Pre-test Estimator Research Articles

Pretest Estimation for the Common Mean of Several Normal Distributions: In Meta-Analysis Context

The estimation of unknown quantities from multiple independent yet non-homogeneous samples has garnered increasing attention in various fields over the past decade. This interest is evidenced by the wide range of applications discussed in recent literature. In this study, we propose a preliminary test estimator for the common mean (μ) with unknown and unequal variances. When there exists prior information regarding the population mean with consideration that μ might be equal to the reference value for the population mean, a hypothesis test can be conducted: H0:μ=μ0 versus H1:μ≠μ0. The initial sample is used to test H0, and if H0 is not rejected, we become more confident in using our prior information (after the test) to estimate μ. However, if H0 is rejected, the prior information is discarded. Our simulations indicate that the proposed preliminary test estimator significantly decreases the mean squared error (MSE) values compared to unbiased estimators such as the Garybill-Deal (GD) estimator, particularly when μ closely aligns with the hypothesized mean (μ0). Furthermore, our analysis indicates that the proposed test estimator outperforms the existing method, particularly in cases with minimal sample sizes. We advocate for its adoption to improve the accuracy of common mean estimation. Our findings suggest that through careful application to real meta-analyses, the proposed test estimator shows promising potential.

Exogeneity tests and weak identification in IV regressions: Asymptotic theory and point estimation

This paper provides new insights on exogeneity tests in linear IV models and their use for estimation, when identification fails or may not be strong. We make two main contributions. First, we show that Durbin-Wu-Hausman (DWH) and Revankar-Hartley (RH) exogeneity tests have correct level asymptotically, even when the first-stage coefficient matrix (which controls identification) is rank-deficient. We provide necessary and sufficient conditions under which these tests are consistent. In particular, we show that test consistency can hold even when identification fails, provided at least one component of the structural parameter vector is identifiable. Second, we study point estimation after estimator (or model) selection, when the outcome of a DWH/RH test determines whether OLS or an IV method is employed in the second-stage. For this purpose, we use (non-local) concepts of asymptotic bias, asymptotic mean squared error (AMSE), and asymptotic relative efficiency (ARE), which remain applicable even when the estimators considered do not have moments (as can happen for 2SLS) or may be inconsistent. We study the asymptotic properties of OLS, 2SLS, and pretest estimators which select OLS or 2SLS based on the outcome of a DWH/RH test. We show that: (i) OLS typically dominates 2SLS estimator asymptotically for MSE across a broad spectrum of cases, including weak identification and moderate endogeneity; (ii) exogeneity-pretest estimators exhibit consistently good performance and asymptotically dominate both OLS and 2SLS. The proposed theoretical findings are documented by Monte Carlo simulations.

A Pretest Estimator for the Two-Way Error Component Model

For a panel data linear regression model with both individual and time effects, empirical studies select the two-way random-effects (TWRE) estimator if the Hausman test based on the contrast between the two-way fixed-effects (TWFE) estimator and the TWRE estimator is not rejected. Alternatively, they select the TWFE estimator in cases where this Hausman test rejects the null hypothesis. Not all the regressors may be correlated with these individual and time effects. The one-way Hausman-Taylor model has been generalized to the two-way error component model and allow some but not all regressors to be correlated with these individual and time effects. This paper proposes a pretest estimator for this two-way error component panel data regression model based on two Hausman tests. The first Hausman test is based upon the contrast between the TWFE and the TWRE estimators. The second Hausman test is based on the contrast between the two-way Hausman and Taylor (TWHT) estimator and the TWFE estimator. The Monte Carlo results show that this pretest estimator is always second best in MSE performance compared to the efficient estimator, whether the model is random-effects, fixed-effects or Hausman and Taylor. This paper generalizes the one-way pretest estimator to the two-way error component model.

Confidence interval for normal means in meta-analysis based on a pretest estimator

Confidence interval for normal means in meta-analysis based on a pretest estimator

Forecasting vector autoregressions with mixed roots in the vicinity of unity

This article evaluates the forecast performance of model averaging forecasts in a nonstationary vector autoregression with mixed roots in the vicinity of unity. The deviation from unit root allows for local to unity, moderate deviation from unity and strong unit root, and the direction of such deviation could be from either the stationary or the explosive side. We provide a theoretical foundation for comparison among various forecasts, including the least squares estimator, the constrained estimator imposing the unit root constraint, and the selection or average over these two basic estimators. Furthermore, three new types of estimators are constructed, i.e., the bagging versions of the pretest estimator, the Mallows-pretest estimator that marries the Mallows averaging criterion and the Wald test, and the Mallows-bagging estimator that combines the Mallows averaging criterion and bagging technique. The asymptotic risks are shown to depend on the local parameters, which are not consistently estimable. Via Monte Carlo simulations, graphic comparisons indicate that the Mallows averaging estimator has both robust and outstanding forecasting performance. Model averaging over the vector autoregressive lag order is further considered to address the issue of model uncertainty in the lag specification. Finite sample simulations show that the Mallows averaging estimator performs superior to other frequently used selection and averaging methods. The application to forecasting the financial indices popularly used in the predictive regression further illustrates the practical merit of the proposed estimator.

Pretest estimation in combining probability and non-probability samples

Pretest estimation in combining probability and non-probability samples

Efficient estimation method for generalized ARFIMA models

This paper focuses on pretest and shrinkage estimation strategies for generalized autoregressive fractionally integrated moving average (GARFIMA) models when some of the regression parameters are possible to restrict to a subspace. These estimation strategies are constructed on the assumption that some covariates are not statistically significant for the response. To define the pretest and shrinkage estimators, we fit two models: one includes all the covariates and the others are subject to linear constraint based on the auxiliary information of the insignificant covariates. The unrestricted and restricted estimators are then combined optimally to get the pretest and shrinkage estimators. We enlighten the statistical properties of the shrinkage and pretest estimators in terms of asymptotic bias and risk. We examine the comparative performance of pretest and shrinkage estimators with respect to unrestricted maximum partial likelihood estimator (UMPLE). We show that the shrinkage estimators have a lower relative mean squared error as compared to the UMPLE when the number of significant covariates exceeds two. Monte Carlo simulations are conducted to examine the relative performance of the proposed estimators to the UMPLE. An empirical application is used for the usefulness of our proposed estimation strategies.

Meta.shrinkage: An R Package for Meta-Analyses for Simultaneously Estimating Individual Means

Meta-analysis is an indispensable tool for synthesizing statistical results obtained from individual studies. Recently, non-Bayesian estimators for individual means were proposed by applying three methods: the James–Stein (JS) shrinkage estimator, isotonic regression estimator, and pretest (PT) estimator. In order to make these methods available to users, we develop a new R package meta.shrinkage. Our package can compute seven estimators (named JS, JS+, RML, RJS, RJS+, PT, and GPT). We introduce this R package along with the usage of the R functions and the “average-min-max” steps for the pool-adjacent violators algorithm. We conduct Monte Carlo simulations to validate the proposed R package to ensure that the package can work properly in a variety of scenarios. We also analyze a data example to show the ability of the R package.

Estimation of the slope parameter in a linear regression model under a bounded loss function

The estimation of the slope parameter of a simple linear regression model in the presence of nonsample prior information under the reflected normal loss function is considered. Usually, the traditional estimation methods such as the least squared (LS) error are used to estimate the slope parameter. Sometimes the researcher has information about the unknown slope parameter from experience as a point guess, the nonsample prior information. In this paper, the shrinkage pretest estimators are introduced and their risk functions are derived under the reflected normal loss function. Several methods of finding distrust coefficient of the shrinkage pretest estimators are proposed. The behavior of the estimators are compared using a simulation study. The results show that the shrinkage pretest estimator outperforms the LS estimator when nonsample prior information is close to the real value. A real data set is analyzed for illustrating the results.

Random Effects Endogeneity: A Variable Pretest

Abstract The appealing but complex Hausman and Taylor (1981) random effects (instrumental variable) estimator requires prior knowledge that certain explanatory variables in a panel are uncorrelated with the latent group effects. The purpose of this examination is to outline a tractable variable pretest that facilitates the initial sorting of regressors as likely exogenous or endogenous. The variable pretest proposed herein builds on the pretest estimator suggested by Baltagi et al (2003) by providing the necessary foundation for regressor identification. Extensions are suggested for the two-way error components construct. Keywords: Panel data, Random effects, Variable pretest, Hausman-Taylor. JEL Classification: C12, C13, C23.

Pretest and shrinkage estimation strategies in accelerated failure time model

This paper focuses on the accelerated failure time model which is often suggested as an alternative to the Cox proportional hazards model. We propose pretest and shrinkage estimation methods for estimating the regression parameters in the accelerated failure time model for right-censored data when some of the parameters shrink to a restricted subspace. Shrinkage estimators take information from the unrestricted model and provide an efficient estimate of the regression parameters by shrinking the unrestricted estimate toward the restricted model estimate. This technique increases the bias in the estimation process but reduces the overall mean-squared error, which offsets the bias. We show that if the shrinkage dimension exceeds two, the risk of the shrinkage estimator is strictly less than that of the maximum-likelihood estimator. Extensive numerical studies are conducted to evaluate the performance of the pretest and shrinkage estimators. An empirical application is provided to illustrate the practical usefulness of the proposed estimators.

A Meta-Analysis for Simultaneously Estimating Individual Means with Shrinkage, Isotonic Regression and Pretests

Meta-analyses combine the estimators of individual means to estimate the common mean of a population. However, the common mean could be undefined or uninformative in some scenarios where individual means are “ordered” or “sparse”. Hence, assessments of individual means become relevant, rather than the common mean. In this article, we propose simultaneous estimation of individual means using the James–Stein shrinkage estimators, which improve upon individual studies’ estimators. We also propose isotonic regression estimators for ordered means, and pretest estimators for sparse means. We provide theoretical explanations and simulation results demonstrating the superiority of the proposed estimators over the individual studies’ estimators. The proposed methods are illustrated by two datasets: one comes from gastric cancer patients and the other from COVID-19 patients.

Sparse Estimation Strategies in Linear Mixed Effect Models for High-Dimensional Data Application.

In a host of business applications, biomedical and epidemiological studies, the problem of multicollinearity among predictor variables is a frequent issue in longitudinal data analysis for linear mixed models (LMM). We consider an efficient estimation strategy for high-dimensional data application, where the dimensions of the parameters are larger than the number of observations. In this paper, we are interested in estimating the fixed effects parameters of the LMM when it is assumed that some prior information is available in the form of linear restrictions on the parameters. We propose the pretest and shrinkage estimation strategies using the ridge full model as the base estimator. We establish the asymptotic distributional bias and risks of the suggested estimators and investigate their relative performance with respect to the ridge full model estimator. Furthermore, we compare the numerical performance of the LASSO-type estimators with the pretest and shrinkage ridge estimators. The methodology is investigated using simulation studies and then demonstrated on an application exploring how effective brain connectivity in the default mode network (DMN) may be related to genetics within the context of Alzheimer’s disease.

A class of general pretest estimators for the univariate normal mean

In this paper, we propose a class of general pretest estimators for the univariate normal mean. The main mathematical idea of the proposed class is the adaptation of randomized tests, where the randomization probability is related to a shrinkage parameter. Consequently, the proposed class includes many existing estimators, such as the pretest, shrinkage, Bayes, and empirical Bayes estimators as special cases. Furthermore, the proposed class can be easily tuned for users by adjusting significance levels and probability function. We derive theoretical properties of the proposed class, such as the expressions for the distribution function, bias, and MSE. Our expressions for the bias and MSE turn out to be simpler than those previously derived for some existing formulas for special cases. We also conduct simulation studies to examine our theoretical results and demonstrate the application of the proposed class through a real dataset.

Density estimation on a network

A novel approach is proposed for density estimation on a network. Nonparametric density estimation on a network is formulated as a nonparametric regression problem by binning. Nonparametric regression using local polynomial kernel-weighted least squares have been studied rigorously, and its asymptotic properties make it superior to kernel estimators such as the Nadaraya–Watson estimator. When applied to a network, the best estimator near a vertex depends on the amount of smoothness at the vertex. Often, there are no compelling reasons to assume that a density will be continuous or discontinuous at a vertex, hence a data driven approach is proposed. To estimate the density in a neighborhood of a vertex, a two-step procedure is proposed. The first step of this pretest estimator fits a separate local polynomial regression on each edge using data only on that edge, and then tests for equality of the estimates at the vertex. If the null hypothesis is not rejected, then the second step re-estimates the regression function in a small neighborhood of the vertex, subject to a joint equality constraint. Since the derivative of the density may be discontinuous at the vertex, a piecewise polynomial local regression estimate is used to model the change in slope. The special case of local piecewise linear regression is studied in detail and the leading bias and variance terms are derived using weighted least squares theory. The proposed approach will remove the bias near a vertex that has been noted for existing methods, which typically do not allow for discontinuity at vertices. For a fixed network, the proposed method scales sub-linearly with sample size and it can be extended to regression and varying coefficient models on a network. The working of the proposed model is demonstrated by simulation studies and applications to a dendrite network dataset.

Portfolio Pretesting with Machine Learning

This paper exploits the idea of pretesting to choose between competing portfolio strategies. We propose a strategy that optimally trades off between the risk of going for a false positive strategy choice versus the risk of making a false negative choice. Various different data-driven approaches are proposed based on an optimal choice of the pretested certainty equivalent and Sharpe Ratio. Our approach belongs to the class of shrinkage portfolio estimators. However, contrary to previous approaches the shrinkage intensity is continuously updated to incorporate the most recent information in the rolling window forecasting set-up. We show that the bagged pretest estimator performs exceptionally well, especially when combined with adaptive smoothing. The resulting strategy allows for a flexible and smooth switch between the underlying strategies and is shown to outperform the corresponding stand-alone strategies.

On Using the t-Ratio as a Diagnostic

The t-ratio has not one but two uses in econometrics, which should be carefully distinguished. It is used as a test and also as a diagnostic. I emphasize that the commonly-used estimators are in fact pretest estimators, and argue in favor of an improved (continuous) version of pretesting, called model averaging.

Estimation strategy of multilevel model for ordinal longitudinal data

This paper considers the shrinkage estimation of multilevel models that are appropriate for ordinal longitudinal data. These models can accommodate multiple random effects and, additionally, allow for a general form of model covariates that are related to the overall level of the responses and changes to the response over time. The likelihood inference for multilevel models is computationally burdensome due to intractable integrals. A maximum marginal likelihood (MML) method with Fisher’s scoring procedure is therefore followed to estimate the random and fixed effects parameters. In real life data, researchers may have collected many covariates for the response. Some of these covariates may satisfy certain constraints which can be used to produce a restricted estimate from the unrestricted likelihood function. The unrestricted and restricted MMLs can then be combined optimally to form the pretest and shrinkage estimators. Asymptotic properties of these estimators including biases and risks will be discussed. A simulation study is conducted to assess the performance of the estimators with respect to the unrestricted MML estimator. Finally, the relevance of the proposed estimators will be illustrated with a real data set.

PMSE performance of two different types of preliminary test estimators under a multivariate t error term

In this paper, assuming that the error terms follow a multivariate t distribution, we derive the exact formula for the predictive mean squared error (PMSE) of two different types of pretest estimators. It is shown analytically that one of the pretest estimator dominates the SR estimator if a critical value of the pretest is chosen appropriately. Also, we compare the PMSE of the pretest estimators with the MMSE, AMMSE, SR and PSR estimators by numerical evaluations. Our results show that the pretest estimators dominate the OLS estimator for all combinations when the degrees of freedom is not more than 5. Key Words: Predictive mean squared error; Homogeneous preliminary test estimator; Heterogeneous preliminary test estimator; Multivariate t error term.

Shrinkage estimation in linear mixed models for longitudinal data

This paper is concerned with the selection and estimation of fixed effects in linear mixed models while the random effects are treated as nuisance parameters. We propose the non-penalty James–Stein shrinkage and pretest estimation methods based on linear mixed models for longitudinal data when some of the fixed effect parameters are under a linear restriction. We establish the asymptotic distributional biases and risks of the proposed estimators, and investigate their relative performance with respect to the unrestricted maximum likelihood estimator (UE). Furthermore, we investigate the penalty (LASSO and adaptive LASSO) estimation methods and compare their relative performance with the non-penalty pretest and shrinkage estimators. A simulation study for various combinations of the inactive covariates shows that the shrinkage estimators perform better than the penalty estimators in certain parts of the parameter space. This particularly happens when there are many inactive covariates in the model. It also shows that the pretest, shrinkage, and penalty estimators all outperform the UE. We further illustrate the proposed procedures via a real data example.