S-estimator

The goal of S-estimators is to have a simple high-breakdown regression estimator, which share the flexibility and nice asymptotic properties of M-estimators. The name 'S-estimators' was chosen as they are based on estimators of scale. The goal of S-estimators is to have a simple high-breakdown regression estimator, which share the flexibility and nice asymptotic properties of M-estimators. The name 'S-estimators' was chosen as they are based on estimators of scale. We will consider estimators of scale defined by a function ρ {displaystyle ho } , which satisfy R1 - ρ {displaystyle ho } is symmetric, continuously differentiable and ρ ( 0 ) = 0 {displaystyle ho (0)=0} . R2 - there exists c > 0 {displaystyle c>0} such that ρ {displaystyle ho } is strictly increasing on [ c , ∞ [ {displaystyle [c,infty [} For any sample { r 1 , . . . , r n } {displaystyle {r_{1},...,r_{n}}} of real numbers, we define the scale estimate s ( r 1 , . . . , r n ) {displaystyle s(r_{1},...,r_{n})} as the solution of 1 n ∑ i = 1 n ρ ( r i / s ) = K { extstyle {frac {1}{n}}sum _{i=1}^{n} ho (r_{i}/s)=K} , where K {displaystyle K} is the expectation value of ρ {displaystyle ho } for a standard normal distribution. (If there are more solutions to the above equation, then we take the one with the smallest solution for s; if there is no solution, then we put s ( r 1 , . . . , r n ) = 0 {displaystyle s(r_{1},...,r_{n})=0} .)