Density estimation for grouped data with application to line transect sampling

Line transect sampling is a method used to estimate wildlife populations, with the resulting data often grouped in intervals. Estimating the density from grouped data can be challenging. In this paper we propose a kernel density estimator of wildlife population density for such grouped data. Our method uses a combined cross-validation and smoothed bootstrap approach to select the optimal bandwidth for grouped data. Our simulation study shows that with the smoothing parameter selected with this method, the estimated density from grouped data matches the true density more closely than with other approaches. Using smoothed bootstrap, we also construct bias-adjusted confidence intervals for the value of the density at the boundary. We apply the proposed method to two grouped data sets, one from a wooden stake study where the true density is known, and the other from a survey of kangaroos in Australia.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS307 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonparametric causal effects based on incremental propensity score interventions

Most work in causal inference considers deterministic interventions that set each unit's treatment to some fixed value. However, under positivity violations these interventions can lead to non-identification, inefficiency, and effects with little practical relevance. Further, corresponding effects in longitudinal studies are highly sensitive to the curse of dimensionality, resulting in widespread use of unrealistic parametric models. We propose a novel solution to these problems: incremental interventions that shift propensity score values rather than set treatments to fixed values. Incremental interventions have several crucial advantages. First, they avoid positivity assumptions entirely. Second, they require no parametric assumptions and yet still admit a simple characterization of longitudinal effects, independent of the number of timepoints. For example, they allow longitudinal effects to be visualized with a single curve instead of lists of coefficients. After characterizing these incremental interventions and giving identifying conditions for corresponding effects, we also develop general efficiency theory, propose efficient nonparametric estimators that can attain fast convergence rates even when incorporating flexible machine learning, and propose a bootstrap-based confidence band and simultaneous test of no treatment effect. Finally we explore finite-sample performance via simulation, and apply the methods to study time-varying sociological effects of incarceration on entry into marriage

Point estimation with exponentially tilted empirical likelihood

Parameters defined via general estimating equations (GEE) can be estimated by maximizing the empirical likelihood (EL). Newey and Smith [Econometrica 72 (2004) 219--255] have recently shown that this EL estimator exhibits desirable higher-order asymptotic properties, namely, that its $O(n^{-1})$ bias is small and that bias-corrected EL is higher-order efficient. Although EL possesses these properties when the model is correctly specified, this paper shows that, in the presence of model misspecification, EL may cease to be root n convergent when the functions defining the moment conditions are unbounded (even when their expectations are bounded). In contrast, the related exponential tilting (ET) estimator avoids this problem. This paper shows that the ET and EL estimators can be naturally combined to yield an estimator called exponentially tilted empirical likelihood (ETEL) exhibiting the same $O(n^{-1})$ bias and the same $O(n^{-2})$ variance as EL, while maintaining root n convergence under model misspecification.Comment: Published at http://dx.doi.org/10.1214/009053606000001208 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Entropy balancing is doubly robust

Covariate balance is a conventional key diagnostic for methods used estimating causal effects from observational studies. Recently, there is an emerging interest in directly incorporating covariate balance in the estimation. We study a recently proposed entropy maximization method called Entropy Balancing (EB), which exactly matches the covariate moments for the different experimental groups in its optimization problem. We show EB is doubly robust with respect to linear outcome regression and logistic propensity score regression, and it reaches the asymptotic semiparametric variance bound when both regressions are correctly specified. This is surprising to us because there is no attempt to model the outcome or the treatment assignment in the original proposal of EB. Our theoretical results and simulations suggest that EB is a very appealing alternative to the conventional weighting estimators that estimate the propensity score by maximum likelihood.Comment: 23 pages, 6 figures, Journal of Causal Inference 201