of
- (1.15) when
*m*= 3,4.

In a society, money changes hands rather continuously from one group of individuals to other groups of individuals. Thus, in a dynamic economy, comparing each person's income with every other person's income may not be the only important consideration to come up with a measure of inequality. A more detailed quantification of income disparity may be desirable which will take into account the absolute differences between average incomes of every subgroups of sizes *к* and *l,k + l = m* with *к, l, m* fixed.

Given this sentiment, it is our belief that the new ИМ Ни introduced via (1.15) fits well. Next, we move forward to highlight a series of crucial properties of Ни stated in Theorem 3.1.

## Selected Properties of the New IIMs

In what follows, we consider the general ИМ estimator H_{w} from (1.15) which is constituted by using a symmetric kernel of degree *m(= k +* /). Now, we state our main results.

Theorem 1.1. *With* F *held fixed, the IlM* **H^i **= Н_{И}(Т, X) *defined in* **(1.15) ***satisfies the following properties: (i) anonymity, (ii) scale independence, and (iii) population independence.*

We outline the properties:

i. Anonymity. This property, also referred to as symmetry, requires that the inequality measure under consideration be independent of any characteristic of individuals other than their income. One may note that for any permutation Y obtained from X we have Hw(F,Y) = Hh(F,X).

ii. Scale invariance. We check that the index H_{W}(F, X) remains unchanged if the incomes of the individuals are increased or decreased at the same rate. Suppose that the original income data Х],Хг,... ,X„ from *n* individuals are changed to the data Yi = сХь Y2 = *cXi*,..., Y„ = cX„ respectively with some fixed c > 0. For example, each employee may receive a 0.1% yearending raise, that is *с =* 1.001. Then, it is easily seen that both sets of data X, Y will come up with the same value for the inequality measure, Ни-

iii. Population independence. This requires inequality measures to be invariant to population replications. In other words, merging two identical distributions should not change the existing inequality measure. It is straightforward to observe that for any c > 0, we will have Hi./(F,X) = H_{W}(F,Y) where Y is a c-times concatenation of the vector X, that is Y = cX.

Remark 1.2. In addition to part (ii) in Theorem 1.1, suppose that a positive constant (= *d)* is added to everyone's original income Xi, X2,..., X„ from *n *individuals to come up with a new dataset Yi = Xi + *d,* Y2 = X2 + *d,*..., Y„ = X„ + *d* respectively. Then, for all *к* and /, we note that (a) Uju from (1.15) will remain the same whether we use X or Y data, but (b) Ни obtained from Y data will be smaller than Ни obtained from X data.

## Addressing the Pigou-Dalton Transfer Property

With F held fixed, our proposed IIM H_{W} = H_{W}(F,X) defined in (1.15) may or may not satisfy the Pigou-Dalton transfer property. Verifying the Pigou-Dalton transfer property amounts to a sound mathematical validation of the following principle. Suppose that X denotes the income vector of *n* individuals written down in an increasing order. Let *e,* denote an «-dimensional vector whose only non-zero element is 1 at the ith position, *i* = 1,..., *n.* Suppose that A,В represent two income distributions and we denote *В* >- *A* if Situation В is preferred over Situation A, that is, the inequality in Situation A is more than the inequality in Situation B. The Pigou-Dalton transfer principle requires that if *i < j* and *c >* 0, then one must have X + (ej - ej)c >- X given that X, + c < X,_{+}i and X._i < X.- — *c* are satisfied.

Suppose that in region A with *n* individuals, the data Xi, X2,..., X„ correspond to their increasingly ordered incomes whereas in region B, the incomes of *n* individuals in ascending order are Xi,.. . ,X_{p} + c,...,Хд — c,.. *,,X„. *Then, according to the Pigou-Dalton transfer principle, the inequality in region В should be less than that in region A.

In Section 1.3.2.1, we examine the Pigou-Dalton transfer property via simulations for our class of IIMs Ни- We add that the empirical evidence makes us lean in a direction to claim a sense of practical data-validation of the Pigou-Dalton transfer property for Ни- However, at this time, we do not debate whether or not a proof of the Pigou-Dalton transfer property may hold for the class of proposed IIMs H_{w}.

### Empirical Validation of Pigou-Dalton Transfer

We are not yet in a position to prove mathematically whether or not the Pigou-Dalton transfer property holds for Ни- We believe that it does.

A simulation study was designed to gain some practical feelings about the possibility of whether or not the Pigou-Dalton transfer property may hold in the context of the class of IIMs H_{k)}. We generated samples (income data) of size *n =* 10 from a gamma distribution with the shape parameter 2.649 and rate 0.84 and then we multiplied each observation by 10,000. We pretended to have such income data from 10 individuals which corresponded to Situation A.

After arranging those 10 observations in ascending order, we subtracted 50 (that is, c = 50) from the 5th ordered observation (that is, *j* = 5), but then added 50 to the second ordered observation (that is, *i = 2)* so that their ranks remained unchanged. We called this Situation B.

Thus, we had two different situations and we evaluated each IIM under both situations A,B. We repeated this whole process 20,000 times, providing 20,000 values for all IIMs from Section 1.2.1 under both situations. Recall that 1д (Ib) denotes an inequality measure corresponding to Situation A(B). Table 1.9 shows the average (mean) of the differences 1д- Ib of the inequality measure values with its standard error and the minimum value (Min) of the differences obtained from 20,000 replications.

If the Pigou-Dalton transfer property holds, we should expect all differences 1д-1в for every inequality measure to remain positive. In other words, the minimum value of all differences should be positive in the case of every inequality measure. The minimum values clearly show that the income values in Situation В are more equal than those in Situation A.

We repeated similar exercises with a number of other kinds of generated datasets. We observed empirical validation of the Pigou-Dalton transfer property for the class of IIMs Н_{ы} every time.