Suppose we maximize the following program: \begin{equation} \mathbf{v}_1^* = \arg\max_{\mathbf{v}} \sum_{\mathbf{x}} 1_{\{S(\mathbf{x})\geq \gamma_1\}} f(\mathbf{x;u})\ln f(\mathbf{ x; v}), \end{equation} where $f(\cdot; \mathbf{v})$ is a Natural Exponential Family (NEF) parametrized by its mean vector $\mathbf{v}$. Suppose further, \begin{equation} \mathbf{v}_2^* = \arg\max_{\mathbf{v}} \sum_{\mathbf{x}} 1_{\{S(\mathbf{x})\geq \gamma_2\}} f(\mathbf{x;u})\ln f(\mathbf{x; v}), \end{equation} where $\gamma_2>\gamma_1$. In general, $\gamma_i>\gamma_{i-1}, \; \forall i>1$. Moreover, we know that,
\begin{equation} \sum_{\mathbf{x}}1_{\{S(\mathbf{x})\geq \gamma_i\}}f(\mathbf{x;u}) \leq \sum_{\mathbf{x}}1_{\{S(\mathbf{x})\geq \gamma_i\}}f(\mathbf{x;v_i^*}), \; \forall i=1,2,\dots,n. \end{equation}
where $\{\gamma_i\}_{i=1}^{n}$ is strictly increasing in $i$.
I need to prove that,
\begin{equation} \sum_{\mathbf{x}} 1_{\{S(\mathbf{x})\geq \gamma_i\}}f(\mathbf{x;v_i^*}) \geq \sum_{\mathbf{x}} 1_{\{S(\mathbf{x})\geq \gamma_i\}}f(\mathbf{x;v_{i-1}^*}) \; \forall i=1,2,\dots, n. \end{equation}