Nonparametric Estimation Procedure Research Articles

Assessing variable importance in survival analysis using machine learning

SUMMARY Given a collection of features available for inclusion in a predictive model, it may be of interest to quantify the relative importance of a subset of features for the prediction task at hand. For example, in HIV vaccine trials, participant baseline characteristics are used to predict the probability of HIV acquisition over the intended follow-up period, and investigators may wish to understand how much certain types of predictors, such as behavioral factors, contribute toward overall predictiveness. Time-to-event outcomes such as time to HIV acquisition are often subject to right censoring, and existing methods for assessing variable importance are typically not intended to be used in this setting. We describe a broad class of algorithm-agnostic variable importance measures for prediction in the context of survival data. We propose a nonparametric efficient estimation procedure that incorporates flexible learning of nuisance parameters, yields asymptotically valid inference, and enjoys double-robustness. We assess the performance of our proposed procedure via numerical simulations and analyze data from the HVTN 702 vaccine trial to inform enrollment strategies for future HIV vaccine trials.

Active mutual conjoint estimation of multiple contrast sensitivity functions.

Recent advances in nonparametric contrast sensitivity function (CSF) estimation have yielded a new tradeoff between accuracy and efficiency not available to classical parametric estimators. An additional advantage of this new framework is the ability to independently tune multiple aspects of the estimator to seek further improvements. Machine learning CSF estimation with Gaussian processes allows for design optimization in the kernel, acquisition function, and underlying task representation, to name a few. This article describes a novel kernel for CSF estimation that is more flexible than a kernel based on strictly functional forms. Despite being more flexible, it can result in a more efficient estimator. Further, trial selection for data acquisition that is generalized beyond pure information gain can also improve estimator quality. Finally, introducing latent variable representations underlying general CSF shapes can enable simultaneous estimation of multiple CSFs, such as from different eyes, eccentricities, or luminances. The conditions under which the new procedures perform better than previous nonparametric estimation procedures are presented and quantified.

Migration and Remittances in Senegal: Effect on Households Members Left Behind

ABSTRACT This article investigates how migration and remittances affect labor market, and how remittances influence human capital in Senegal. Using parametric and nonparametric estimation procedures for the correction of selection bias, the results reveal that migration and remittances reduce labor market participation of household members left behind, and that remittances increase human capital development of the left-behind. The study suggests that both the migration status and the level of remittances are relevant in understanding labor market participation and human capital formation.

Identification and estimation of panel semiparametric conditional heteroskedastic frontiers with dynamic inefficiency

We study a semiparametric panel stochastic frontier model with nonseparable unobserved heterogeneity, which allows for time-varying conditional heteroskedastic productivity components. It does not require distributional assumptions on random noise except conditional symmetry. We utilize conditional characteristic functions from Kotlarski’s Lemma to derive new moment conditions that yield the identification of the heteroskedastic variance parameters of inefficiency and random noise. Identification only requires a panel with three periods for serially correlated inefficiency. A nonparametric estimation procedure is also developed for the conditional variance of inefficiency, and its convergence rate is established. Monte Carlo simulation shows that the estimator is robust to misspecification of inefficiency distributions.

Factor Overnight GARCH-Itô Models

Abstract This article introduces a unified factor overnight GARCH-Itô model for large volatility matrix estimation and prediction. To account for whole-day market dynamics, the proposed model has two different instantaneous factor volatility processes for the open-to-close and close-to-open periods, while each embeds the discrete-time multivariate GARCH model structure. To estimate latent factor volatility, we assume the low rank plus sparse structure and employ nonparametric estimation procedures. Then, based on the connection between the discrete-time model structure and the continuous-time diffusion process, we propose a weighted least squares estimation procedure with the non-parametric factor volatility estimator and establish its asymptotic theorems.

Nonparametric density estimation with nonuniform B-spline bases

Recently, Kirkby et al. introduced a general and efficient nonparametric density estimation procedure for local bases that is based on the statistical Galerkin method. However, the resulting density estimators using the statistical Galerkin method are not guaranteed to be proper, i.e., the density functions are not ensured to be nonnegative and normalized. To address this issue, we present a constrained Galerkin method with B-spline functions for nonparametric density estimation. Furthermore, we also provide a procedure to generate the optimal knot vectors for density estimations with nonuniform B-spline bases, which are based on the cumulative distribution of the feature function. Extensive numerical experiments are conducted to demonstrate the efficiency and accuracy of our framework.

Cost-effectiveness analysis under multiple effectiveness outcomes: A probabilistic approach.

Probability based criteria are proposed for the assessment of cost-effectiveness of a new treatment compared to a standard treatment when there are multiple effectiveness measures. Depending on the preferences of a policy maker, there are several options to define such criteria. Two such metrics are investigated in detail. One metric is defined as the conditional probability that a new treatment is more effective with respect to the multiple effectiveness measures for patients having lower costs under the new treatment. A second metric is defined as the conditional probability that a new treatment is less costly for patients having greater health benefits under the new treatment. The metrics offer considerable flexibility to a policy maker as thresholds for cost and effectiveness can be incorporated into the metrics. Parametric confidence limits are developed using a percentile bootstrap approach assuming multivariate normality for the joint distribution of the log(cost) and effectiveness measures. A non-parametric estimation procedure is also developed using the theory of U-statistics. Numerical results indicate that the proposed confidence limits accurately maintain coverage probabilities. The methodologies are illustrated using a study on the treatment of type two diabetes. Code implementing the proposed methods are provided in the supporting information.

A Three-Stage Nonparametric Kernel-Based Time Series Model Based on Fuzzy Data

In this paper, a nonlinear time series model is developed for the case when the underlying time series data are reported by LR fuzzy numbers. To this end, we present a three-stage nonparametric kernel-based estimation procedure for the center as well as the left and right spreads of the unknown nonlinear fuzzy smooth function. In each stage, the nonparametric Nadaraya–Watson estimator is used to evaluate the center and the spreads of the fuzzy smooth function. A hybrid algorithm is proposed to estimate the unknown optimal bandwidths and autoregressive order simultaneously. Various goodness-of-fit measures are utilized for performance assessment of the fuzzy nonlinear kernel-based time series model and for comparative analysis. The practical applicability and superiority of the novel approach in comparison with further fuzzy time series models are demonstrated via a simulation study and some real-life applications.

Assessing treatment effect through compliance score in randomized trials with noncompliance

A randomized trial is the gold standard for assessing the benefit of a treatment versus a control. When noncompliance is present, treatment effect depends on the tendency to comply—an attribute that is not directly measurable. Though the principal causal effect has been the most important for handling noncompliance, it is not immediately applicable to clinical decision-making as it targets the average effect in the latent strata of potential compliance. In this work, we propose the concept of compliance score, a linear combination of baseline characteristics, that uncovers the inherent attribute of compliance. We then assess the heterogeneous causal effect, namely, the causal effect of treatment as a function of baseline characteristics through the compliance score. A pseudo-response, along with a nonparametric estimation procedure, is proposed to ensure consistent and optimally efficient estimation. Compare to principal causal effect, the proposed effect is actionable and allows prediction of treatment effect at individual level. This work is motivated by and applied to a clinical trial to evaluate the benefit of antiretroviral regimens in HIV-infected patients.

Identification of unobserved distribution factors and preferences in the collective household model

This paper develops a non-parametric collective random utility model with a continuous choice of consumption and sharing. We allow for a heterogeneous population by introducing unobserved distribution factors and preferences that are non-separable in the stochastic household choice system. We provide necessary and sufficient conditions for invertibility with respect to the vector of unobserved heterogeneity. Further, we show non-parametric identification of each individual’s idiosyncratic preferences and Pareto weights from the distribution of observed choices. Based on this result we develop a non-parametric two-step estimation procedure and estimate individual counterfactuals in a collective labour supply model for households with children using the LISS panel.

Bayesian nonparametric analysis of restricted mean survival time.

The restricted mean survival time (RMST) evaluates the expectation of survival time truncated by a prespecified time point, because the mean survival time in the presence of censoring is typically not estimable. The frequentist inference procedure for RMST has been widely advocated for comparison of two survival curves, while research from the Bayesian perspective is rather limited. For the RMST of both right- and interval-censored data, we propose Bayesian nonparametric estimation and inference procedures. By assigning a mixture of Dirichlet processes (MDP) prior to the distribution function, we can estimate the posterior distribution of RMST. We also explore another Bayesian nonparametric approach using the Dirichlet process mixture model and make comparisons with the frequentist nonparametric method. Simulation studies demonstrate that the Bayesian nonparametric RMST under diffuse MDP priors leads to robust estimation and under informative priors it can incorporate prior knowledge into the nonparametric estimator. Analysis of real trial examples demonstrates the flexibility and interpretability of the Bayesian nonparametric RMST for both right- and interval-censored data.

A General Framework for Inference on Algorithm-Agnostic Variable Importance

In many applications, it is of interest to assess the relative contribution of features (or subsets of features) toward the goal of predicting a response—in other words, to gauge the variable importance of features. Most recent work on variable importance assessment has focused on describing the importance of features within the confines of a given prediction algorithm. However, such assessment does not necessarily characterize the prediction potential of features, and may provide a misleading reflection of the intrinsic value of these features. To address this limitation, we propose a general framework for nonparametric inference on interpretable algorithm-agnostic variable importance. We define variable importance as a population-level contrast between the oracle predictiveness of all available features versus all features except those under consideration. We propose a nonparametric efficient estimation procedure that allows the construction of valid confidence intervals, even when machine learning techniques are used. We also outline a valid strategy for testing the null importance hypothesis. Through simulations, we show that our proposal has good operating characteristics, and we illustrate its use with data from a study of an antibody against HIV-1 infection. Supplementary materials for this article are available online.

Assessing conditional causal effect via characteristic score.

Observational studies usually include participants representing the wide heterogeneous population. The conditional causal effect, treatment effect conditional on baseline characteristics, is of practical importance. Its estimation is subject to two challenges. First, the causal effect is not observable in any individual due to counterfactuality. Second, high-dimensional baseline variables are involved to satisfy the ignorable treatment selection assumption and to attain better estimation efficiency. In this work, a nonparametric estimation procedure, along with a pseudo-response, is proposed to estimate the conditional treatment effect through "characteristic score"-a parsimonious representation of baseline variable influence on treatment benefit. Adopting sparse dimension reduction with variable prescreening in the proposed estimation, we aim to identify the key baseline variables that impact the conditional treatment effect and to uncover the characteristic score that best predicts the treatment effect. This approach is applied to an HIV study for assessing the benefit of antiretroviral regimens and identifying the beneficiary subpopulation.

Uncovering Latent Biases in Text: Method and Application to Peer Review

Quantifying systematic disparities in numerical quantities such as employment rates and wages between population subgroups provides compelling evidence for the existence of societal biases. However, biases in the text written for members of different subgroups (such as in recommendation letters for male and non-male candidates), though widely reported anecdotally, remain challenging to quantify. In this work, we introduce a novel framework to quantify bias in text caused by the visibility of subgroup membership indicators. We develop a nonparametric estimation and inference procedure to estimate this bias. We then formalize an identification strategy to causally link the estimated bias to the visibility of subgroup membership indicators, provided observations from time periods both before and after an identity-hiding policy change. We identify an application wherein “ground truth” bias can be inferred to evaluate our framework, instead of relying on synthetic or secondary data. Specifically, we apply our framework to quantify biases in the text of peer reviews from a reputed machine-learning conference before and after the conference adopted a double-blind reviewing policy. We show evidence of biases in the review ratings that serves as “ground truth”, and show that our proposed framework accurately detects the presence (and absence) of these biases from the review text without having access to the review ratings.

Nonparametric density estimation and bandwidth selection with B-spline bases: A novel Galerkin method

A general and efficient nonparametric density estimation procedure for local bases, including B-splines, is proposed, which employs a novel statistical Galerkin method combined with basis duality theory. To select the bandwidth, an efficient cross-validation procedure is introduced, based on closed-form expressions in terms of the primal and dual B-spline basis. By utilizing a closed-form expression for the dual basis, the least-squares cross validation formula is calculated in closed-form, enabling an efficient estimation of the optimal bandwidth. The full computational procedure achieves optimal complexity, and is very accurate in comparison with existing estimation procedures, including state-of-the-art kernel density estimators. The presented theoretical results are supported by extensive numerical experiments, which demonstrate the efficiency and accuracy of the new methodology. This new approach provides a complete and optimally efficient framework for density estimation with a B-spline basis, based on simple and elegant closed-form estimators with theoretical convergence results that are substantiated in numerical experiments.

Nonparametric estimation of aggregated Sobol’ indices: Application to a depth averaged snow avalanche model

Avalanche models are increasingly employed for elaborating land-use maps and designing defense structures, but they rely on poorly known parameters. Careful uncertainty assessment is thus required but difficulty arises from the nature of the outputs of these models, which are commonly both functional and scalar. Hence, so far in the avalanche field, few sensitivity analyses have been performed. In this work, we propose to determine the most influential inputs of an avalanche model by estimating aggregated first-order Sobol’ indices. We propose a nonparametric estimation procedure based on the Nadaraya–Watson kernel smoother, which allows to estimate the aggregated Sobol’ indices from a given random sample of small to moderate size. Due to the limited size of the sample, the kernel estimation is biased. Therefore, we propose a bootstrap based bias correction before selecting the bandwidth by cross-validation. To estimate the aggregated Sobol’ indices, we reduce the dimension of the output using principal components analysis. After different test cases showing the efficiency of our approach, it is applied to a real avalanche case. Results show that the friction parameters and the snow depth in the release zone are the most influential parameters determining the avalanche characteristics.

Nonparametric Density Estimation and Bandwidth Selection with B-spline bases: a Novel Galerkin Method

We study a general and efficient nonparametric density estimation procedure for local bases, including B-splines, using a novel statistical Galerkin method, combined with basis duality theory. We provide an efficient cross-validation procedure to select the bandwidth, based on closed-form expressions in terms of the primal and dual basis. By utilizing a closed-form expression for the dual basis, we are able to compute the least-squares cross validation formula in closed-form, which allows us to effectively determine the bandwidth. The full computational procedure achieves optimal complexity, and is very accurate in comparisons with existing estimation procedures, including state-of-the-art kernel density estimators. Our theoretical results are supported by extensive numerical experiments, which demonstrate the efficiency and accuracy of the new methodology.

Estimation and inference for distribution and quantile functions in endogenous treatment effect models

Given a standard endogenous treatment effect model, we propose nonparametric estimation and inference procedures for the distribution and quantile functions of the potential outcomes among compliers, as well as the local quantile treatment effect function. The preliminary distribution function estimator is a weighted average of indicator functions, but is not monotonically increasing in general. We therefore propose a simple monotonizing method for proper distribution function estimation, and obtain the quantile function estimator by inversion. Our monotonizing method is an alternative to Chernozhukov et al. (2010) and is arguably preferable when the outcome has unbounded support. We show that all the estimators converge weakly to Gaussian processes at the parametric rate, and propose a multiplier bootstrap for uniform inference. Our uniform results thus generalize the pointwise theory developed by Frölich and Melly (2013). Monte Carlo simulations and an application to the effect of fertility on family income distribution illustrate the use of the methods. All results extend to the subpopulation of treated compliers as well.

Efficient Multiple Imputation for Sensitivity Analysis of Recurrent Events Data with Informative Censoring

Missing data are commonly encountered in clinical trials due to dropout or nonadherence to study procedures. In trials in which recurrent events are of interest, the observed count can be an undercount of the events if a patient drops out before the end of the study. In many applications, the data are not necessarily missing at random and it is often not possible to test the missing at random assumption. Consequently, it is critical to conduct sensitivity analysis. We develop a control-based multiple imputation method for recurrent events data, where patients who drop out of the study are assumed to have a similar response profile to those in the control group after dropping out. Specifically, we consider the copy reference approach and the jump to reference approach. We model the recurrent event data using a semiparametric proportional intensity frailty model with the baseline hazard function completely unspecified. We develop nonparametric maximum likelihood estimation and inference procedures. We then impute the missing data based on the large sample distribution of the resulting estimators. The variance estimation is corrected by a bootstrap procedure. Simulation studies demonstrate the proposed method performs well in practical settings. We provide applications to two clinical trials.

Copula estimation through wavelets

Recently some nonparametric estimation procedures have been proposed using kernels and wavelets to estimate the copula function. In this context, knowing that a copula function can be expanded in a wavelet basis, we propose a new nonparametric copula estimation procedure through wavelets for independent data and times series under an $\alpha $-mixing condition. The main feature of this estimator is that we make no assumptions on the data distribution and there is no need to use ARMA–GARCH modelling before estimating the copula. Convergence rates for the estimator were computed, showing the estimator consistency. Some simulation studies are presented, as well as analysis of real data sets.