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Quantiles Equivariance

It is widely claimed that the quantile function is equivariant under increasing transformations. We show by a counterexample that this is not true (even for strictly increasing transformations). However, we show that the quantile function is equivariant under left continuous increasing transformations. We also provide an equivariance relation for continuous decreasing transformations. In the case that the transformation is not continuous, we show that while the transformed quantile at p can be arbitrarily far from the quantile of the transformed at p (in terms of absolute difference), the probability mass between the two is zero. We also show by an example that weighted definition of the median is not equivariant under even strictly increasing continuous transformations.