Title: Nonlinear Statistical Inverse Problems and Instrumental Variables.
Abstract: We consider general inverse problems where an unknown quantity x is related to an observed quantity y by a known injective, nonlinear operator F, i.e. the inverse problem can be formulated as an operator equation F(x)=y. We are particularly interested in the case that this operator equation is ill-posed in the sense that the inverse of F is not continuous. In this case special care must be taken to avoid potentially infinite amplification of errors resulting from imperfect knowledge of either y or F, which will be modeled stochastically. Stability is achieved by regularization, which may be interpreted as incorporating a-priori knowledge of x into the reconstruction method.
As an application we will study instrumental variables in econometrics. Nonlinear integral equations with imperfectly known kernels occur ininstrumental regression if independence of errors and instruments is assumed.
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