This paper studies non-identified ill-posed inverse models with estimated operator. Leading examples are the nonparametric IV regression and the functional linear IV regression. We argue that identification of infinite-dimensional parameters is less crucial than identification of finite-dimensional parameters. We show that in the case of identification failures , a very general family of continuously-regularized estimators is consistent for the best approximation of the parameter of interest and obtain L_2 and L_\infty finite-sample risk bounds. This class includes Tikhonov, iterated Tikhonov, spectral cut-off, and Landweber-Fridman as special cases. We show that in many cases the best approximation coincides with the structural parameter and can be a useful and tractable object to infer relation between structural variables otherwise. Unlike in the identified case, estimation of the operator may have a non-negligible impact on the estimation accuracy and inference. We develop inferential methods for linear functionals in non-identified models as well as honest uniform confidence sets for the best approximation. Lastly, we demonstrate the discontinuity in the asymptotic distribution for extreme cases of identification failures where we observe a degenerate U-statistics asymptotics .
This paper documents the consequences of the identification failures for a class of linear ill-posed inverse models. The Tikhonov-regularized estimator converges to a well-defined limit equal to the best approximation of the structural parameter in the orthogonal complement to the null space of the operator. We illustrate that in many cases the best approximation may coincide with the structural parameter or at least may reasonably approximate it. We characterize the nonasymptotic Hilbert space norm and the uniform norm convergence rates for the best approximation. Nonidentification has important implications for the large sample distribution of the Tikhonov-regularized estimator, and we document the transition between the Gaussian and the weighted chi-squared limits. The theoretical results are illustrated for the nonparametric IV and the functional linear IV regressions and are further supported by the Monte Carlo experiments .
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studies non-identified
documents the consequences of the identification failures for a class of linear
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inverse models with estimated operator. Leading examples are the nonparametric IV regression and the functional linear IV regression. We argue that identification of infinite-dimensional parameters is less crucial than identification of finite-dimensional parameters. We show that in the case of identification failures , a very general family of continuously-regularized estimators is consistent for
inverse models. The Tikhonov-regularized estimator converges to a well-defined limit equal to
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parameter of interest and obtain L_2 and L_\infty finite-sample risk bounds. This class includes Tikhonov, iterated Tikhonov, spectral cut-off, and Landweber-Fridman as special cases. We show
structural parameter in the orthogonal complement to the null space of the operator. We illustrate
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coincides
may coincide
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and can be a useful and tractable object to infer relation between structural variables otherwise. Unlike in the identified case, estimation of the operator may have a non-negligible impact on the estimation accuracy and inference. We develop inferential methods for linear functionals in non-identified models as well as honest uniform confidence sets
or at least may reasonably approximate it. We characterize the nonasymptotic Hilbert space norm and the uniform norm convergence rates
R
Lastly, we demonstrate the discontinuity in the asymptotic distribution for extreme cases of identification failures where we observe a degenerate U-statistics asymptotics
Nonidentification has important implications for the large sample distribution of the Tikhonov-regularized estimator, and we document the transition between the Gaussian and the weighted chi-squared limits. The theoretical results are illustrated for the nonparametric IV and the functional linear IV regressions and are further supported by the Monte Carlo experiments