A Note on Minimax Testing and Confidence Intervals in Moment Inequality Models

This note uses a simple example to show how moment inequality models used in the empirical economics literature lead to general minimax relative efficiency comparisons. The main point is that such models involve inference on a low dimensional parameter, which leads naturally to a definition of "distance" that, in full generality, would be arbitrary in minimax testing problems. This definition of distance is justified by the fact that it leads to a duality between minimaxity of confidence intervals and tests, which does not hold for other definitions of distance. Thus, the use of moment inequalities for inference in a low dimensional parametric model places additional structure on the testing problem, which leads to stronger conclusions regarding minimax relative efficiency than would otherwise be possible

Jesus and Tiberius: An Examination of Source Reliability

Since the introduction to the critical method of studying the Old and New Testament in the nineteenth century, doubt has been thrown on the historical reliability of the biblical narrative accounts, especially the four Gospels. Yet, far less scrutiny and denigration have been applied to historical sources written during the time of the Roman Empire. A comparison, then, is proposed. It would be beneficial to compare the sources that detailed the life and ministry of Jesus of Nazareth, namely, Matthew, Mark, Luke, and John and the four sources which chronicled the life of Tiberius, emperor of the Roman Empire during the Ministry of Jesus. How do the sources compare as to their composition in proximity to their subject? Do the sources agree with one another? Is there a level of objectivity in the sources that allowed them to present the correct details of their subject? These questions will determine the reliability of the documents in question and whether the four Gospels measure up to critical examination

A Minimal Poset Resolution of Stable Ideals

We use the theory of poset resolutions to construct the minimal free resolution of an arbitrary stable monomial ideal in the polynomial ring whose coefficients are from a field. This resolution is recovered by utilizing a poset of Eliahou-Kervaire admissible symbols associated to a stable ideal. The structure of the poset under consideration is quite rich and in related analysis, we exhibit a regular CW complex which supports a minimal cellular resolution of a stable monomial ideal.Comment: 25 pages, 2 figure

Optimal inference in a class of regression models

We consider the problem of constructing confidence intervals (CIs) for a linear functional of a regression function, such as its value at a point, the regression discontinuity parameter, or a regression coefficient in a linear or partly linear regression. Our main assumption is that the regression function is known to lie in a convex function class, which covers most smoothness and/or shape assumptions used in econometrics. We derive finite-sample optimal CIs and sharp efficiency bounds under normal errors with known variance. We show that these results translate to uniform (over the function class) asymptotic results when the error distribution is not known. When the function class is centrosymmetric, these efficiency bounds imply that minimax CIs are close to efficient at smooth regression functions. This implies, in particular, that it is impossible to form CIs that are tighter using data-dependent tuning parameters, and maintain coverage over the whole function class. We specialize our results to inference on the regression discontinuity parameter, and illustrate them in simulations and an empirical application.Comment: 39 pages plus supplementary material