Simulation extrapolation estimation in parametric models with Laplace measurement error

Abstract

The normal-error-based simulation extrapolation (N-SIMEX) procedure provides a simulation-based method to remove or reduce the bias in estimators of parameters in measurement error models. This paper shows that the N-SIMEX procedure only works for the normal measurement error and does not work for Laplace measurement error. A new L-SIMEX procedure is particularly designed to remove or reduce the Laplace measurement errors in parametric models. Unlike in the N-SIMEX procedure where the measurement error is removed or reduced by adding independent normal errors controlled by the scale parameter, the proposed procedure removes or reduces the Laplace measurement error by adding a noise variable which is a difference between two independent gamma random variables, and where the noise level is governed by the shape parameter. Heuristic and rigorous arguments are provided to justify the proposed method and a Jackknife-type estimation procedure is provided to estimate the variance of the L-SIMEX estimate. Simulation studies and a real data example are presented to demonstrate the proposed estimation procedure. A finite sample comparison with some revised moment estimators of Hong and Tamer is also included.