Small‐scale Inference: Empirical Bayes and Confidence Methods for as Few as a Single Comparison

Empirical Bayes methods of estimating the local false discovery rate (LFDR) by maximum likelihood estimation (MLE), originally developed for large numbers of comparisons, are applied to a single comparison. Specifically, when assuming a lower bound on the mixing proportion of true null hypotheses, the LFDR MLE can yield reliable hypothesis tests and confidence intervals given as few as one comparison. Simulations indicate that constrained LFDR MLEs perform markedly better than conventional methods, both in testing and in confidence intervals, for high values of the mixing proportion, but not for low values. (A decision‐theoretic interpretation of the confidence distribution made those comparisons possible.) In conclusion, the constrained LFDR estimators and the resulting effect‐size interval estimates are not only effective multiple comparison procedures but also they might replace p‐values and confidence intervals more generally. The new methodology is illustrated with the analysis of proteomics data.     

Estimating Mann–Whitney‐type Causal Effects

Mann–Whitney‐type causal effects are generally applicable to outcome variables with a natural ordering, have been recommended for clinical trials because of their clinical relevance and interpretability and are particularly useful in analysing an ordinal composite outcome that combines an original primary outcome with death and possibly treatment discontinuation. In this article, we consider robust and efficient estimation of such causal effects in observational studies and clinical trials. For observational studies, we propose and compare several estimators: regression estimators based on an outcome regression (OR) model or a generalised probabilistic index (GPI) model, an inverse probability weighted estimator based on a propensity score model and two doubly robust (DR), locally efficient estimators. One of the DR estimators involves a propensity score model and an OR model, is consistent and asymptotically normal under the union of the two models and attains the semiparametric information bound when both models are correct. The other DR estimator has the same properties with the OR model replaced by a GPI model. For clinical trials, we extend an existing augmented estimator based on a GPI model and propose a new one based on an OR model. The methods are evaluated and compared in simulation experiments and applied to a clinical trial in cardiology and an observational study in obstetrics.     

Asymptotic comparison at optimal levels of reduced‐bias extreme value index estimators

In this article we are interested in the asymptotic comparison, at optimal levels, of a set of semi‐parametric reduced‐bias extreme value (EV) index estimators, valid for a wide class of heavy‐tailed models, underlying the available data. Again, as in the classical case, there is not any estimator that can always dominate the alternatives, but interesting clear‐cut patterns are found. Consequently, and in practice, a suitable choice of a set of EV index estimators will jointly enable us to better estimate the EV index γ, the primary parameter of extreme events.     

Dynamic Panel Data Models With Irregular Spacing: With an Application to Early Childhood Development

With the increased availability of longitudinal data, dynamic panel data models have become commonplace. Moreover, the properties of various estimators of such models are well known. However, we show that these estimators break down when the data are irregularly spaced along the time dimension. Unfortunately, this is an increasingly frequent occurrence as many longitudinal surveys are collected at non‐uniform intervals and no solution is currently available when time‐varying covariates are included in the model. In this paper, we propose two new estimators for dynamic panel data models when data are irregularly spaced and compare their finite‐sample performance to the näive application of existing estimators. We illustrate the practical importance of this issue in an application concerning early childhood development. Copyright © 2016 John Wiley & Sons, Ltd.     

On solving bias‐corrected non‐linear estimation equations with an application to the dynamic linear model

In a seminal paper, Mak, Journal of the Royal Statistical Society B, 55, 1993, 945, derived an efficient algorithm for solving non‐linear unbiased estimation equations. In this paper, we show that when Mak's algorithm is applied to biased estimation equations, it results in the estimates that would come from solving a bias‐corrected estimation equation, making it a consistent estimator if regularity conditions hold. In addition, the properties that Mak established for his algorithm also apply in the case of biased estimation equations but for estimates from the bias‐corrected equations. The marginal likelihood estimator is obtained when the approach is applied to both maximum likelihood and least squares estimation of the covariance matrix parameters in the general linear regression model. The new approach results in two new estimators when applied to the profile and marginal likelihood functions for estimating the lagged dependent variable coefficient in the dynamic linear regression model. Monte Carlo simulation results show the new approach leads to a better estimator when applied to the standard profile likelihood. It is therefore recommended for situations in which standard estimators are known to be biased.     

Benchmarked Estimators for a Small Area Mean Under a Onefold Nested Regression Model

In this paper, we modify small area estimators, based on the unit‐level model, so that they add up to reliable higher‐level estimates of population totals. These modifications result in benchmarked small area estimators. We consider two benchmarking procedures. One is based on augmenting the unit‐level model with a suitable variable. The other one uses the calibrated weights of the direct estimators that are reliable at the higher levels. These weights are used in estimators that are based on the aggregation of the unit‐level model for each small area. The mean squared error estimators of the proposed benchmarked estimators are obtained by suitably modifying those associated with the corresponding non benchmarked estimators. The properties of the estimators are evaluated via simulation.     

A general framework for frontier estimation with panel data

The main objective of the paper is to present a general framework for estimating production frontier models with panel data. A sample of firms i = 1, ..., N is observed on several time periods t = 1, ... T. In this framework, nonparametric stochastic models for the frontier will be analyzed. The usual parametric formulations of the literature are viewed as particular cases and the convergence of the obtained estimators in this general framework are investigated. Special attention is devoted to the role of N and of T on the speeds of convergence of the obtained estimators. First, a very general model is investigated. In this model almost no restriction is imposed on the structure of the model or of the inefficiencies. This model is estimable from a nonparametric point of view but needs large values of T and of N to obtain reliable estimates of the individual production functions and estimates of the frontier function. Then more specific nonparametric firm effect models are presented. In these cases, only NT must be large to estimate the common production function; but again both large N and T are needed for estimating individual efficiencies and for estimating the frontier. The methods are illustrated through a numerical example with real data.     

Ridge regression estimation of the Rotterdam model

The best guesses of unknown coefficients specified in Theil's model of introspection are like predictions and not like de Finetti's prevision and therefore not the values taken by random variables. Constrained least squares procedures can be formulated which are free of these difficulties. The ridge estimator is a simple version of a constrained least squares estimator which can be made operational even when little prior information is available. Our operational ridge estimators are nearly minimax and are not less stable than least squares in the presence of high multicollinearity. Finally, we have presented the ridge estimates for the Rotterdam demand model.     

Spurious Fixed Effects Regression*

This article shows that spurious regression results can occur for a fixed effects model with weak time series variation in the regressor and/or strong time series variation in the regression errors when the first‐differenced and Within‐OLS estimators are used. Asymptotic properties of these estimators and the related t‐tests and model selection criteria are studied by sending the number of cross‐sectional observations to infinity. This article shows that the first‐differenced and Within‐OLS estimators diverge in probability, that the related t‐tests are inconsistent, that R²s converge to zero in probability and that AIC and BIC diverge to ?∞ in probability. The results of the article warn that one should not jump to the use of fixed effects regressions without considering the degree of time series variations in the data.     

Estimating the Kullback–Liebler risk based on multifold cross‐validation

This paper concerns a class of model selection criteria based on cross‐validation techniques and estimative predictive densities. Both the simple or leave‐one‐out and the multifold or leave‐m‐out cross‐validation procedures are considered. These cross‐validation criteria define suitable estimators for the expected Kullback–Liebler risk, which measures the expected discrepancy between the fitted candidate model and the true one. In particular, we shall investigate the potential bias of these estimators, under alternative asymptotic regimes for m. The results are obtained within the general context of independent, but not necessarily identically distributed, observations and by assuming that the candidate model may not contain the true distribution. An application to the class of normal regression models is also presented, and simulation results are obtained in order to gain some further understanding on the behavior of the estimators.     

Estimation of multivariate models for time series of possibly different lengths

We consider the problem of estimating parametric multivariate density models when unequal amounts of data are available on each variable. We focus in particular on the case that the unknown parameter vector may be partitioned into elements relating only to a marginal distribution and elements relating to the copula. In such a case we propose using a multi‐stage maximum likelihood estimator (MSMLE) based on all available data rather than the usual one‐stage maximum likelihood estimator (1SMLE) based only on the overlapping data. We provide conditions under which the MSMLE is not less asymptotically efficient than the 1SMLE, and we examine the small sample efficiency of the estimators via simulations. The analysis in this paper is motivated by a model of the joint distribution of daily Japanese yen–US dollar and euro–US dollar exchange rates. We find significant evidence of time variation in the conditional copula of these exchange rates, and evidence of greater dependence during extreme events than under the normal distribution. Copyright © 2006 John Wiley & Sons, Ltd.     

Model‐Assisted Regression Estimators for Longitudinal Data with Nonignorable Dropout

Estimation with longitudinal Y having nonignorable dropout is considered when the joint distribution of Y and covariate X is nonparametric and the dropout propensity conditional on (Y,X) is parametric. We apply the generalised method of moments to estimate the parameters in the nonignorable dropout propensity based on estimating equations constructed using an instrument Z, which is part of X related to Y but unrelated to the dropout propensity conditioned on Y and other covariates. Population means and other parameters in the nonparametric distribution of Y can be estimated based on inverse propensity weighting with estimated propensity. To improve efficiency, we derive a model‐assisted regression estimator making use of information provided by the covariates and previously observed Y‐values in the longitudinal setting. The model‐assisted regression estimator is protected from model misspecification and is asymptotically normal and more efficient when the working models are correct and some other conditions are satisfied. The finite‐sample performance of the estimators is studied through simulation, and an application to the HIV‐CD4 data set is also presented as illustration.     

Efficient estimation of factor models with time and cross‐sectional dependence

This paper studies the efficient estimation of large‐dimensional factor models with both time and cross‐sectional dependence assuming (N,T) separability of the covariance matrix. The asymptotic distribution of the estimator of the factor and factor‐loading space under factor stationarity is derived and compared to that of the principal component (PC) estimator. The paper also considers the case when factors exhibit a unit root. We provide feasible estimators and show in a simulation study that they are more efficient than the PC estimator in finite samples. In application, the estimation procedure is employed to estimate the Lee–Carter model and life expectancy is forecast. The Dutch gender gap is explored and the relationship between life expectancy and the level of economic development is examined in a cross‐country comparison.     

A generalization of the Binomial distribution based on the dependence ratio

We propose a generalization of the Binomial distribution, called DR‐Binomial, which accommodates dependence among units through a model based on the dependence ratio (Ekholm et al., Biometrika, 82, 1995, 847). Properties of the DR‐Binomial are discussed, and the constraints on its parameter space are studied in detail. Likelihood‐based inference is presented, using both the joint and profile likelihoods; the usefulness of the DR‐Binomial in applications is illustrated on a real dataset displaying negative unit‐dependence, and hence under‐dispersion compared with the Binomial. Although the DR‐Binomial turns out to be a reparameterization of Altham's Additive‐Binomial and Kupper–Haseman's Correlated‐Binomial distribution, we believe its introduction is useful, both in terms of interpretability and mathematical tractability and in terms of generalizability to the Multinomial case.     

On Maximum Likelihood Estimation of Dynamic Panel Data Models

We analyse the finite sample properties of maximum likelihood estimators for dynamic panel data models. In particular, we consider transformed maximum likelihood (TML) and random effects maximum likelihood (RML) estimation. We show that TML and RML estimators are solutions to a cubic first‐order condition in the autoregressive parameter. Furthermore, in finite samples both likelihood estimators might lead to a negative estimate of the variance of the individual‐specific effects. We consider different approaches taking into account the non‐negativity restriction for the variance. We show that these approaches may lead to a solution different from the unique global unconstrained maximum. In an extensive Monte Carlo study we find that this issue is non‐negligible for small values of T and that different approaches might lead to different finite sample properties. Furthermore, we find that the Likelihood Ratio statistic provides size control in small samples, albeit with low power due to the flatness of the log‐likelihood function. We illustrate these issues modelling US state level unemployment dynamics.     

On Estimators of the Nearest Neighbour Distance Distribution Function for Stationary Point Processes

There are three approaches for the estimation of the distribution function D(r) of distance to the nearest neighbour of a stationary point process: the border method, the Hanisch method and the Kaplan-Meier approach. The corresponding estimators and some modifications are compared with respect to bias and mean squared error (mse). Simulations for Poisson, cluster and hard-core processes show that the classical border estimator has good properties; still better is the Hanisch estimator. Typically, mse depends on r, having small values for small and large r and a maximum in between. The mse is not reduced if the exact intensity λ (if known) or intensity estimators from larger windows are built in the estimators of D(r); in contrast, the intensity estimator should have the same precision as that of λ D(r). In the case of replicated estimation from more than one window the best way of pooling the subwindow estimates is averaging by weights which are proportional to squared point numbers.     

Common correlated effect cross‐sectional dependence corrections for nonlinear conditional mean panel models

This paper provides an approach to estimation and inference for nonlinear conditional mean panel data models, in the presence of cross‐sectional dependence. We modify Pesaran's (Econometrica, 2006, 74(4), 967–1012) common correlated effects correction to filter out the interactive unobserved multifactor structure. The estimation can be carried out using nonlinear least squares, by augmenting the set of explanatory variables with cross‐sectional averages of both linear and nonlinear terms. We propose pooled and mean group estimators, derive their asymptotic distributions, and show the consistency and asymptotic normality of the coefficients of the model. The features of the proposed estimators are investigated through extensive Monte Carlo experiments. We also present two empirical exercises. The first explores the nonlinear relationship between banks' capital ratios and riskiness. The second estimates the nonlinear effect of national savings on national investment in OECD countries depending on countries' openness.     

Smoothed binary regression quantiles

This paper extends results regarding smoothed median binary regression to general smoothed binary quantile regression, discusses the interpretation of the resulting estimators under alternative assumptions, and shows how they may be used to obtain semiparametric estimates of counterfactual probabilities. The estimators are applied to a model of labour force participation of married women in the USA. We find that the elasticity with respect to non‐labour income is significantly negative only for women that belong to the middle of the conditional willingness‐to‐participate (WTP) distribution. In comparing the quantile models with parametric logit and semiparametric single‐index specifications, we find that the models agree closely for women around the centre of the WTP distribution, but there are considerable disagreements as we move towards the tails of the distribution. Copyright © 2006 John Wiley & Sons, Ltd.     

Estimating Nonlinear Models with Multiple Fixed Effects: A Computational Note*

In this paper we consider estimation of nonlinear panel data models that include multiple individual fixed effects. Estimation of these models is complicated both by the difficulty of estimating models with possibly thousands of coefficients and also by the incidental parameters problem; that is, noisy estimates of the fixed effects when the time dimension is short contaminate the estimates of the common parameters due to the nonlinearity of the problem. We propose a simple variation of existing bias‐corrected estimators, which can exploit the additivity of the effects for numerical optimization. We exhibit the performance of the estimators in simulations.     

Estimating Precision and Recall for Deterministic and Probabilistic Record Linkage

Linking administrative, survey and census files to enhance dimensions such as time and breadth or depth of detail is now common. Because a unique person identifier is often not available, records belonging to two different units (e.g. people) may be incorrectly linked. Estimating the proportion of links that are correct, called Precision, is difficult because, even after clerical review, there will remain uncertainty about whether a link is in fact correct or incorrect. Measures of Precision are useful when deciding whether or not it is worthwhile linking two files, when comparing alternative linking strategies and as a quality measure for estimates based on the linked file. This paper proposes an estimator of Precision for a linked file that has been created by either deterministic (or rules‐based) or probabilistic (where evidence for a link being a match is weighted against the evidence that it is not a match) linkage, both of which are widely used in practice. This paper shows that the proposed estimators perform well.