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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Policy on AI Tool Use: Please adhere to the AI tool policy specified in the course syllabus. The direct copying of AI-generated content is strictly prohibited. All submitted work must reflect your own understanding; where external tools are consulted, content must be thoroughly rephrased and synthesized in your own words.

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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.

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

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

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

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.