For the general parametric regression models with covariates contaminated with normal measurement errors, this paper proposes an accelerated version of the classical simulation extrapolation algorithm to estimate the unknown parameters in the regression function. By applying the conditional expectation directly to the target function, the proposed algorithm successfully removes the simulation step, by generating an estimation equation either for immediate use or for extrapolating, thus significantly reducing the computational time. Large sample properties of the resulting estimator, including the consistency and the asymptotic normality, are thoroughly discussed. Potential wide applications of the proposed estimation procedure are illustrated by examples, simulation studies, as well as a real data analysis.
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Truncated conditional expectation functions are objects of interest in a wide range of economic applications, including income inequality measurement, financial risk management, and impact evaluation. They typically involve truncating the outcome variable above or below certain quantiles of its conditional distribution. In this paper, based on local linear methods, a novel, two-stage, nonparametric estimator of such functions is proposed. In this estimation problem, the conditional quantile function is a nuisance parameter that has to be estimated in the first stage. The proposed estimator is insensitive to the first-stage estimation error owing to the use of a Neyman-orthogonal moment in the second stage. This construction ensures that inference methods developed for the standard nonparametric regression can be readily adapted to conduct inference on truncated conditional expectations. As an extension, estimation with an estimated truncation quantile level is considered. The proposed estimator is applied in two empirical settings: sharp regression discontinuity designs with a manipulated running variable and randomized experiments with sample selection.
Wavelet shrinkage estimators are widely applied in several fields of science for denoising data in wavelet domain by reducing the magnitudes of empirical coefficients. In nonparametric regression problem, most of the shrinkage rules are derived from models composed by an unknown function with additive gaussian noise. Although gaussian noise assumption is reasonable in several real data analysis, mainly for large sample sizes, it is not general. Contaminated data with positive noise can occur in practice and nonparametric regression models with positive noise bring challenges in wavelet shrinkage point of view. This work develops bayesian shrinkage rules to estimate wavelet coefficients from a nonparametric regression framework with additive and strictly positive noise under exponential and lognormal distributions. Computational aspects are discussed and simulation studies to analyse the performances of the proposed shrinkage rules and compare them with standard techniques are done. An application in winning times Boston Marathon dataset is also provided.
In this paper, a functional partial quantile regression approach, a quantile regression analog of the functional partial least squares regression, is proposed to estimate the function-on-function linear quantile regression model. A partial quantile covariance function is first used to extract the functional partial quantile regression basis functions. The extracted basis functions are then used to obtain the functional partial quantile regression components and estimate the final model. In our proposal, the functional forms of the discretely observed random variables are first constructed via a finite-dimensional basis function expansion method. The functional partial quantile regression constructed using the functional random variables is approximated via the partial quantile regression constructed using the basis expansion coefficients. The proposed method uses an iterative procedure to extract the partial quantile regression components. A Bayesian information criterion is used to determine the optimum number of retained components. The proposed functional partial quantile regression model allows for more than one functional predictor in the model. However, the true form of the proposed model is unspecified, as the relevant predictors for the model are unknown in practice. Thus, a forward variable selection procedure is used to determine the significant predictors for the proposed model. Moreover, a case-sampling-based bootstrap procedure is used to construct pointwise prediction intervals for the functional response. The predictive performance of the proposed method is evaluated using several Monte Carlo experiments under different data generation processes and error distributions. Through an empirical data example, air quality data are analyzed to demonstrate the effectiveness of the proposed method.
Correlated data are ubiquitous in today's data-driven society. A fundamental task in analyzing these data is to understand, characterize and utilize the correlations in them in order to conduct valid inference. Yet explicit regression analysis of correlations has been so far limited to longitudinal data, a special form of correlated data, while implicit analysis via mixed-effects models lacks generality as a full inferential tool. This paper proposes a novel regression approach for modelling the correlation structure, leveraging a new generalized z-transformation. This transformation maps correlation matrices that are constrained to be positive definite to vectors with un-restricted support, and is order-invariant. Building on these two properties, we develop a regression model to relate the transformed parameters to any covariates. We show that coupled with a mean and a variance regression model, the use of maximum likelihood leads to asymptotically normal parameter estimates, and crucially enables statistical inference for all the parameters. The performance of our framework is demonstrated in extensive simulation. More importantly, we illustrate the use of our model with the analysis of the classroom data, a highly unbalanced multilevel clustered data with within-class and within-school correlations, and the analysis of the malaria immune response data in Benin, a longitudinal data with time-dependent covariates in addition to time. Our analyses reveal new insights not previously known.
The persistent Betti numbers are used in topological data analysis to infer the scales at which topological features appear and disappear in the filtration of a topological space. Most commonly by means of the corresponding barcode or persistence diagram. While this approach to data science has been very successful, it suffers from sensitivity to outliers, and it does not allow for additional filtration parameters. Such parameters naturally appear when a cloud of data points comes together with additional measurements taken at the locations of the data. For these reasons, multiparameter persistent homology has recently received significant attention. In particular, the multicover and \v{C}ech bifiltration have been introduced to overcome the aforementioned shortcomings. In this work, we establish the strong consistency and asymptotic normality of the multiparameter persistent Betti numbers in growing domains. Our asymptotic results are established for a general framework encompassing both the marked \v{C}ech bifiltration, as well as the multicover bifiltration constructed on the null model of an independently marked Poisson point process. In a simulation study, we explain how the asymptotic normality can be used to derive tests for the goodness of fit. The statistical power of such tests is illustrated through different alternatives exhibiting more clustering, or more repulsion than the null model.
In high-dimensional data analysis, bi-level sparsity is often assumed when covariates function group-wisely and sparsity can appear either at the group level or within certain groups. In such cases, an ideal model should be able to encourage the bi-level variable selection consistently. Bi-level variable selection has become even more challenging when data have heavy-tailed distribution or outliers exist in random errors and covariates. In this paper, we study a framework of high-dimensional M-estimation for bi-level variable selection. This framework encourages bi-level sparsity through a computationally efficient two-stage procedure. In theory, we provide sufficient conditions under which our two-stage penalized M-estimator possesses simultaneous local estimation consistency and the bi-level variable selection consistency if certain nonconvex penalty functions are used at the group level. Both our simulation studies and real data analysis demonstrate satisfactory finite sample performance of the proposed estimators under different irregular settings.
Plausibility is a formalization of exact tests for parametric models and generalizes procedures such as Fisher's exact test. The resulting tests are based on cumulative probabilities of the probability density function and evaluate consistency with a parametric family while providing exact control of the $\alpha$ level for finite sample size. Model comparisons are inefficient in this approach. We generalize plausibility by incorporating weighing which allows to perform model comparisons. We show that one weighing scheme is asymptotically equivalent to the likelihood ratio test (LRT) and has finite sample guarantees for the test size under the null hypothesis unlike the LRT. We confirm theoretical properties in simulations that mimic the data set of our data application. We apply the method to a retinoblastoma data set and demonstrate a parent-of-origin effect. Weighted plausibility also has applications in high-dimensional data analysis and P-values for penalized regression models can be derived. We demonstrate superior performance as compared to a data-splitting procedure in a simulation study. We apply weighted plausibility to a high-dimensional gene expression, case-control prostate cancer data set. We discuss the flexibility of the approach by relating weighted plausibility to targeted learning, the bootstrap, and sparsity selection.
This paper gives a new approach for the maximum likelihood estimation of the joint of the location and scale of the Cauchy distribution. We regard the joint as a single complex parameter and derive a new form of the likelihood equation of a complex variable. Based on the equation, we provide a new iterative scheme approximating the maximum likelihood estimate. We also handle the equation in an algebraic manner and derive a polynomial containing the maximum likelihood estimate as a root. This algebraic approach provides another scheme approximating the maximum likelihood estimate by root-finding algorithms for polynomials, and furthermore, gives non-existence of closed-form formulae for the case that the sample size is five. We finally provide some numerical examples to show our method is effective.
The rapid finding of effective therapeutics requires the efficient use of available resources in clinical trials. The use of covariate adjustment can yield statistical estimates with improved precision, resulting in a reduction in the number of participants required to draw futility or efficacy conclusions. We focus on time-to-event and ordinal outcomes. A key question for covariate adjustment in randomized studies is how to fit a model relating the outcome and the baseline covariates to maximize precision. We present a novel theoretical result establishing conditions for asymptotic normality of a variety of covariate-adjusted estimators that rely on machine learning (e.g., l1-regularization, Random Forests, XGBoost, and Multivariate Adaptive Regression Splines), under the assumption that outcome data is missing completely at random. We further present a consistent estimator of the asymptotic variance. Importantly, the conditions do not require the machine learning methods to converge to the true outcome distribution conditional on baseline variables, as long as they converge to some (possibly incorrect) limit. We conducted a simulation study to evaluate the performance of the aforementioned prediction methods in COVID-19 trials using longitudinal data from over 1,500 patients hospitalized with COVID-19 at Weill Cornell Medicine New York Presbyterian Hospital. We found that using l1-regularization led to estimators and corresponding hypothesis tests that control type 1 error and are more precise than an unadjusted estimator across all sample sizes tested. We also show that when covariates are not prognostic of the outcome, l1-regularization remains as precise as the unadjusted estimator, even at small sample sizes (n = 100). We give an R package adjrct that performs model-robust covariate adjustment for ordinal and time-to-event outcomes.
This study considers the 3D human pose estimation problem in a single RGB image by proposing a conditional random field (CRF) model over 2D poses, in which the 3D pose is obtained as a byproduct of the inference process. The unary term of the proposed CRF model is defined based on a powerful heat-map regression network, which has been proposed for 2D human pose estimation. This study also presents a regression network for lifting the 2D pose to 3D pose and proposes the prior term based on the consistency between the estimated 3D pose and the 2D pose. To obtain the approximate solution of the proposed CRF model, the N-best strategy is adopted. The proposed inference algorithm can be viewed as sequential processes of bottom-up generation of 2D and 3D pose proposals from the input 2D image based on deep networks and top-down verification of such proposals by checking their consistencies. To evaluate the proposed method, we use two large-scale datasets: Human3.6M and HumanEva. Experimental results show that the proposed method achieves the state-of-the-art 3D human pose estimation performance.
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