DATA604_Discussion_Work_8

Dilip Ganesan

Antithetic Variables

Let us consider the two independent identically distributed random variables U1 and U2, the Variance is given by

but in reality,

The variance of (U1+U2)/2 will be smaller if U1 and U2 are negatively correlated.

Hence we can say that for negatively correlated variables can reduce variance.

If U is uniformly distributed on [0,1], then 1-U has the same distribution but they are negatively correlated then the variance will reduce.

Let us take a example from the Rizzo book.

# This is a function, which has a flag with a default antithetic as True.
# Based on the flag negative correlations are set.
MC.Phi <- function(x, R = 10000, antithetic = TRUE) 
{ 
    u <- runif(R/2) 
    if (!antithetic) 
        v <- runif(R/2) 
    else 
        v <- 1 - u 
    u <- c(u, v) 
    cdf <- numeric(length(x)) 
    for (i in 1:length(x)) 
    { 
        g <- x[i] * exp(-(u * x[i])^2 / 2) 
        cdf[i] <- mean(g) / sqrt(2 * pi) + 0.5 
    } 
    cdf 
}


x <- seq(.1, 2.5, length=5) 
Phi <- pnorm(x) 
set.seed(123) 
MC1 <- MC.Phi(x, anti = FALSE) 
set.seed(123) 
MC2 <- MC.Phi(x) 
print(round(rbind(x, MC1, MC2, Phi), 5))

##        [,1]    [,2]    [,3]    [,4]    [,5]
## x   0.10000 0.70000 1.30000 1.90000 2.50000
## MC1 0.53983 0.75825 0.90418 0.97311 0.99594
## MC2 0.53983 0.75805 0.90325 0.97132 0.99370
## Phi 0.53983 0.75804 0.90320 0.97128 0.99379

# From the range of value between .1 and 2.5 let us take the value of x as 1.50
m <- 1000 
MC1 <- MC2 <- numeric(m) 
x <- 1.50 
for (i in 1:m) 
{ 
  MC1[i] <- MC.Phi(x, R = 1000, anti = FALSE) 
  MC2[i] <- MC.Phi(x, R = 1000) 
}

print(sd(MC1))

## [1] 0.004035471

print(sd(MC2))

## [1] 0.0007043272

print(round((var(MC1)- var(MC2))/var(MC1)*100),2)

## [1] 97

There is a reduction of 96.9% almost 97% reduction in variance using the Antithetic variables, compared to just Monte Carlo Integration.