A Review of Statistical Methods for Handling Nonignorable Missing Data using Instrument Approach

While model selection is a well-studied topic in parametric and nonparametric regression or density estimation, selection of possibly high-dimensional nuisance parameters in semiparametric problems is far less developed. In this paper, we propose a selective machine learning framework for making inferences about a finite-dimensional functional defined on a semiparametric model, when the latter admits a doubly robust estimating function and several candidate machine learning algorithms are available for estimating the nuisance parameters. We introduce a new selection criterion aimed at bias reduction in estimating the functional of interest based on a novel definition of pseudo-risk inspired by the double robustness property. Intuitively, the proposed criterion selects a pair of learners with the smallest pseudo-risk, so that the estimated functional is least sensitive to perturbations of a nuisance parameter. We establish an oracle property for a multi-fold cross-validation version of …

Controlled Direct Effect (CDE) is one of the causal estimands used to evaluate both exposure and mediation effects on an outcome. When there are unmeasured confounders existing between the mediator and the outcome, the ordinary identification assumption does not work. In this manuscript, we consider an identification condition to identify CDE in the presence of unmeasured confounders. The key assumptions are: 1) the random allocation of the exposure, and 2) the existence of instrumental variables directly related to the mediator. Under these conditions, we propose a novel doubly robust estimation method, which work well if either the propensity score model or the baseline outcome model is correctly specified. Additionally, we propose a Generalized Information Criterion (GIC)-based model selection criterion for CDE that ensures model selection consistency. Our proposed procedure and related methods are applied to both simulation and real datasets to confirm the performance of these meth …

This paper outlines a new nonparametric estimation procedure for unobserved $\Phi$-mixing processes. It is assumed that the only information on the stationary hidden states $(X_{k})_{k\ge0}$ is given by the process $(Y_{k})_{k\ge0}$, where $Y_{k}$ is a noisy observation of $f_{\star}(X_{k})$. The paper introduces a maximum pseudo-likelihood procedure to estimate the function $f_{\star}$ and the distribution $\nu_{b,\star}$ of $(X_{0},\ldots,X_{b-1})$ using blocks of observations of length $b$. The identifiability of the model is studied in the particular cases $b=1$ and $b=2$ and the consistency of the estimators of $f_{\star}$ and of $\nu_{b,\star}$ as the number of observations grows to infinity is established.

The semiparametric estimation approach, which includes inverse-probability-weighted and doubly robust estimation using propensity scores, is a standard tool in causal inference, and it is rapidly being extended in various directions. On the other hand, although model selection is indispensable in statistical analysis, an information criterion for selecting an appropriate regression structure has just started to be developed. In this paper, based on the original definition of Akaike information criterion (AIC; \citealt{Aka73}), we derive an AIC-type criterion for propensity score analysis. Here, we define a risk function based on the Kullback-Leibler divergence as the cornerstone of the information criterion and treat a general causal inference model that is not necessarily a linear one. The causal effects to be estimated are those in the general population, such as the average treatment effect on the treated or the average treatment effect on the untreated. In light of the fact that th …

With nonignorable nonresponse, an effective method to construct valid estimators of population parameters is to use a covariate vector called instrument that can be excluded from the nonresponse propensity but are still useful covariate even when other covariates are conditioned. The existing work in this approach assumes such an instrument is given, which is frequently not the case in applications. In this paper we investigate how to search for an instrument from a given set of covariates. The method for estimation we apply is the pseudo likelihood proposed by Tang et al. (2003) and Zhao and Shao (2015), which assumed that an instrument is given and the distribution of response given covariates is parametric and the propensity is nonparametric. Thus, in addition to the challenge of searching an instrument, we also need to do variable and model selection simultaneously. We propose a method for instrument, variable, and model selection and show that our method produces consistent instru …