In this paper, we first introduce two new entropy estimators. These estimators are obtained by correcting Corea(1995)'s estimator in the extreme points and also assigning different weights to the end points.We then make a comparison among our proposed new entropy estimators and the entropy estimators proposed by Vasicek (1976), Ebrahimi, et al. (1994) and Corea(1995). We also introduce goodness of fit tests for exponentiality and normality based on our proposed entropy estimators. Results of a simulation study show that the proposed estimators and goodness of fit tests have good performances in comparison with the leading competitors.

In this paper the class of all equivariant is characterized functions. Then two conditions for the proof of the existence of equivariant estimators are introduced. Next the Lehmann's method is generalized for characterization of the class of equivariant location and scale function in terms of a given equivariant function and invariant function to an arbitrary group family. This generalized method has applications in mathematics, but to make it useful in statistics, it is combined with a suitable function to make an equivariant estimator. This of course is usable only for unique transitive groups, but fortunately most statistical examples are of this sort. For other group equivariant estimators are directly obtained.