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- estimators and OLS

Hello everyone,

I am trying to close the loop on this important topic of estimators.

An estimator is really just a function to calculate point statistics that are close estimates (with low variance) of population parameters. For example, given a set of data, we can compute the mean and the mode. Both the mean and the median are estimators of the center of the data (the mean is better).

I am trying to close the loop on this important topic of estimators.

An estimator is really just a function to calculate point statistics that are close estimates (with low variance) of population parameters. For example, given a set of data, we can compute the mean and the mode. Both the mean and the median are estimators of the center of the data (the mean is better).

- My understanding is that in the case of linear regression, if the key assumptions are met (assumptions mostly about how residuals behave), we can use OLS to find the "best" regression sample coefficients that approximates the regression population coefficients. OLS, if the conditions are met, represent the best, linear, unbiased estimator (BLUE). In the case of linear regression, we have OLS estimators, i.e. a functions to calculate the sample intercept, the sample slope, and the sample correlation coefficient....We could also solve for those same coefficients using Maximum Likelihood (ML), which is another important estimator, but the estimated coefficients would not be the "best" in that case, correct?
- OLS is BLUE, assuming the assumptions are met, as long as the model is "linear in the parameters" so OLS would give the best estimates also for polynomial regression models like ##Y=\beta_0 + \beta_1 X+ \beta_2 X^2##, correct? But would we need to first "convert" the polynomial data into linear data, before applying OLS, taking the ##\sqrt() ## of the independent variable ##X## so the data in the scatterplot follows some sort of linear trend? If the untransformed data follows a quadratic model, the OLS assumptions about the residuals are not met so the OLS estimators would not perform best... So in what sense is OLS generally applicable to linear models?
- What are other commonly used estimators? Maximum likelihood is the important estimator used to estimate coefficients for generalized linear models...Which other estimator is important to know? I read about weighted least-squares which I guess is a variant of OLS