Using quantiles to estimate the parameters of a distribution: adjusting for unobserved extreme values

I with to estimate the parameters of a specified semi-infinite distributional family based on a sample drawn from that distribution. It is plausible that my sample median converges to the population median. More generally I would like to choose parameters such that the observed values of the quantiles from the sample lie near the calculated values of the same quantiles given the distribution parameters, for some metric of distance, appropriately weighted.

However, it is clear that the highest observed value does not represent the 100 percent quantile, and the lowest observation does not represent the zero quantile. As a result, the true values even of the sample quantiles are unknown (although the sample order statistics are of course known). Here the out-of-sample values I am referring to represent only sample variability, not censorship or truncation.

Recognizing this, how does one calculate the sample quantiles to match with the calculated quantiles? Does one discard a fraction of the observations at each end of the distribution? Does one push all the values away from the median by some adjustment factor? Or is there some other procedure that is widely used to adjust for this problem?