TESTING LORENZ CURVES WITH NON-SIMPLE RANDOM SAMPLES.

The article focuses aims to derive the variance-covariance formulae for testing Lorenz and generalized Lorenz curves when samples are not simple random. Specifically, it considers the three most commonly used non-simple random sampling methods: stratified sampling, cluster sampling, and multistage sampling. Consistent and asymptotically unbiased sample estimators of the Lorenz and generalized Lorenz curves come directly from the definitions of the sampling procedures. Using the Bahadur representation, we are able to derive the asymptotic covariance structures of the Lorenz and generalized Lorenz curve estimates. Comparing these structures with those for simple random samples reveals that different sampling methods will lead to quite different variance-covariance formulae. Whether a given sample is treated as simple random or not may greatly affect inference testing; it is thus important to use the correct variance-covariance formulae in performing inference tests. Simple random sampling has been assumed in most statistical inference procedures developed to test Lorenz dominance, concentration dominance, and stochastic dominance. This assumption is highly unrealistic since simple random economic samples are rarely available. As this note has shown, different assumptions about the sampling method may lead to different estimation procedures and different variance-covariance formulae.