There are many ways of estimate parameters from a sample. Sometimes it’s difficult to calculate variance and bias estimation. Here I want to show one amazing technique which can solve this problem: **Jacknife estimation**. This method is one of the most common resampling techniques such as boostrap. The parameter of interest is estimated from the subsamples omitting the $i$-th observation and the variance of the full sample estimator from the variability between the subsample estimates.

We are going to use a default R dataset called *precip* to show a simple example*. *Data refers to the average amount of precipitation (rainfall) in inches for each of 70 United States and Puerto Rico cities. This is our population and imagine we only have access to 25 observations.

precip # Data N <- length(precip) # Total Population n <- 25 # Sample Size s <- sample(precip, n) # Sample

would be our sample from which we want to estimate parameter . First we generate subsamples whithout the -th element and compute the statistic, in this case the mean.

Then, we calculate the pseudovalues

and the jacknife estimator is the mean of the pseudovalues

k <- 25 # Number of subsamples sJ <- matrix(nrow=n-1,ncol=k) # Subsamples Matrix for(i in 1:k) sJ[,i] <- s[-i] theta.i <- apply(sJ, 2, mean) theta <- mean(s) # Global Estimator theta.Ji <- n*theta - (n-1)*theta.i # Pseudovalues theta.J <- mean(theta.Ji) # Jacknife Estimator

Where the jacknife estimator variance is the variance of global mean estimated.

var.J <- var(theta.Ji) # Var Jacknife Estimator IC.up <- theta.J + qt(0.975, n-1)*sqrt(var.J/n) IC.low <- theta.J - qt(0.975, n-1)*sqrt(var.J/n) real.theta <- mean(precip) # Real mean library(ggplot2) dt <- data.frame(s, real.theta, theta.J, IC.up, IC.low) ggplot(dt, aes(s)) + geom_density(aes(y=..count..), col='grey', fill='grey') + geom_vline(xintercept = c(real.theta, theta.J, IC.up, IC.low), color=c('#e41a1c',rep('#377eb8',3)), linetype=c(rep('dashed',2), rep('solid',2)),size=1) + geom_text(aes(real.theta+2, 0.001, label = "Real Mean"), size = 4, hjust = 0, angle=90, vjust = 0, color='#e41a1c') + geom_text(aes(theta.J-2,0.001, label = "Jacknife Mean"), size = 4, hjust = 0, angle=90, vjust = 0, color='#377eb8')