ABSTRACT
Early writers on inequality rarely distinguish clearly between sample and population statistics. A distribution for them might refer to some generic random variable or to the sample distribution of the finite number of observations actually available. We will try to be precise with regard to distinguishing between the “theoretical” distribution and the sample distribution. This will result in some repetition, since certainly the sample c.d.f. is a well defined distribution function. In many cases, one definition of a measure of inequality could serve in both settings. The sample measure of inequality arises when the sample c.d.f. is used, and the parent or theoretical measure of inequality arises when the evaluation is made with respect to the theoretical distribution. The increase in clarity is judged to be worth the price in redundancy. Actually, some early measures of inequality really only make sense when applied to sample data, and no amount of ingenuity seems to yield a plausible population version of the measure. In this class will be found measures which are slopes of certain fitted lines (sometimes fitted by eye!). When discussing population measures, we will speak of a single non-negative random variable X with distribution function F(x) and survival function F(x). When we speak of sample measures of inequality, we will deal with n quantities X1,X2, . . . ,Xn and their corresponding sample distribution function:
Fn(x) = 1 n
I(x,Xi) (4.1.1)
where
I(a,b) = 0 if a< b 1 if a≥ b.
