The bootstrap (see Bootstrap Methods), introduced by Efron (1979), merges simulation with formal model-based statistical inference. A statistical model for a sample X n of size n is a family of distributions {P θ, n : θ ∈ Θ}. The parameter space Θ is typically metric, possibly infinite-dimensional. The value of θ that identifies the true distribution from which X n is drawn is unknown. Suppose that \({\hat{\theta }}_{n} ={ \hat{\theta }}_{n}({X}_{n})\) is a consistent estimator of θ. The bootstrap idea is
- (a)
Create an artificial bootstrap world in which the true parameter value is \({\hat{\theta }}_{n}\) and the sample X ∗ n is generated from the fitted model \({P}_{{\hat{\theta }}_{n},n}\). That is, the conditional distribution of X ∗ n , given the data X n , is \({P}_{{\hat{\theta }}_{n},n}\).
- (b)
Act as if a sampling distribution computed in the fully known bootstrap world is a trustworthy approximation to the corresponding, but unknown, sampling distribution in the model world.
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References and Further Reading
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Beran, R. (2011). Bootstrap Asymptotics. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_149
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