I have been given with a set of data, which is supposedly come from an unknown distribution F. And I am asked to propose a suitable parametric or nonmparametric estimator for quantiles q(α) =F⁻¹(α) for the series of α. From plotting the data as a histogram, I see that the data has a positively skewed distribution, but I`m stuck from there.
The simplest estimator would simply be the empirical quantiles from your dataset. Alternatively, you could feed your data into a kernel density and take quantiles from that. Or finally, you could fit a known distribution (if possible, guided by substantive knowledge of your data - if you know it's nonnegative, then there are likely better alternatives than the normal distribution).
If your data depend on covariates, you could try analogues: quantile regression, functional data analysis, or an appropriate regression model.