A general class of estimators is developed for improving upon best scale invariant estimators of two or more arbitrary scale parameters or powers thereof for arbitrary positive distributions with sufficient moments under weighted squared error loss function. The technique is to compute the risk difference in terms of moments of the distribution. Some conditions are obtained under which the maximum improvement is possible, and the form of the estimator can be chosen to achieve this maximum along any specified ray. The result is then extended to the estimation of a linear transform of the parameter vector. Finally, some examples are given with numerical calculations to obtain the amount of risk improvement.