Ensemble Conditional Variance Estimation (ECVE) is a novel sufficient dimension reduction (SDR) method in regressions with continuous response and predictors. ECVE applies to general non-additive error regression models. It operates under the assumption that the predictors can be replaced by a lower dimensional projection without loss of information. It is a semiparametric forward regression model based exhaustive sufficient dimension reduction estimation method that is shown to be consistent under mild assumptions. It is shown to outperform central subspace mean average variance estimation (csMAVE), its main competitor, under several simulation settings and in a benchmark data set analysis.
When evaluating and comparing models using leave-one-out cross-validation (LOO-CV), the uncertainty of the estimate is typically assessed using the variance of the sampling distribution. Considering the uncertainty is important, as the variability of the estimate can be high in some cases. An important result by Bengio and Grandvalet (2004) states that no general unbiased variance estimator can be constructed, that would apply for any utility or loss measure and any model. We show that it is possible to construct an unbiased estimator considering a specific predictive performance measure and m…
It is well-known that kernel regression estimators do not produce a constant estimator variance over a domain. To correct this problem, Nishida and Kanazawa (2015) proposed a variance-stabilizing (VS) local variable bandwidth for Local Linear (LL) regression estimator. In contrast, Choi and Hall (1998) proposed the skewing (SK) methods for a univariate LL estimator and constructed a convex combination of one LL estimator and two SK estimators that are symmetrically placed on both sides of the LL estimator (the convex combination (CC) estimator) to eliminate higher-order terms in its asymptotic…
A number of popular estimators of the between-study variance, $τ^2$, are based on the Cochran's $Q$ statistic for testing heterogeneity in meta analysis. We introduce new point and interval estimators of $τ^2$ for log-odds-ratio. These include new DerSimonian-Kacker-type moment estimators based on the first moment of $Q_F$, the $Q$ statistic with effective-sample-size weights, and novel median-unbiased estimators. We study, by simulation, bias and coverage of these new estimators of $τ^2$ and, for comparative purposes, bias and coverage of a number of well-known estimators based on the $Q$ sta…
Variance estimation is important for statistical inference. It becomes non-trivial when observations are masked by serial dependence structures and time-varying mean structures. Existing methods either ignore or sub-optimally handle these nuisance structures. This paper develops a general framework for the estimation of the long-run variance for time series with non-constant means. The building blocks are difference statistics. The proposed class of estimators is general enough to cover many existing estimators. Necessary and sufficient conditions for consistency are investigated. The first as…
This paper presents two novel classes of variance estimators with superior properties, in the absence of parametric or semi-parametricassumptions. The first new class of estimator is the Oblozene Chlebizky (OC) variance estimators as a novel alternative to the generalized sandwich in Paper 1 of 4. That the OC concept is unlikely to arise from other, more standard, frameworks is manifestly true in light of the 40 year lacuna since White (1980). For any member of the generalized sandwich variance estimator class, there is an OC with the same expected value. The this alternative replaces a random…
For small area estimation of area-level data, the Fay-Herriot model is extensively used as a model based method. In the Fay-Herriot model, it is conventionally assumed that the sampling variances are known whereas estimators of sampling variances are used in practice. Thus, the settings of knowing sampling variances are unrealistic and several methods are proposed to overcome this problem. In this paper, we assume the situation where the direct estimators of the sampling variances are available as well as the sample means. Using these information, we propose a Bayesian yet objective method pro…
Mendelian randomization (MR) has become a popular approach to study the effect of a modifiable exposure on an outcome by using genetic variants as instrumental variables. A challenge in MR is that each genetic variant explains a relatively small proportion of variance in the exposure and there are many such variants, a setting known as many weak instruments. To this end, we provide a theoretical characterization of the statistical properties of two popular estimators in MR, the inverse-variance weighted (IVW) estimator and the IVW estimator with screened instruments using an independent select…
Variance estimation in the linear model when $p > n$ is a difficult problem. Standard least squares estimation techniques do not apply. Several variance estimators have been proposed in the literature, all with accompanying asymptotic results proving consistency and asymptotic normality under a variety of assumptions. It is found, however, that most of these estimators suffer large biases in finite samples when true underlying signals become less sparse with larger per element signal strength. One estimator seems to be largely neglected in the literature: a residual sum of squares based estima…
Conditional density estimation (CDE) models can be useful for many statistical applications, especially because the full conditional density is estimated instead of traditional regression point estimates, revealing more information about the uncertainty of the random variable of interest. In this paper, we propose a new methodology called Odds Conditional Density Estimator (OCDE) to estimate conditional densities in a supervised learning scheme. The main idea is that it is very difficult to estimate $p_{x,y}$ and $p_{x}$ in order to estimate the conditional density $p_{y|x}$, but by introducin…
We consider inference procedures, conditional on an observed ancillary statistic, for regression coefficients under a linear regression setup where the unknown error distribution is specified nonparametrically. We establish conditional asymptotic normality of the regression coefficient estimators under regularity conditions, and formally justify the approach of plugging in kernel-type density estimators in conditional inference procedures. Simulation results show that the approach yields accurate conditional coverage probabilities when used for constructing confidence intervals. The plug-in ap…
In practice rarely (if ever) is the spatial covariance known in spatial prediction problems. Often, prediction is performed after estimated spatial covariance parameters are plugged into the prediction equation. The estimated spatial association parameters arealso plugged into the prediction variance of the spatial predictor. However, simply plugging in spatial covariance parameter estimates into the prediction variance of the spatial predictor does not take into account the uncertainty in the true values of the spatial covariance parameters. Therefore the plug-in prediction variance estimate …
Random-effects meta-analysis requires an estimate of the between-study variance, $τ^2$. We study methods of estimation of $τ^2$ and its confidence interval in meta-analysis of odds ratio, and also the performance of related estimators of the overall effect. We provide results of extensive simulations on five point estimators of $τ^2$ (the popular methods of DerSimonian-Laird, restricted maximum likelihood, and Mandel and Paule; the less-familiar method of Jackson; and the new method (KD) based on the improved approximation to the distribution of the Q statistic by Kulinskaya and Dollinger (201…
We propose nonparametric estimators for conditional value-at-risk (CVaR) and conditional expected shortfall (CES) associated with conditional distributions of a series of returns on a financial asset. The return series and the conditioning covariates, which may include lagged returns and other exogenous variables, are assumed to be strong mixing and follow a nonparametric conditional location-scale model. First stage nonparametric estimators for location and scale are combined with a generalized Pareto approximation for distribution tails proposed by Pickands (1975) to give final estimators fo…
Missing data is a common challenge in biomedical research. This fact, along with growing dataset volumes of the modern era, make the issue of computationally-efficient analysis with missing data of crucial practical importance. A general computationally-efficient estimation framework for dealing with missing data is the pseudo-expected estimating equations (PEEE) approach. The method is applicable with any parametric model for which estimation involves the solution of a set of estimating equations, such as likelihood score equations. A key limitation of the PEEE approach is that there is curre…
Conditional copula graphic estimator for semi-competing risks data
Children’s Hospital Research Institute of Manitoba, University of Manitoba（曼尼托巴儿童医院研究所，曼尼托巴大学）；Department of Mathematics and Statistics, University of Guelph（圭尔夫大学数学与统计系）；Hospital for Sick Children（多伦多多伦多儿童医院）；Department of Statistics, University of Manitoba（曼尼托巴大学统计系）
Reference based multiple imputation methods have become popular for handling missing data in randomised clinical trials. Rubin's variance estimator is well known to be biased compared to the reference based imputation estimator's true repeated sampling variance. Somewhat surprisingly given the increasingly popularity of these methods, there has been relatively little debate in the literature as to whether Rubin's variance estimator or alternative (smaller) variance estimators targeting the repeated sampling variance are more appropriate. We review the arguments made on both sides of this debat…
A single primary sampling unit (PSU) per stratum design is a popular design for estimating the parameter of interest. Although, the point estimator of the design is unbiased and efficient, an unbiased variance estimator does not exist. A common practice to solve this is to collapse or combine the two adjacent strata, but the attained estimator of variance is not design-unbiased, and the bias increases as the population means of the collapsed strata become more variant. Therefore, the one PSU per stratum design with collapsed stratum variance estimator might not be a good choice, and some stati…
This article proposes a novel variance estimator for within-cluster resampling (WCR) and modified within-cluster resampling (MWCR) - two existing methods for analyzing longitudinal data. WCR is a simple but computationally intensive method, in which a single observation is randomly sampled from each cluster to form a new dataset. This process is repeated numerous times, and in each resampled dataset (or outputation), we calculate beta using a generalized linear model. The final resulting estimator is an average across estimates from all outputations. MWCR is an extension of WCR that can accoun…
It often happens that the same problem presents itself to different communities and the solutions proposed or adopted by those communities are different. We take the case of the variance estimation of the population average treatment effect in cluster-randomized experiments. The econometrics literature promotes the cluster-robust variance estimator (Athey and Imbens, 2017), which can be dated back to the study of linear regression with clustered residuals (Liang and Zeger, 1986). The A/B testing or online experimentation literature promotes the delta method (Kohavi et al., 2010, Deng et al., 2…
The paper concerns small-area estimation in the Fay-Herriot type area-level model with random dispersions, which models the case that the sampling errors change from area to area. The resulting Bayes estimator shrinks both means and variances, but needs numerical computation to provide the estimates. In this paper, an approximated empirical Bayes (AEB) estimator with a closed form is suggested. The model parameters are estimated via the moment method, and the mean squared error of the AEB is estimated via the single parametric bootstrap method. The benchmarked estimator and a second-order unbi…
This paper provides a design-based framework for variance (bound) estimation in experimental analysis. Results are applicable to virtually any combination of experimental design, linear estimator (e.g., difference-in-means, OLS, WLS) and variance bound, allowing for unified treatment and a basis for systematic study and comparison of designs using matrix spectral analysis. A proposed variance estimator reproduces Eicker-Huber-White (aka. "robust", "heteroskedastic consistent", "sandwich", "White", "Huber-White", "HC", etc.) standard errors and "cluster-robust" standard errors as special cases.…
Computing the variance of a conditional expectation has often been of importance in uncertainty quantification. Sun et al. has introduced an unbiased nested Monte Carlo estimator, which they call $1\frac{1}{2}$-level simulation since the optimal inner-level sample size is bounded as the computational budget increases. In this letter we construct unbiased non-nested Monte Carlo estimators based on the so-called pick-freeze scheme due to Sobol'. An extension of our approach to compute higher order moments of a conditional expectation is also discussed.
In the analysis of observational studies, inverse probability weighting (IPW) is commonly used to consistently estimate the average treatment effect (ATE) or the average treatment effect in the treated (ATT). The variance of the IPW ATE estimator is often estimated by assuming the weights are known and then using the so-called "robust" (Huber-White) sandwich estimator, which results in conservative standard error (SE) estimation. Here it is shown that using such an approach when estimating the variance of the IPW ATT estimator does not necessarily result in conservative SE estimates. That is, …
In a completely randomized experiment, the variances of treatment effect estimators in the finite population are usually not identifiable and hence not estimable. Although some estimable bounds of the variances have been established in the literature, few of them are derived in the presence of covariates. In this paper, the difference-in-means estimator and the Wald estimator are considered in the completely randomized experiment with perfect compliance and noncompliance, respectively. Sharp bounds for the variances of these two estimators are established when covariates are available. Further…
Conditional Variance Estimation (CVE) is a novel sufficient dimension reduction (SDR) method for additive error regressions with continuous predictors and link function. It operates under the assumption that the predictors can be replaced by a lower dimensional projection without loss of information. In contrast to the majority of moment based sufficient dimension reduction methods, Conditional Variance Estimation is fully data driven, does not require the restrictive linearity and constant variance conditions, and is not based on inverse regression. CVE is shown to be consistent and its objec…
Analytic evaluation of heteroskedasticity consistent covariance matrix estimates (HCCME) is difficult because of the complexity of the formulae currently available. We obtain new analytic formulae for the bias of a class of estimators of the covariance matrix of OLS in a standard linear regression model. These formulae provide substantial insight into the properties and performance characteristics of these estimators. In particular, we find a new estimator which minimizes the maximum possible bias and improves substantially on the standard Eicker-White estimate.
The logistic regression analysis proposed by Schouten et al. (Stat Med. 1993;12:1733-1745) has been a standard method in current statistical analysis of case-cohort studies, and it enables effective estimation of risk ratio from selected subsamples. Schouten et al. (1993) also proposed the standard error estimate of the risk ratio estimator can be calculated by the robust variance estimator. In this article, however, we show that the robust variance estimator does not account for the duplications of case and subcohort samples and generally has certain bias, i.e., inaccurate confidence interval…
In this paper we study the cumulative conditional expectation function (CCEF) in the copula context. It is shown how to compute CCEF in terms of the cumulative copula function, this natural representation allows to deduce some useful properties, for instance with applications to convex combination of copulas. We introduce approximations of CCEF based on Bernstein polynomial copulas. We introduce estimators for CCEF, which were constructed through Bernstein polynomial estimators for copulas. The estimators are asymptotically normal and biased for CCEF.
The two-level normal hierarchical model (NHM) has played a critical role in the theory of small area estimation (SAE), one of the growing areas in statistics with numerous applications in different disciplines. In this paper, we address major well-known shortcomings associated with the empirical best linear unbiased prediction (EBLUP) of a small area mean and its mean squared error (MSE) estimation by considering an appropriate model variance estimator that satisfies multiple properties. The proposed model variance estimator simultaneously (i) improves on the estimation of the related shrinkag…
We consider the problem of estimating the conditional distribution of a post-model-selection estimator where the conditioning is on the selected model. The notion of a post-model-selection estimator here refers to the combined procedure resulting from first selecting a model (e.g., by a model selection criterion such as AIC or by a hypothesis testing procedure) and then estimating the parameters in the selected model (e.g., by least-squares or maximum likelihood), all based on the same data set. We show that it is impossible to estimate this distribution with reasonable accuracy even asymptoti…
The Doubly Robust (DR) estimation of ATE can be carried out in 2 steps, where in the first step, the treatment and outcome are modeled, and in the second step the predictions are inserted into the DR estimator. The model misspecification in the first step has led researchers to utilize Machine Learning algorithms instead of parametric algorithms. However, existence of strong confounders and/or Instrumental Variables (IVs) can lead the complex ML algorithms to provide perfect predictions for the treatment model which can violate the positivity assumption and elevate the variance of DR estimator…
Conformal methods create prediction bands that control average coverage under no assumptions besides i.i.d. data. Besides average coverage, one might also desire to control conditional coverage, that is, coverage for every new testing point. However, without strong assumptions, conditional coverage is unachievable. Given this limitation, the literature has focused on methods with asymptotical conditional coverage. In order to obtain this property, these methods require strong conditions on the dependence between the target variable and the features. We introduce two conformal methods based on …
This paper explores bias in the estimation of sampling variance in Respondent Driven Sampling (RDS). Prior methodological work on RDS has focused on its problematic assumptions and the biases and inefficiencies of its estimators of the population mean. Nonetheless, researchers have given only slight attention to the topic of estimating sampling variance in RDS, despite the importance of variance estimation for the construction of confidence intervals and hypothesis tests. In this paper, we show that the estimators of RDS sampling variance rely on a critical assumption that the network is First…
