Concept

Critical value — where it appears

The threshold a statistic must pass for a test to reject, fixed by the level and by the statistic's distribution under the null. Reading a statistic against a critical value from a different procedure is the single most common way a correct computation produces a false statement.

Named by 40 essays across 25 fields — each of them below, with the objects they name alongside it.

A break that was looked for

A two-regime whitening finds its change point by maximising a profile, and then reads a criterion that counts parameters. Under no break there is no parameter to count, because every position describes the same model.

charged · Break point

A quantile is the dearer reading, everywhere. The error each rule and window delivers on the two error readings, over 400 draws. The lower pair of lines is the implied long-run variance — the instrument the earlier field uses — and the upper pair is the 95% point of the standardised resampled mean, read against the finite-sample truth of 3.889 found by simulating the law directly. The quantile costs more at every one of the eight cells: at the plug-in rule it is 59.1% against 45.3% for the taper. That is not a defect in the bootstrap; a quantile is a statement about the shape of a distribution as well as its scale, and a fixed number of resamples estimates a tail worse than a variance. What matters for the comparison is that the two orderings between the windows are not the same, which the margins figure is about.

The instrument and the reading

Every comparison between two block windows in this collection is an error in an implied long-run variance. Nobody reads a long-run variance. Read on the 95% point a test uses, the same bootstrap costs half as much again.

readout · Bootstrap

Two searches, one sample

A searched break in a regression manufactures 34.7 of likelihood ratio where a count of coefficients says 11.1. A searched window manufactures 84.0. The two together manufacture 99.4, not 118.7.

twice · Break point

Two instruments, two block lengths. The block length that would actually have been best on each draw, for each of the two error readings, averaged over 400 samples of 120 rows. For the rectangular window the implied long-run variance wants 18.92 and the 95% point wants 16.05; for the tapered window, 21.82 against 17.74. The quantile wants a shorter block under both windows — a ratio of 0.848 and 0.813. That is the mechanism the whole field turns on: a rule for choosing a block length is a way of guessing a target, and the two instruments do not have the same target. A rule tuned to one is systematically long for the other, and the two windows do not pay the same price for being long.

A length for each instrument

The block length that is best for an implied variance is 18.92; the one best for the 95% point of the same resamples is 16.05. A rule is a way of guessing a target, and there are two targets.

readout · Bootstrap

A null with a model in it

The distribution to read the winner of a table against cannot be resampled from the data, because the data does not contain the null. It has to be generated from a model — which is the assumption the resampling was chosen to avoid.

search · Bootstrap

How far each reference distribution's 95% point falls short. Seven constructions on rows that repeat each other, at a block length of 5 and 200 draws, against the statistic's own 95% point of 3.0224 computed from three thousand draws of the same world. Reading down: a multiplier on every row keeps no dependence at all and is 44% short; a multiplier shared along a block keeps the triangle; a fixed block keeps the same triangle and is 8% closer, which is the pair that says a taper is not what decides this; the stationary bootstrap; the two tapered blocks, both further short than the untapered one at this block length; and errors generated from a fitted model, which is the only construction here not bounded by what the residuals report.

A taper and a critical value

Two constructions whose tapers visibly differ give the same critical value, and two that share a taper exactly do not. Adding a construction whose taper is a decision rather than an accident says which half of that is true.

taper · Reference

The charge that is not a sum

Charging two searches what each costs on its own is conservative, and conservative here means the test never fires. At the largest break measured it declares nothing, on every draw, while a calibrated threshold reaches 29%.

twice · Break point

Three readings, one verdict. Every tolerance of a fourteen-unit trial, with three comparisons on each. The first is between a chain started at an assignment and a chain started at its complement, which is what a mirror split separates. The second is between two chains started at the same assignment on different streams, which nothing about the set can separate — so a large reading there says the run is too short and not that the set is in pieces. The third is the same comparison on the statistic's absolute value, which is symmetric under the complement and therefore blind to the split by construction. The verdict is the pattern rather than any one line: the split is called only where the first fires and the other two do not, which happens at exactly the tolerances the enumeration calls disconnected — 0.8, 0.75, 0.7.

The statistic that changes sign

A test for an unreachable half needs a quantity that tells one half from the other. Every symmetric reading of a mirror pair is identical, and a magnitude is the natural thing to reach for.

reach · Randomisation

What a search costs in parameters

An information criterion's penalty is an estimate of the optimism a fit carries. For a break point the optimism can be measured and cannot be counted, and it comes to about two and a half parameters.

charged · Break point

The same data, one regression per choice of left-hand side. The two-step procedure has to put one series on the left, and with 3 series there are 3 ways to do it. Each returns a relation and a residual test; the 5% point is -3.71, simulated. Here they do not agree: 2 of 3 reject, and the relations they report are written with a 1 in the position of whichever series was on the left, so they can be compared. Nothing in a printed output records which regression was run.

Which series goes on the left

The two-step procedure has to pick a series to regress the others on, and nothing in its output records which. With a pair that choice never changes the verdict. With three series and one relation between them, the three choices disagree about whether the system is cointegrated at all 98.0% of the time.

systems · Rank

A charge that depends on the rule

The break search's charge is 34.7 on its own and 15.4 once a window has been chosen from the same sample. Most of what a break search finds under correlated errors is the correlation, and a whitening has taken it already.

twice · Break point

The trace statistic under the null, and the 5% point it needs. 600 systems of 3 unrelated random walks, each put through the reduced-rank regression, with the statistic for "rank ≤ 0" collected. The 5% point is 31.91. There is no standard table to look that up in: the distribution depends on the number of common trends under the null and is not a chi-square, so the value is simulated on one set of seeds and applied on another — exactly the position the pair's residual test was in one field ago.

Counting what is still wandering

The statistic that turns a spectrum into an integer has one name and three distributions. Its 5% point is 8.12, 18.64 or 31.74 depending only on how many series are left wandering under the null being tested — and read against the wrong one of those three, it calls unrelated random walks cointegrated most of the time.

systems · Rank

The best of 8 arms, tested as though it were the only one. 8,000 trials with no effect in any arm. Stage one runs 8 arms at 60 each, the best is carried forward, and stage two adds 30 more to it and to the control. The histogram is where the final statistic lands and the curve is the standard normal it is being read against — shifted right, because the arm was chosen for being ahead. 10.5% of these trials clear 1.96 against a claimed 5%, and the value that actually holds the rate for this design is 2.313.

Dropping the losers

Carrying the best of eight arms forward and testing it at 1.96 rejects a true null 10.3% of the time — the hypothesis was chosen by looking at the data, so the statistic is a maximum wearing a single comparison's clothes. The value that holds the rate is 2.313, and it has to be solved for.

adaptive · Stopping

What each instrument costs to read. The number of draws each instrument needs to separate a rectangular block from a trapezoidal one at two standard errors, at a block length of 20 and 120 rows — measured from each instrument's own spread on the same draws. The implied variance needs 7.0 and the 95% point needs 20.2, a factor of 2.90 at this block length. There is a closed form beside it and it does not depend on either the scale or the size of the gap: the standard error of a p-quantile is √(p(1−p))/f(q) over √B where a standard deviation's is σ/√(2B), which at the 95% point of a nearly normal reference distribution is 3.30 times as many draws for the same statement. And the quantile route needs every one of those draws resampled, where the variance route needs none.

Measuring a variance rather than a quantile

A resample's implied long-run variance can be computed from the sample with no resampling in it at all. A critical value cannot, and the difference is a factor of three in the draws before any of the resampling is counted.

crossing · Bootstrap

Residuals that keep their own variance

A reference distribution for a search has to be generated from a fitted model, and the generator draws residuals. Four ways of drawing them keep four different things — and the one this site has reached for three times repairs nothing at all here.

select · Bootstrap

Four rules of four change sign. The margin between the two block windows in points of coverage, under each of four rules, on each of three intervals built from the same resamples, over 300 draws. Positive is the tapered window covering better. On the percentile interval the taper wins at all four rules, by 5.33, 1.67, 5.00 and 4.00 points, which is the earlier field's own reading. On the studentised interval the rectangle wins at all four, by 4.33, 7.00, 4.33 and 2.67. And a normal interval, which uses no resampling at all, puts the two within a third of a point at every rule — so the disagreement is manufactured entirely by what is done with the resamples.

The ordering reverses again

One field found two of four rules changing sign between two readings of one resampling. Turn the same resamples into a studentised interval instead of a percentile one and all four change sign.

student · Bootstrap

Where the +1 matters, and why nobody has noticed that it does. The true size of the two rules at every B, computed rather than simulated: under the null the count of re-randomisations reaching the observed statistic is uniform over {0 … B}, so both sizes are integer arithmetic. With the +1 the size is (⌊α(B+1)⌋)/(B+1), which never exceeds 5%. Without it the size is (⌊αB⌋+1)/(B+1), which is larger except at B = 19, 39, 59 — the values with B + 1 a multiple of 1/α, and the values everybody uses. At B = 19 the two rules are the same rule; at B = 20 the uncorrected one is a 9.5% test. The marks are simulated on 500 trials of 120 patients, as the second route to the same numbers.

The plus one and the round number

A sampled randomisation test counts the observed allocation as one of its own reference draws, and the correction is invisible at B = 19, 39, 59 and 999 — every value anybody uses. At B = 20 the version without it is an 8.00% test where the corrected one is 3.80%, and the convention protecting everybody is a preference for round numbers minus one.

exact · Reference

The reversal that was the instrument's

On an implied variance the rectangle wins at a protocol length and at the rule of thumb. On the 95% point a test reads, and on the coverage an interval delivers, the taper wins at all four rules.

readout · Bootstrap

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.

banded · Bootstrap

Three quarters of the way to one search

The pair that started this reads 0.762 on a scale whose one is containment. And the pair that shares nothing but its response reads −0.306, so the sign the earlier field found does not transport at all.

apart · Criterion

Two defects and one resampling

Four resamplings, each the repair for one defect and wrong about the other. Put both defects in the same world and the statistic's 5% point is 3.8028, where the best of the four reaches 2.8326 — until a multiplier that stays on its own row and shares a sign with its neighbours reaches 2.9988.

proxy · Bootstrap

The interval that over-covers when the instrument fails. Counted coverage of two nominal 95.0% intervals for the same causal effect, read off the same 2000 draws of 200 rows at each first stage. The exact Anderson–Rubin set covers 95.3% at every setting — flat, because the statistic it inverts is built from y − tβ, which contains no π at all, and is therefore the same number on the same draw whatever the instrument is worth. The conventional interval covers 99.1% at π = 0.02 and 95.6% at π = 0.6: it goes wrong at the weak end by covering too MUCH, at a median width of 7.320, because its standard error is computed from residuals taken at an estimate that has itself gone wrong. A weak instrument does not make this interval lie about its coverage; it makes it useless while telling the truth.

What the first stage does not know

A single weak instrument does not make the conventional interval undercover — it makes it cover 99.1% at a width of 7.320. Where the promise actually breaks is many instruments — coverage falls from 97.2% to 51.5% while the median width falls from 1.454 to 0.583.

instrument · Exclusion

What a fixed-width interval covers, by the number of blocks the trial ran before it stopped. Two thousand runs of each rule, the modelled weighting, a promise of 0.34. Reading its report: 4–8 blocks, 22.3% of runs, 78.2%; 9–12 blocks, 16.6% of runs, 90.4%; 13–16 blocks, 18.4% of runs, 96.2%; 17–20 blocks, 17.4% of runs, 96.0%; 21–28 blocks, 17.9% of runs, 96.4%; 29–36 blocks, 7.4% of runs, 99.3% — 91.45% overall. Reading the arms: 4–8 blocks, 0.0%, none; 9–12 blocks, 0.9%, 94.4%; 13–16 blocks, 30.4%, 95.6%; 17–20 blocks, 50.0%, 93.9%; 21–28 blocks, 18.0%, 94.4%; 29–36 blocks, 0.7%, 92.3% — 94.50% overall.

The trials that stopped early

A fixed-width trial that stops when its own interval is short enough covers 91.45% — an average of 78.2% among the 22.3% of runs that stop within eight blocks and 96% to 99% among those that run longer. Widening every interval by 17.1% brings the average to 95% and leaves the early stops at 85.6%, while 92.8% of runs now report an interval wider than the width they promised. Even doubling every interval leaves the early stops short.

stop · Width

A second break on a flat profile

Searching a hundred and twenty rows for one change point manufactures five units of likelihood. Searching for a second manufactures four more, on a series that has at most one — and on a profile whose whole range is under seven.

charged · Break point

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.

joint · Reference

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.

banded · Reference

A bias against a variance, with the answer in between. How wrong one sample's reference distribution is, split into the two things it is wrong by. Sharing the multiplier over more rows keeps more of the dependence and closes the bias from 1.688 to 0.835; every row it is shared over also removes an independent sign from the 101 the sample started with, and the spread of the resulting quantile rises from 1.307 to 2.172. The distance a practitioner with one sample is actually exposed to is the two together, and it is smallest at ℓ = 5.

How long a block a multiplier shares

Sharing a sign over more rows keeps more of the dependence and leaves fewer independent signs to build a distribution from. The bias falls from 1.6885 to 0.8479 and the spread rises from 1.3073 to 2.1716, and the rejection rate walks straight through its nominal level on the way from 11.3% to 1.3%.

proxy · Reference

What the diagnostic says at two hundred units. The same test run 8 times on independent streams, at five tolerances of a two-hundred-unit trial, 40,000 steps each. At the loosest tolerance every run says the same thing — the walk reaches the whole set — and it keeps saying it as the set is thinned. Past a point the runs stop agreeing with each other: at the tightest tolerance here 6 of 8 report that the chains have not mixed and 2 report a split, which is a diagnostic disagreeing with itself rather than a property of the set. That disagreement is the honest answer at this size, and it is one nothing in this collection could give before: an enumeration stops at about twenty-four units.

The diagnostic at two hundred

Pointed at a trial size no enumeration reaches, the test gives three answers rather than one — and past a certain thinness it stops agreeing with itself, which is the honest reading and the one nothing could give before.

reach · Randomisation

A thin enough set is not one set. Every admissible set of 14 units this table can enumerate, by how much of the assignment space it admits and how many pieces it falls into under single swaps. A walk is uniform on the piece it starts in and never leaves it. The pieces are not fragments: at 522 admissible assignments the set splits into 3 halves of exactly 520 each, and every assignment's complement is in the other half — no sequence of admissible single swaps takes an assignment to its own mirror image. Two-swap proposals reconnect four of the six disconnected sets here — the two they do not are the thinnest, where a two-unit move rarely lands anywhere admissible either — which makes a bigger proposal a correctness repair rather than the speed dial it was measured as.

The walk that cannot cross

A thin enough admissible set is not one set. It splits into an assignment and its mirror image, no sequence of admissible single swaps joins them, and the walk that samples it is uniform on half the reference distribution for ever.

dict · Randomisation

What a reference distribution costs to sample

A randomisation test on a trial too large to enumerate has to sample its reference distribution, at 1/p attempts per draw and a p-value resolved to 1/(B + 1). Six constraints cost 9,878 attempts per thousand draws, and a thousand draws resolve p to 9.99·10⁻⁴ and not one digit finer.

product · Reference

The cheap repair needs a number nobody has. The obvious alternative to re-randomising is to simulate the design under its null once and use the critical value that comes out — which is what the arm-dropping design does, where the critical value has to be solved for and is 2.313. It does not transfer here. The rule chases outcomes, so how imbalanced the allocation gets depends on how often anything succeeds, and the critical value moves from 1.668 at a success rate of 0.05 to 2.718 at 0.8. Calibrated at 0.3 and used at 0.8 the test's real size is 12.4%; used at 0.05 it is 0.12%. The randomisation test needs none of this, because it conditions on the outcomes that happened rather than on a rate they were supposed to come from.

What the exactness buys

Against a z test calibrated to reject exactly 5% of true nulls on this design, the randomisation test loses nineteen points of power. What it buys is that the calibration needs the success rate — which moves the critical value from 1.668 to 2.718 and is the quantity the trial was run to find out.

exact · Nuisance

What the interval covers

Eight rules and windows, and not one of them reaches its promised 95%. The range is 80.8% to 91.0%, and the choice between two block windows is a choice inside a shortfall that is four times larger.

readout · Bootstrap

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'.

effective · Reference

The same error, caught or invisible, by how it is arranged. The overidentification test's rejection rate against the error the violation actually puts into the estimate, so the two rows are the same estimate being equally wrong. With the whole violation on one instrument the test keeps its size at 5.0% under the null and reaches 86.4% by an error of 0.800. With both instruments violating in the same ratio as their first stages the two Wald ratios are identical, the test has nothing to compare, and it rejects at 5.8% at that same error — its own size. Over 1000 draws of 300 rows at each setting, at a nominal 5.0%. The test is a comparison between instruments and it was never a check on either.

Two instruments that disagree

The overidentification test keeps its size at 5.0% and reaches 86.4% power against a violation carried by one instrument. Against the same error carried by both in proportion to their first stages it rejects on 4.6% of draws — its own size — while the estimate is wrong by 0.3000, which is 94.2% of the confounding the instruments were brought in to remove.

instrument · Exclusion

How long an honest forecaster looks broken for. The calibration error shown by a forecaster with no miscalibration in it at all, at five record lengths, drawn against one over the square root of the length so that the closed form is a straight line through the origin. It is 0.1252 at fifty forecasts and 0.0090 at ten thousand, against a closed form of √(2K/πn) times the mean root bin variance which gives 0.1257 and 0.0089. The threshold drawn across it is 0.02, a figure routinely read as evidence that something is wrong; the mean falls under it at 1976 forecasts and the 95th percentile at about 4111. Below that, an honest forecaster and a miscalibrated one are being told apart by a statistic that is mostly the sample size.

The miscalibration a perfect forecaster shows

A forecaster whose true reliability is exactly zero shows a calibration error of 0.1252 on fifty forecasts and 0.0090 on ten thousand. Every one of 1,200 blameless hundred-forecast records exceeds the 0.02 routinely read as evidence of a problem, and the mean does not fall under it until 1,976 forecasts.

calibrate · Calibration

The damage and the warning, against the same dial. Two readings at each persistence. In the darker colour, how often a regression between two independent series of 200 steps is called significant at 5%: 4.9% at φ = 0, 34.2% at 0.8, 52.4% at 0.9, 83.4% at a unit root. In the lighter, how often the standard unit-root test refuses a unit root on one of those series — the chance the analyst is told the series is stationary and may be regressed: 87.2% at φ = 0.9 and 31.9% at 0.95. At φ = 0.9 both are high at once, which is a correct diagnostic licensing a regression that is wrong half the time.

The cliff that is a slope

A regression between two independent series is called significant 4.9% of the time at no persistence, 52.4% at a lag-one correlation of 0.9, and 83.4% at a unit root. The rule the field offers asks whether the last of those holds, and at 0.9 the unit-root test correctly refuses one 87.2% of the time.

timeseries · Spurious

The bounded error and the unbounded one. How the sequential trace procedure's answer is distributed, against the sample length, for a three-series system with 2 genuine relations. Over-counting — claiming a stationary combination that is a random walk — reads 4.9%, 7.2%, 5.7%, 6.2%, 5.9%, 4.2% across the six lengths, never far from the 5% of a single test. Under-counting reads 69.5%, 40.2%, 14.0%, 0.5%, 0.0%, 0.0%. The procedure is described as a 5% rule and the 5% applies to one of those columns.

The rank is a decision

The sequential procedure's 5% bounds one of its two errors. Over-counting reads between 4.2% and 7.2% at every sample length from fifty observations to three hundred; under-counting reads 69.5% at fifty and 0.0% at three hundred, and nothing in the procedure bounds it.

systems · Rank

The difference in restricted mean survival at every horizon, in three worlds. Treatment minus control, in closed form, with dropout irrelevant to the truth. The proportional treatment's difference grows to 0.4766 at τ = 3 and the waning treatment's to 0.2675. The crossing treatment's rises to 0.1776 at τ = 2, near where the two survival curves cross, and falls back to 0.1366 at τ = 3. The ticks along the bottom are the eleven horizons, from 0.5 to 3 in quarters, at which a trial below reads its differences.

A horizon chosen after looking

A difference in restricted mean survival read at whichever of eleven horizons looks most convincing rejects 11.24% of trials in which the treatment does nothing, against 4.70% at a horizon fixed in advance. The correlation of the differences across horizons is closed, and the Gaussian process it defines prices the choice at a critical value of 2.317 — which brings the counted size back to 4.99% and keeps 96.92% of the power that a horizon nobody could have known to fix would have had.

survival · Censoring

How slow a return a sample can see

At two hundred observations the test finds a gap that halves in five steps four times in five, one that halves in eight 37.3% of the time, and one that halves in fifty 4.95% of the time — which is the rate at which it finds pairs with no mechanism at all. The boundary moves with the sample, not with its square root.

timeseries · Spurious

Three detectors for one departure, all at 5%. How often each of three checks on the calibration scores fires, against the size of the drift, with every critical value simulated under no drift so that all three sit at 5.0% exactly. The incumbent — a rank comparison of the first half of the scores against the second — reaches four-in-five power at a growth factor of 4.31. Reading each score's rank against its position reaches it at 2.65, and the largest running departure of the scores from their mean at 2.12. The ordering of the three is the ordering by how much of the sample's arrangement each one uses.

A detector built for the ordering

The best of three checks for a drifting scale fires at half the growth factor the standard one needs — 2.12 against 4.31 — and still leaves 6.50 points of coverage gone before it does, against 0.51 for serial correlation. The reversal was not a property of the test.

conformal · Exchangeability

Named alongside it

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

Monte Carlo Reference distribution Block bootstrap Error rate Dependence Resampling Specification search Long-run variance Statistical power Persistence Autocorrelation Heteroskedasticity

All concepts