Further Remarks on Estimating the Size Distribution of a Given Aggregate Income - Clark, Eugene; Fishman, Leo (The Review of Economics and Statistics, 1950)
IN reply ' to the present writers' appraisal of his method of estimating the size distribution of a given aggregate income,2 David Durand attempts to justify the validity and practical usefulness of his method.3 In the process, however, he has shifted the emphasis from the issues we raised to other matters. Dr. Durand's initial method necessitated graphic interpolation from a cumulative frequency curve. Given an income distribution at one level of aggregate income, it yielded a distribution of a new aggregate, with the same relative inequality as measured by the Lorenz curve, and with income size classes having the same limits. He suggested that this method was useful in investigating the effect of income variations on demand, savings, tax yields, etc. In working with Durand's derived distribution, we discovered that average incomes within the income size classes, as calculated from the table in his initial article, differed sharply from the corresponding average incomes in the original NRC distribution. Noting these discrepancies, we pointed out that it was impossible for theoretical reasons to maintain simultaneously the same Lorenz curve, class limits, and average incomes in the derived as in the original distribution. Durand admits the theoretical validity of this point, but claims that it lacks significance because most problems involving income distributions . . . do not require great precision. 4 We disagree with Durand in this matter. The originator of a statistical technique cannot foretell all the uses to which it may be put or the degree of accuracy which may be expected of the results. In describing the technique, therefore, he should indicate its shortcomings, and the degree of accuracy of the results it yields. Durand did not mention in his initial article, that the published table representing his derived distribution yielded average incomes substantially different from the original NRC averages, and that in four of the seventeen size classes the indicated averages were outside the limits of their respective classes. Durand did not even mention the possibility that results of this character might be produced by his method. In his later article, Durand demonstrates that by numerical interpolation he can derive a new distribution with averages that come much closer to those in the original NRC distribution, although small differences still exist. This modified method does not possess the advantage of simplicity which he claimed for his initial method. We are more seriously concerned, however, that the main points made in our original article shall not be lost sight of in what is essentially a discussion of highly refined techniques of numerical interpolation. Our criticism was concerned with the basic issue of the consistency of graphic (or other) methods for estimating the size distribution of different aggregate incomes with the objectives for which those methods are used. Most persons engaged in statistical estimation presumably would consider that consistency with its objectives is one of the prime requisites of an acceptable and useful method, even when the underlying data are rough. We pointed out that there are two fundamental approaches to this problem of statistical estimation. The approach presented by Durand results in a distribution retaining the original income size classes and Lorenz curve. With this method it is impossible to retain the same average income within the original 1 David Durand, An of the Errors Involved in Estimating the Size Distribution of a Given Aggregate Income, this REVIEW, XXX (1948), pp. 63-68. 2 Eugene Clark and Leo Fishman, Appraisal of Methods for Estimating the Size Distribution of a Given Aggregate Income, this REVIEW, XXIX (I947), PP. 43-46. 'David Durand, A Simple Method for Estimating the Size Distribution of a Given Aggregate Income, this REVIEW, xxV (I943), Pp. 227-30. 4 David Durand, An of the Errors .. op. cit., p. 63.