Asymptotic Standard Errors Research Articles

Confidence intervals estimator of the kinetic parameters: do its reliability depend on the assembling method of the oxygen uptakes?

Gas exchange data acquired repeatedly under the same exercise conditions are assembled together to improve the kinetic parameters of breath-by-breath oxygen uptake. The latter are provided by the non-linear regression procedure, together with the corresponding estimators of the width of the Confidence Intervals (i.e., the Asymptotic Standard Errors; ASEs). We tested, for two different assembling procedures, whether the range of values identified by the ASE actually correspond to the 95% Confidence Interval. Ten O2 uptake responses were acquired on 10 healthy volunteers performing a square-wave moderate-intensity exercise. Kinetic parameters were estimated running the non-linear regression with a mono-exponential model on an increasingly greater number of responses (Nr, from 1 to 10), assembled together using the "stacking" and the "1-s-bins" procedures. Kinetic values obtained assembling together the 10 repetitions were assumed as "true" values. The time constant was not affected by Nr or by the assembling procedure (ANOVA; p>0.54 and p>0.16, respectively). The corresponding ASE decreased according to Nr (ANOVA; p=0.000), being significantly smaller for the "1-s-bins" procedure compared to the "stacking" one (ANOVA; p<0.001). Excluding 20s at the start of the fitting window, the range of values identified with the ASE provided by the "1-s-bins" and the "stacking" procedures included the "true" value in 85% and in 95% of cases, respectively. The "stacking" procedure should be preferred since it yielded ASEs for the time constant that provided a range of values satisfying the statistical meaning of the width of the Confidence Intervals, at the given degree of probability.

Introducing Ph.D. students to asymptotic inference for two‐stage M‐estimators: Easing analytic and coding demands via the use of numerical derivatives

AbstractApplications of two‐stage M‐estimators (2SMEs) abound in empirical economics. Asymptotic theory for 2SMEs (correct formulation of the asymptotic standard errors [ASE]) has been available for decades. Nevertheless, due to the daunting nature of the requisite matrix formulations, when conducting statistical inference based on two‐stage estimates, applied researchers often implement bootstrapping methods or ignore the two‐stage nature of the estimator and report the uncorrected second‐stage outputs from packaged statistical software. In the present paper, we offer teachers of econometrics a pedagogical approach for introducing Ph.D. students to asymptotic inference for 2SMEs, with a view toward easier software implementation and empirical application. We seek to demonstrate to students (and their teachers) that the analytic and coding demands for calculating correct ASEs for the 2SME need not be burdensome (or prohibitive). The main instructional (and practical) innovation that we offer in this regard is our suggested use of numerical derivative (ND) software for calculating the most challenging components of the ASE formulations. An exercise demonstrates to the student that, by implementing ND software, one can overcome the analytic and coding impediments to conducting inference based on 2SMEs, without abandoning rigor.

The odd log- logistic Power Inverse Lindley distribution: Model, Properties and Applications

In this article, we introduce a new three-parameter odd log-logistic power inverse Lindley distribution and discuss some of its properties. These include the shapes of the density and hazard rate functions, mixture representation, the moments, the quantile function, and order statistics. Maximum likelihood estimation of the parameters and their estimated asymptotic standard errors are derived. Three algorithms are proposed for generating random data from the proposed distribution. A simulation study is carried out to examine the bias and root mean square error of the maximum likelihood estimators of the parameters. An application of the model to three real data sets is presented finally and compared with the fit attained by some other well-known two and three-parameter distributions for illustrative purposes. It is observed that the proposed model has some advantages in analyzing lifetime data as compared to other popular models in the sense that it exhibits varying shapes and shows more flexibility than many currently available distributions.

Clustering asymmetrical data with outliers: Parsimonious mixtures of contaminated mean-mixture of normal distributions

Mixture modeling has emerged as a statistical tool to perform unsupervised model-based clustering for heterogeneous data. A framework of using contaminated mean-mixture of normal distributions as the components of the mixture model is designed to accommodate asymmetric data with outliers. Fourteen parsimonious variants of the postulated model are introduced by employing an eigenvalue decomposition of the component scale matrices. Simultaneously clustering and outliers detection is an outstanding advantage of the proposed model in analyzing non-normally distributed data. A computationally feasible and flexible EM-type algorithm is outlined for obtaining maximum likelihood parameter estimates. Moreover, the score vector and empirical information matrix for calculating asymptotic standard errors of the parameter estimates are derived by offering an information-based approach. The applicability of the proposed method is demonstrated through the analysis of simulated and real datasets with varying proportions of outliers.

개정 명명반응 모형을 위한 IRT 척도 연계 방법과 연계 계수의 표준오차

Under item response theory (IRT), common-item linking methods are used to develop a common ability scale between two test forms administered to examinee groups from different populations. The nominal response (NR) model, proposed first by Bock, was reparameterized by Thissen and his colleagues. This paper presents three types of IRT linking methods, the direct least squares (DLS), mean/least squares (MLS), and item category response function (ICRF) methods, for the reparameterized NR model and investigates their performance through computer simulations. The presentation assumes that the highest response categories are specified for all linking items. This paper also presents analytic formulas for computing the asymptotic standard errors (SEs) of linking coefficient estimates for the three methods. Important findings were obtained from a simulation study. Overall, the ICRF method outperformed the DLS and MLS methods in linking accuracy, and the DLS and MLS methods performed almost equally. The linking coefficients for the DLS and MLS methods should be estimated using item parameter estimates only for the highest response categories whereas those for the ICRF method should be estimated by the criterion function that is defined using all ICRFs across linking items. The analytic formulas for the asymptotic SEs worked properly for the three linking methods, and the SEs were, approximately, inversely proportional to the square root of the sample size.

The Community Stress-Drop Validation Study—Part I: Source, Propagation, and Site Decomposition of Fourier Spectra

AbstractAs part of the community stress-drop validation study initiative, we apply a spectral decomposition approach to isolate the source spectra of 556 events occurred during the 2019 Ridgecrest sequence (Southern California). We perform multiple decompositions by introducing alternative choices for some processing and model assumptions, namely: three different S-wave window durations (i.e., 5 s, 20 s, and variable between 5 and 20 s); two attenuation models that account differently for depth dependencies; and two different site amplification constraints applied to restore uniqueness of the solution. Seismic moment and corner frequency are estimated for the Brune and Boatwright source models, and an extensive archive including source spectra, site amplifications, attenuation models, and tables with source parameters is disseminated as the main product of the present study. We also compare different approaches to measure the precision of the parameters expressed in terms of 95% confidence intervals (CIs). The CIs estimated from the asymptotic standard errors and from Monte Carlo resampling of the residual distribution show an almost one-to-one correspondence; the approach based on model selection by setting a threshold for misfit chosen with an F-ratio test is conservative compared to the approach based on the asymptotic standard errors. The uncertainty analysis is completed in the companion article in which the outcomes from this work are used to compare epistemic uncertainty with precision of the source parameters.

Application of Mixture Models for Doubly Inflated Count Data

In health and social science and other fields where count data analysis is important, zero-inflated models have been employed when the frequency of zero count is high (inflated). Due to multiple reasons, there are scenarios in which an additional count value of k &gt; 0 occurs with high frequency. The zero- and k-inflated Poisson distribution model (ZkIP) is more appropriate for such situations. The ZkIP model is a mixture distribution with three components: degenerate distributions at 0 and k count and a Poisson distribution. In this article, we propose an alternative and computationally fast expectation–maximization (EM) algorithm to obtain the parameter estimates for grouped zero and k-inflated count data. The asymptotic standard errors are derived using the complete data approach. We compare the zero- and k-inflated Poisson model with its zero-inflated and non-inflated counterparts. The best model is selected based on commonly used criteria. The theoretical results are supplemented with the analysis of two real-life datasets from health sciences.

Modifying the linear two-step Windmeijer correction for the presence of spatial error dependence

The aim in the paper is to show how the presence of spatial dependence affects the often-adopted Windmeijer (J Econom 126:25–51, 2005) finite sample correction (For example it is an option facilitating robust estimation in the software package Stata, which is used by many applied econometricians and data analysts.), which corrects the downward bias in estimated parameter standard errors. Windmeijer (2005) explains why, with numerous instruments, the estimated asymptotic standard errors of the efficient, two-step, GMM estimator are downward biased in small samples. GMM estimation is based on an estimated optimal weight matrix, which is the inverse of the covariance of the sample moments, and the bias results from the weight matrix being evaluated at estimated, rather than the true values of parameters. Hwang et al. (J Econom 229(2):276–298, 2022) provide a correction to the Windmeijer (2005) finite sample correction to allow for over-identification bias. The novel contribution of the current paper is to show how the Windmeijer (2005) correction can be modified given spatial dependence in the error term of a model with moments conditions that are linear in parameters estimated by GMM, leading to corrected standard errors and therefore more accurate inference. Monte Carlo simulations are used to demonstrate the effect of the modification and two examples using real data shows how inference may be affected by ignoring the effect of spatial error dependence.

Regression with an imputed dependent variable

SummaryResearchers are often interested in the relationship between two variables, with no single data set containing both. A common strategy is to use proxies for the dependent variable that are common to two surveys to impute the dependent variable into the data set containing the independent variable. We show that commonly employed regression or matching‐based imputation procedures lead to inconsistent estimates. We offer a consistent and easily implemented two‐step estimator, “rescaled regression prediction.” We derive the correct asymptotic standard errors for this estimator and demonstrate its relationship to alternative approaches. We illustrate with empirical examples using data from the US Consumer Expenditure Survey (CE) and the Panel Study of Income Dynamics (PSID).

Asymptotic Standard Errors of IRT Linking Coefficients for the Bifactor Extension of the 3PL Model

The bifactor extension of the three-parameter logistic (B3PL) model has been used in applications of multidimensional item response theory (IRT) such as test equating and vertical scaling. Developing a common multidimensional IRT scale (that is, multidimensional coordinate system) is critical in those applications. Three common-item scale linking methods, the direct least squares (DLS), mean/least squares (MLS), and item response function (IRF) methods, for the B3PL model have been found to be effective in developing a common multidimensional IRT ability scale between two test forms to be linked. In this paper, the asymptotic standard errors (SEs) of IRT linking coefficients estimated by the DLS, MLS, and IRF methods are derived assuming that the B3PL model holds and the asymptotic variance-covariance matrix of item parameter estimates from separate calibrations is available. The delta method is used for the derivations. Computer simulations which investigate the accuracy of the derivations under various conditions are given, showing that the derivations are reasonably accurate when sufficiently large samples are used and that in general the SEs of the IRF method are smaller than those of the DLS and MLS methods. The simulation results also suggest that the SEs of linking coefficient estimates are, approximately, inversely proportional to the square root of the sample size when two test forms are administered to the same number of examinees.

Bayesian and Non-Bayesian Reliability Estimation of Stress-Strength Model for Power-Modified Lindley Distribution.

A two-parameter continuous distribution, namely, power-modified Lindley (PML), is proposed. Various structural properties of the new distribution, including moments, moment-generating function, conditional moments, mean deviations, mean residual lifetime, and mean past lifetime, are provided. The reliability of a system is discussed when the strength of the system and the stress imposed on it are independent. Maximum-likelihood estimation of the parameters and their estimated asymptotic standard errors are derived. Bayesian estimation methods of the parameters with independent gamma prior are discussed based on symmetric and asymmetric loss functions. We proposed using the MCMC technique with the Metropolis–Hastings algorithm to approximate the posteriors of the stress-strength parameters for Bayesian calculations. The confidence interval for likelihood and the Bayesian estimation method is obtained for the parameter of the model and stress-strength reliability. We prove empirically the importance and flexibility of the new distribution in modeling with real data applications.

Geometric ergodicity of Gibbs samplers for the Horseshoe and its regularized variants

The Horseshoe is a widely used and popular continuous shrinkage prior for high-dimensional Bayesian linear regression. Recently, regularized versions of the Horseshoe prior have also been introduced in the literature. Various Gibbs sampling Markov chains have been developed in the literature to generate approximate samples from the corresponding intractable posterior densities. Establishing geometric ergodicity of these Markov chains provides crucial technical justification for the accuracy of asymptotic standard errors for Markov chain based estimates of posterior quantities. In this paper, we establish geometric ergodicity for various Gibbs samplers corresponding to the Horseshoe prior and its regularized variants in the context of linear regression. First, we establish geometric ergodicity of a Gibbs sampler for the original Horseshoe posterior under strictly weaker conditions than existing analyses in the literature. Second, we consider the regularized Horseshoe prior introduced in [18], and prove geometric ergodicity for a Gibbs sampling Markov chain to sample from the corresponding posterior without any truncation constraint on the global and local shrinkage parameters. Finally, we consider a variant of this regularized Horseshoe prior introduced in [15], and again establish geometric ergodicity for a Gibbs sampling Markov chain to sample from the corresponding posterior.

Analyzing relevance vector machines using a single penalty approach

AbstractRelevance vector machine (RVM) is a popular sparse Bayesian learning model typically used for prediction. Recently it has been shown that improper priors assumed on multiple penalty parameters in RVM may lead to an improper posterior. Currently in the literature, the sufficient conditions for posterior propriety of RVM do not allow improper priors over the multiple penalty parameters. In this article, we propose a single penalty relevance vector machine (SPRVM) model in which multiple penalty parameters are replaced by a single penalty and we consider a semi‐Bayesian approach for fitting the SPRVM. The necessary and sufficient conditions for posterior propriety of SPRVM are more liberal than those of RVM and allow for several improper priors over the penalty parameter. Additionally, we also prove the geometric ergodicity of the Gibbs sampler used to analyze the SPRVM model and hence can estimate the asymptotic standard errors associated with the Monte Carlo estimate of the means of the posterior predictive distribution. Such a Monte Carlo standard error cannot be computed in the case of RVM, since the rate of convergence of the Gibbs sampler used to analyze RVM is not known. The predictive performance of RVM and SPRVM is compared by analyzing two simulation examples and three real life datasets.

Model-Assisted Analyses of Cluster-Randomized Experiments

AbstractCluster-randomized experiments are widely used due to their logistical convenience and policy relevance. To analyse them properly, we must address the fact that the treatment is assigned at the cluster level instead of the individual level. Standard analytic strategies are regressions based on individual data, cluster averages and cluster totals, which differ when the cluster sizes vary. These methods are often motivated by models with strong and unverifiable assumptions, and the choice among them can be subjective. Without any outcome modelling assumption, we evaluate these regression estimators and the associated robust standard errors from the design-based perspective where only the treatment assignment itself is random and controlled by the experimenter. We demonstrate that regression based on cluster averages targets a weighted average treatment effect, regression based on individual data is suboptimal in terms of efficiency and regression based on cluster totals is consistent and more efficient with a large number of clusters. We highlight the critical role of covariates in improving estimation efficiency and illustrate the efficiency gain via both simulation studies and data analysis. The asymptotic analysis also reveals the efficiency-robustness trade-off by comparing the properties of various estimators using data at different levels with and without covariate adjustment. Moreover, we show that the robust standard errors are convenient approximations to the true asymptotic standard errors under the design-based perspective. Our theory holds even when the outcome models are misspecified, so it is model-assisted rather than model-based. We also extend the theory to a wider class of weighted average treatment effects.

Comparison of the Standard Errors of Equating by the IRT and Equipercentile Methods Under the Random Groups Design

검사 동등화 연구에서 자료 수집 설계로 랜덤집단 설계가 종종 사용된다. 최근 들어, 두 검사형 간 동등점수 관계를 찾기 위해 전통적인 동백분위 동등화 방법보다는 문항반응이론(IRT) 기반 동등화 방법이 더 많이 활용되고 있다. 본 연구의 첫째 목적은 랜덤집단 설계하에서 IRT 동등화 방법에 대한 동등화 표준오차의 점근 공식을 혼합형 검사에 적용할 수 있도록 일반적으로 제시하는 데 있다. 둘째 목적은 점근 공식에 기초하여 IRT 방법과 동백분위 방법에 의해 산출되는 동등화 표준오차가 어떻게 다른지를 경험적 자료를 통해 검토하는 것이다. 둘째 목적과 관련하여, 다양한 동등화 조건들을 포함하여 실시한 비교 연구를 통해 다음의 주요 결과를 얻었다. 첫째, IRT 관찰점수 동등화 방법은 IRT 진점수 동등화 방법보다 평균적으로 더 작은 동등화 표준오차를 산출하였으며, 두 IRT 동등화 방법은 동백분위 동등화 방법보다 더 작은 동등화 표준오차를 산출하였다. 둘째, 각 동등화 방법은 검사의 길이가 증가할수록 더 큰 동등화 표준오차를 산출하였다. 셋째, 각 동등화 방법은 표본의 크기가 커질수록 더 작은 동등화 표준오차를 산출하였으며, 평균 동등화 표준오차는 대략적으로 표본 크기의 제곱근에 반비례하였다.The random groups design is often used for collecting sample data for test equating. Recently, the item response theory (IRT) based equating methods have been more widely used than the traditional equipercentile equating method. One of the two purposes of this study is to present general formulas for the asymptotic standard errors of equated scores estimated by the IRT true score and observed score equating methods under the random groups design so that they can be used for equating with mixed-format tests. The other purpose is to use empirical test data to examine how the standard errors of equating produced by the IRT equating methods differ from those by the equipercentile equating method. Concerning the second purpose, a comparative study, including various equating conditions, was conducted and the following were found. First, on the average, the IRT observed score equating method resulted in slightly smaller standard errors of equating than the IRT true score equating method, and the two IRT equating methods produced smaller standard errors than the equipercentile equating method which was based on unsmoothed score distributions. Second, the standard errors produced by the three equating methods increased as the lengths of the test forms equated increased and thus the relative frequencies for test scores were lowered. Third, the mean standard errors of equating for the three equating methods decreased as the sample size increased and the amounts of the errors were inversely proportional to the square root of the sample size.

A skew‐t quantile regression for censored and missing data

Quantile regression has emerged as an important analytical alternative to the classical mean regression model. However, the analysis could be complicated by the presence of censored measurements due to a detection limit of equipment in combination with unavoidable missing values arising when, for instance, a researcher is simply unable to collect an observation. Another complication arises when measures depart significantly from normality, for instance, in the presence of skew heavy‐tailed observations. For such data structures, we propose a robust quantile regression for censored and/or missing responses based on the skew‐t distribution. A computationally feasible EM‐based procedure is developed to carry out the maximum likelihood estimation within such a general framework. Moreover, the asymptotic standard errors of the model parameters are explicitly obtained via the information‐based method. We illustrate our methodology by using simulated data and two real data sets.

A unified model-implied instrumental variable approach for structural equation modeling with mixed variables

The model-implied instrumental variable (MIIV) estimator is an equation-by-equation estimator of structural equation models that is more robust to structural misspecifications than full information estimators. Previous studies have concentrated on endogenous variables that are all continuous (MIIV-2SLS) or all ordinal . We develop a unified MIIV approach that applies to a mixture of binary, ordinal, censored, or continuous endogenous observed variables. We include estimates of factor loadings, regression coefficients, variances, and covariances along with their asymptotic standard errors. In addition, we create new goodness of fit tests of the model and overidentification tests of single equations. Our simulation study shows that the proposed MIIV approach is more robust to structural misspecifications than diagonally weighted least squares (DWLS) and that both the goodness of fit model tests and the overidentification equations tests can detect structural misspecifications. We also find that the bias in asymptotic standard errors for the MIIV estimators of factor loadings and regression coefficients are often lower than the DWLS ones, though the differences are small in large samples. Our analysis shows that scaling indicators with low reliability can adversely affect the MIIV estimators. Also, using a small subset of MIIVs reduces small sample bias of coefficient estimates, but can lower the power of overidentification tests of equations.

Asymptotic Standard Errors of Generalized Partial Credit Model True Score Equating Using Characteristic Curve Methods.

In this study, the delta method was applied to estimate the standard errors of the true score equating when using the characteristic curve methods with the generalized partial credit model in test equating under the context of the common-item nonequivalent groups equating design. Simulation studies were further conducted to compare the performance of the delta method with that of the bootstrap method and the multiple imputation method. The results indicated that the standard errors produced by the delta method were very close to the criterion empirical standard errors as well as those yielded by the bootstrap method and the multiple imputation method under all the manipulated conditions.

Robust clustering of multiply censored data via mixtures of t factor analyzers

Mixtures of t factor analyzers (MtFA) have been well recognized as a prominent tool in modeling and clustering multivariate data contaminated with heterogeneity and outliers. In certain practical situations, however, data are likely to be censored such that the standard methodology becomes computationally complicated or even infeasible. This paper presents an extended framework of MtFA that can accommodate censored data, referred to as MtFAC in short. For maximum likelihood estimation, we construct an alternating expectation conditional maximization algorithm in which the E-step relies on the first-two moments of truncated multivariate-t distributions and CM-steps offer tractable solutions of updated estimators. Asymptotic standard errors of mixing proportions and component mean vectors are derived by means of missing information principle, or the so-called Louis’ method. Several numerical experiments are conducted to examine the finite-sample properties of estimators and the ability of the proposed model to downweight the impact of censoring and outlying effects. Further, the efficacy and usefulness of the proposed method are also demonstrated by analyzing a real dataset with genuine censored observations.

Asymptotic standard errors of intraclass correlation coefficients for two-way model

Intraclass correlation coefficients are estimated using appropriate analysis of variance models that are commonly used in behavioral measurement, biometric, and psychometric. The ICCs estimation accuracy is quantified by the degree of their sampling variability using the asymptotic standard errors or confidence intervals, which facilitates conducting hypothesis testing on ICCs. The article derived the asymptotic standard errors of absolute agreement ICC and relative consistency ICC for the two-way model. Monte Carlo simulations were performed for both normal data and nonnormal data under different conditions. Results supported that the proposed asymptotic standard errors for all ICCs were converging to the simulated true standard errors under different test conditions.