This article presents a new family of the estimating functions related with minimum distance estimations, and discusses its relationship to the family of the minimum density power divergence estimating equations. Two representative minimum distance estimations; the minimum $L_2$ distance estimation and the minimum Hellinger distance estimation are studied in the light of the theory of estimating equations. Despite of the desirable properties of minimum distance estimations, they are not widely used by general researchers, because theories related with them are complex and are hard to be computationally implemented in real problems. Hopefully, this article would be a help for understanding the minimum distance estimations better.