I am considering the next polynomial regression model, i.e $X_i\in \mathbb{R}$ $$Y_i=f(X_i)+\epsilon_i$$
with $f(x)=\sum_{i=1}^{d+1}\theta_ix^{j-1}$ and $\theta\in\mathbb{R^{d+1}}$. I need to calculate the Gauss Markov Estimator of $$\frac{1}{n}\sum_{i-1}^{n}f'(X_i)$$ in the form of $\Psi^T\theta$ with suitable vectors $\psi\in\mathbb{R^{d+1}}$.
Well, according to this, I have the next example. Supposing that I am interested in determine the Gauss Markov Estimator of the derivative of $f(x)=\sum_{i=1}^{d+1}\theta_ix^{j-1}$, the is easy to find the gauss markov estimator, I can take:
$$\psi=(0,1,2x,...,dx^{d-1})$$
And if I multiplicate $\psi^T\theta=f'(x)$.
Now, in order to describe the Gauss Markov estimator for the desired function, I analyzed the next:
$$\frac{1}{n}\sum_{i-1}^{n}f'(X_i)=f´(\frac{1}{n}\sum_{i=2}^{d+1}X_i)$$, it means that I need to describe the Gauss Markov Estimator of the derivative of the mean of the vector $X_i$.
Now if I take a a fixed location of $X$, $x_i\in\mathbb{R}$. I take the next estimator:
I will supposse that $Y_i=\frac{1}{n}\sum_{i=1}^{n}X_i$ then I need to describe the estimator of $f´(y)$, with $y$ a fixed point such that $y=\sum_{i=1}^{n}x$, but I don´t understand what to do now, I think that $y=\sum_{i=1}^{n}x$ doesn´t have sense, because $x$ will be constant.
How should I describe the estimator Gauss Markov Estimator of the function given?