Normal Distribution
nothing is called 95% until it has been counted
a-band-with-no-parameter-to-fit

Where the general fit becomes the parametric one

Where the general fit becomes the parametric one. The band family's objective at the autoregression's own geometric sequence, cut off at each width, on one sample of 60 rows. The horizontal line is the profile likelihood the parametric fit maximises, written independently through a different whitening. At the full width the two are the same number to 3e-14, which is what says the general construction contains the parametric one rather than resembling it. Below 10 lags there is no line at all: the geometric sequence cut off short is not a covariance matrix, so the objective has nothing to evaluate. Between the two the truncation is briefly above the parametric likelihood — a wrong covariance can fit one sample better than the right one, which is the whole reason a width has to be charged for rather than chosen.

The band family's objective at the autoregression's own geometric sequence, cut off at each width, on one sample of 60 rows. The horizontal line is the profile likelihood the parametric fit maximises, written independently through a different whitening. At the full width the two are the same number to 3e-14, which is what says the general construction contains the parametric one rather than resembling it. Below 10 lags there is no line at all: the geometric sequence cut off short is not a covariance matrix, so the objective has nothing to evaluate. Between the two the truncation is briefly *above* the parametric likelihood — a wrong covariance can fit one sample better than the right one, which is the whole reason a width has to be charged for rather than chosen.

A covariance with no parameterwide3 views

What else it draws

The same object, drawn to answer the other questions the essays put to 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 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.

Three defensible rules and three different widths. The error of the fitted line against the width of the band, under AR(1) at 0.8, with the widths four rules choose marked on it. The criterion charging a unit a lag picks 6.2; a heavier charge picks 3.8; the rule of thumb every applied long-run variance uses picks 4; the width that actually minimises the error is 13.3, and it is available to nobody — chosen per draw it has a standard deviation of 10.2, which is most of the range on offer. What the three rules deliver differs by less than a hundredth of the error they are all paying: 1.1104, 1.1141 and 1.1133 against the best available 1.1031. The curve is nearly flat between eight and twenty lags, which is why three rules can disagree by a factor of three and cost almost nothing.

The error of the fitted line against the width of the band, under AR(1) at 0.8, with the widths four rules choose marked on it. The criterion charging a unit a lag picks 6.2; a heavier charge picks 3.8; the rule of thumb every applied long-run variance uses picks 4; the width that actually minimises the error is 13.3, and it is available to nobody — chosen per draw it has a standard deviation of 10.2, which is most of the range on offer. What the three rules deliver differs by less than a hundredth of the error they are all paying: 1.1104, 1.1141 and 1.1133 against the best available 1.1031. The curve is nearly flat between eight and twenty lags, which is why three rules can disagree by a factor of three and cost almost nothing.

Where it is used

5 essays draw this figure, each at the numbers its own argument is about, so the same picture answers 5 different questions.

All 80 figures

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