On construction of smallest one-sided confidence intervals for the response rate in adaptive two- or multi-stage designs

To assess the effectiveness of a treatment in phase II clinical trial for cancer study, an adaptive multi-stage design, especially a two-stage one, is commonly used. This type of design allows an on-going study to end early for the participant's safety and design's efficiency. Since a large value of the response rate p for the treatment is wanted, lower one-sided confidence intervals for p are of interest. Due to the limited sample size, exact intervals with a guaranteed confidence level are derived using a rank function that is based on the Clopper–Pearson lower confidence limit. When the sample size in stage 2 is a constant, two kinds of smallest intervals are constructed with or without using the sufficiency principle. The proposed intervals outperform the existing exact intervals, and the intervals not based on minimal sufficient statistic are recommended for practice due to their small expected lengths. When the sample size in stage 2 varies, the smallest interval is also proposed.

Curved Exponential Models in Econometrics

Curved exponential models have the property that the dimension of the minimal sufficient statistic is larger than the number of parameters in the model. Many econometric models share this feature. The first part of the paper shows that, in fact, econometric models with this property are necessarily curved exponential. A method for constructing an explicit set of minimal sufficient statistics, based on partial scores and likelihood ratios, is given. The difference in dimension between parameterand statistic and the curvature of these models have important consequences for inference. It is not the purpose of this paper to contribute significantly to the theory of curved exponential models, other than to show that the theory applies to many econometric models and to highlight some multivariate aspects. Using the methods developed in the first part, we show that demand systems, the single structural equation model, the seemingly unrelated regressions, and autoregressive models are all curved exponential models.