Abstract
At the beginning of this book we said that we wanted to lay out the key features of what we called, “the statistical paradigm,” which consists of broadly applicable concepts that guide reasoning from data in diverse contexts. One of its foundations is the idea that data may be used to express knowledge and uncertainty about unknown values of model parameters, especially through confidence intervals.
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Notes
- 1.
We are not intending \(f(x)\) to be a pdf. We are here, in this chapter, using the notation \(y=f(x)\) to refer to some general function.
- 2.
The “law of propagation of error,” as it was called, is mentioned as a standard technique by Schultz (1929).
- 3.
The bootstrap approximate 95 % CI based on percentiles in Eq. (9.26) has the property that as \(n \rightarrow \infty \) the probability of coverage is \(.95 + \eta _n\) where \(\eta _n\) vanishes at the rate of \(1/\sqrt{n}\). The \(BC_a\) intervals have the analogous property with \(\eta _n\) vanishing at the rate \(1/n\), which means the theoretical coverage probability should be closer to .95.
- 4.
Actually, a stronger result is needed, and it is stated in terms of the supremum (also known as the least upper bound). The supremum of a set of numbers \(S(x)\), written \(\sup _x S(x)\), is the smallest value \(c\) such that \(S(x) < c\). (Thus the alternative name, “least upper bound.”) It is used when \(S(x)\) is bounded but does not reach a maximum across the range of \(x\). The stronger version of the result in the theorem is that the convergence is uniform in the sense that
$$\sup _x |\hat{F}_n(x)-F(x)| \mathop {\rightarrow }\limits ^{P} 0.$$This holds when \(F(x)\) is a continuous cdf, and in many other cases.
- 5.
With this convention, if \(G={\text {1,000}}\) then there are 24 values smaller than \(r_{.025}\) and 24 values larger than \(r_{.975}\). If \(G=100\) there are 2 values smaller than \(r_{.025}\) and 2 values larger than \(r_{.975}\).
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© 2014 Springer Science+Business Media New York
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Kass, R.E., Eden, U.T., Brown, E.N. (2014). Propagation of Uncertainty and the Bootstrap. In: Analysis of Neural Data. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9602-1_9
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