sensitivity (version 1.28.0)

squaredIntEstim: Squared integral estimate

Description

This function provides two estimators of a squared expectation. The first one, naive, is the square of the sample mean. It is positively biased. The second one is a U-statistics, and unbiased. The two are equivalent for large sample sizes.

Usage

squaredIntEstim(x, method = "unbiased")

Value

A real number, corresponding to the estimated value of the squared integral.

Arguments

x

A vector of observations supposed to be drawn independently from a square integrable random variable

method

If "unbiased", computes the U-statistics, otherwise the square of the sample mean is computed

Author

O. Roustant

Details

Let X1, ..., Xn be i.i.d. random variables. The aim is to estimate t = E(Xi)^2. The naive estimator is the square of the sample mean: T1 = [(X1 + ... + Xn)/n]^2. It is positively biased, and the bias is equal to s^2/n, where s^2 = var(X1). The U-statistics estimator is the average of Xi * Xj over all unordered pairs (i,j). Equivalently, it is equal to T1 minus the (unbiased) sample variance divided by n.

References

O. Roustant, F. Gamboa and B. Iooss, Parseval inequalities and lower bounds for variance-based sensitivity indices, Electronic Journal of Statistics, 14:386-412, 2020

Van der Vaart, A. W. Asymptotic statistics. Vol. 3. Cambridge university press, 2000.

Examples

n <- 100  # sample size
nsim <- 100 # number of simulations
mu <- 0

T <- Tunb <- rep(NA, nsim)
theta <- mu^2  # E(X)^2, with X following N(mu, 1)

for (i in 1:nsim){
  x <- rnorm(n, mean = mu, sd = 1)
  T[i] <- squaredIntEstim(x, method = "biased")
  Tunb[i] <- squaredIntEstim(x, method = "unbiased")
}

par(mfrow = c(1, 1))
boxplot(cbind(T, Tunb))
abline(h = theta, col = "red")
abline(h = c(mean(T), mean(Tunb)), col = c("blue", "cyan"), lty = "dotted")
# look at the difference between median and mean