Testing Distributional Assumptions : A L-Moment Approach

Stein (1972, 1986) provides a flexible method for measuring the deviation of any probability distribution from a given distribution, thus effectively giving the upper bound of the approximation error which can be represented as the expectation of a Stein's operator. Hosking (1990, 1992) proposes the concept of L-moment which better summarizes the characteristics of a distribution than conventional moments (C-moments). The purpose of the paper is to propose new tests for conditional parametric distribution functions with weakly dependent and strictly stationary data generating processes (DGP) by constructing a set of the Stein equations as the L-statistics of conceptual ordered sub-samples drawn from the population sample of distribution; hereafter are referred to as the L-moment (GMLM) tests. The limiting distributions of our tests are nonstandard, depending on test criterion functions used in conditional L-statistics restrictions; the covariance kernel in the tests reflects parametric dependence specification. The GMLM tests can resolve the choice of orthogonal polynomials remaining as an identification issue in the GMM tests using the Stein approximation (Bontemps and Meddahi, 2005, 2006) because L-moments are simply the expectations of quantiles which can be linearly combined in order to characterize a distribution function. Thus, our test statistics can be represented as functions of the quantiles of the conditional distribution under the null hypothesis. In the broad context of goodness-of-fit tests based on order statistics, the methodologies developed in the paper differ from existing methods such as tests based on the (weighed) distance between empirical distribution and a parametric distribution under the null or the tests based on likelihood ratio of Zhang (2002) in two respects: 1) our tests are motivated by the L-moment theory and Stein's method; 2) offer more flexibility because we can select an optimal number of L-moments so that the sample size necessary for a test to attain a given level of power is minimal. Finally, we provide some Monte-Carlo simulations for IID data to examine the size, the power and the robustness of the GMLM test and compare with both existing moment-based tests and tests based on order statistics