I'm currently exploring a problem related to a two-parameter estimator, and I've simplified it for ease of discussion, eliminating the need for an extensive probability background. The problem is as follows:
Consider the expression: $H := f\left(\frac{A_1 + A_2}{2} + \frac{B_1 + B_2}{2}\right) - f\left(\frac{A_1 + A_2}{2}\right) - \frac{1}{2}\left[f(A_1 + B_1) - f(A_1) + f(A_2 + B_2) - f(A_2)\right]$
where $f$ is a $C^{\infty}$function, and all of its derivatives are $\alpha$ Hölder continuous.
I am looking to expand $ f $ using Taylor's Expansion and simplify the result with the following conditions:
- Each term in the expansion should have the factor $B_1 - B_2 $.
- For terms that include the factor $f $, this factor should appear as "matching differences", such as $ f(A_1) - f(A_2) $, $ f(B_1) - f(B_2) $, $ f\left(\frac{A_1 + A_2}{2} + \lambda \frac{B_1 + B_2}{2}\right) - f(A_1 + \lambda B_1) $. The key objective here is to ensure that when applying the Hölder continuous property, $A_1,A_2,B_1,B_2$ appear in pairs $A_1 - A_2$ and $ B_1 - B_2 $ , rather than individually.
Additional notes:
- You may perform multiple expansions, even expanding terms within an expansion, to achieve the correct form.
- The use of an integral remainder is permissible, provided the integrand meets the above two criteria.
I'm curious if such an expansion is feasible and would greatly appreciate any guidance or insights on this matter. Thank you in advance for your time and help!