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, \({d} = \tilde{X}.\)