User:Amyxz/sandbox

Not keeping the FCR means $FCR>q$ when $q={\frac {V}{R}}={\frac {\alpha *m_{0}}{R}}$, where $m_{0}$ is the number of true null hypotheses and $R$ is the number of rejected hypothesis. Intervals with simultaneous coverage probability $1-q$ can control the FCR to be bounded by $q$.

Problems

Selection

Selectioncauses reduced average coverage. Selection can be presented as conditioning on an event defined by the data and may affect the coverage probability of a CI for a single parameter. Equivalently, the problem of selection changes the basic sense of P-values which is solved with FDR procedures. Solution: the goal of conditional coverage following any selection rule for any set of (unknown) values for the parameters is impossible to achieve. A weaker property when it comes to selective CIs is possible and will avoid false coverage statements. FCR is a measure of interval coverage following selection. Therefore, even though a 1−α CI does not offer selective (conditional) coverage, the probability of constructing a no covering CI is at most α, where

$Pr[\theta \in CI,CIconstructed]\leq Pr[\theta \in CI]\leq \alpha$

Selection and Multiplicity

When facing both multiplicity (inference about multiple parameters) and selection, not only is the expected proportion of coverage over selected parameters at 1−α not equivalent to the expected proportion of no coverage at α, but also the latter no longer can be ensured by constructing marginal CIs for each selected parameter. Solution: taking the expected proportion of parameters not covered by their CIs among the selected parameters, where the proportion is 0 if no parameter is selected. This false coverage-statement rate (FCR) is a property of any procedure that is defined by the way in which parameters are selected and the way in which the multiple intervals are constructed.