@staticmethod
def ftilde(logx, settings):
    """
        Helper function for finding the analytic value of the estimator.
        See "check_by_integrating" docstring for more details.

        ftilde(X) is mean of f(theta) on the iso-likelihood contour
        L(theta) = L(X).
        """
    r = settings.r_given_logx(logx)
    return r ** 2 / settings.n_dim
