Diagnosing holographic type dark energy models with the Statefinder hierarchy, composite null diagnostic and $w-w'$ pair

The main purpose of this work is to distinguish various holographic type dark energy (DE) models, including the $\Lambda$HDE, HDE, NADE and RDE model, by using various diagnostic tools. The first diagnostic tool is the Statefinder hierarchy, in which the evolution of Statefinder hierarchy parmeter $S^{(1)}_3(z)$ and $S^{(1)}_4(z)$ are studied. The second is composite null diagnostic (CND), in which the trajectories of $\{S^{(1)}_3, \epsilon\}$ and $\{S^{(1)}_4, \epsilon\}$ are investigated, where $\epsilon$ is the fractional growth parameter. The last is $w-w'$ analysis, where $w$ is the equation of state for DE and the prime denotes derivative with respect to $ln a$. In the analysis we consider two cases: varying current fractional DE density $\Omega_{de0}$ and varying DE model parameter $C$. We find that: (1) Both the Statefinder hierarchy and the CND have qualitative impact on $\Lambda$HDE, but only have quantitative impact on HDE. (2) $S_4^{(1)}$ can lead to larger differences than $S_3^{(1)}$, while the CND pair has a stronger ability to distinguish different models than the Statefinder hierarchy. (3) For the case of varying $C$, the $\{w, w'\}$ pair has qualitative impact on $\Lambda$HDE; for the case of varying $\Omega_{de0}$, the $\{w, w'\}$ pair only has quantitative impact; these results are different from the cases of HDE, RDE and NADE, in which the $\{w, w'\}$ pair only has quantitative impact on these models. In conclusion, compared with HDE, RDE and NADE, the $\Lambda$HDE model can be easily distinguished by using these diagnostic tools.