Katkat
Back

Explain Bootstrap and Prove Uniformity

GoogleData Scientist · Technical Screen
HardData ScientistTechnical ScreenStatistics & Math

Last updated

This question tests the conceptual understanding of bootstrap resampling for variance estimation and the mathematical proof of probability distributions using the properties of the Uniform distribution.

Part 1: The Bootstrap

Explain the concept of bootstrapping in the context of estimating the standard error of the sample median. Why is the bootstrap distribution often used when the underlying population distribution is unknown, and what are the primary assumptions regarding the sample size and data independence?

Part 2: Uniformity Proof

Let X be a continuous random variable with cumulative distribution function (CDF) F(x). Prove that the random variable Y = F(X) follows a Uniform(0, 1) distribution, provided that F is strictly increasing and continuous.

Consider the implication of this result: If you have a black-box function that samples from an unknown distribution F, how can you use this transformation to generate random variables from any target distribution G, assuming G is invertible?

Hints

Hint 1

For the bootstrap, consider the difference between parametric and non-parametric approaches and why the median lacks a closed-form standard error formula.

Hint 2

For the proof, use the definition of the CDF: P(Y <= y) = P(F(X) <= y).

Hint 3

To generate from G, consider the inverse transform sampling method using the result Y ~ Uniform(0, 1) as an intermediate step.

Follow-up questions

  • How does the bootstrap performance change if the data are autocorrelated, and what modification would you suggest?
  • Under what specific conditions might the Probability Integral Transform fail to produce a perfectly Uniform(0, 1) distribution?
  • How would you construct a confidence interval for the median using the bootstrap, and how do you handle potential ties in the data?