This paper analyzes several different biases that emerge from the (possibly) low-precision nonparametric ingredient in a semiparametric model. We show that both the variance part and the bias part of the nonparametric ingredient can lead to some biases in the semiparametric estimator, under conditions weaker than typically required in the literature. We then propose two bias-robust inference procedures, based on multi-scale jackknife and analytical bias correction, respectively. We also extend our framework to the case where the semiparametric estimator is constructed by some discontinuous functionals of the nonparametric ingredient. Simulation study shows that both bias-correction methods have good finite-sample performance.
Analysis of survival data with biased samples caused by left-truncation or length-biased sampling has received extensive interest. Many inference methods have been developed for various survival models. These methods, however, break down when survival data are typically error-contaminated. Although error-prone survival data commonly arise in practice, little work has been available in the literature for handling length-biased data with measurement error. In survival analysis, the transformation model is one of the frequently used models. However, methods of analyzing the transformation model w…
This paper highlights a tension between semiparametric efficiency and bootstrap consistency in the context of a canonical semiparametric estimation problem, namely the problem of estimating the average density. It is shown that although simple plug-in estimators suffer from bias problems preventing them from achieving semiparametric efficiency under minimal smoothness conditions, the nonparametric bootstrap automatically corrects for this bias and that, as a result, these seemingly inferior estimators achieve bootstrap consistency under minimal smoothness conditions. In contrast, several "debi…
In this article, the weighted empirical likelihood is applied to a general setting of two-sample semiparametric models, which includes biased sampling models and case-control logistic regression models as special cases. For various types of censored data, such as right censored data, doubly censored data, interval censored data and partly interval-censored data, the weighted empirical likelihood-based semiparametric maximum likelihood estimator $(\tildeθ_n,\tilde{F}_n)$ for the underlying parameter $θ_0$ and distribution $F_0$ is derived, and the strong consistency of $(\tildeθ_n,\tilde{F}_n)$…
It was shown that when one disposes of a parametric information of the truncation distribution, the semiparametric estimator of the distribution function for truncated data (Wang, 1989) is more efficient than the nonparametric one. On the basis of this estimation method, we derive an estimator for the tail index of Pareto-type distributions that are randomly right-truncated and establish its consistency and asymptotic normality. The finite sample behavior of the proposed estimator is carried out by simulation study. We point out that, in terms of both bias and root of the mean squared error, o…
This article examines density estimation by combining a parametric approach with a nonparametric factor. The plug-in parametric estimator is seen as a crude estimator of the true density and is adjusted by a nonparametric factor. The nonparametric factor is derived by a criterion called local L_2-fitting. A class of estimators that have multiplicative adjustment is provided, including estimators proposed by several authors as special cases, and the asymptotic theories are developed. Theoretical comparison reveals that the estimators in this class are better than, or at least competitive with, …
Deviations from the center within a robust neighborhood of a parametric model distribution may naturally be considered an infinite dimensional nuisance parameter. Thus, the semiparametric method may be tried, which is to compute the scores function for the main parameter minus its orthogonal projection on the closed linear tangent space for the nuisance parameter, and then rescale for Fisher consistency. In this paper, we derive such a semiparametric influence curve by nonlinear projection on the tangent balls arising in robust statistics. This semiparametric influence curve is then compared w…
During the past few decades, missing-data problems have been studied extensively, with a focus on the ignorable missing case, where the missing probability depends only on observable quantities. By contrast, research into non-ignorable missing data problems is quite limited. The main difficulty in solving such problems is that the missing probability and the regression likelihood function are tangled together in the likelihood presentation, and the model parameters may not be identifiable even under strong parametric model assumptions. In this paper we discuss a semiparametric model for non-ig…
We examine the performance of efficient and AIPW estimators under two-phase sampling when the complete-data model is nearly correctly specified, in the sense that the misspecification is not reliably detectable from the data by any possible diagnostic or test. Asymptotic results for these nearly true models are obtained by representing them as sequences of misspecified models that are mutually contiguous with a correctly specified model. We find that for the least-favourable direction of model misspecification the bias in the efficient estimator induced can be comparable to the extra variabili…
We study a well-known estimator of the fractal index of a stochastic process. Our framework is very general and encompasses many models of interest; we show how to extend the theory of the estimator to a large class of non-Gaussian processes. Particular focus is on clarity and ease of implementation of the estimator and the associated asymptotic results, making it easy for practitioners to apply the methods. We additionally show how measurement noise in the observations will bias the estimator, potentially resulting in the practitioner erroneously finding evidence of fractal characteristics in…
This paper proposes a multi-stage projection-based Lasso procedure for the semiparametric sample selection model in high-dimensional settings under a weak nonparametric restriction on the selection correction. In particular, the number of regressors in the main equation, p, and the number of regressors in the selection equation, d, can grow with and exceed the sample size n. The analysis considers the exact sparsity case and the approximate sparsity case. The main theoretical results are finite-sample bounds from which sufficient scaling conditions on the sample size for estimation consistency…
University of Virginia（弗吉尼亚大学）；University of California, Berkeley（加州大学伯克利分校）；Microsoft Research（微软研究院）
This paper proposes a Lasso-type estimator for a high-dimensional sparse parameter identified by a single index conditional moment restriction (CMR). In addition to this parameter, the moment function can also depend on a nuisance function, such as the propensity score or the conditional choice probability, which we estimate by modern machine learning tools. We first adjust the moment function so that the gradient of the future loss function is insensitive (formally, Neyman-orthogonal) with respect to the first-stage regularization bias, preserving the single index property. We then take the l…
Many economic and causal parameters depend on nonparametric or high dimensional first steps. We give a general construction of locally robust/orthogonal moment functions for GMM, where moment conditions have zero derivative with respect to first steps. We show that orthogonal moment functions can be constructed by adding to identifying moments the nonparametric influence function for the effect of the first step on identifying moments. Orthogonal moments reduce model selection and regularization bias, as is very important in many applications, especially for machine learning first steps. We gi…
In this article we study a class of parameters with the so-called `mixed bias property'. For parameters with this property, the bias of the semiparametric efficient one step estimator is equal to the mean of the product of the estimation errors of two nuisance functions. In non-parametric models, parameters with the mixed bias property admit so-called rate doubly robust estimators, i.e. estimators that are consistent and asymptotically normal when one succeeds in estimating both nuisance functions at sufficiently fast rates, with the possibility of trading off slower rates of convergence for t…
In this paper, local linear estimators are adapted for the unknown infinitesimal coefficients associated with continuous-time asset return model with jumps, which can correct the bias automatically due to their simple bias representation. The integrated diffusion models with jumps, especially infinite activity jumps are mainly investigated. In addition, under mild conditions, the weak consistency and asymptotic normality is provided through the conditional Lindeberg theorem. Furthermore, our method presents advantages in bias correction through simulation whether jumps belong to the finite act…
Measurement error is a pervasive issue which renders the results of an analysis unreliable. The measurement error literature contains numerous correction techniques, which can be broadly divided into those which aim to produce exactly consistent estimators, and those which are only approximately consistent. While consistency is a desirable property, it is typically attained only under specific model assumptions. Two techniques, regression calibration and simulation extrapolation, are used frequently in a wide variety of parametric and semiparametric settings. However, in many settings these me…
We investigate the joint asymptotic behavior of so-called blocks estimator of the extremal index, that determines the mean length of clusters of extremes, based on the exceedances over different thresholds. Due to the large bias of these estimators, the resulting estimates are usually very sensitive to the choice of the threshold and thus difficult to interpret. We propose and examine a bias correction that asymptotically removes the leading bias term while the rate of convergence of the random error is preserved.
The $α$-Brownian bridge, or scaled Brownian bridge, is a generalization of the Brownian bridge with a scaling parameter that determines how strong the force that pulls the process back to 0 is. The bias of the maximum likelihood estimator of the parameter $α$ is derived and a bias-correction that improves the estimator substantially is proposed. The properties of the bias-corrected estimator and four Bayesian estimators based on non-informative priors are evaluated in a simulation study.
This paper establishes a global bias-correction divide-and-conquer (GBC-DC) rule for biased estimation under the case of memory constraint. In order to introduce the new estimation, a closed representation of the local estimators obtained by the data in each batch is adopted, aiming to formulate a pro forma linear regression between the local estimators and the true parameter of interest. Least square method is then used within this framework to composite a global estimator of the parameter. Thus, the main advantage over the classical DC method is that the new GBC-DC method can absorb the info…
University of California, Berkeley（加州大学伯克利分校）；Stanford University（斯坦福大学）
We analyze bias correction methods using jackknife, bootstrap, and Taylor series. We focus on the binomial model, and consider the problem of bias correction for estimating $f(p)$, where $f \in C[0,1]$ is arbitrary. We characterize the supremum norm of the bias of general jackknife and bootstrap estimators for any continuous functions, and demonstrate the in delete-$d$ jackknife, different values of $d$ may lead to drastically different behaviors in jackknife. We show that in the binomial model, iterating the bootstrap bias correction infinitely many times may lead to divergence of bias and va…
Estimation error has plagued quantitative finance since Harry Markowitz launched modern portfolio theory in 1952. Using random matrix theory, we characterize a source of bias in the sample eigenvectors of financial covariance matrices. Unchecked, the bias distorts weights of minimum variance portfolios and leads to risk forecasts that are severely biased downward. To address these issues, we develop an eigenvector bias correction. Our approach is distinct from the regularization and eigenvalue shrinkage methods found in the literature. We provide theoretical guarantees on the improvement our c…
An important challenge in statistical analysis lies in controlling the bias of estimators due to the ever-increasing data size and model complexity. Approximate numerical methods and data features like censoring and misclassification often result in analytical and/or computational challenges when implementing standard estimators. As a consequence, consistent estimators may be difficult to obtain, especially in complex and/or high dimensional settings. In this paper, we study the properties of a general simulation-based estimation framework that allows to construct bias corrected consistent est…
