Efficient estimation for partially linear varying coefficient models when coefficient functions have different smoothing variables

Current methods used to analyze time to event data either rely on highly parametric assumptions which result in biased estimates of parameters which are purely chosen out of convenience, or are highly unstable because they ignore the global constraints of the true model. By using Targeted Maximum Likelihood Estimation (TMLE) one may consistently estimate parameters which directly answer the statistical question of interest. Targeted Maximum Likelihood Estimators are substitution estimators, which rely on estimating the underlying distribution. However, unlike other substitution estimators, the underlying distribution is estimated specifically to reduce bias in the estimate of the parameter of interest. We will present here an extension of TMLE for observational time to event data, the Collaborative Targeted Maximum Likelihood Estimator (C-TMLE) for the treatment specific survival curve. Through the use of a simulation study we will show that this method improves on commonly used methods in both robustness and efficiency. In fact, we will show that in certain situations the C-TMLE produces estimates whose mean square error is lower than the semi-parametric efficiency bound. We will also demonstrate that a semi-parametric efficient substitution estimator (TMLE) outperforms a semi-parametric efficient non-substitution estimator (the Augmented Inverse Probability Weighted estimator) in sparse data situations. Lastly, we will show that the bootstrap is able to produce valid 95 percent confidence intervals in sparse data situations, while influence curve based inference breaks down.

The authors consider semiparametric efficient estimation of parameters in the conditional mean model for a simple incomplete data structure in which the outcome of interest is observed only for a random subset of subjects but covariates and surrogate (auxiliary) outcomes are observed for all. They use optimal estimating function theory to derive the semiparametric efficient score in closed form. They show that when covariates and auxiliary outcomes are discrete, a Horvitz‐Thompson type estimator with empirically estimated weights is semiparametric efficient. The authors give simulation studies validating the finite‐sample behaviour of the semiparametric efficient estimator and its asymptotic variance; they demonstrate the efficiency of the estimator in realistic settings.

A common and important problem in clustered sampling designs is that the effect of within-cluster exposures (i.e., exposures that vary within clusters) on outcome may be confounded by both measured and unmeasured cluster-level factors (i.e., measurements that do not vary within clusters). When some of these are ill/not accounted for, estimation of this effect through population-averaged models or random-effects models may introduce bias. We accommodate this by developing a general theory for the analysis of clustered data, which enables consistent and asymptotically normal estimation of the effects of within-cluster exposures in the presence of cluster-level confounders. Semiparametric efficient estimators are obtained by solving so-called conditional generalized estimating equations. We compare this approach with a popular proposal by Neuhaus and Kalbfleisch (1998, Biometrics 54, 638-645) who separate the exposure effect into a within- and a between-cluster component within a random intercept model. We find that the latter approach yields consistent and efficient estimators when the model is linear, but is less flexible in terms of model specification. Under nonlinear models, this approach may yield inconsistent and inefficient estimators, though with little bias in most practical settings.

To address the objective in a clinical trial to estimate the mean or mean difference of an expensive endpoint Y, one approach employs a two-phase sampling design, wherein inexpensive auxiliary variables W predictive of Y are measured in everyone, Y is measured in a random sample, and the semiparametric efficient estimator is applied. This approach is made efficient by specifying the phase two selection probabilities as optimal functions of the auxiliary variables and measurement costs. While this approach is familiar to survey samplers, it apparently has seldom been used in clinical trials, and several novel results practicable for clinical trials are developed. We perform simulations to identify settings where the optimal approach significantly improves efficiency compared to approaches in current practice. We provide proofs and R code. The optimality results are developed to design an HIV vaccine trial, with objective to compare the mean 'importance-weighted' breadth (Y) of the T-cell response between randomized vaccine groups. The trial collects an auxiliary response (W) highly predictive of Y and measures Y in the optimal subset. We show that the optimal design-estimation approach can confer anywhere between absent and large efficiency gain (up to 24 % in the examples) compared to the approach with the same efficient estimator but simple random sampling, where greater variability in the cost-standardized conditional variance of Y given W yields greater efficiency gains. Accurate estimation of E[Y | W] is important for realizing the efficiency gain, which is aided by an ample phase two sample and by using a robust fitting method.

Efficient local IV estimation of an empirical auction model

A semiparametric efficient estimator in case-control studies for gene–environment independent models

Dimension-reduced semiparametric estimation of distribution functions and quantiles with nonignorable nonresponse

Summary A class of cohort sampling designs, including nested case–control, case–cohort and classical case–control designs involving survival data, is studied through a unified approach using Cox’s proportional hazards model. By finding an optimal sample reuse method via local averaging, a closed form estimating function is obtained, leading directly to the estimators of the regression parameters that are relatively easy to compute and are more efficient than some commonly used estimators in case–cohort and nested case–control studies. A semiparametric efficient estimator can also be found with some further computation. In addition, the class of sampling designs in this study provides a variety of sampling options and relaxes the restrictions of sampling schemes that are currently available.

Robins et al. (2008, 2016b) applied the theory of higher order infuence functions (HOIFs) to derive an estimator of the mean of an outcome Y in a missing data model with Y missing at random conditional on a vector X of continuous covariates; their estimator, in contrast to previous estimators, is semiparametric efficient under minimal conditions. However the Robins et al. (2008, 2016b) estimator depends on a non-parametric estimate of the density of X. In this paper, we introduce a new HOIF estimator that has the same asymptotic properties as their estimator but does not require non-parametric estimation of a multivariate density, which is important because accurate estimation of a high dimensional density is not feasible at the moderate sample sizes often encountered in applications. We also show that our estimator can be generalized to the entire class of functionals considered by Robins et al. (2008) which include the average effect of a treatment on a response Y when a vector X suffices to control confounding and the expected conditional variance of a response Y given a vector X.

Stresses that control function estimators explicitly recognize that nonrandom selection into a program may give rise to an endogeneity problem—nonrandom selection could cause participation to correlate with unobserved factors that influence the outcome variable. When randomized controlled trials prove unusable to estimate program impacts, different types of estimators can work for nonexperimental data. An important issue when applying the various methods remains whether they assume that program participation could depend on unobserved factors. If this proves possible, one alternative remains instrumental variables (IV) methods and control function methods. Economists have used IV methods for many decades, but control function methods remain a relatively new approach. Control function methods remove bias caused by selection on unobservables by explicitly modeling the selection process and how it relates to the observed outcomes. The major challenge in applying these methods for estimating program impacts comes from the need to separately identify the intercept of the control function from the treatment effect.

Longitudinal studies are widely used in various fields, such as public health, clinic trials and financial data analysis. A major challenge for longitudinal studies is the repeated measurements from each subject, which cause time dependent correlations within subjects. Generalized Estimating Equations (GEE) can deal with correlated outcomes for longitudinal data through marginal effect. Our proposed model will be based on GEE, with a semi-parametric approach, to provide a flexible structure for regression models: coefficients for parametric covariates will be estimated and nuisance covariates will be fitted in kernel smoothers for the non-parametric part. The profile kernel estimator and the seemingly unrelated kernel estimator (SUR) will be used to obtain consistent and efficient semi-parametric estimators. We provide simulation results for estimating semi-parametric models with one or multiple non-parametric terms. Financial market data is a major component of data analysis; thus, we focus on the financial market in the application part. Credit card loan data will be used with the payment information for each customer across six months to investigate whether gender, income, age, or other factors will influence payment status significantly. Furthermore, we propose model comparisons to evaluate whether different models should be fitted for different subgroups of consumers, such as male and female.

We develop a general semisupervised framework for statistical inference in the two‐sample comparison setting. Although the supervised Mann–Whitney statistic outperforms many estimators in the two‐sample problem for nonnormally distributed responses, it is excessively inefficient because it ignores large amounts of unlabelled information. To borrow strength from unlabelled data, we propose a class of efficient and adaptive estimators that use two‐step semiparametric imputation. The probabilistic index model is adopted primarily to achieve dimension reduction for multivariate covariates, and a follow‐up reweighting step balances the contributions of labelled and unlabelled data. The asymptotic properties of our estimator are derived with variance comparison through a phase diagram. Efficiency theory shows our estimators achieve the semiparametric variance lower bound if the probabilistic index model is correctly specified, and are more efficient than their supervised counterpart when the model is not degenerate. The asymptotic variance is estimated through a two‐step perturbation resampling procedure. To gauge the finite sample performance, we conducted extensive simulation studies which verify the adaptive nature of our methods with respect to model misspecification. To illustrate the merits of our proposed method, we analyze a dataset concerning homelessness in Los Angeles.

Causal attribution, which seeks to explain the reasons behind events or behaviors, plays a critical role in causal inference and deepens our understanding of cause-and-effect relationships in scientific research. The probabilities of necessary causation (PN) and sufficient causation (PS) are two of the most common quantities for attribution in causal inference. While several works have explored the identification or bounds of PN and PS, efficient estimation remains unaddressed. To fill this gap, this paper focuses on obtaining semiparametric efficient estimators of PN and PS under two sets of identifiability assumptions: strong ignorability and monotonicity, and strong ignorability and conditional independence. We derive efficient influence functions and semiparametric efficiency bounds for PN and PS under the two sets of identifiability assumptions, respectively. Based on this, we propose efficient estimators for PN and PS and show their large sample properties. Extensive simulations validate the superiority of our estimators compared to competing methods. We apply our methods to a real-world dataset to assess various risk factors affecting stroke.

I consider the problem of estimating an additive partially linear model using general series estimation methods with polynomial and splines as two leading cases. I show that the finite‐dimensional parameter is identified under weak conditions. I establish the root‐n‐normality result for the finite‐dimensional parameter in the linear part of the model and show that it is asymptotically more efficient than a semiparametric estimator that ignores the additive structure. When the error is conditional homoskedastic, my finite‐dimensional parameter estimator reaches the semiparametric efficiency bound. Efficient estimation when the error is conditional heteroskedastic is also discussed.

This chapter studies the estimation problem for a linear model. The first four sections are fairly classical and the presented results are based on the direct analysis of the linear estimation procedures. Sections 4.5 and 4.6 reproduce in a very short form the same results but now based on the likelihood analysis. The presentation is based on the celebrated chi-squared phenomenon which appears to be the fundamental fact yielding the exact likelihood-based concentration and confidence properties. The further sections are complementary and can be recommended for a more profound reading. The issues like regularization, shrinkage, smoothness, and roughness are usually studied within the nonparametric theory; here we try to fit them to the classical linear parametric setup. A special focus is on semiparametric estimation in Sect. 4.9. In particular, efficient estimation and chi-squared result are extended to the semiparametric framework.The main tool of the study is the quasi maximum likelihood method. We especially focus on the validity of the presented results under possible model misspecification. Another important issue is the way of measuring the estimation loss and risk. We distinguish below between response estimation or prediction and the parameter estimation. The most advanced results like chi-squared result in Sect. 4.6 are established under the assumption of a Gaussian noise. However, a misspecification of noise structure is allowed and addressed.