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When a p-dimensional parameter θ is defined through the moment condition Em(X,θ) = 0, a simple estimation procedure of θ is proposed by Hong and Tamer when X, a k-dimensional random vector, is contaminated with Laplace measurement error U, that is, we can only observe Z = X + U. However, the estimation procedure was designed particularly for the cases where the components of the measurement error vector U are independent. We shall introduce a general multivariate Laplace distribution, then extend the Hong-Tamer moment estimation procedure to a more general multivariate scenario. Moreover, the Hong-Tamer moment estimation procedure is based on the unconditional expectation Em(X,θ) = EH(Z,θ) for some function H. Example shows this techniques does not work in some cases. We will further discuss an estimation procedure based on the condition expectation E(m(X,θ)|Z), which can be treated as an extension of the regression calibration technique. Large sample properties of the proposed estimation procedure will be investigated. Next, I will try to extend the above extended regression technique to nonparametric setup, particularly focusing on the normal and Laplace measurement error.