On $I$-statistical and $I$-lacunary statistical convergence of order $\alpha$

In this paper‎, ‎following a very recent‎ ‎and new approach‎, ‎we further generalize recently introduced‎ ‎summability methods‎, ‎namely‎, ‎$I$-statistical convergence and‎ ‎$I$-lacunary statistical convergence (which extend the important‎ ‎summability methods‎, ‎statistical convergence and lacunary‎ ‎statistical convergence using ideals of $\mathbb{N}$) and‎ ‎introduce the notions of $I$-statistical convergence of order‎ ‎$\alpha$ and $I$-lacunary statistical convergence of order‎ ‎$\alpha$‎, ‎where $0 < \alpha < 1$‎. ‎We mainly investigate their‎ ‎relationship and also make some observations about these classes‎ ‎and in the way try to give an answer to an open problem posed By‎ ‎Das‎, ‎Savas and Ghosal in 2011‎. ‎The study leaves a lot of‎ ‎interesting open problems‎.