Poisson regression with Laplace measurement error

Abstract

In this dissertation, novel estimation procedures are proposed for a class of Poisson linear regression when the covariate is contaminated with Laplace measurement error. This dissertation contains two research projects. In the first project, we propose a weighted least squares estimation procedure that incorporates the first two conditional moments of the response variable given the observed surrogate, and the weight function is intentionally chosen to avoid the complexity caused by the random denominator and to increase the estimation efficiency. To solve for the conditional moments, a Tweedie-type formula for the conditional expectation of the likelihood function given the observed surrogate has been adopted. Instead of assuming the distribution of the unobserved covariate is known, we assume that the distribution of that latent variable is unknown. Large sample properties of the proposed estimator, including the consistency and the asymptotic normality, are discussed. The finite sample performance of the proposed estimation procedure is evaluated by simulation studies, showing that the proposed estimator is more efficient than the existing ones. In the second project, we propose a corrected maximum likelihood estimation procedure based upon the Tweedie-type formula. Two situations, the distribution of the latent variable is known as well as unknown, are considered. Large sample properties of the proposed estimator are discussed, and simulation study shows that the estimator is more efficient than the existing estimation procedures. Besides, further simulation studies are also conducted to compare our proposed two estimation procedures. And sensitivity analysis has been done to examine the robustness of our methods in real data. Although the discussion is conducted for univariate cases, the proposed estimation procedure can be readily extended to the multivariate cases by using multivariate Tweedie-type formulae.