# Tests for stochastic dominance

In the area of income distributions and tax analysis, it is important to look at Lorenz curves and similar comparisons. In practice, a finite number of ordinates of the desired curves or functions are compared. These ordinates are typically represented by quantiles and/or conditional interval means. Thus, the distribu­tion theory of the proposed tests are typically derived from the existing asymp­totic theory for ordered statistics or conditional means and variances. A most up-to-date outline of the required asymptotic theory is Davidson and Duclos

(1998) . To control for the size of a sequence of tests at several points the union intersection (UI) and Studentized Maximum Modulus technique for multiple comparisons is generally favored in this area. In this line of inquiry the inequality nature of the order relations is not explicitly utilized in the manner described above for parametric tests. Therefore, procedures that do so may offer power gain. Some alternatives to these multiple comparison techniques have been sug­gested, which are typically based on Wald-type joint tests of equality of the same ordinates; e. g. see Anderson (1996). These alternatives are somewhat problematic since their implicit null and alternative hypotheses are typically not a satisfac­tory representation of the inequality (order) relations that need to be tested. For instance, Xu et al. (1995) take proper account of the inequality nature of such hypotheses and adapt econometric tests for inequality restrictions to testing for FSD and SSD, and to GL dominance, respectively. Their tests follow the x2 theory outlined earlier.

McFadden (1989) and Klecan et al. (1991) have proposed tests of first – and second-order "maximality" for stochastic dominance which are extensions of the

Kolmogorov-Smirnov statistic. McFadden (1989) assumes iid observations and independent variates, allowing him to derive the asymptotic distribution of his test, in general, and its exact distribution in some cases. He provides a Fortran and a GAUSS program for computing his tests. Klecan et al. generalize this ear­lier test by allowing for weak dependence in the processes and replace indepen­dence with exchangeability. They demonstrate with an application for ranking investment portfolios. The asymptotic distribution of these tests cannot be fully characterized, however, prompting Klecan et al. to propose Monte Carlo methods for evaluating critical levels. Similarly, Maasoumi et al. (1997) propose bootstrap – KS tests with several empirical applications. In the following subsections some definitions and results are summarized which help to describe these tests.