Poisson regression models with errors-in-variables: implication and treatment

Overdispersion has been a common phenomenon in count data and usually treated with the negative binomial model. This paper shows that measurement errors in covariates in general also lead to overdispersion on the observed data if the true data generating process is indeed the Poisson regression. This kind of overdispersion cannot be treated using the negative binomial model, as otherwise, biases will occur. To provide consistent estimates, we propose a new type of corrected score estimator assuming that the distribution of the latent variables is known. The consistency and asymptotic normality of the proposed estimator are established. Simulation results show that this estimator has good finite sample performance. We also illustrate that the Akaike information criterion and Bayesian information criterion work well for selecting the correct model if the true model is the errors-in-variables Poisson regression.     

CORRECTED SCORE FUNCTIONS IN CLASSICAL ERROR-IN-VARIABLES AND INCIDENTAL PARAMETER MODELS

Nakamura (1990) introduced an approach to estimation in measurement error models based on a corrected score function, and claimed that the estimators obtained are consistent for functional models. Proof of the claim essentially assumed the existence of a corrected log-likelihood for which differentiation with respect to model parameters can be interchanged with conditional expectation taken with respect to the measurement error distributions, given the response variables and true covariates. This paper deals with simple yet practical models for which the above assumption is false, i.e. a corrected score function for the model may not be obtained through differentiating a corrected log-likelihood although it exists. Alternative regularity conditions with no reference to log-likelihood are given, under which the corrected score functions yield consistent and asymptotically normal estimators. Application to functional comparative calibration yields interesting results.     

Asymptotic Properties of M-estimators Based on Estimating Equations and Censored Data

Properties of Huber's M-estimators based on estimating equations have been studied extensively and are well understood for complete (i.i.d.) data. Although the concepts of M-estimators and influence curves have been extended for some time by Reid (1981) to incomplete data that are subject to right censoring, results on the general behavior of M-estimators based on incomplete data remain scattered and restrictive. This paper establishes a general large sample theory for M-estimators based on censored data. We show how to extend any asymptotic result available for M-estimators based on complete data to the case of censored data. The extensions are usually straightforward and include the multiparameter situation. Both the lifetime and censoring distributions may be discontinuous. We illustrate several extensions which provide simple and tractable sufficient conditions for an M-estimator to be strongly consistent and asymptotically normal. The influence curves and asymptotic variance of the M-estimators are also derived. The applicability of the new sufficient conditions is demonstrated through several examples, including location and scale M-estimators.     

Bayesian inference for the proportion of true null hypotheses using minimum Hellinger distance

It is important that the proportion of true null hypotheses be estimated accurately in a multiple hypothesis context. Current estimation methods, however, are not suitable for high-dimensional data such as microarray data. First, they do not consider the (strong) dependence between hypotheses (or genes), thereby resulting in inaccurate estimation. Second, the unknown distribution of false null hypotheses cannot be estimated properly by these methods. Third, the estimation is affected strongly by outliers. In this paper, we find out the optimal procedure for estimating the proportion of true null hypotheses under a (strong) dependence based on the Dirichlet process prior. In addition, by using the minimum Hellinger distance, the estimation should be robust to any model misspecification as well as to any outliers while maintaining efficiency. The results are confirmed by a simulation study, and the newly developed methodology is illustrated by a real microarray data.     

Maximum likelihood estimation of Burr XII distribution parameters under random censoring

In this paper, we consider the maximum likelihood estimation of the parameters of Burr XII distribution using randomly right censored data. We provide necessary and sufficient conditions for the existence and uniqueness of the maximum likelihood estimates. Under such conditions, it is shown that the maximum likelihood estimates are strongly consistent for the true values of the parameters and are asymptotically bivariate normal. An application to leukemia free-survival times for allogeneic and autologous transplant patients is given.     

A note on the adaptive control of false discovery rates

Summary.  The use of a fixed rejection region for multiple hypothesis testing has been shown to outperform standard fixed error rate approaches when applied to control of the false discovery rate. In this work it is demonstrated that, if the original step-up procedure of Benjamini and Hochberg is modified to exercise adaptive control of the false discovery rate, its performance is virtually identical to that of the fixed rejection region approach. In addition, the dependence of both methods on the proportion of true null hypotheses is explored, with a focus on the difficulties that are involved in the estimation of this quantity.     

OPTIMAL TESTS OF SIGNIFICANCE

To perform a test of significance of a null hypothesis, a test statistic is chosen which is expected to be small if the hypothesis is false. Then the significance level of the test for an observed sample is the probability that the test statistic, under the assumptions of the hypothesis, is as small, or smaller than, its observed value. A "good" test statistic is taken to be one which is stochastically small when the null hypothesis is false. Optimal test statistics are defined using this criterion and the relationship of these methods to the Neyman-Pearson theory of hypothesis testing is considered.     

Letter to the editor comments on Groparu-Cojocaru and Doray (2013)

Although estimating the five parameters of an unknown Generalized Normal Laplace (GNL) density by minimizing the distance between the empirical and true characteristic functions seems appealing, the approach cannot be advocated in practice. This conclusion is based on extensive numerical simulations in which a fast minimization procedure delivers deceiving estimators with values that are quite far away from the truth. These findings can be predicted by the very large values obtained for the true asymptotic variances of the estimators of the five parameters of the true GNL density.     

A condition for best asymptotic normality in a regular family of distributions

A theorem is presented which provides a simple sufficient condition for a weakly consistent estimator of a parameter in a regular family of distributions to be best asymptotically normal (B.A.N.). As a corollary the B.A.N. property of a maximum likelihood estimator is established under weaker conditions than those of Zacks (1971). Two examples are provided to illustrate the technique.     

A Proposed Hybrid Effect Size Plus p-Value Criterion: Empirical Evidence Supporting its Use

ABSTRACT

When the editors of Basic and Applied Social Psychology effectively banned the use of null hypothesis significance testing (NHST) from articles published in their journal, it set off a fire-storm of discussions both supporting the decision and defending the utility of NHST in scientific research. At the heart of NHST is the p-value which is the probability of obtaining an effect equal to or more extreme than the one observed in the sample data, given the null hypothesis and other model assumptions. Although this is conceptually different from the probability of the null hypothesis being true, given the sample, p-values nonetheless can provide evidential information, toward making an inference about a parameter. Applying a 10,000-case simulation described in this article, the authors found that p-values’ inferential signals to either reject or not reject a null hypothesis about the mean (α?=?0.05) were consistent for almost 70% of the cases with the parameter’s true location for the sampled-from population. Success increases if a hybrid decision criterion, minimum effect size plus p-value (MESP), is used. Here, rejecting the null also requires the difference of the observed statistic from the exact null to be meaningfully large or practically significant, in the researcher’s judgment and experience. The simulation compares performances of several methods: from p-value and/or effect size-based, to confidence-interval based, under various conditions of true location of the mean, test power, and comparative sizes of the meaningful distance and population variability. For any inference procedure that outputs a binary indicator, like flagging whether a p-value is significant, the output of one single experiment is not sufficient evidence for a definitive conclusion. Yet, if a tool like MESP generates a relatively reliable signal and is used knowledgeably as part of a research process, it can provide useful information.     

The Multiplicative Definition of Interaction

The theory of the multiplicative definition of second order interaction is considered. Necessary and sufficient conditions in terms of the correlations in the marginal two-dimensional distributions are found for the existence of a “perfect” set of probabilities. Some progress is reported on the proof of the uniqueness of the multiplicative set of probabilities for a given set of two-dimensional distributions. The multiplicative property of such sets is preserved under pooling of marginal sets only in trivial cases, and so the multiplicative definition cannot be said to be a straightforward generalization from the definition of no interaction in two-dimensional distributions.     

Robust empirical likelihood for linear models under median constraints

In this paper, we consider the application of the empirical likelihood for

linear models under median constraints in view of robustness. For two simple median constraints, it is shown that conditions to ensure the consistency of the empirical likelihood confidence regions can be surprisingly relaxed compared with the normal approach under L norm. However, the coverage accuracy of the empirical likelihood confidence regions based on simple median constrains cannot be improved because of the discontinuity of the constraints. Therefore, a smoothed version of median constraint is proposed and a general theory is established to ensure its validity.     

Comparisons of estimators of the number of true null hypotheses and adaptive FDR procedures in multiplicity testing

Many exploratory studies such as microarray experiments require the simultaneous comparison of hundreds or thousands of genes. It is common to see that most genes in many microarray experiments are not expected to be differentially expressed. Under such a setting, a procedure that is designed to control the false discovery rate (FDR) is aimed at identifying as many potential differentially expressed genes as possible. The usual FDR controlling procedure is constructed based on the number of hypotheses. However, it can become very conservative when some of the alternative hypotheses are expected to be true. The power of a controlling procedure can be improved if the number of true null hypotheses (m ₀) instead of the number of hypotheses is incorporated in the procedure [Y. Benjamini and Y. Hochberg, On the adaptive control of the false discovery rate in multiple testing with independent statistics, J. Edu. Behav. Statist. 25(2000), pp. 60–83]. Nevertheless, m ₀ is unknown, and has to be estimated. The objective of this article is to evaluate some existing estimators of m ₀ and discuss the feasibility of these estimators in incorporating into FDR controlling procedures under various experimental settings. The results of simulations can help the investigator to choose an appropriate procedure to meet the requirement of the study.     

Estimation of Models With Multiple-Valued Explanatory Variables

ABSTRACT

We study estimation and inference when there are multiple values (“matches”) for the explanatory variables and only one of the matches is the correct one. This problem arises often when two datasets are linked together on the basis of information that does not uniquely identify regressor values. We offer a set of two intuitive conditions that ensure consistent inference using the average of the possible matches in a linear framework. The first condition is the exogeneity of the false match with respect to the regression error. The second condition is a notion of exchangeability between the true and false matches. Conditioning on the observed data, the probability that each match is correct is completely unrestricted. We perform a Monte Carlo study to investigate the estimator’s finite-sample performance relative to others proposed in the literature. Finally, we provide an empirical example revisiting a main area of application: the measurement of intergenerational elasticities in income. Supplementary materials for this article are available online.     

Large-Scale Replication Projects in Contemporary Psychological Research

ABSTRACT

Replication is complicated in psychological research because studies of a given psychological phenomenon can never be direct or exact replications of one another, and thus effect sizes vary from one study of the phenomenon to the next—an issue of clear importance for replication. Current large-scale replication projects represent an important step forward for assessing replicability, but provide only limited information because they have thus far been designed in a manner such that heterogeneity either cannot be assessed or is intended to be eliminated. Consequently, the nontrivial degree of heterogeneity found in these projects represents a lower bound on the true degree of heterogeneity. We recommend enriching large-scale replication projects going forward by embracing heterogeneity. We argue this is the key for assessing replicability: if effect sizes are sufficiently heterogeneous—even if the sign of the effect is consistent—the phenomenon in question does not seem particularly replicable and the theory underlying it seems poorly constructed and in need of enrichment. Uncovering why and revising theory in light of it will lead to improved theory that explains heterogeneity and increases replicability. Given this, large-scale replication projects can play an important role not only in assessing replicability but also in advancing theory.     

A unified empirical likelihood approach for testing MCAR and subsequent estimation

For an estimation with missing data, a crucial step is to determine if the data are missing completely at random (MCAR), in which case a complete‐case analysis would suffice. Most existing tests for MCAR do not provide a method for a subsequent estimation once the MCAR is rejected. In the setting of estimating means, we propose a unified approach for testing MCAR and the subsequent estimation. Upon rejecting MCAR, the same set of weights used for testing can then be used for estimation. The resulting estimators are consistent if the missingness of each response variable depends only on a set of fully observed auxiliary variables and the true outcome regression model is among the user‐specified functions for deriving the weights. The proposed method is based on the calibration idea from survey sampling literature and the empirical likelihood theory.     

Combining Inverse Probability Weighting and Multiple Imputation to Improve Robustness of Estimation

Inverse probability weighting (IPW) and multiple imputation are two widely adopted approaches dealing with missing data. The former models the selection probability, and the latter models data distribution. Consistent estimation requires correct specification of corresponding models. Although the augmented IPW method provides an extra layer of protection on consistency, it is usually not sufficient in practice as the true data‐generating process is unknown. This paper proposes a method combining the two approaches in the same spirit of calibration in sampling survey literature. Multiple models for both the selection probability and data distribution can be simultaneously accounted for, and the resulting estimator is consistent if any model is correctly specified. The proposed method is within the framework of estimating equations and is general enough to cover regression analysis with missing outcomes and/or missing covariates. Results on both theoretical and numerical investigation are provided.     

Beta-Riesz distributions on symmetric cones

The Riesz distributions on a symmetric cone are used to introduce a class of beta-Riesz distributions. Some fundamental properties of these distributions are established. In particular, we study the effect of a projection on a beta-Riesz distribution and we give some properties of independence. We also calculate the expectation of a beta-Riesz random variable. As a corollary, we give the regression on the mean of a Riesz random variable; that is, we determine the conditional expectation $E(U\mid U+V)$ where $U$ and $V$ are two independent Riesz random variables.     

On False Discovery and Non‐discovery Proportions of the Dynamic Adaptive Procedure

This paper studies the asymptotic behaviour of the false discovery and non‐discovery proportions of the dynamic adaptive procedure under some dependence structure. A Bahadur‐type representation of the cut point in simultaneously performing a large scale of tests is presented. The asymptotic bias decompositions of the false discovery and non‐discovery proportions are given under some dependence structure. In addition to existing literatures, we find that the randomness due to the dynamic selection of the tuning parameter in estimating the true null rate serves as a source of the approximation error in the Bahadur representation and enters into the asymptotic bias term of the false discovery proportion and those of the false non‐discovery proportion. The theory explains to some extent why some seemingly attractive dynamic adaptive procedures do not outperform the competing fixed adaptive procedures substantially in some situations. Simulations justify our theory and findings.