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In the present context we can envision situations where the propensity of an individual to file an income tax form may well depend on the size of the individual's income. Thus, rather than observing random variables with density /(#) we actually observe random variables with density proportional to g(x)f(x) where g{x) is a weighting function (the particular case of size biased sampling has received much attention in the literature, here g(x)(xx). It is quite clear that weightings will affect inequality as measured by the Lorenz order.

D. with common distribution function Fχ(x)). 1 (using the fact that the Lorenz order is scale invariant). 4) are not transparently amenable to a proof using Strassen's theorem. 3). n < U2:n < ••• < Un:n and an additional independent observation Un+ι. Combined they form a sample of size n +1 whose n + 1 —jth order statistic will be denoted by ί7 n +i_j : n +i. Note that the event {i7n+i > f7n+i_j:rι+i} depends only on the rank of {7n+i and is independent of the vector of order statistics > #n+i:n+i} Observe that { + Un-j:n+l and that Pr(tΓ n + 1 > J7n+l-j:n+l) = (j + l)/(n + 1).

1971). A general definition of the Lorenz curve. Econometrica 39, 1037-1039. GASTWIRTH, S. and RINOTT, Y. (1988). Medians-means comparisons. Paper #24 (private communication). KARLIN, M. (1980). On mixtures from exponential families. J. Roy. Statist. Soc, Ser. B 42, 192-198. SHARED, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 24, 423-439. STRASSEN, B. (1990). Concepts of Lorenz-ordering with a special regard to the Lorenz ordering of order statistics.

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