# A different objective function in liner regression analysis

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?

• A link or a reference would help – Matt F. Feb 8 '17 at 13:23
• The integration of the so called Mann-Whitney-type estimating equation is $U_n(\beta)$. (p5, Z. Ying, W. Yu, Z. Zhao, M. Zheng, Regression analysis of doubly truncated data, preprint, [link]https://arxiv.org/abs/1701.00902) – R. Qian Feb 8 '17 at 14:03
• That paper suggests that it is considering a variant of a minimand from a 2001 paper (columbia.edu/~zj7/resampling.pdf), so the answer should be in that 2001 paper....but I can't actually see the connection between the two papers. In any case, the scientific motivation seems to be the double truncation of quasar data: quasars too dim can't be observed well, and quasars too bright look too much like other objects. – Matt F. Feb 8 '17 at 15:26