A stochastic process \(z(t)\) is observed over a given interval of time, where \(z(t) = x(t) + F(t,\xi)\), \(\xi\) is an unknown parameter, \(F(t,\xi)\) a real-valued function of time \(t\) and \(x(t)\) is a Gaussian process with known covariance function \(\psi_x(s, t)\). Estimates of a parameter \(f(\xi)\) are considered. \(z(t)\) is written in the form \(\sum z_\nu \psi_\nu(t) \chi_\nu^{-\frac12}\), where \(z_1, z_2,\ldots\) are observable coordinates which are independent and normally distributed with variance 1, and \(\psi_\nu\), and \(\chi_\nu\) are eigenfunctions and eigenvalues of the homogeneous integral equation with nucleus \(\psi_x(s,t)\). The object of the paper is for a chosen \(\xi_0\) to find greatest lower bounds for \(E [\varphi - f(\xi_0)]^2\) under variation of \(\varphi\) subject to \(E \varphi = f(\xi)\) for all \(\xi\). Application is made of a technique due to \textit{E. W. Barankin} (this Zbl. 34, 230) for construction of the lower bound. Two special cases are studied, (i) \(F(t,\xi)) = \alpha F(t - \tau)\) with amplitude \(\alpha\) and time-delay \(\tau\) subject to estimation, and (ii) \(F(t,\xi) = e^{\xi/2} F(e^\xi t)\) with the ``Doppler shift'' subject to estimation.