Least absolute deviations estimate

Though the maximum likelihood principle is not less intuitive than the least squares method itself, it enables the statisticans to derive estimation criteria for any known distribution, and to generally prove that the estimates have nice properties such as asymptotic unbiasedness (ref. 1). In particular, the method of least absolute deviations introduced in Section 1.8.2 is also a maximum likelihood estimator assuming a different distribution for the error. 

Firstly, the krlglng estimator is optimal only for the least square criterion. Other criteria are known which yield no more complicated estimators such as the minimization of the mean absolute deviation (mAD), E P(2c)-P (3c), yielding median-type regression estimates. 

Li and Lin (2002) used a form of penalized least squares with the smoothly clipped absolute deviation penalty proposed by Fan and Li (2001). This method estimates the parameters, /3, by minimizing not the usual residual sum of squares, but... 

Another factor to be taken into account is the degree of over determination, or the ratio between the number of observations and the number of variable parameters in the least-squares problem. The number of observations depends on many factors, such as the X-ray wavelength, crystal quality and size, X-ray flux, temperature and experimental details like counting time, crystal alignment and detector characteristics. The number of parameters is likewise not fixed by the size of the asymmetric unit only and can be manipulated in many ways, like adding parameters to describe complicated modes of atomic displacements from their equilibrium positions. Estimated standard deviations on derived bond parameters are obtained from the least-squares covariance matrix as a measure of internal consistency. These quantities do not relate to the absolute values of bond lengths or angles since no physical factors feature in their derivation. 

---