Moment-Based Tests for Discrete Distributions

In this paper, we develop moment-based tests for parametric discrete distributions. Momentbased test techniques are attractive as they provide easy-to-implement test statistics. We propose a general transformation that makes the moments of interest insensitive to the parameter estimation uncertainty. This transformation is valid in some extended family of non differentiable moments that are of great interest in the case of discrete distributions. We compare this strategy with the one which consists in correcting for the parameter uncertainty considering the power function under local alternatives. The special example of the backtesting of VaR forecasts is treated in detail, and we provide simple moments that have good size and power properties in Monte Carlo experiments. Additional examples considered are discrete counting processes and the geometric distribution. We finally apply our method to the backtesting of VaR forecasts derived from a T-GARCH(1,1) model estimated on foreign exchange rate data.