After reading this post, I still don't thoroughly understand what an estimator is. Suppose the samples $D_i={(x_1,y_1),...(x_n,y_n)}$ are drawn randomly from function $$f(x)=sin(2\pi x),$$ so my ultimate goal is to come up with an estimated function $h(x)$, which should approximate $f$ as close as possible, right? Since I don't know what the true $f$ looks like, so I might choose different models to do the approximation, here I consider the linear regression model, which is $$h(x)=\beta^Tx,$$ then I try to estimate the model parameters $\beta$ via OLS, and finally given training sample $D_i$, I have the estimates $\hat\beta_{|D_i}$, and absolutely dffierent $\hat\beta$ for different training set $D_i$.
Here is my problem,
What is the estimator here? According to the post I read, I think that the process of choosing the linear regression model and estimating $\hat\beta$ via OLS are components of the estimator, right?
How to check the estimator is biased or not? Since I chose the linear regression model $h(x)$, and now have the estimates $\hat\beta$, I think to check the estimator is biased or not, is to find out $$B(\hat\beta)=\beta_{true}-E[\hat\beta],$$ right? Now this is the thing confuses me badly, since we don't have this $\beta_{true}$, because the true $f$ is actually $sin(\theta x)$ with $\theta=2\pi$, we only have $\theta_{true}$, no such $\beta_{true}$, right?
Hope you can help me to get things right.
UPDATE
To address @whuber's comment, my question originates from this lecture note (from page 11)
1.how do you "randomly draw" values from a function?
Let's say the target function $f(x)=sin(2\pi x)$ is defined on $x\in [-1,1]$, I could randomly (via a uniform distribution over $[-1,1]$) draw $N$ points $\{x_1,...,x_N\}$, where $N$ is the size of the sample, after that I could compute the output value $y_i=f(x_i)$, no noise added. Now the sample is $\{(x_1,y_1),...,(x_N,y_N)\}$.
2.are either (or both) of the $xi$ or $yi$ measured? what is the expected statistical behavior of the measurement errors?
Here I only measure the training/test error of output value $y_i$. I mean, with different training sample, I have different $\hat\beta$, then for a fixed point $x_0$, the predicted output $\hat y_0=h(x_0)=\beta^Tx_0$ will be different, but the true value is $y_0=f(x_0)$.
3.why are you choosing a linear estimator β′x to approximate a function sin(2πx) that is a fortiori strongly nonlinear?
As what is done in the lecture note, what I have is just the sample, which looks nonlinear, maybe it's not wise to choose a linear estimator, and maybe I should choose this one $$h(x)=\beta_0+\beta_1x+\beta_2x^2$$, a polynomial estimator, but it's still far from the target $sin(2\pi x)$.