Integral formula using R

Question

I have to implement this formula:

$K(x) = \int_{0}^{0.5}q_{\theta}(x)d{\theta}$

where $q_{\theta}(x)$´s are the conditional quantiles in some $\theta$.

using a range of $\theta = [0.45; 0.40; 0.35; 0.3; 0.25; 0.2; 0.15; 0.1; 0.05]$

Note that i am integrating in $\theta$ not in $x$.

The idea is, when im in $\theta=.05$ i automatically have $q_{.05}(x)$ through:

q_theta<-rq(Y~x,tau = .05)

and so on through in the interval $\theta = [0.45; 0.40; 0.35; 0.3; 0.25; 0.2; 0.15; 0.1; 0.05]$.

q_theta<-rq(Y~x,tau = .45)
q_theta<-rq(Y~x,tau = .35)
q_theta<-rq(Y~x,tau = .3)
q_theta<-rq(Y~x,tau = .25)
q_theta<-rq(Y~x,tau = .2)
q_theta<-rq(Y~x,tau = .15)
q_theta<-rq(Y~x,tau = .1)

And i proceed to the integration, where i have no idea how to do:

My data:

set.seed(33)
N<-100
Y<- rnorm(N,0,3)
x<- runif(N)

And then, i calculate: $K(x) = \int_{0}^{0.5}q_{\theta}(x)d{\theta}$

I hope I have been clear, otherwise I redo the question.

I dont know how to build this integral function.

Any sugestion?

Thanks.

Answer 1

I guessed $d\theta$ to be $dx$. Because I can't conclude Y to be a fixed vector, I used x and y as integral function's arguments.

integrate(f, lower, upper, ...) is adaptive quadrature of functions of one variable over a finite or infinite interval. f is an R function taking a numeric first argument and returning a numeric vector of the same length.

library(quantreg)

# explanatory simple function (this function can treat only one theta)
K0 <- function(x, y){
  temp <- rq(y ~ x, tau = theta)$coef - rq(y ~ x, tau = .50)$coef
  f <- function(x) temp[1] + temp[2] * x
  integrate(f, 0, 0.5)
}
# rq(Y ~ x, tau = theta)$coef are it's coefficients. 
# Because of a primary expression,  ~[1] is a intercept, ~[2] is a slope.
# The same is true of (q_theta$coef - q_theta_50$coef), so make it a function(x) to integrate.

set.seed(33)
N<-100
Y<- rnorm(N,0,3)
x<- runif(N)        # the same length of x and y is essential
one_theta <- 0.9
K0(x = x, y = Y)
# 1.604942 with absolute error < 1.8e-14

# improvement to treat multi theta
K <- function(x, y){
  temp <- NULL
  for(i in seq.int(theta)) temp <- rbind(temp, rq(y ~ x, tau = theta[i])$coef - rq(y ~ x, tau = .50)$coef)
  temp2 <- apply(temp, 1, function(a) {
    f <- function(x) a[1] + a[2] * x
    integrate(f, 0, 0.5)
  })
  names(temp2) <- paste0("theta = ", theta)
  temp2
}

theta <- c(0.99, 0.975, 0.95, 0.90, 0.85, 0.80, 0.75)
K(x = x, y = Y)
#  :
# $`theta = 0.9`
# 1.604942 with absolute error < 1.8e-14

# $`theta = 0.85`
# 1.148646 with absolute error < 1.3e-14
# :