I have a copula representing the dependence between two variables X and Y. I want to compute the following formula: E(X|Y≤1%). It is the expected value of X conditional on Y being lower than 1%. I see that a somewhat similar question was asked there but the R code provided does not give the value I am looking for. Below are some details about the copula and marginal distribution.
library(VineCopula)
library(copula)
#I estimate my Copula and assumes normal distribution for the two marginals
copula_dist <- mvdc(copula=claytonCopula(param=1.0), margins=c("norm","norm"),
paramMargins=list(list(mean=0, sd=5),list(mean=0, sd=5)))
#I take a sample of 500 events
sim <- rMvdc(500,copula_dist)
# Compute the density
pdf_mvd <- dMvdc(sim, my_dist)
# Compute the CDF
cdf_mvd <- pMvdc(sim, my_dist)
You have to evaluate this double integral: integral of x*pdf(x,y), -oo < x < +oo, -oo < y < 1%, and divide it by Pr(Y < 1%). This is done below. I also perform an approximation by simulations to have a check.
library(copula)
# the distribution
copula_dist <- mvdc(copula=claytonCopula(param=1.0), margins=c("norm","norm"),
paramMargins=list(list(mean=0, sd=5),list(mean=0, sd=5)))
### we will calculate E[X | Y < y0]
y0 <- 1/100
### approximation of E[X | Y < y0] using simulations
sim <- rMvdc(100000, copula_dist)
mean(sim[sim[,2]<y0,1])
# [1] -1.967642
### approximation of E[X | Y < y0] using numerical integration
### this is E[X * 1_{Y<y0}] / P(Y < y0)
library(cubature)
# PDF of the distribution
pdf <- function(xy) dMvdc(xy, copula_dist)
# P(Y < y0)
denominator <- pnorm(y0, mean=0, sd=5)
# integrand
f <- function(xy) xy[1] * pdf(xy)
# integral
integral <- hcubature(f, lowerLimit = c(-Inf, -Inf), upperLimit = c(Inf, y0))
integral$integral / denominator
# [1] -1.942691