MLR

library(unbiasedgoodness)

Experiments that were not performed are irrelevant Models that were not selected? https://en.wikipedia.org/wiki/Conditionality_principle

https://en.wikipedia.org/wiki/Hypothetico-deductive_model

The biggest obstacle for us to say that we have novel theory

cite{MostPowerMS} already defines selective unbiasedness, Lehmann and Scheffe UMPU theory for exponential families

F-distribution is not an exponential family

So what about F, and other tests that are not covered by the selective version of Lehmann and Scheffe theorem?

Part of our theoretical contribution: showing how to apply some broader conditions including monotone likelihood ratio (MLR) and stochastic ordering relations to establish selective unbiasedness

Includes as special cases the (non-central) t- and F-tests, justifying for our goodness-of-fit procedures

TBD: Selective Karlin-Rubin theorem?

Stochastic ordering and selective unbiasedness

shaked2007stochastic remark after Theorem 1.C.6 (page 45) shows that if stochastic order is preserved for all measurable truncations then the original (non-truncated) random variables are likelihood ratio ordered

shaked2007stochastic

Monotone likelihood ratio and selective UMP

shaked2007stochastic Theorem 1.C.6 (page 45) shows likelihood ratio ordering is preserved by general truncations (to Borel measurable sets)