Bootstrap

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General

Nonparametric bootstrap

This is the most common bootstrap method

The upstrap Crainiceanu & Crainiceanu, Biostatistics 2018

Parametric bootstrap

Examples

Standard error

  • Standard error from a mean
    foo <- function() mean(sample(x, replace = TRUE))
    set.seed(1234)
    x <- rnorm(300)
    set.seed(1)
    sd(replicate(10000, foo()))
    # [1] 0.05717679
    sd(x)/sqrt(length(x)) # The se of mean is s/sqrt(n)
    # [1] 0.05798401
    
    set.seed(1234)
    x <- rpois(300, 2)
    set.seed(1)
    sd(replicate(10000, foo()))
    # [1] 0.08038607
    sd(x)/sqrt(length(x)) # The se of mean is s/sqrt(n)
    # [1] 0.08183151
    
  • Difference of means from two samples (cf 8.3 The two-sample problem from the book "An introduction to Bootstrap" by Efron & Tibshirani)
    # Define the two samples
    sample1 <- 1:10
    sample2 <- 11:20
    
    # Define the number of bootstrap replicates
    nboot <- 100000
    
    # Initialize a vector to store the bootstrap estimates
    boot_estimates <- numeric(nboot)
    
    # Run the bootstrap
    set.seed(123)
    for (i in seq_len(nboot)) {
      # Resample the data with replacement
      resample1 <- sample(sample1, replace = TRUE)
      resample2 <- sample(sample2, replace = TRUE)
      
      # Compute the difference of means
      boot_estimates[i] <- mean(resample1) - mean(resample2)
    }
    
    # Compute the standard error of the bootstrap estimates
    se_boot <- sd(boot_estimates)
    
    # Print the result
    cat("Bootstrap SE estimate of difference of means:", se_boot, "\n")
    # 1.283541
    
    sd1 <- sd(sample1)
    sd2 <- sd(sample2)
    
    # Calculate the sample sizes
    n1 <- length(sample1)
    n2 <- length(sample2)
    
    # Calculate the true standard error of the difference of means
    se_true <- sqrt((sd1^2/n1) + (sd2^2/n2))
    
    # Print the result
    cat("True SE of difference of means:", se_true, "\n") \
    # 1.354006

Bootstrapping Extreme Value Estimators

Bootstrapping Extreme Value Estimators de Haan, 2022