Chapter 2Maximum concentration statistics
The goal of this chapter is to present general results on maximum concentration (MC) statistics for the one-dimensional parameter. The multidimensional case is discussed in Chapter 6. First, we formulate the necessary assumptions. Then we introduce the minimum length (short) two-sided confidence interval (CI) and the implied MC estimator as the limit point when the confidence level goes to zero. In the spirit of MC-statistics, we derive the density level (DL) test as the test with the short acceptance interval. Finally, we discuss the efficiency and its connection with sufficient statistics. We use the term maximum concentration statistics because the MC estimator is where the concentration probability of the coverage of the true parameter reaches its maximum.
2.1 Assumptions
Our M-statistics relies on the existence of statistic 
having cdf 
known up to unknown one-dimensional parameter 
. Here we assume that 
is a continuous random variable. Moreover, we assume that the pdf is positive for all and unimodal for every . In the following sections, ...
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