Testing Lorenz curves with non-simple random samples

An attempt is made to derive the variance-covariance formulae for testing Lorenz and generalized Lorenz curves when samples are not simple random. In particular, the three most commonly used non-simple random sampling methods are considered: stratified sampling, cluster sampling, and multistage sampling. Using the Bahadur representation, the asymptotic covariance structures of the Lorenz and generalized Lorenz curve estimates are derived. Comparing these structures with those for simple random samples reveals that different sampling methods will lead to quite different variance-covariance formulae.