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Suppose I have some data $x_N$ of `size' $N$ which were drawn according to some measure $P_N(x_N)$.

I'm imagining $x_N$ as being any of:

  • an exchangeable / iid sequence of length $N$
  • a stationary sequence of length $N$
  • a stationary random field with $N$ sites
  • (other examples with some sort of symmetry / stationarity assumption, such that a form of bootstrapping is available)

Suppose now that, for some downstream task (e.g. computing approximate confidence intervals), I'm interested in bootstrapping $x_N$, i.e. I have some conditional distribution $\beta_N ( x_N \to y_N)$ which takes as input my data $x_N$, and outputs a bootstrapped version of the data, $y_N$, which has the same type as the original data.

The marginal law of $y_N$ is then given by

\begin{align} Q_N ( y_N ) = \int P_N ( x_N ) \beta_N ( x_N \to y_N) dx_N. \end{align}

Now, typically it will be the case that $Q_N \neq P_N$. However, as $N$ grows, it intuitively seems that one should have $Q_N \approx P_N$ in some appropriate metric. This would be one way of justifying using bootstrap samples for approximate inference; in the large-sample regime, you are somehow not perturbing things too badly.

What I am looking for is a more precise mathematical statement of the above intuition, i.e. for a given class of data-generating processes $\{ P_N \}_{N \geqslant 1}$, and a family of bootstrap resampling mechanisms $\{ \beta_N \}_{N \geqslant 1}$, I would like conditions under which a bound of the form

$$D ( Q_N, P_N ) \leqslant \varepsilon(N),$$

where $D$ is some discrepancy measure (e.g. some transport distance), and $\varepsilon(N)$ is some relatively-explicit sequence which tends to $0$.

If my intuition is false, i.e. $\lim\inf_{N \to \infty} D ( Q_N, P_N ) = \delta > 0$ for most useful $D$, then it would be useful to understand the failure modes, and to possibly revise my question to adjust for them. Understanding the qualitative differences between $Q_N$ and $P_N$ is of general interest to me; the bounds are just a specific manifestation of this.

As a final comment, I'm particularly interested in how one proves results of this form for non-iid $x_N$, e.g. the case of stationary time series, stationary random fields, etc.

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