## Confidence Intervals in the Presence of Nuisance Parameters

We consider the problem of constructing confidence intervals in the presence of nuisance parameters. In the first part of the talk we discuss generalizations of the unified method of Feldman and Cousins (1998) with nuisance parameters. We demonstrate our methods with an important example arising in High Energy Physics, namely, the Poisson signal plus noise model.

In the second part of the talk we study an important extension of the previous model, namely, the marked signal plus noise model (also known as the two-component mixture model in the statistics literature), which has applications in Astronomy and multiple testing problems. We consider estimation and inference in this model where the distribution of one component is completely unknown. We develop methods for estimating the mixing proportion and the unknow distribution nonparametrically, given i.i.d. data from the mixture model. We use ideas from shape restricted function estimation and develop "tuning parameter-free" estimators that are easily implementable and have good finite sample performance. Distribution-free finite sample lower confidence bounds are developed for the mixing proportion.