A linear adjustment for more efficient versions of the Bernstein-type esimators proposed in Pallini (2005) is asymptotically studied, as the sample size becomes increasingly large. The Bernstein-type approximation in Pallini (2005) originate from the well-known definitions of univariate and multivariate Bernstein polynomials in the theory of approximation of continuous functions. The Bernstein-type approximations are defined through a constructive coefficient that conveniently characterizes the velocity of their uniform convergence. The statistical estimation of a smooth function of population mrans is considered, with conclusions about the asymptotic behaviour of the linear adjustment, as the sample size becomes increasingly large. Monte Carlo simulations on the ratio of means example are detailed and discussed.