求解包含非参数密度和分布的积分的最佳方法

假设我想解一个包含两个积分的函数,比如(这是一个例子,实际函数更难看)

我做到了:

# Create the data
val <- runif(300, min=1, max = 10) #use the uniform distribution
CDF <- (val - 1)/(10 - 1)
pdf <- 1 / (10 - 1)
data <- data.frame(val = val, CDF = CDF, pdf = pdf)

c = 2
d = 1

# Inner integral
integrand1 <- function(x) {
  i <- which.min(abs(x - data$val))
  FF <- data$CDF[i]
  ff <- data$pdf[i]
  (1 - FF)^(c/d) * ff
}

# Vectorize the inner integral
Integrand1 <- Vectorize(integrand1)

# Outer integral
integrand2 <- function(x){
  i <- which.min(abs(x - data$val))
  FF <- data$CDF[i]
  ff <- data$pdf[i]
  (quadgk(Integrand1, x, 10) / FF) * c * ff
}

# Vectorize the outer integral
Integrand2 <- Vectorize(integrand2)

# Solve
require(pracma)
quadgk(Integrand2, 1, 10)

积分非常慢。有更好的方法解决这个问题吗?非常感谢。

---------编辑---------

v 具体如下:

# Create the original data
v <- runif(300, min = 1, max = 10)
require(np)

# Compute the CDF and pdf
v.CDF.bw <- npudistbw(dat = v, bandwidth.compute = TRUE, ckertype = "gaussian")
v.pdf.bw <- npudensbw(dat = v, bandwidth.compute = TRUE, ckertype = "gaussian")

# Extend v on a grid (I add this step because the v vector in my data
# is not very large. In this way I approximate the estimated pdf and CDF
# on a grid)
val <- seq(from = min(v), to = max(v), length.out = 1000)
data <- data.frame(val)
CDF <- npudist(bws = v.CDF.bw, newdata = data$val, edat = data )
pdf <- npudens(bws = v.pdf.bw, newdata = data$val, edat = data )
data$CDF <- CDF$dist
data$pdf <- pdf$dens