Double-blind randomized controlled trials are traditionally seen as the gold standard for causal inferences as the difference-in-means estimator is an unbiased estimator of the average treatment effect in the experiment. The fact that this estimator is unbiased over all possible randomizations does not, however, mean that any given estimate is close to the true treatment effect. Similarly, while predetermined covariates will be balanced between treatment and control groups on average, large imbalances may be observed in a given experiment and the researcher may therefore want to condition on such covariates using linear regression. This article studies the theoretical properties of both the difference-in-means and OLS estimators conditional on observed differences in covariates. By deriving the statistical properties of the conditional estimators, we can establish guidance for how to deal with covariate imbalances.

No abstract uploaded!

We introduce two exponential-type integrators for the “good” Boussinesq equation. They are of orders one and two, respectively, and they require lower spatial regularity of the solution compared to classical exponential integrators. For the first integrator, we prove first-order convergence in H^r for solutions in H^{r+1} with r>1/2. This new integrator even converges (with lower order) in H^r for solutions in H^r, i.e., without any additional smoothness assumptions. For the second integrator, we prove second-order convergence in H^r for solutions in H^{r+3} with r>1/2 and convergence in L^2 for solutions in H^3. Numerical results are reported to illustrate the established error estimates. The experiments clearly demonstrate that the new exponential-type integrators are favorable over classical exponential integrators for initial data with low regularity.

No abstract uploaded!

No abstract uploaded!