When we apply the item response models to the responses to estimate ability, we always assume that the responses are true, without guessing, cheating or carelessness. However, it is not the case in the real world. The responses always consist of various disturbances. Consequently, the maximum-likelihood estimator may produce biased ability estimation and the sampling variability because of the incorrect likelihood function.
Two robust approaches were investigated by downweighting the effects of aberrant observations (guessing and random answer) to the log likelihood. The bisquare approach (in BILOG) eliminates the observation entirely from the estimation and thus only makes use of valid the response. However, the approach are prone to produce negative infinity estimates for unusual response when the correct answers are sparse. Also, it may encounter nonconvergence problem.
The Huber approach, by contrast, is more practical and realistic in dealing with the unusual response. It does not downweight the response at all for the response within a certain range which could be treated as the normal response. In addition, it does not downweighte the doubted response as severe as the bisquare approach. Therefore, the Huber approach is believed to be a compromise between the maximum-likelihood and the bisquare approach.
Questions:
(1) The decision of the turning constant (B and H) in the Huber approach is arbitrary. What are the arguments for the values?
(2) The rational of Mallows-type weights seems counterintuitive.