How to Compare Bootstrap and Delta Method Variance Estimates in Statistics

Understanding Variance Estimation Comparison

What We're Solving:

We're comparing two different methods for estimating the variance of a transformed statistic p̂(1 - p̂) from a sample: the Delta Method (an analytical approach) and Bootstrap (a computational resampling approach). The twist is that we need different Delta Method formulas depending on whether p̂ equals 1/2 or not!

The Approach:

Think of this as a "methods showdown" - we're testing how well two different variance estimation techniques perform. The Delta Method uses calculus and approximations, while Bootstrap uses computer simulation. We're specifically looking at the function g(p̂) = p̂(1 - p̂), which represents the variance of a Bernoulli distribution.

Step-by-Step Solution:

Step 1: Understand why p̂ = 1/2 is special

• The function g(p) = p(1-p) has its maximum at p = 1/2
• At this point, the first derivative g'(p) = 1-2p equals zero
• When the first derivative is zero, the first-order Delta Method breaks down, so we need the second-order version

Step 2: Set up your Delta Method calculations

• For p̂ ≠ 1/2: Use first-order Delta Method

- Find g'(p̂) = 1 - 2p̂ - Variance estimate: [g'(p̂)]² × Var(p̂) - Since Var(p̂) = p̂(1-p̂)/n, this becomes: (1-2p̂)² × p̂(1-p̂)/24

• For p̂ = 1/2: Use second-order Delta Method (from Theorem 5.5.26)

- This involves the second derivative g''(p) = -2 - The formula will be more complex, involving terms with 1/n²

Step 3: Implement Bootstrap procedure

• Generate B = 1000 bootstrap samples from your original sample
• For each bootstrap sample, calculate p̂ and then g(p̂) = p̂(1-p̂)
• Compute the sample variance of these 1000 g(p̂*) values
• This gives you the bootstrap variance estimate

Step 4: Compare the methods

• Calculate both estimates for various values of p̂
• Pay special attention to how they behave when p̂ is close to 1/2
• Look for patterns: Which method gives larger/smaller estimates? How do they differ?

The Framework:

Since this appears to be asking you to work through a computational comparison, here's your analysis structure:

• 1. Setup Phase: Define your sample parameters (n=24, B=1000)
• 2. Computational Phase:

- Implement both variance estimation methods - Test with different p̂ values, including cases near 1/2

• 3. Comparison Phase: Create tables or plots showing both estimates
• 4. Analysis Phase: Discuss patterns, differences, and when each method works better

Memory Tip:

Remember "First fails at the peak" - when you're at the maximum of a function (like p̂(1-p̂) at p̂=1/2), the first derivative is zero, so first-order Delta Method fails and you need second-order. Bootstrap doesn't care about derivatives - it just resamples, making it more robust but computationally intensive!

The beauty of this problem is seeing how mathematical theory (Delta Method needing different orders) plays out in practice against the brute-force approach of Bootstrap!