This article describes non-monotonic estimators of a location parameter from a noisy measurement Z = Ɵ + V when the possible values of e have the form (0, ± 1, ± 2,. . . , ± n}. If the noise V is Cauchy, then the estimator is a non-monotonic step function. The shape of this rule reflects the non-monotonic shape of the likelihood ratio of a Cauchy random variable. If the noise V is Gaussian with one of two possible scales, then the estimator is also a nonmonotonic step function. The shape this rule reflects the non-monotonic shape of the likelihood ratio of the marginal distribution of Z given Ɵ under a least-favorable prior distribution.