In this paper, loss is taken to be proportional to squared error with the constant of proportionality equal to the square of the inverse of a scale parameter, and an invariant estimator is defined to be one with risk invariant under transformations of location and scale. For certain classes of estimators, best (minimum-mean-squared-error) invariant estimators are found for specified linear functions of an unknown scale parameter and one or more unknown location parameters. Even when the specified function is equal to a single location parameter, the best invariant estimator is not equal to the best unbiased estimator in the class except for complete samples from certain distributions such as the Gaussian.
Nancy R. Mann. "Optimum Estimators for Linear Functions of Location and Scale Parameters." Ann. Math. Statist. 40 (6) 2149 - 2155, December, 1969. https://doi.org/10.1214/aoms/1177697292