| dc.description | In Pericchi and Pereira (2012)it is argued against the traditional way on which
testing is based on xed signi cance level, either using p-values (with xed levels of
evidence, like the 5% rule) or values. We instead, follow an approach put forward
by DeGroot (1975), on which an optimal test is chosen by minimizing type I and type
II errors.
Morris DeGroot in his authoritative book (1975), Probability and Statistics 2nd Edi-
tion, stated that it is more reasonable to minimize a weighted sum of Type I and Type
II error than to specify a value of type I error and then minimize Type II error. He
showed it beyond reasonable doubt, but only in the very restrictive scenario of simple
VS simple hypothesis, and it is not clear how to generalize it. We propose here a very
natural generalization for composite hypothesis, by using general weight functions in
the parameter space. This was also the position taken by Pereira (1985, 1993, 2010).
We show, in a parallel manner to DeGroot's proof and Pereira's discussion, that the
optimal test statistics are Bayes Factors, when the weighting functions are priors with
mass on the whole parameter space. On the other hand when the weight functions are
point masses in speci c parameter values of practical signi cance, then a procedure is
designed for which the sum of Type I error and Type II error in the speci ed points of
practical signi cance is minimized. This can be seen as bridge between Bayesian Statis-
tics and a new version | |
