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

Every essay — page 15

Essays 337 to 360 of 613, in the same order.

A promise about two arms

The fixed-width interval whose coverage is exact was built for one mean, and almost nothing anybody runs an experiment for is one mean. The construction survives with the harmonic effective size in place of the block size, and four things do not: a unit of that size costs four observations rather than one, the second variance puts Neyman's allocation in reach of a blinded rule, the conservation identity comes up one degree of freedom per block short, and there is an exact estimator under either of two conditions and none under both.

The sums, pooled as they are and made independent. Coverage on four designs at a requirement of 0.9, 20,000 runs each, about 3.0, 3.6, 3.4, 3.0 blocks. Refusing the sums: 94.91%, 95.06%, 94.96%, 94.86%. Pooling them as they are: 94.98%, 93.67%, 96.17%, 94.93% — exact at balance, short when the allocation is unbalanced and the same in every block, long when it alternates. Pooling the residualised sums, each block's sum less the part of it its own difference predicts: 94.98%, 95.16%, 95.06%, 94.70%, on every design. Each bar is ±0.31%.

The sums made independent

The block sums are a free second copy of a two-arm trial's variance only at balance. With ten and two in every block, pooling them covers 93.67% on four blocks; swap the arms each block and it covers 96.17%. Take from each sum the part its own difference predicts and the copy is independent at every allocation.

4 figures · Blocking, part 8 Coverage as the effect moves with the level. Coverage on ten and two in every block, with the level between blocks moving at a spread of 0.5 and the effect moving with it at each slope, 4000 runs each. At -3: refusing the sums 94.4%, pooling them 88.8%, residualised 97.0%; At -2.5: refusing the sums 94.3%, pooling them 88.9%, residualised 95.6%; At -2: refusing the sums 94.9%, pooling them 90.0%, residualised 93.8%; At -1.5: refusing the sums 95.1%, pooling them 91.7%, residualised 92.7%; At -1: refusing the sums 95.1%, pooling them 94.8%, residualised 94.0%; At -0.5: refusing the sums 95.1%, pooling them 97.2%, residualised 97.7%; At 0: refusing the sums 95.0%, pooling them 98.8%, residualised 99.6%; At 0.5: refusing the sums 95.0%, pooling them 99.4%, residualised 99.9%; At 1: refusing the sums 95.0%, pooling them 99.4%, residualised 99.9%.

The sums under a moving level

A trial's block sums are a free second estimate of its variance only while they see what the differences see, and a level that moves between blocks with an effect moving against it breaks that: pooled raw at a ceiling they cover 87.6% to 90.0%. Residualising the sums on an unbalanced design does not make them safe. It moves the failure. On ten and two, 90.0% at the ceiling becomes 93.8%, the trough leaves for a slope of about −1.5, and over the rest of the range the same estimate over-covers, at 99.9% on an interval half as wide again as it needs to be.

3 figures · Blocking, part 9

Counting what is independent

Akaike's penalty is 2q because the optimism of a fit is a trace and the trace collapses to the parameter count when the rows are independent. Repair the trace and the criterion gets worse, on either scale, because the row count entered twice and a penalty is the second place: whiten the fit and the ordinary penalty is correct again, which recovers 88% of what counting rows gives up. Estimating the dependence from the rows being selected on costs 8% of that. And a multiplier resampling cannot keep more dependence than the residuals have, which is a ceiling rather than a tuning problem.

The correction is not a property of the sample. tr(HΩ)/q for each of fifteen candidates, at ρ = 0.7. Two candidates that fit the same number of coefficients need corrections that differ by as much as 1.49, because one of them is fitting the persistent predictors and the other is not — so no single number can be right for both, and the scalar n/n_eff = 5.537 is above every one of them. The four predictors carry persistences 0.9, 0.6, 0.3, 0; at one persistence for every column the whole spread collapses and a scalar looks exactly as good as the trace.

A penalty is a trace

Akaike's 2q is not a count of coefficients. It is the answer a trace collapses to when the rows are independent — and once they are not, the trace is still the right object and is no longer the count.

3 figures · Order selection, part 4 The one candidate an effective sample size is right about. n/n_eff with the finite-sample inflation Σ(1 − |k|/n)ρ^|k| is not an approximation to tr(HΩ) for a fit with only an intercept — it is that trace, to machine precision, because the hat matrix of a constant column is 1/n everywhere and its trace against Ω is the mean of Ω. The quoted limit form n(1 − ρ)/(1 + ρ) is not even right about that one. And the average correction the table's fifteen candidates actually need is 3.318 per parameter, well below the scalar, so applying it to all of them over-charges every one.

One number for a table of candidates

An effective sample size is a real quantity, it is exactly right about one thing, and that thing is a mean. Substituted into Akaike's criterion it changes nothing at all, because the penalty it is meant to fix has no sample size in it.

4 figures · Dependence, part 3 The row count entered twice, and a penalty is one place. Regret against the best available model as the errors are made persistent. Counting rows more than quadruples; both penalty repairs — the trace, and the scalar effective sample size — are worse than it at every persistence measured; and whitening the sample and keeping the ordinary penalty falls, recovering 86.9% of what counting rows gives up at ρ = 0.85. Doing it at an estimated ρ recovers 80.6%, so having to estimate the dependence from the rows being selected on costs 7.3% of what knowing it is worth. Mallows' forms are drawn beside the logarithmic ones and behave the same, which is what rules the linearisation out.

The repair that was exact and made it worse

A penalty computed from the trace is exactly the optimism it estimates, and selecting with it gives up a fifth more than not correcting anything. The row count entered the criterion twice, and a penalty is the second place.

3 figures · Forecast, part 9 A fit takes the low frequencies out of what it leaves behind. The autocorrelation of the errors, of the residuals of a fitted benchmark, and of those residuals rescaled by their own leverage. (I − H) removes the component of the errors lying in a column space that is itself slow-moving, so the residuals are less persistent at every lag — by 5.9% at the first and 26.6% by the fourth. The leverage correction is the standard repair for what a fit does to a residual's size; drawn here against what it does to a residual's dependence, it does nothing.

The residuals are not the errors

A fit removes the part of the errors lying in its own column space, and a persistent design's column space is itself slow — so what is left behind is smoother than what went in, at every lag, by an amount that grows with the lag.

4 figures · Bootstrap, part 8 The ceiling a multiplier cannot reach past. A wild-type resampling forms e*_t = e_t·w_t with the multiplier independent of the residual, so what comes out has autocovariance γ_resid(k)·γ_w(k) — the residuals' own, multiplied by the multiplier's. Since |γ_w| ≤ 1 the reference distribution's dependence is bounded above by the residuals', and the residuals' is already below the errors'. The two shortfalls compose. For a block of ℓ the multiplier's autocorrelation is exactly the triangle (1 − k/ℓ)⁺, drawn here as the dashed prediction against the realised resamples at ℓ = 5; the bound is attained only at ℓ = n, where the reference distribution is built from one sign.

What a multiplier cannot keep

Two reasons were named for the quarter a blocked resampling falls short, and taking either away makes the gap larger. What is left is a bound — a multiplier can only take dependence out, and the residuals' own is already below the errors'.

3 figures · Reference, part 7

When the two are not independent

The dictionary that is an outer product needs the covariates independent, and the count that is a rate times a binomial coefficient needs the draws independent. Neither holds in a real trial. A linearisation turns the missing fourth-order expectation into Mehler's formula applied to products, which makes the whole geometry closed at any correlation — and the result it was built to test does not survive: what a rule removes of a pure interaction is exactly nothing at independence and 4ρ²/(1+ρ²)² everywhere else. A walk on the admissible set is exactly uniform, needs a burn-in, and is dearer than hunting until almost nothing is admissible.

The fourth-order expectation, by two routes. Every inner product in an eight-term slice of the dictionary at ρ = 0.5, computed from the linearisation and Mehler's formula and counted from two hundred thousand draws of a correlated pair. The entries that matter are the ones off the main effects: ⟨f(X)u(Y), g(X)v(Y)⟩ is a fourth-order expectation, which the independent-covariate field could not write down. The worst departure is 1.99 standard errors over 36 pairs, measured in each pair's own error because the entries differ in size by two orders of magnitude.

The fourth moment that was missing

Mehler's formula makes the main effects exact at any correlation and stops there, because the interactions need an expectation of four Hermite functions rather than two. A linearisation turns the four into two, and the whole geometry becomes closed again.

4 figures · Blocking, part 8 The zero was a fact about independence. What a balancing rule handed every main effect of both covariates removes of a pure interaction, as the covariates are allowed to move together. At ρ = 0 it is exactly nothing — at machine precision, at any number of main effects — which is the independent-covariate result and is correct. It is not small anywhere else: the product of the two covariates loses 64.0% of itself by ρ = 0.5, because h₁h₁ = h₀ + √2·h₂ and Mehler pairs h₂ with h₂ at ρ². Four interactions are drawn and none of them keeps the zero.

A zero that was an assumption

A rule handed every main effect of both covariates removes exactly none of a pure interaction. That is true at machine precision, it is a fact about independence, and it dies as the square of the correlation.

5 figures · Criterion, part 9 How long a walk has to be given. Every equal split of twelve units is enumerated, the 410 admissible ones are found, the transition matrix is built, and the distance from uniform is computed exactly at each step — no simulation anywhere. The walk is started at the least balanced admissible assignment, which is the state a rejection sampler is least likely to have handed it and the one a burn-in has to cover. It is 0.0849 away after twenty steps and 0.00008 after ninety. A real cost, and a small one, and naming it is what stops it being assumed to be zero.

Walking the admissible set

A rerandomisation test hunts for admissible assignments and throws away the rest. A walk visits them instead — and it is exactly uniform only because it stands still when a proposal fails, which is the step that looks like waste.

3 figures · Randomisation, part 7 Where a walk is cheaper than a hunt. Both costs in the same unit. A rejection sampler evaluates 1/p assignments per independent draw and does not care how large the trial is; a walk evaluates one per step and yields an effective draw every τ steps, and τ is a property of the constraint and the statistic together. They cross at a tolerance of 0.194 standard deviations, where about one assignment in 396 is admissible — far tighter than any trial is designed at. And the walk does not remove the acceptance cost; it pays it once, hunting for somewhere to start.

Draws that repeat each other

A hunt costs 1/p evaluations per independent draw. A walk costs one per step and yields an effective draw every τ steps. Both are counted in the same unit, and the walk is dearer at every tolerance a trial is designed at.

4 figures · Reference, part 8 What exact balance on the main effects removes of the interaction, in a sample. The share of the interaction's in-sample variance explained by the eight main effects — what exact balance on them would remove — averaged over 4,000 samples, at 25, 50, 100, 200, 400 units and correlations of 0, 0.3, 0.5. At 0, whose closed form is 0.0%: 62.1%, 40.3%, 24.5%, 14.9%, 8.4%. At 0.3, whose closed form is 30.3%: 69.8%, 54.2%, 44.4%, 38.4%, 34.8%. At 0.5, whose closed form is 64.0%: 80.7%, 73.1%, 69.2%, 66.8%, 65.4%.

The share a sample removes

A rule balancing eight main effects of two independent covariates removes exactly none of their interaction — in the population. A trial is a sample, and in a sample of a hundred units exact balance on the same eight functions removes 24% of the interaction's variance on average, between 10% and 42% from one trial to the next. At twenty-five units it removes 62%. The zero is the centre of nothing: it is the limit the in-sample share falls towards, by about two fifths with every doubling of the trial, carried there by an interaction whose spread grows with the very functions it is regressed on. At a correlation of a half the closed form says 64%; a trial of a hundred gets 69% on average and anything from 53% to 83%.

3 figures · Criterion, part 10

The weights the corner needs

A fixed-width interval about a difference is exact when the arms share a variance or the allocation ratio is constant, and exact under neither when both fail. It is exact there too, with h_b(λ) = (1/m_A + λ/m_B)⁻¹ — the inverse variances, written as a function of the variance ratio alone, which is a contrast and so is readable by a blinded rule. The estimated precision weights that had no theorem behind them turn out to be that rule at an estimated ratio. And an interval's own scale estimate is right in exactly two cases: inverse-variance weights, and equal ones.

Exact in the corner, where nothing was. Coverage of a nominal 95% interval on five designs, at a required half-width of 0.3. The first four are the two-arm field's own and the fifth is its corner — two variances, block sizes that swing by eight, and an allocation that alternates between five to one and one to five — where neither of that field's two conditions holds. The effective-size weights over-cover there at 98.40%; the weights h_b(λ) = (1/m_A + λ/m_B)⁻¹ cover at 94.84%, and at 94.84% when λ is estimated from the within-arm contrasts rather than known. Nothing here is supposed to move.

Weights that need only a ratio

A fixed-width interval about a difference is exact under either of two conditions and under neither in the corner. It is exact there too, and the only thing it needs is how much larger one arm's variance is than the other's.

3 figures · Nuisance, part 5 How wrong the ratio is allowed to be. λ enters only through the weights, so misstating it leaves the estimate unbiased and moves two things — the interval's calibration and its efficiency — both of which are closed forms of the design. Coverage stays at its level over a factor of two in either direction (94.93% at half the truth, 94.27% at twice it) and starts to go at a factor of five. An estimate on hundreds of within-arm degrees of freedom is never wrong by anything like that, which is what makes the feasible rule usable rather than merely definable.

Blinded, and still exact

The one number the exact interval needs is a ratio of within-arm spreads, which is a contrast and contains no mean — so a rule forbidden to look at the effect may compute it, on more degrees of freedom than the interval itself has.

3 figures · Width, part 6 The weights may not read the block they weight. A weighted least squares decomposition needs weights that are constants, or at least independent of the differences they multiply. One λ̂ pooled across the trial is estimated on hundreds of degrees of freedom and is effectively a constant; a λ̂ estimated inside each block is estimated on that block's own two or three, and is correlated with the difference it weights. Coverage falls from 94.68% to 82.76% — and the interval gets wider while doing it, 0.5163 against 0.3024, which is the signature of weights that are noise.

The condition that cannot be dropped

The weights may not read the block they weight. Estimate the variance ratio inside each block rather than across the trial and the coverage falls to 83% — on an interval that is at the same time seventy per cent wider.

3 figures · Allocation, part 7 One pooled ratio, two periods. The calibration factor of weights built from one variance ratio pooled across the trial — the factor by which the interval's own scale estimate overstates the estimate's variance — when arm B's ratio is F times higher in one kind of period than the other, closed form on the corner's block compositions. With the small A-heavy blocks the noisier, the factor rises: 1.00 at F = 1, 1.50 at F = 1.5, 2.07 at F = 2, 3.12 at F = 3, 3.91 at F = 4, 4.86 at F = 6, 5.44 at F = 8. With the large B-heavy blocks the noisier, it falls: 1.00 at F = 1, 0.74 at F = 1.5, 0.65 at F = 2, 0.58 at F = 3, 0.55 at F = 4, 0.53 at F = 6, 0.53 at F = 8. A ratio estimated per period gives weights that are the inverse variances again, and its factor is exactly 1 at every F.

A ratio that changes with the period

The corner's exact interval needs one variance ratio, and the corner's own alternating allocation makes two kinds of period that need not share one. When they differ the pooled ratio is precise and wrong: at a fourfold difference the interval covers 100.0% one way round and 85.1% the other. A ratio estimated per period is exact again, blind to the effect, and costs under a tenth of the width — and a ratio that merely wanders at random costs width and no coverage at all.

5 figures · Nuisance, part 11 What each stopping rule spends. The average number of observations the corner design takes before it stops, with the stopping rule reading pooled within-arm spreads (the upper bar) and each kind of period's own spreads (the lower), at a promised half-width of 0.3, 3,000 trials each. one ratio: 1,656 and 1,649; 2-fold, small blocks noisier: 1,020 and 1,015; 2-fold, large blocks noisier: 2,976 and 2,614; 4-fold, small blocks noisier: 751 and 669; 4-fold, large blocks noisier: 5,642 and 3,368; 8-fold, small blocks noisier: 711 and 490; 8-fold, large blocks noisier: 10,989 and 3,019. With one ratio the two rules are the same rule. With the large blocks noisier the pooled rule over-spends by a factor that grows with the difference, to 3.6 times at eightfold; with the small blocks noisier it over-spends less.

A stopping rule that reads the periods

A fixed-width trial in the corner stops when its estimated precision reaches the promise, and it estimated that precision from spreads pooled across a trial whose two kinds of period need not share a variance ratio. Read each kind of period's own spreads instead and the trial stops on 3,019 observations where it took 10,989, at an eightfold difference, with the interval it reports still covering 95.1%. The stopping time reads nothing but within-arm spreads, per period now as well as pooled, and that is all the exactness needed.

5 figures · Nuisance, part 12

Estimating the dependence, not naming it

Whitening a sample repairs a criterion, and the whitening that repairs it is told the dependence is a first-order autoregression and left to find one number. A real dependence has no parameter. The obvious estimate — the sample autocovariances, cut off at some lag — is not a covariance matrix on half the draws there are, so the rule built on it does not exist; the tapered estimate that is always a covariance matrix costs a further seven points of what the repair is worth. And the two constructions named as escaping the resampling's ceiling turn out to be one construction and one identity: a moving block attenuates exactly as a blocked multiplier does, because the attenuation is the join.

The price of each thing the rule is not told. What each rule gives up against the best model available, at a persistence of 0.85 on a fifteen-candidate table, over 400 draws. Reading down: least squares with the ordinary penalty; the whitening at the true ρ; the same at a ρ̂ estimated per candidate; that rule with the term the Gaussian likelihood carries and it omits; a Bartlett-tapered Ω̂ estimated once from the fullest candidate at L = 8; the same estimated per candidate; and the truncated Ω̂, which exists on only 45.0% of draws and is averaged over those. Knowing ρ recovers 89.9% of what counting rows gives up, estimating it 83.4%, and estimating a whole covariance 75.6%.

A covariance with no parameter in it

The whitening that repairs a criterion is told the dependence is a first-order autoregression and left to find one number. A real dependence is not one number, and the obvious estimate of it is not a covariance matrix.

4 figures · Dependence, part 5 A window with two wrong ends. The regret of a rule whitened by a Bartlett-tapered Ω̂, as the window widens, against three rules that need no window at all. At L = 0 the estimate is the identity and the rule is exactly least squares — 0.08556, the same number to five places. It falls to 0.01754 at L = 20 and rises again by L = 30, because a quarter of the sample's lags are then being estimated from it. The automatic bandwidth a practitioner would reach for, 4(n/100) to the power 2/9, which at this sample size is 4, gives 0.02963 — 69% above the best available window. The rule told the dependence is an AR(1) sits at 0.01440 throughout, which is the price of not knowing the form.

The window that has to be chosen, and the term that was dropped

An estimated covariance has a bandwidth in it, and both ends of the dial are wrong for different reasons. The rule a practitioner would reach for is two thirds worse than the best window there is.

3 figures · Order selection, part 5 Two constructions on one triangle, and a third that is not. Three resamplings that all keep runs of neighbours, on the same residuals at a block length of 5, with the lags running past ℓ so that the tapers separate. A blocked multiplier never moves a residual; a fixed-length moving block moves every one; and they attenuate identically, worst gap 1.4 standard errors, both sitting on γ_resid(k)(1 − k/ℓ)⁺ and both exactly zero past ℓ — so the attenuation is the block boundary rather than the multiplier. The third is the stationary bootstrap, whose runs are geometric rather than fixed: its taper is γ_resid(k)(1 − 1/ℓ)^k, it agrees with the other two at the first lag and at no other, and at lag 6 it still carries 0.0081 where they carry -0.0005.

The triangle that was not the multiplier's

A resampling that leaves each residual on its own row can keep only what the residuals have, times a triangle. A construction that moves every one of them has the same triangle — and the one in this collection's own table has a different taper entirely.

3 figures · Bootstrap, part 9 What each construction carries, against what there was. The autocorrelation of a resampled error series at five lags, averaged over 60 samples of 40 resamples each. Three facts are in the picture. The residuals lie below the errors at every lag, which is the ceiling a multiplier cannot exceed. The blocked multiplier and the fixed-length block lie on top of each other below it — they attenuate identically, because the attenuation is the join — while the stationary bootstrap, whose runs are geometric rather than fixed, sits above them both. And the sieve is the exception in kind rather than in degree: at lag six it carries 0.0638 where the residuals have 0.0300 and the multiplier has -0.0011, because a fitted model extrapolates past the lags it was told about and a truncated sample sequence cannot.

Errors generated from a fitted model

The one construction that is not bounded by the residuals, because a model extrapolates past the lags it was told about and a truncated sample sequence cannot. It is nearly exact where the only defect is dependence, and it pays for it where there are two.

3 figures · Reference, part 9 The floor a repair has to choose. Regret of the truncated covariance repaired by flooring its eigenvalues, at a persistence of 0.85 on 120 rows, 120 draws, by floor, on a log scale. Window 4: 1.5649 at 0.001, 0.0945 at 0.01, 0.0385 at 0.05, 0.0285 at 0.1, 0.0226 at 0.2, 0.0235 at 0.4, 0.0292 at 0.7; the taper at the same window 0.0275. Window 8: 1.1919 at 0.001, 0.0980 at 0.01, 0.0334 at 0.05, 0.0258 at 0.1, 0.0197 at 0.2, 0.0209 at 0.4, 0.0249 at 0.7; the taper at the same window 0.0178. Window 12: 0.8692 at 0.001, 0.0621 at 0.01, 0.0239 at 0.05, 0.0209 at 0.1, 0.0212 at 0.2, 0.0239 at 0.4, 0.0282 at 0.7; the taper at the same window 0.0150. Window 20: 1.7110 at 0.001, 0.2495 at 0.01, 0.0399 at 0.05, 0.0238 at 0.1, 0.0202 at 0.2, 0.0219 at 0.4, 0.0259 at 0.7; the taper at the same window 0.0135. Least squares gives up 0.0890 and the whitening told the truth 0.0075.

A covariance repaired

A truncated autocovariance sequence fails to be a covariance matrix on most draws of a persistent series, and this line of essays has always scored it on the draws where it survives. The textbook repair is to floor its eigenvalues. At a floor of 0.001 that moves the matrix by four per cent and gives up 1.56 of regret, seventeen times what least squares gives up, because a whitening reads an inverse and an inverse reads the smallest eigenvalues. At a floor of 0.2 it beats the taper at the shortest windows. So the repair is not a repair: it is a second dial.

3 figures · Dependence, part 15

A cut point, at a correlation

The geometry of a balancing dictionary over two dependent covariates is closed for polynomials and was taken to be asymptotic for cut points, because a threshold's Hermite coefficients never terminate. Conditioning on the second variable closes it exactly: every mixed inner product is ρ^j times a one-variable answer and the only two-dimensional object left is an orthant probability, which at the median is (2/π) arcsin ρ. The truncation that was feared falls geometrically in the correlation rather than algebraically in the order — and the interaction guarantee a correlation destroys for powers survives it exactly for median splits, because a two-valued function squares to a constant.

The expansion that never terminates. The Hermite coefficients of a median split, in magnitude, against the reference j to the power −3/4, anchored at the first one. Every even order is exactly zero because sign is an odd function, and every odd order is not, so no truncation is exact — where a polynomial of degree d is exact at any order past d. Summed, the tail past J falls like 1/√J: sixty orders still leave 6.6% of the variance outside. That statement is what made a cut dictionary's geometry unavailable in closed form, and it is a statement about the function against itself. What it is not is the accuracy of an inner product between two correlated variables, where every term past J carries a factor of ρ^m as well.

A cut is not a polynomial, and it does not have to be

A threshold's expansion never terminates, which is why a balancing dictionary's geometry was closed for powers and taken to draws for cut points. Conditioning on the second variable closes it for both.

4 figures · Blocking, part 9 Sheppard's arcsine, by two routes. Corr(sign X, sign Y) as the covariates' correlation runs from zero to one, drawn twice. One route is a sixty-four-node quadrature of the orthant probability over the correlation — the general construction, which works at any pair of cut points; the other is (2/π) arcsin ρ, which is elementary and works only at the median. They agree to 3.3e-16 at every one of 81 correlations, which is what licenses the quadrature everywhere else. The curve is above the diagonal at small ρ and below it at large: two signs agree with probability ½ + arcsin(ρ)/π, so a correlation of 0.5 gives exactly ⅓ and a correlation of 0.8 gives 0.5903.

The arcsine that closes it, and the error that was overstated

Two median splits of a correlated pair agree with probability ½ + arcsin(ρ)/π, exactly. And the truncation the field was avoiding falls geometrically in the correlation, not algebraically in the order.

3 figures · Routes, part 3

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