I'm reading a paper about linear regression and in some point they define: $$ w_{ij}=\frac{h^{2}_{ji}}{ph_{ii}(1-h_{jj})^{2}}$$, where $h_{ij}$ are the elements of the hat matrix.
The problem is that they propose a bound for $w_{ij}$ and is not clear why the bound and the approximation are valid. $$w_{ij} \leq \frac{h_{jj}}{p(1-h_{jj})^{2}} \simeq \frac{1}{p}h_{jj}(1+2h_{jj})$$
We know that $$h_{ii}= h^{2}_{ii} + \sum_{i \neq j} h^{2}_{ij}$$, so $h_{ii} \ge h_{ij}^{2} $, but the bound proposed imply that, $ h_{ii}h_{jj}\ge h_{ij}^{2}$. ¿ that's valid?.
Please. i hope your help
Yes, $h_{ii}h_{jj}\geq h_{ij}^2$ is true. Note that $h_{ii} = \mathbf{e}_i^T H \mathbf{e}_{i}$, $h_{jj} = \mathbf{e}_j^T H \mathbf{e}_{j}$, and $h_{ij} = \mathbf{e}_i^T H \mathbf{e}_{j}$, where $\mathbf{e}_{k}$ refers to the $k$-th standard basis vector of $\mathbb{R}^{n}$ and $H$ is your hat matrix. Since $H$ is symmetric positive semi-definite, the result follows by using the Cauchy-Schwarz inequality (see Is this equivalent to Cauchy-Schwarz Inequality?).