Let the stationary stochastic process X (t) with unknown mean m and covariance B(t) = E (x (s)-m) (x (s+t)--m) be observed on the interval 0 ≤ t ≤ T. In this paper we shall give a formula for the mean square error (m.s.e.) of an (estimator B2(0) of B (0) whose distribution function W(t) belongs to the class of functions with bounded variations and has two equal jumps at the terminal values of the interval [0, T]. A comparison between this estimate and an Equi distributed one (B1 (0)) will be considered to show that B2 (0) is better than B1 (0) (in the sense of the least m.s.e.). To illustrate this, we shall consider some examples of covariance functions of stationary processes having rational spectral density. This is not a great restriction since any spectral density can be approximated as a sum of rational spectral densities.