Statistical inferences for testing marginal rank and (generalized) Lorenz dominances.

• Author: Zheng, Buhong

1. Introduction

   Rank dominance, Lorenz dominance, and generalized Lorenz dominance are the three most commonly used tools in ranking income distributions; rank dominance and generalized Lorenz dominance yield social welfare rankings of income distributions, while Lorenz dominance provides inequality rankings. In their important contributions, Kolm (1969) and Atkinson (1970) establish that Lorenz dominance implies and is implied by all inequality measures satisfying the Pigou-Dalton principle of transfers; Saposnik (1981) proves that rank dominance is equivalent to welfare dominance by all increasing welfare functions; Shorrocks (1983) shows that generalized Lorenz dominance is equivalent to welfare dominance by all increasing and concave welfare functions.

   The empirical applications of these dominance methods have been greatly enhanced by the important contributions of Beach and Davidson (1983), Sendler (1979), and Gail and Gastwirth (1978), who provide the Lorenz curve with (asymptotically) distribution-free statistical inference procedures. Beach and Davidson's results also lead directly to the statistical inference of the generalized Lorenz curve, which was formally stated by Bishop, Chakraborti, and Thistle (1989). Although the asymptotic distribution of sample quantiles were well-known in the statistical literature (e.g., Cramer 1946), Bishop, Chow, and Formby (1991) were the first to formally test rank dominance.

   The applicability of these inference procedures, however, is limited by the requirement that the samples drawn from different distributions must be independent.(1) Although this requirement is not very restrictive in many cross-sectional or cross-time studies, it certainly cannot be fulfilled in addressing marginal changes in income quantiles and in Lorenz and generalized Lorenz curves. Marginal changes in, say, a Lorenz curve refer to the changes in the Lorenz curve of the same distribution after an exogenous shock or an endogenous change has occurred to the distribution. The dominance methods applied to the comparison of the distributions before and after the marginal change are referred to as marginal dominances. An example of interest is the impact of wives' participation in the labor force on family income inequality. It is commonly believed that wives' participation in the labor force reduced family income inequality during the 1950s and 1960s in the U.S. but has increased inequality in recent years. Many recent empirical studies, however, have revealed that working wives still reduce family income inequality (Cancian, Danziger, and Gottschalk [1993] and Treas [1987] provide surveys on these studies). All of these empirical works employ samples to estimate marginal changes, but none of them applies statistical inference tests. It is also worth noting that none of them uses Lorenz curve dominance.

   The present paper extends the existing statistical inferences of rank dominance and (generalized) Lorenz dominance to testing marginal dominances. It advances upon Beach and Davidson (1983) by deriving the full (asymptotic) joint variance - covariance structure for marginal changes in the ordinates of Lorenz and generalized Lorenz curves. It also provides inference for testing marginal changes in income quantiles. In proving the major results, I adopt a different yet more tractable approach (the Bahadur representation) than that used in either Sendler (1979) or Beach and Davidson (1983). As a consequence, the covariance structure can be derived in a straightforward manner and the property that the structure can be consistently estimated can be seen immediately.

   The rest of the paper is organized as follows. The next section defines marginal rank and (generalized) Lorenz dominances. Section 3 provides large sample properties of the estimates of the marginal changes. The full (asymptotic) variance-covariance structures are also provided. Section 4 illustrates the inference procedures by examining the issue of working wives and income distribution in the U.S., Section 5 shows that the developed inferences can be modified and applied to more general cases where samples are partially dependent.

2. Marginal Changes and Marginal Dominances

   Consider a joint distribution between two variables x [element of] [0, [infinity]) and y [element of] [0, [infinity]) with a continuous cumulative distribution function (c.d.f.) F(x, y). Without loss of generality, we may interpret x as family income before wives' participation in the labor force and y as family income after wives' participation in the labor force. The marginal distributions of x and y are denoted as H(x) and K(y), that is, H(x) [equivalent to] F(x, [infinity]) and K(y) [equivalent to] F([infinity], y). For convenience, we further assume that functions H and K are strictly monotonic and the first two moments of x and y exist and are finite. Thus, for a given population share p, which is the same for both x and y, there exist unique and finite income quantiles [Xi](p) and [Zeta](p) such that H([Xi](p)) = p and K([Zeta](p)) = p.

   The Lorenz and generalized Lorenz curve ordinates of H(x) and K(y) corresponding to p are usually defined as

   [Phi](p) [equivalent to] 1/[[Mu].sub.x] [integral of] xdH(x) between limits [Xi](p) and 0 and [Psi](p) [equivalent to] 1/[[Mu].sub.y] [integral of] ydK(y) between limits [Zeta](p) and 0 (2.1)

   and

   [Theta](p) [equivalent to] [integral of] xdH(x) between limits [Xi](p) and 0 = [[Mu].sub.x][Phi](p) and [Theta](p) [equivalent to] [integral of] ydK(y) between limits [Zeta](p) and 0 = [[Mu].sub.y][Psi](p), (2.2)

   where [[Mu].sub.x] and [[Mu].sub.y] are the mean incomes of x and y, respectively.

   With these notations, we can formally define marginal changes and marginal dominances.

   DEFINITION 2.1. Given a joint distribution F (x, y) and a population share p, the marginal change in the quantile is defined as the difference between [Xi](p) and [Zeta](p), that is, [[Delta].sup.Q](p) = [Zeta](p) [Xi](p); the marginal change in the Lorenz ordinate is [[Delta].sup.L](p) = [Psi](p) - [Phi](p); and the marginal change in the generalized Lorenz ordinate is [[Delta].sup.G](p) = [Theta](p) - [Theta](p). Marginal rank dominance holds if [[Delta].sup.Q](p) does not change sign for all p [element of] [0, 1] and is nonzero for some p [element of] [0, 1]; marginal Lorenz dominance holds if [[Delta].sup.L](p) does not change sign for all p [element of] [0, 1] and is nonzero for some p [element of] (0, 1); marginal generalized Lorenz dominance holds if [[Delta].sup.G](p) does not change sign for all p [element of] [0, 1] and is nonzero for some p [element of] [0, 1].

   In empirical studies, population quantiles and Lorenz and generalized Lorenz curves are usually characterized by a set of ordinates corresponding to the abscissae {[p.sub.i] [where] i = 1, 2, . . ., K} and [p.sub.K+1] = 1. Assuming 0 [less than] [p.sub.1] [less than] [p.sub.2] [less than] . . . [less than] [p.sub.K] [less than] 1, we have two sets of (K + 1) population quantiles {[[Xi].sub.i]} and {[[Zeta].sub.i]}, two sets of K population Lorenz curve ordinates {[[Phi].sub.i]} and {[[Psi].sub.i]}, and two sets of (K + 1) population generalized Lorenz curve ordinates {[[Theta].sub.i]} and {[[Theta].sub.i]}. For each i, i = 1, 2, . . ., K, these ordinates (quantiles) are related as shown in Equation 2.2; also [[Phi].sub.K+ 1] = [[Mu].sub.x] and [[Psi].sub.K+1] = [[Mu].sub.y].

   Assume a paired sample of size n, ([x.sub.1], [y.sub.1]), ([x.sub.2], [y.sub.2]), . . ., ([x.sub.n], [y.sub.n]), is independently and identically drawn from population with c.d.f. F(x, y). Then for each [p.sub.i], consistent sample estimates of [[Xi].sub.i] and [[Zeta].sub.i] are [x.sub.([r.sub.i])] and [y.sub.([r.sub.i])] (Settling 1980, Theorem 2.3.1), where [x.sub.(l)] and [y.sub.(l)] are the lth order statistics of {[x.sub.i]}...