I'm an undergraduate student who is green in statistics. I have a problem in the chose of objective function when estimating the parameters.
Let $Y = \beta^TX + \epsilon $ be the standard liner regression model, where $Y$ is the response, $X$ the $p$-dimensional covariate vector with $\beta$ the corresponding regression parameter vector and $\epsilon$ the error term that is independent of covariate. And let $Y_i$, $X_i$, $i = 1, 2, ..., n$, be the observations.
Some authors define an objective function as $$ U_n(\beta) = \sum_{i=1}^{n}\sum_{j=1}^{n}|e_i(\beta)-e_j(\beta)|, $$ where $e_i(\beta) = Y_i - \beta^TX_i$, the residual of $i$th observation.
Like OLS, $\hat\beta$, the fitting value of $\beta$, is the minimiser of the objective function $U_n(\beta)$. In OLS method, the meaning of objective function is clear. It is the sum of "absolute errors" (absolute values of residuals).
But what's the meaning of $U_n(\beta)$ here, and why authors choose this function?
