In a heteroscedastic linear model, it is known that if the variances are a parametric function of the design, then one can construct an estimate of the regression parameter which is asymptotically equivalent to the weighted least squares estimate with known variances. We show that the same is true when the only thing known about the variances is that they are determined by an unknown but smooth function of the design or the mean response.
"Adapting for Heteroscedasticity in Linear Models." Ann. Statist. 10 (4) 1224 - 1233, December, 1982. https://doi.org/10.1214/aos/1176345987