On M-estimators of Approximate Quantiles and Approximate Conditional Quantiles

M-estimators introduced in Huber (1964) provide a class of robust estimators of a center of symmetry of a symmetric probability distribution which also have very high eeciency at the model. However it is not clear what they do estimate when the probability distributions are nonsymmetric. In this paper we rst show that in the case of arbitrary, not necessarily symmetric probabilty distributions, some M-functionals coincide with quantiles of a convolution of the parent distribution and a probability distribution determined by the M-function. Next, we consider linear combinations of two such M-functionals. These combinations reduce the contribution coming from the convoluting factor and related to the use of the M-functionals and provide good approximation of normal quantiles. We show that use of particular M-functions, which we call probit M-functions, results in recovering exactly the quantiles of the normal probability distribution. Our estimators of quantiles are the empirical functionals. We obtain asymptotic distributions of these estimators. Similarly to the case of estimation of the symmetry center by the classical M-estimators we show that very low asymptotic variances can be obtained so that the variance of the maximum likelihood estimator can be, for every p, approached arbitrarily close. Hence, when compared with the empirical quantiles, a considerable reduction of variance can be achieved in a vicinity of the normal distribution. However, in contrast to the maximum likelihood estimator, the new estimators are robust. They also have the Local Asymptotic Minimax (LAM) property, like M-estimators. In the nal part of the paper we show that applying our estimators locally and using the kernel method, one can obtain a class of M-estimators of quantiles of conditional distributions when the marginal probability distribution is close to the normal distribution.