测量误差为Laplace分布的非线性统计推断

NONLINEAR STATISTICAL INFERENCES WITH LAPLACE MEASUREMENT ERROR

当p-维参数θ通过矩条件Em(X,θ)=0定义,且X带有Laplace测量误差时,即我们只能观测到Z=X+U,文献中提出了一种基于无条件期望关系Em(X,θ)=EH(Z,θ)的估计方法,其中H为某个形式已知的函数.然而该方法仅适用于U的各分量服从Laplace分布且相互独立的情况.文章将介绍一种一般的多元Laplace分布,并将基于无条件期望的估计方法推广到具有这种多元Laplace分布的测量误差模型中.另外,基于无条件期望关系的估计方法对一些统计推断问题并不适用.文章将构造一种基于条件期望E[m(X,θ)|Z]的估计方法.当X为一维时,我们对这些估计的大样本性质进行了讨论.

When a p-dimensional parameter θ is defined through the moment condition Em（X,θ） = 0,a simple estimation procedure of θ was proposed in literature based on the unconditional expectation Em（X,θ） = EH（Z,θ） for some function H,when X,a k-dimensional random vector,are contaminated with Laplace measurement error U,that is,only Z = X ＋ U can be observed.However,the estimation procedure was designed particularly for the cases where the components of the measurement error vector U are independent.In this paper,we first introduce a general multivariate Laplace distribution,then extend the existing method to the general multivariate scenario.However,an example shows that the estimation procedure based on the unconditional expectation does not work in some cases.In this paper,we will propose an estimation procedure based on the condition expectation E（m（X,θ）｜Z）.Large sample properties of the proposed estimation procedure when X is one-dimensional are discussed.