In this paper, following a very recent and new approach of [1], we further generalize recently introduced summability methods in [13] and introduce new notions, namely, I-statistical convergence of weight and I-lacunary statistical convergence of weight , where $g\colon \mathbb N \rightarrow [0, \infty)$ is a function satisfying $\lim_{n\rightarrow \infty} g(n) = \infty$ and $\frac {n}{g(n)}\not\rightarrow 0$ as $n \rigtharrow 0$, for sequences of sets. We mainly investigate their relationship and also make some observations about these classes. The study leaves a lot of interesting open problems.