This paper discusses a new superfamily of divergences that is similar in spirit to the S-divergence family introduced by Ghosh et al. (Bernoulli, 23, 2746–2783. 2017). This new family serves as an umbrella that contains the logarithmic power divergence family (Renyi 1961; Maji et al. RASHI, 2, 39–51. 2017) and the logarithmic density power divergence family (Jones et al. Biometrika, 88, 865–873. 2001) as special cases. Various properties of this new family and the corresponding minimum distance procedures are discussed with particular emphasis on the robustness issue; these properties are demonstrated both theoretically as well as through simulation studies. In particular the method demonstrates the limitation of the first order influence function in assessing the robustness of the corresponding minimum distance procedures. In this respect, we examine the necessity and usefulness of the third order influence functions for the divergence based test statistics, which is studied in the literature for the first time.
