Estimation and Testing For Dominance Relations Partial but unambiguous rankings of distributions.

Anderson’s Method (can also be used for FGT indices). Based upon Trapezoidal Rule for approximating integrals Anderson (1996) Proposed a technique for estimating and testing differences between higher order integrals. Consider f(x) and f(y) the distributions of two independent variables x and y each defined over the common range (a,b), partition this range into k mutually exclusive and exhaustive intervals of size d1, d2,…, dk and consider the k long vectors p(x) and p(y) of sample proportions falling into the pre-defined k intervals.

Form the following Integrating Matrices

The Testing Procedure N’th order dominance of x over y can be examined via the non-positivity of all of the elements of the vector IF(N-1)If(p(x)-p(y)). Inference is facilitated by noting that under a null of no difference between the two distributions the vector IF(N-1)If(p(x)-p(y)) is normally distributed since (p(x)-p(y)) ~ N(0,Ω) where Ω is of a well known form. Testing the joint non-positivity of the individual elements of IF(N-1)If(p(x)-p(y)) can be tricky (Stoline and Ury maximum modulus distribution or Wolak’s simulation method).

Incomplete Moments Method [note these can also be used for FGT indices and can also deal with data dependent cuttoffs. (Davidson and Duclos (2002))]

Incomplete Moments Method Continued

Integrated Squared Difference Method (Hall and Yatchew (2004))

K-S Methods An Alternative approach, first advocated by McFadden and followed by Barrett and Donald () and Maasoumi, Linton and Wang () is based upon the Kolmonogorov-Smirnov approach to comparing two distributions. K-S compared two empirical cumulative densities over their whole range and considered the maximum vertical (absolute) distance between them and derived their distributions. McFadden and others did the same for Fi-Gi i>1 and used various simulation techniques to derive their distributions.

The K-S Results

The Test Inconsistency Issue The Anderson and Davidson – Duclos methods choose a selection of points in the domain of x at which to evaluate and compare Fi(x) and Gi(x), the K-S and Hall - Yatchew methods compare Fi(x) and Gi(x) over all values of x. Leads to the criticism that the first class of tests could be inconsistent. An inconsistent test is one which, under certain circumstances, has a probability of 0 of rejecting a false null asymptotically. Imagine Fi(x) and Gi(x) intersect at two points on the interior of their range and that x* is chosen so that the integral of the difference between them up to x* and the integral of the difference between them from x* up to ∞ both equal 0. If x* and ∞ are chosen as the evaluation points, Anderson and Davidson and Duclos tests will be inconsistent. Problem can be circumvented by choosing a sufficient number of comparison points.

Poverty Tests An attractive feature of the Anderson and Davidson and Duclos approaches (not available with the expected squared difference and K-S based approaches) when considering poverty analysis is that if one of the elements chosen in the vector of cut off points corresponds to the poverty cut off x* then Fi(x*) and Gi(x*) and their corresponding variances correspond to the FGT poverty measures for the respective value of i. Thus (Fi(x*)-Gi(x*))/(VAR(Fi(x*)-Gi(x*))0.5 corresponds to the ‘t’ test for a headcount ratio comparison when i=1, to a depth of poverty comparison when i=2, to an intensity of poverty comparison when i=3.

Multivariate Dominance Problems Multivariate Poverty, Inequality, Polarization and mobility indices have all been developed in the literature and practically implemented with little problem, however dominance comparisons have not received the same coverage in the literature. When considering dominance issues with many dimensions the non-parametric “curse of dimensionality” problem rears its ugly head. Essentially for Anderson’s and Davidson and Duclos methods the # of comparison points or cells (n) increases by the power of the # of dimensions (k) as in nk The K-S method turns out to be a way out since it does not depend upon cell probability calculations but merely upon the maximal and minimal distance between two surfaces. Unfortunately the distribution of D is not known when k > 1, though P(√nD > d) for k=1 does provide a conservative (i.e. larger) estimate when k > 1.