Estimating quantiles using quantile regression

Question

I understand that quantile regression estimates the conditional quantile of some measured variable (call the variable $y$), but can you use quantile regression to estimate an unconditional quantile of a distribution? For example, if I had 100 data points

$$y_1,...,y_{100}\sim N(\mu, \sigma^2)$$

then I could use some method to try to estimate the true quantile of the distribution, i.e., $q = \mu + z_\alpha\sigma$. But if I don't have any covariate information, can I still use quantile regression to estimate some quantile of interest from my data $y$?

Answer 1

YES

This is just a quantile regression on an intercept and nothing else. The theory is similar to how OLS linear regression on just an intercept gives the mean.

Minimizing the sum of squared deviations gives the mean, and minimizing the sum of quantile loss values gives the quantile.

Answer 2

Why not try it and see?

set.seed(1234)
library(quantreg)

x <- rnorm(1000)
m1 <- rq(x~1, c(0.1, 0.5, 0.9))
m1                            #-1.2121   -0.03965, 1.33418
quantile(x, c(0.1, 0.5, 0.9)) #-1.21322  -0.03979   1.3344

Interestingly, the estimates from quantile regression are not exactly the actual quantiles, although they are very close. I don't know why.

I also tried some other distributions, and some were a little less close than the one above.