Normal Distribution
nothing is called 95% until it has been counted
Concept

Local optimality — where it appears

Optimality computed at a guess about the parameter values, which is the only kind available when the information depends on the answer. It is why a design for a non-linear model has to be defended over a range of guesses rather than at one, and the defence is a maximin problem.

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

Also named here as support points — the same set of essays touches all of them, so they are one junction rather than several.

One curve, three designs, and the disagreement is the weights. The compartmental response exp(−θ₁t) − exp(−θ₂t) at the guess θ = (0.2, 1.2), with three designs underneath it. Each mark is a setting and its height is the share of the experiment spent there. The D-optimal design splits the runs equally between two settings — that is the determinant's answer and it is equal for every model of this kind. The design for the first parameter alone puts 1.9% of the experiment at its early setting and the rest at its late one, because the early runs are there to identify the nuisance and nothing more. The design that protects a 4-fold rectangle needs 3 settings and spends 9.1% at the earliest of them.

The guess with two numbers in it

Every optimal design for a non-linear model is optimal at a guess. Where the model has one parameter that moves the settings, that guess is a number and everything about it comes out in closed form; where it has two, three constants become functions and one of them becomes zero.

blind · Local design Four designs, and what each of them guarantees. The worst Ds-efficiency each design achieves anywhere in a rectangle of parameter values 4 times wide in each coordinate. The design built at the guess guarantees 7.3%: it is perfect where it was built and nearly useless at one corner. The D-optimal design at the same guess guarantees 25.6% — it answers the wrong question everywhere and is therefore not concentrated on being right anywhere. The third is the one this field exists to measure: maximin over the first parameter's whole range, with the second held at its guess. It guarantees 7.0%, which is no better than the design that protects nothing. Protecting both is worth 45.9%, and it needs 3 settings to do it.

The worst case in two directions

A design that protects a range of one parameter is robust. Protect the range of one parameter while holding the other at a guess and the design is still robust, still has a guarantee, and guarantees no more than a design that protects nothing at all.

blind · Criterion

Named alongside it

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

Compartmental modelDesign measureDs-optimalityEfficiencyEquivalence theoremThe non-linear modelNuisance parameterOptimal designSupport pointsD-optimalityDesign weightsEqualisation

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