Some work in political methodology recommends that applied researchers obtain point estimates of quantities of interest by simulating model coefficients, transforming these simulated coefficients into simulated quantities of interest, and then averaging the simulated quantities of interest (e.g., CLARIFY). But other work advises applied researchers to directly transform coefficient estimates to estimate quantities of interest. I point out that these two approaches are not interchangeable and examine their properties. I show that the simulation approach compounds the transformation-induced bias identified by Rainey (2017), adding bias with direction and magnitude similar to the transformation-induced bias. I refer to this easily avoided additional bias as “simulation-induced bias.” Even if researchers use simulation to estimate standard errors, they should directly transform maximum likelihood estimates of coefficient estimates to obtain point estimates of quantities of interest.

In small samples, maximum likelihood (ML) estimates of logit model coefficients have substantial bias away from zero. As a solution, we remind political scientists of Firth's (1993, Biometrika, 80, 27–38) penalized maximum likelihood (PML) estimator. Prior research has described and used PML, especially in the context of separation, but its small sample properties remain under-appreciated. The PML estimator eliminates most of the bias and, perhaps more importantly, greatly reduces the variance of the usual ML estimator. Thus, researchers do not face a bias-variance tradeoff when choosing between the ML and PML estimators—the PML estimator has a smaller bias and a smaller variance. We use Monte Carlo simulations and a re-analysis of George and Epstein (1992, American Political Science Review, 86, 323–337) to show that the PML estimator offers a substantial improvement in small samples (e.g., 50 observations) and noticeable improvement even in larger samples (e.g., 1000 observations).

We developed a maximum likelihood estimator corresponding to the predicted hazard rate that emerges from a continuous time game of incomplete information with a fixed time horizon (i.e., Kreps and Wilson, 1982, Journal of Economic Theory27, 253–279). Such games have been widely applied in economics and political science and involve two players engaged in a war of attrition contest over some prize that they both value. Each player can be either a strong or weak competitor. In the equilibrium of interest, strong players do not quit whereas weak players play a mixed strategy characterized by a hazard rate that increases up to an endogenous point in time, after which only strong players remain. The observed length of the contest can therefore be modeled as a mixture between two unobserved underlying durations: one that increases until it abruptly ends at an endogenous point in time and a second involving two strong players that continues indefinitely. We illustrate this estimator by studying the durations of Senate filibusters and international crises.

Signaling games are central to political science but often have multiple equilibria, leading to no definitive prediction. We demonstrate that these indeterminacies create substantial problems when fitting theory to data: they lead to ill-defined and discontinuous likelihoods even if the game generating the data has a unique equilibrium. In our experiments, currently used techniques frequently fail to uncover the parameters of the canonical crisis-signaling game, regardless of sample size and number of equilibria in the data generating process. We propose three estimators that remedy these problems, outperforming current best practices. We fit the signaling model to data on economic sanctions. Our solutions find a novel U-shaped relationship between audience costs and the propensity for leaders to threaten sanctions, which current best practices fail to uncover.

Navigation methods, traditional and modern, use lines of position in the plane. Standard Gaussian assumptions about errors leads to a constant sum of squared distances from the lines defining a probability contour. It is well known these contours are a family of ellipses centred on the most probable position and they can be computed using algebraic methods. In this paper we show how the most probable position, the axes and foci of ellipses can be found using geometric methods. This results in a ruler and compasses construction of these points and this gives insight into the way the shape and orientation of the probability contours depend on the angles between the lines of position. We start with the classical case of three lines of position with equal variances, we show how this can be extended to the case where the variances in the errors in the lines of position differ, and we go on to consider the case of four lines of position using a methodology that generalises to an arbitrary number of lines.

The estimation of GPS repeater jamming plays an important role in modern military applications. In this paper, based on the maximum likelihood estimation (MLE) method, a new approach for the estimation of GPS repeater jamming using Kalman filtering with coloured observation noise is proposed. The state equation and observation equation of the Kalman filter with the coloured observation noise are deduced and the generation process explained. The recursive formula for the Kalman filter used to iteratively filter for the MLE output is also derived. Experimental results show that the new design has a better performance than the MLE; hence it can be used for the precise estimation of repeater jamming signals under the condition that the carrier-to-noise density ratio is below 20 dB-Hz.

In this paper, we analyze the celebrated EM algorithm from the point of view of proximal point algorithms. More precisely, we study a new type of generalization of the EM procedure introduced in [Chretien and Hero (1998)] and called Kullback-proximal algorithms. The proximal framework allows us to prove new results concerning the cluster points. An essential contribution is a detailed analysis of the case where some cluster points lie on the boundary of the parameter space.