3  Resampling, Jackknife and Bootstrap

3.1 Introduction

This chapter covers resampling methods including the jackknife and bootstrap techniques.

3.2 Jackknife

The jackknife is a resampling technique used to estimate the bias and variance of a statistic.

Jackknife is like a leave-one-out cross-validation. Let \(\mathbf{x}= (x_1,\dots,x_n)\) be an observed random sample, and denote the \(i\)th jackknife sample by \(\mathbf{x}_{-i} = (x_1,\dots,x_{i-1},x_{i+1},\dots,x_n)\), that is, a subset of \(\mathbf{x}\).

For the parameter of interest \(\theta\), if the statistics is \(T(\mathbf{x})=:\hat{\theta}\) is computed on the full

3.2.1 When does jackknife not work?

Jackknife does not work when the function \(T(\cdot)\) is not a smooth functional!

3.3 Bootstrap

The bootstrap is a resampling method that allows estimation of the sampling distribution of almost any statistic using random sampling methods.

3.4 Applications

These methods are widely used in statistical inference and have applications in various fields.

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