Understanding and implementing numerical integration with a quantile function in R

Question

I need to calculate this integral below, using R:

The q_theta(x) function I managed to do in R with quantile regression (package: quantreg).

<

Answer 1

Recall your previous post Building a function by defining X and Y and then Integrating in R, we build a linear interpolation function

## note `rule = 2` to enable "extrapolation";
## otherwise `rule = 1` gives `NA` outside [0.01, 0.5]
integrand <- approxfun(mat[, 1], y, rule = 2)

Then we can perform numeric integration on [0, 0.5]:

integrate(integrand, lower = 0, upper = 0.5)
# -5.594405 with absolute error < 4e-04

Now for a>, let's have a proof first.

Note, your quantile function is not for normal distribution, so this result does not hold. You can actually verify this

quant <- approxfun(mat[, 1], mat[, 2], rule = 2)
integrate(quant, lower = 0, upper = 0.5)
# -3.737973 with absolute error < 0.00029

Compared with previous integration result -5.594405, the difference is not a factor of 2.