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I have used the "quantreg" package in R to find quantiles for my data. All my data, both predictors and responses are limited between 0 and 1, while a number of quantiles given by "rq" or "rqss" functions have values less than 0 or more than one. I was wondering if there is a solution for this problem, maybe a different function which gives truncated quantiles. Thanks

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Quantile regression is an additive model. No purely additive linear model can prevent predicted values from going outside any given range. You can either hope to find a transformation of $Y$ that does not have a limited range that will fit the model, or consider the use of ordinal regression. For example the R rms package orm function handles continuous $Y$ and allows estimation of quantiles, the mean, and exceedance probabilities

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Thanks to the property of equivariance to monotonic transformations, shared by all quantile methods, one can simply operate a logit transformation and run a linear quantile regression on the transformed outcome.

For further information you might consider referring to this paper on quantile regression for bounded outcomes (Bottai & McKeown 2010).

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