This paper assumes that cause-effect relationships in the process can be described by a linear structural equation model and the corresponding directed acyclic graph. When unmeasured confounders exist, we consider a situation where the total effect can be estimated by both the instrumental variable estimator and the front-door estimator. In this situation, there is no qualitative superiority/inferiority relationship between these two estimators in terms of the estimation accuracy of the total effects. Taking this into account, in order to estimate the total effect with better estimation accuracy, we propose a novel integrated estimator based on these two estimators. In addition, through numerical experiments, from the viewpoint of the estimation accuracy of the total effects, we show that (i) the integrated estimator is better than the individual estimators, and (ii) in some situations, the integrated estimator is better than the back-door estimator even when the back-door estimator is better than the individual estimators.

Construction of the confidence intervals for the proportion parameter of Binomial population is one of the fundamental problems in the theory of the statistical inference. As for the Jeffreys interval, one of the widely used intervals by Bayesian approach, it is pointed out that the coverage probability is sometimes far less than the confidence level 1-α. In this paper, we consider the confidence intervals by the hierarchical Bayesian models. We introduce hyper prior distributions for the hyper parameters of the prior for the binomial population, and construct confidence intervals by the Markov chain Monte Carlo calculations. We find that the performance of our proposed interval is better than Jeffreys interval in some settings.