Small signal with background: objective confidence intervals and regions for physical parameters from the principle of maximum likelihood

It is argued that the choice between one-sided and two-sided confidence intervals must be made according to a rule prior to and independent of the data and it is shown that such a rule was found in principle by a statistician about half a century ago. The novel problem with unphysical estimates of a parameter in presence of background is solved in the realm of classical statistics by applying this rule and the principle of maximum likelihood. Optimal confidence intervals are given for the measurement of a bounded magnitude with normal errors, most effective in discriminating a signal next to the bound, and it is shown how to get them in any single case for a bounded discrete variable with background, in general and specifically for Poisson and binomial variables, with two examples of application. The upper limit provided by this method, when the data are consistent with no signal, does not decrease with unphysical estimates going far off the physical values, so removing the last claimed support of Bayesian inference in physics. Procedures are given extending the method to several parameters.