2 Appendix

2.1 Statistical criteria for location of measures

Least Squares (LS) criterion: Let $X_1, X_2, \dots, X_n$ be a sample and define $$SS(c) = \sum_{i=1}^n (X_i-c)^2.$$ The value of $c$ which minimizes $SS(c)$ is the mean, that is, ${c} = \overline{X}.$

Proof: $\sum_{i=1}^n (X_i-c)^2 = \sum_{i=1}^n (X_i^2 - 2cX_i + c^2) = \sum_{i=1}^n X_i^2 - 2c\sum_{i=1}^n X_i + nc^2$, which has a minimum at $c=2\sum_{i=1}^n X_i/2n = \overline{X}.$

Least Absolute Distance (LAD) criterion: Let $X_1, X_2, \dots, X_n$ be a sample and define $$LAD(d) = \sum_{i=1}^n |X_i-d|.$$ The value of $d$ which minimizes $LAD(D)$ is the median, that is, ${d} = \tilde{X}.$