Concept

Parametric bootstrap — where it appears

A resampling method that fits a model, simulates new datasets from the fitted model, and refits each to see how an estimate varies. It can reach quantities no formula gives, and it is only as good as the fitted model it simulates from.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Most of the rise is the optimiser's, and under one law it is not. The rise in log-likelihood from the tapered plug-in to the maximum over the same eight-lag band, beside what the same optimiser produces on a sample generated from the plug-in's own covariance — where the family is correctly specified by construction and there is nothing to find. Under AR(1) at 0.8 the raw rise is 5.72 and the manufactured baseline is 4.79, leaving 0.93 at 1.8 standard errors; under long memory the excess is 0.14, at 0.2. Under the moving average it is 11.87 at 19.4 standard errors, on every draw. The taper is a shrinkage, and it costs nothing where the sequence decays smoothly and a great deal where it stops dead.

The plug-in and the maximum

A tapered covariance estimate sits five and a half log-likelihood units below the maximum of the likelihood it is substituted into. Four fifths of that is what the optimiser would have found if nothing were missing.

family · Dependence

Seven ways to report the height at the setting a response-surface analysis recommends, at noise 2. Bias and root mean squared error of the reported height about the true height at the chosen setting, over 2,000 studies with a central composite design of thirteen runs, or the same design run twice. 13 runs, the fit's own height: bias 0.884, error 1.537. 13 runs, optimism bootstrapped away: bias 0.183, error 1.396. One confirmation run alone: bias -0.005, error 2.005. Corrected fit and one confirmation run: bias 0.056, error 1.121. 26 runs, the fit's own height: bias 0.475, error 0.998. 26 runs, optimism bootstrapped away: bias 0.050, error 0.924. 26 runs split: choose on 13, estimate on 13: bias -0.014, error 1.213.

The height at the chosen setting

The height a response-surface fit predicts at the setting it recommends reads high, because the setting was chosen where the fit was highest. The obvious repair — choose on some runs, estimate on the rest — is impossible in the usual thirteen-run design, which has nine distinct settings where two separate fits need twelve. A parametric bootstrap of the optimism removes four fifths of it at no cost in runs, 0.884 down to 0.183 at twice the noise. With the design run twice, correcting the full fit beats splitting it: an error of 0.924 against 1.213, at a better setting.

surface · Optimum

Named alongside it

The objects these essays reach for when they reach for this one.

Attenuation Bias Bootstrap Central composite design Confirmation run Covariance matrix Degrees of freedom Dependence Experimental design Long memory Mean squared error Monte Carlo

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