Improving Reproducibility Probability estimation, preserving RP-testing

Reproducibility Probability (RP) estimation is improved in a general parametric framework, which includes Z, t, χ2, and F tests. The preservation of RP-testing (i.e. RP estimation based significance testing with threshold at 1/2) is taken into account. Average conservative, weighted conservative, uninformative Bayesian, and Rao-Blackwell RP estimators are introduced, and their relationship studied. Several optimality criteria to define the parameters of weighting functions of conservative RP estimators are adopted. RP-testing holds for average conservative estimators and, under mild conditions, for weighted conservative ones; uninformative Bayesian and Rao-Blackwell RP estimators perform RP-testing only under the location shift model. The performances of RP estimators are compared mainly through MSE. The reduction of MSE given by average conservative estimators is, on average, higher than 20%, and can reach 35%. The performances of optimal weighted RP estimators are even better: on average, the reduction of MSE is higher than 30%