Envelope methodology is succinctly pitched as a class of procedures for increasing efficiency in multivariate analyses without altering traditional objectives \citep[first sentence of page 1]{cook2018introduction}. This description is true with the additional caveat that the efficiency gains obtained by envelope methodology are mitigated by model selection volatility to an unknown degree. The bulk of the current envelope methodology literature does not account for this added variance that arises from the uncertainty in model selection. Recent strides to account for model selection volatility have been made on two fronts: 1) development of a weighted envelope estimator, in the context of multivariate regression, to account for this variability directly; 2) development of a model selection criterion that facilitates consistent estimation of the correct envelope model for more general settings. In this paper, we unify these two directions and provide weighted envelope estimators that directly account for the variability associated with model selection and are appropriate for general multivariate estimation settings for vector valued parameters. Our weighted estimation technique provides practitioners with robust and useful variance reduction in finite samples. Theoretical justification is given for our estimators and validity of a nonparametric bootstrap procedure for estimating their asymptotic variance are established. Simulation studies and a real data analysis support our claims and demonstrate the advantage of our weighted envelope estimator when model selection variability is present.
Envelope methodology can provide substantial efficiency gains in multivariate statistical problems, but in some applications the estimation of the envelope dimension can induce selection volatility that may mitigate those gains. Current envelope methodology does not account for the added variance that can result from this selection. In this article, we circumvent dimension selection volatility through the development of a weighted envelope estimator. Theoretical justification is given for our estimator and validity of the residual bootstrap for estimating its asymptotic variance is established…
Envelope methods offer targeted dimension reduction for various models. The overarching goal is to improve efficiency in multivariate parameter estimation by projecting the data onto a lower-dimensional subspace known as the envelope. Envelope approaches have advantages in analyzing data with highly correlated variables, but their iterative Grassmannian optimization algorithms do not scale very well with ultra high-dimensional data. While the connections between envelopes and partial least squares in multivariate linear regression have promoted recent progress in high-dimensional studies of en…
A novel approach to obtain weighted likelihood estimates of multivariate location and scatter is discussed. A weighting scheme is proposed that is based on the distribution of the Mahalanobis distances rather than the distribution of the data at the assumed model. This strategy allows to avoid the curse of dimensionality affecting non-parametric density estimation, that is involved in the construction of the weights through the Pearson residuals Markatou et al (1998). Then, weighted likelihood based outlier detection rules and robust dimensionality reduction techniques are developed. The effec…
Re-examining and calibrating weighted survival analysis for causal inference
College of Economics, China Jiliang University（中国计量大学经济管理学院）；Department of Statistics, Rutgers University（罗格斯大学统计学系）；Institute for Health, Health Care Policy, and Aging Research, Rutgers University（罗格斯大学健康、医疗政策和老龄化研究所）
Estimation of a precision matrix (i.e., inverse covariance matrix) is widely used to exploit conditional independence among continuous variables. The influence of abnormal observations is exacerbated in a high dimensional setting as the dimensionality increases. In this work, we propose robust estimation of the inverse covariance matrix based on an $l_1$ regularized objective function with a weighted sample covariance matrix. The robustness of the proposed objective function can be justified by a nonparametric technique of the integrated squared error criterion. To address the non-convexity of…
This paper presents a score-based weighted likelihood estimator (SWLE) for robust estimations of generalized linear model (GLM) for insurance loss data. The SWLE exhibits a limited sensitivity to the outliers, theoretically justifying its robustness against model contaminations. Also, with the specially designed weight function to effectively diminish the contributions of extreme losses to the GLM parameter estimations, most statistical quantities can still be derived analytically, minimizing the computational burden for parameter calibrations. Apart from robust estimations, the SWLE can also …
Multivariate sign functions are often used for robust estimation and inference. We propose using data dependent weights in association with such functions. The proposed weighted sign functions retain desirable robustness properties, while significantly improving efficiency in estimation and inference compared to unweighted multivariate sign-based methods. Using weighted signs, we demonstrate methods of robust location estimation and robust principal component analysis. We extend the scope of using robust multivariate methods to include robust sufficient dimension reduction and functional outli…
Dimension reduction provides a useful tool for analyzing high dimensional data. The recently developed \textit{Envelope} method is a parsimonious version of the classical multivariate regression model through identifying a minimal reducing subspace of the responses. However, existing envelope methods assume an independent error structure in the model. While the assumption of independence is convenient, it does not address the additional complications associated with spatial or temporal correlations in the data. In this article, we introduce a \textit{Spatial Envelope} method for dimension redu…
In a Phase II dose-finding study with a placebo control, a new drug with several dose levels is compared with a placebo to test for the effectiveness of the new drug. The main focus of such studies often lies in the characterization of the dose-response relationship followed by the estimation of a target dose that leads to a clinically relevant effect over the placebo. This target dose is known as the minimum effective dose (MED) in a drug development study. Several approaches exist that combine multiple comparison procedures with modeling techniques to efficiently estimate the dose-response m…
This paper establishes unified frameworks of renewable weighted sums (RWS) for various online updating estimations in the models with streaming data sets. The newly defined RWS lays the foundation of online updating likelihood, online updating loss function, online updating estimating equation and so on. The idea of RWS is intuitive and heuristic, and the algorithm is computationally simple. This paper chooses nonparametric model as an exemplary setting. The RWS applies to various types of nonparametric estimators, which include but are not limited to nonparametric likelihood, quasi-likelihood…
An envelope is a targeted dimension reduction subspace for simultaneously achieving dimension reduction and improving parameter estimation efficiency. While many envelope methods have been proposed in recent years, all envelope methods hinge on the knowledge of a key hyperparameter, the structural dimension of the envelope. How to estimate the envelope dimension consistently is of substantial interest from both theoretical and practical aspects. Moreover, very recent advances in the literature have generalized envelope as a model-free method, which makes selecting the envelope dimension even m…
Groupwise Predictor Envelope Models for Multivariate Linear Regression
The Institute of Statistical Mathematics（统计数学研究所）；Graduate School of Mathematics, Kyushu University（九州大学大学院数理学府）；Faculty of Mathematics, Kyushu University（九州大学数学部）
There has been growing attention on how to effectively and objectively use covariate information when the primary goal is to estimate the average treatment effect (ATE) in randomized clinical trials (RCTs). In this paper, we propose an effective weighting approach to extract covariate information based on the empirical likelihood (EL) method. The resulting two-sample empirical likelihood weighted (ELW) estimator includes two classes of weights, which are obtained from a constrained empirical likelihood estimation procedure, where the covariate information is effectively incorporated into the f…
In this article, we present the maximum weighted likelihood estimator (MWLE) for robust estimations of heavy-tail finite mixture models (FMM). This is motivated by the complex distributional phenomena of insurance claim severity data, where flexible density estimation tools such as FMM are needed but MLE often produces unstable tail estimates under FMM. Under some regularity conditions, MWLE is proved to be consistent and asymptotically normal. We further prove that the tail index obtained by MWLE is consistent even if the model is misspecified, justifying the robustness of MWLE in estimating …
Median Based Unit Weibull Distribution (MBUW): Do the Higher Order Probability Weighted Moments (PWM) Add More Information over the Lower Order PWM in Parameter Estimation
Compared to the conditional mean as a simple point estimator, the conditional density function is more informative to describe the distributions with multi-modality, asymmetry or heteroskedasticity. In this paper, we propose a novel parametric conditional density estimation method by showing the connection between the general density and the likelihood function of inhomogeneous Poisson process models. The maximum likelihood estimates can be obtained via weighted logistic regressions, and the computation can be significantly relaxed by combining a block-wise alternating maximization scheme and …
We revisit the problem of estimating the local average treatment effect (LATE) and the local average treatment effect on the treated (LATT) when control variables are available, either to render the instrumental variable (IV) suitably exogenous or to improve precision. Unlike previous approaches, our doubly robust (DR) estimation procedures use quasi-likelihood methods weighted by the inverse of the IV propensity score - so-called inverse probability weighted regression adjustment (IPWRA) estimators. By properly choosing models for the propensity score and outcome models, fitted values are ens…
A constrained multivariate linear model is a multivariate linear model with the columns of its coefficient matrix constrained to lie in a known subspace. This class of models includes those typically used to study growth curves and longitudinal data. Envelope methods have been proposed to improve estimation efficiency in the class of unconstrained multivariate linear models, but have not yet been developed for constrained models that we develop in this article. We first compare the standard envelope estimator based on an unconstrained multivariate model with the standard estimator arising from…
Precise estimation of expected Darwinian fitness, the expected lifetime number of offspring of organism, is a central component of life history analysis. The aster model serves as a defensible statistical model for distributions of Darwinian fitness. The aster model is equipped to incorporate the major life stages an organism travels through which separately may effect Darwinian fitness. Envelope methodology reduces asymptotic variability by establishing a link between unknown parameters of interest and the asymptotic covariance matrices of their estimators. It is known both theoretically and …
Optimal Adjustment Sets for Nonparametric Estimation of Weighted Controlled Direct Effect
The University of Science and Technology of China（中国科学技术大学）；University of Wisconsin–Madison（威斯康星大学麦迪逊分校）；Cornell Tech, Cornell University（康奈尔大学康奈尔科技校区）
Owing to their capability of summarising interactions between elements of a system, networks have become a common type of data in many fields. As networks can be inhomogeneous, in that different regions of the network may exhibit different topologies, an important topic concerns their local properties. This paper focuses on the estimation of the local degree distribution of a vertex in an inhomogeneous network. The contributions are twofold: we propose an estimator based on local weighted averaging, and we set up a Monte Carlo cross-validation procedure to pick the parameters of this estimator…
In this paper, we propose a reduced-bias estimator of the EVI for Pareto-type tails (heavy-tailed) distributions. This is derived using the weighted least squares method. It is shown that the estimator is unbiased, consistent and asymptotically normal under the second-order conditions on the underlying distribution of the data. The finite sample properties of the proposed estimator are studied through a simulation study. The results show that it is competitive to the existing estimators of the extreme value index in terms of bias and Mean Square Error. In addition, it yields estimates of $γ>0$…
