  The original frequentist approach for computing confidence intervals involves
the construction of the confidence belt which provides a mapping of the
observation in data into a subset of values for the parameter. There are
different prescriptions for constructing the confidence belt, here we use the
one provided by Feldman and Cousins. Alternative methods based on the
frequentist idea exist, including the delta likelihood method, the $CL_s$
method and a method here referred to as the $p$-value method, which have all
been commonly used in high energy experiments. The purpose of this article is
to draw attention to a series of potential problems when applying these
alternative methods to the important case where the predicted signal depends
quadratically on the parameter of interest, a situation which is common in high
energy physics as it covers scenarios encountered in effective theories. These
include anomalous Higgs couplings and anomalous trilinear and quartic gauge
couplings. It is found that the alternative methods, contrary to the original
method using the confidence belt, encode the goodness-of-fit into the
confidence intervals and potentially over-constrain the parameter.
