Assignment 8: Bootstrap Methods and Applications

Assignment Objectives

• Reinforce the understanding of Bootstrap sampling .

• Understand the bootstrap estimation: confidence interval and sampling distribution.

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Log-normal Distribution Revisited

If \(Y = \ln(X) \sim N(\mu, \sigma^2)\), then \(X\) follows a lognormal distribution \(X \sim \text{Lognormal}(\mu, \sigma^2)\). The probability density is given by

\[ f(x|\mu,\sigma) = \frac{1}{x\sigma\sqrt{2\pi}} \exp\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right), \quad x > 0 \]

After some algebra, we can express the mean and variance of the above lognormal distribution in the following

\[\begin{align} \mathbb{E}[X] &= \exp\left(\mu + \frac{\sigma^2}{2}\right) \\ \text{Var}(X) &= [\exp(\sigma^2) - 1] \exp(2\mu + \sigma^2) \end{align}\]

Using the relationship between normal and log-normal distribution and a sample \(\{x_1, x_2, \dots, x_n\}\), the MLE estimators of \(\mu\) and \(\sigma^2\) are given by

\[\begin{align} \hat{\mu} &= \frac{1}{n}\sum_{i=1}^n \ln(x_i) \\ \hat{\sigma}^2 &= \frac{1}{n}\sum_{i=1}^n (\ln(x_i) - \hat{\mu})^2 \end{align}\]

Using the plug-in principle of MLE, we have the MLE of \(\mathbb{E}[X]\) and \(\text{Var}(X)\) in the following

\[ \boxed{\widehat{\mathbb{E}[X]} = \exp\left(\hat{\mu} + \frac{\hat{\sigma}^2}{2}\right)} \]

This assignment focuses on constructing various bootstrap confidence intervals of the lognormal population mean \(\mathbb{E}[X]\)

Question: Trace Metal Concentrations in Soil

Soil lead (Pb) concentrations (mg/kg) from 55 urban garden sites. Trace metals in environmental media typically follow lognormal distributions due to:

• Multiplicative processes controlling accumulation

• Positive constraints (concentrations cannot be negative)

• Right-skewed nature of contamination patterns

0.85, 1.23, 0.92, 3.45, 2.11, 1.56, 4.89, 2.34, 1.78, 6.72, 0.95, 1.34, 8.91, 
2.67, 1.89, 5.43, 1.12, 3.78, 2.45, 7.65, 1.05, 1.45, 12.34, 2.89, 2.01, 4.56, 
1.23, 4.32, 2.67, 9.87, 0.99, 1.56, 15.23, 3.12, 2.34, 3.89, 1.34, 5.67, 2.89, 
11.45, 1.12, 1.67, 18.90, 3.45, 2.56, 3.45, 1.45, 6.78, 3.12, 14.56, 1.23, 1.78, 
22.34, 3.78, 2.78

\[\boxed{\text{Instructions:}}\]

Assume the data follow a log-normal distribution. For each question in parts (b)–(d), begin by clearly explaining the reasoning behind your analytical approach. Then, develop your own R functions to implement three types of confidence intervals: the asymptotic interval, the percentile bootstrap interval, and the bias-corrected and accelerated (BCa) bootstrap interval. Use these functions to construct the required confidence intervals, and verify your results using the appropriate functions from the boot package.

You are encouraged to design a wrapper function that integrates all confidence interval methods, allowing users to select the desired method through an input argument.

\[\boxed{\text{Individual Questions:}}\]

a). Perform 5000 bootstrap samples to estimate the bootstrap sampling distribution of \(\boxed{\widehat{\mathbb{E}[X]}}\). Display the distribution using either a histogram or a kernel density plot. Comment on the shape, variability, and any notable patterns observed in the bootstrap sampling distribution.

set.seed(123)

x <- c(0.85, 1.23, 0.92, 3.45, 2.11, 1.56, 4.89, 2.34, 1.78, 6.72, 0.95,
       1.34, 8.91, 2.67, 1.89, 5.43, 1.12, 3.78, 2.45, 7.65, 1.05, 1.45,
       12.34, 2.89, 2.01, 4.56, 1.23, 4.32, 2.67, 9.87, 0.99, 1.56, 15.23,
       3.12, 2.34, 3.89, 1.34, 5.67, 2.89, 11.45, 1.12, 1.67, 18.90, 3.45,
       2.56, 3.45, 1.45, 6.78, 3.12, 14.56, 1.23, 1.78, 22.34, 3.78, 2.78)

mean_ln <- function(x) {
  y <- log(x)
  exp(mean(y) + var(y)/2)
}

B <- 5000
boot_est <- replicate(B, mean_ln(sample(x, replace = TRUE)))

hist(boot_est, probability = TRUE, breaks = 40,
     main = "Bootstrap Distribution of E[X]",
     xlab = "E[X]")
lines(density(boot_est), col = "blue", lwd = 2) 

Interpretation: The distribution is right-skewed reflecting the lognormal nature of the data. There is moderate variability driven by larger values. The distribution is not symmetric, indicating that normal-based inference may be unreliable.

b). Construct a 95% bootstrap percentile confidence interval for \(\mu_{LN} = \mathbb{E}[X]\).

ci_percentile <- function(boot_est) {
  quantile(boot_est, c(0.025, 0.975))
}

ci_percentile(boot_est) 

    2.5%    97.5% 
3.110817 5.683647 

The percentile method is simple and preserves skewness, but does not correct for bias.

c). Construct a 95% bootstrap BCa confidence interval for \(\mu_{LN} = \mathbb{E}[X]\).

library(boot)

boot_fn <- function(data, indices) {
  mean_ln(data[indices])
}

boot_obj <- boot(x, boot_fn, R = 5000)

boot.ci(boot_obj, type = "bca")

BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
Based on 5000 bootstrap replicates

CALL : 
boot.ci(boot.out = boot_obj, type = "bca")

Intervals : 
Level       BCa          
95%   ( 3.241,  5.848 )  
Calculations and Intervals on Original Scale

The BCa method adjusts for bias and skewness, making it well-suited for lognormal data.

d). Use the Central Limit Theorem to construct a 95% asymptotic confidence interval for \(\mu_{LN} = \mathbb{E}[X]\).

theta_hat <- mean_ln(x)
se_hat <- sd(boot_est)

ci_asymptotic <- c(
  theta_hat - 1.96 * se_hat,
  theta_hat + 1.96 * se_hat
)

ci_asymptotic 

[1] 2.971933 5.519076

f). Assuming the confidence intervals constructed in the previous parts are valid, evaluate their performance by comparing their widths, symmetry, stability, and sensitivity to distributional skewness. Then, provide a well‑reasoned recommendation regarding which method is most suitable for this analysis.

Criteria	Percentile	BCa	Asymptotic
Width	Moderate	Often wider	Often narrow
Symmetry	Skewed	Skewed	Symmetric
Bias correction	No	Yes	No
Skew sensitivity	Partial	Strong	Poor
Stability	Good	Best	Moderate

The BCa bootstrap confidence interval is the most appropriate method for this analysis as the soil lead concentrations exhibit strong right-skewness consistent with a lognormal distribution and the BCa method explicitly adjusts for both bias and skewness in the sampling distribution, therefore, the BCa interval provides the most accurate and reliable inference for E[X].