Maximizing the likelihood over the truncated support always leads to strictly greater probability on the truncated region than original pdf?

[kullbach_liebler]

Suppose ##\mathbf{X}## is a random variable with a finite support ##\Omega## and with some pdf ##f(\cdot; \mathbf{v}_0)## where ##\mathbf{v}_0## is the parameter vector. Define, ##\mathcal{A}:= \{\mathbf{x}:S(\mathbf{x}) \geq \gamma\} \subset \Omega## and ##\tilde{\mathbf{x}}:=S(\tilde{\mathbf{x}}) = \gamma##, ##\mathcal{B}:=\{\mathbf{x} \geq \tilde{\mathbf{x}}\}## where ##S:\mathbf{x} \mapsto \mathbb{R}^n##. Moreover, suppose that,

$$\#(\mathcal{B} \cap \mathcal{A}) > \#(\neg \mathcal{B} \cap \mathcal{A})$$
and
$$!\exists \mathbf{x}^*>\tilde{\mathbf{x}}:=\arg \max_{\mathbf{x}}S(\mathbf{x}).$$

Then, it implies that,

\begin{align}
\max_{\mathbf{v}}\sum_{\mathbf{x} :S(\mathbf{x}) \geq \gamma} f(\mathbf{x};\mathbf{v}) > \sum_{\mathbf{x} :S(\mathbf{x}) \geq \gamma} f(\mathbf{x};\mathbf{v}_0)
\end{align}

My questions:

1) Is the statement true?;
2) How could I improve the presentation of this proposition?;
3) What are the mildest possible conditions under which (1) will hold?

 

[Stephen Tashi]

2) How could I improve the presentation of this proposition?;

I don't know what a statistical journal would want, but for the purpose of getting an answer on an internet forum, you could give a more verbal statement of the proposition before presenting it using only notation.

\begin{align}
\max_{\mathbf{v}}\sum_{\mathbf{x} :S(\mathbf{x}) \geq \gamma} f(\mathbf{x};\mathbf{v}) > \sum_{\mathbf{x} :S(\mathbf{x}) \geq \gamma} f(\mathbf{x};\mathbf{v}_0)
\end{align}

This appears to say the maximum value of a function ( which is a summation rather than an integration) when taken over a set is greater than the value of that function evaluated at the particular element ##v_o## in that set. Is that the general idea?

 

[pbuk]

1) Is the statement true?

Let ## \mathbf{v_max} ## be a value of ## \mathbf{v} ## which maximizes the sum. What happens when ## \mathbf{v_0} = \mathbf{v_max} ##?