I am trying to understand the idea of Loss functions For Regression Task perfectly.

I have read many textbooks and articles, and I came up with questions related to this subject.

Several different uses of loss functions can be distinguished.

a) In prediction problems: a loss function depending on predicted and observed value defines the quality of a prediction.

b) In estimation problems: a loss function depending on the true parameter and the estimated value defines the quality of estimation.

c) Many estimators (such as least squares or M-estimators) are defined as optimizers of certain loss functions which then depend on the data and the estimated value.

*Now, since my focus is on Loss Functions For Regression Task*

My questions are as follows.

1- Should I write the loss function formula as a function of the parameter or of the variables (L($\theta-\hat\theta))$ OR (L($y-\hat\y))$?

2- Should I consider the Loss function formula for one point or Not (with sums or not)?

**Note**

My thought is to introduce Loss function first and then to use the standard notation for all Loss functions (least square, absolute value and Huber Loss, Quntile Loss and so on).

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