The paper considers two examples in the two sample normal problem and finds noniformative priors which satisfy (i) a criterion of matching asymptotically the posterior distribution function of a suitably normalized real-valued parameter of interest with the corresponding frequentist distribution function, and (ii) avoid the marginalization paradox. These two considerations lead to unique priors within certain classes for the two problems under consideration. In the first example, the prior turns out to be the one-at-a time reference prior of Berger and Bernardo (1992a, 1992b), while in the second example, the prior is outside the class of reference priors.