A symmetry that was not enough

The parity field varies six marginals with the joint law of the ranks held Gaussian and arrives at a clean separating case. A mean split’s interaction zero needs the marginal to be odd — symmetric about its centre — and not normal; the heavy-tailed symmetric covariate, whose skewness is zero and whose tails are far from a normal’s, holds every zero exactly. Its conclusion is one sentence: it is not normality the guarantees needed.

That sentence is true and it is about a Gaussian copula.

The row #

The same covariate, at the same rank correlation, under five copulas.

Under the three radially symmetric copulas it leaks 0.000%, as the parity argument says it must. Under the two asymmetric ones it leaks 14.229%, against 7.707% for a normal covariate under the same copula.

A covariate that costs exactly nothing on its own doubles what the copula costs.

For scale: a covariate skewed at 2.26 leaks 21.539% under a Gaussian copula and 12.118% under a Frank, so this doubling is comparable to the whole spread between two symmetric copulas at a substantial skew — and it is produced by a covariate whose skewness is zero.

What the marginal is #

The heavy-tailed symmetric covariate is $\sinh$ of a standard normal, which is the one transformation in the dictionary that is odd and is not linear. Its skewness is 0.000 to three decimals — computed by quadrature rather than assumed — and its tails are far heavier than a normal’s: it is the case the parity field built to separate odd from normal, since a reader told a guarantee holds for a normal covariate will reasonably suspect normality is doing the work.

It separates them cleanly under a Gaussian copula, and that is what makes its behaviour here worth an essay rather than a line. The one covariate constructed to show that a guarantee does not need normality is the one that shows the guarantee needs something else entirely.

Why parity does not see it #

The parity argument is exact and it is about one thing.

A mean split of a covariate is the indicator that it exceeds its own mean, centred. If the marginal is odd, that indicator is a function of the sign of the latent normal and nothing else, so the interaction of two such indicators is an even function of the latent pair — and an even function is orthogonal to both main effects whenever the latent law is symmetric under a joint sign flip. That is the zero, and it holds for any monotone odd relabelling of the covariate, to the last digit, whatever the tails do.

Every step of that uses the latent law’s symmetry under a joint sign flip, which is exactly radial symmetry of the copula. A Gaussian, a t and a Frank copula all have it; a Clayton does not, and neither does its reflection.

So the parity condition on the marginal is necessary and it was never sufficient. The field that established it said as much — its own last line records the copula being held fixed as the thing that isolates the marginal — and what nobody had is the size of what the other half contributes when the marginal is doing everything right.

What a guarantee stated as a condition on one thing is worth #

There is a general shape here and it is worth naming, because this collection keeps arriving at it from different directions.

A guarantee proved under a symmetry is a guarantee about a pair of assumptions that the proof happens to state as one. The parity argument reads the marginal must be odd, and every step of it also uses the latent law’s symmetry — which is not mentioned, because the latent law was fixed before the argument started and a fixed thing does not look like an assumption.

The same shape appears one field over, where a mean’s zero turns out to need the copula’s reflection symmetry as well as the covariate’s, and it appears again in what a searched charge is a charge for, where a critical value derived for one search is used inside a rule that runs two.

The tell is always the same: an argument that varies one thing and holds another fixed cannot report a condition on the thing it held fixed, and the write-up states the conditions the argument found rather than the conditions the theorem needs. Nothing about that is careless; it is what varying one thing means. What it requires is that somebody eventually vary the other, which is what this field is.

Which is not a correction and is a doubling #

The size is the surprise, and it is worth separating from the direction.

If a symmetric marginal simply failed to protect against an asymmetric copula, the leak would sit at the copula’s own 7.707% — the marginal contributing nothing, as it contributes nothing under a Gaussian copula. That is what “the marginal is fine and the copula is broken” would look like.

It is 14.229%, which is 1.846 times the copula’s own number.

So the heavy-tailed covariate is not neutral under an asymmetric copula. It is actively worse than a normal one, by as much again, while having a skewness of exactly zero and satisfying every condition the parity argument asks for.

The reflection figure is where the mechanism shows. Under a symmetric marginal the copula and its reflection leak the same amount, whatever the marginal’s tails are — which is the parity argument’s symmetry still doing its job, at half strength. The two curves separate only once the marginal is skewed. So the heavy tails move the level and the skew moves the direction, and the two failures are doing different things rather than more and less of one thing.

What the covariate’s fourth moment actually is #

“Tails far heavier than a normal’s” is the sort of phrase this collection prefers to attach a number to, and for $\sinh$ of a standard normal the number is available in closed form.

$$\operatorname{E}\!\left[\sinh^2 Z\right] = \frac{e^2 - 1}{2} = 3.194, \qquad \operatorname{E}\!\left[\sinh^4 Z\right] = \frac{2e^8 - 8e^2 + 6}{16} = 369.30,$$

so the kurtosis is 369.30 / 3.194² = 36.2, against a normal’s 3.

Twelve times a normal’s kurtosis, and a skewness of exactly zero. That is the covariate in one line, and it explains why the parity field could use it to separate odd from normal: it is as far from normal as anything in the dictionary and it is perfectly symmetric.

The refutation of additivity, in its cleanest form #

The row is the sharpest test of additivity anywhere in this line of fields, because one of the two components is exactly zero.

Additivity says the leak of a pair is the marginal’s own leak plus the copula’s own leak. Here the marginal’s own leak is 0.000% — measured, at every radially symmetric copula — and the copula’s own leak with a normal covariate is 7.707%. So additivity predicts 7.707%, a pure pass-through.

Measured: 14.229%. A factor of 1.846, and an excess of 6.522 points contributed by a covariate that contributes nothing.

Everywhere else in the table an excess over additivity can be argued about, because both components are non-zero and a reader can wonder whether the shares were computed on comparable scales. Here there is nothing to argue about. A component that is exactly zero cannot be mis-scaled, and the interaction term carries all of the difference.

What a fourth moment buys, against what a third does #

Setting the row beside the skewed covariates prices the two departures against each other.

A covariate skewed at 2.26 leaks 21.539% under a Gaussian copula — on its own, with no help from the dependence. The heavy-tailed symmetric covariate leaks 0.000% under the same copula and 14.229% under an asymmetric one.

So a fourth moment buys about two thirds of what a large third moment buys, and it buys it on completely different terms: the third moment’s leak is unconditional, and the fourth moment’s requires the copula to be asymmetric as well. A trial with symmetric covariates is safe from the first entirely and safe from the second only if its dependence is radially symmetric — which is the one property of a copula that nothing in a rank correlation reports.

A second reading of the same row #

There is a way of stating the finding that makes it less surprising and it is worth having, because it is the version that transports.

Every quantity here is a squared multiple correlation: the share of a target the balancing rule removes. The target is a product of two centred indicators and the rule holds two main effects. What the rule can remove depends on how much of the product lies in the span of the two main effects, and that is a fact about the joint law of the pair rather than about either margin.

So there was never a reason to expect the marginal’s contribution to be separable from the copula’s. The quantity is a projection in a space the two halves jointly define, and the parity argument is the one special case where the geometry decouples — where an odd marginal and a symmetric copula make the target orthogonal to the span for a reason that does not mention either of them beyond their symmetry.

Outside that case there is no decomposition, and the table’s failure to add is what a projection in a jointly defined space looks like when somebody tries to add two of its margins.

The two asymmetric copulas still agree, and that is the evidence #

One check has to be run before the doubling means anything, and it is the check that separates parity buys less than it looked from parity buys nothing.

Under a skewed covariate the lower-tail copula and its reflection leak 3.431% and 36.213% — a factor of ten, because the direction of the copula’s asymmetry and the direction of the marginal’s skew either agree or disagree.

Under the heavy-tailed symmetric covariate they leak 14.229% and 14.229%: the same number, to every digit the quadrature carries.

So the marginal’s symmetry is still doing exactly what the parity argument says it does. It makes the two tails of the covariate interchangeable, so the direction of the copula’s asymmetry cannot matter — which is why a copula and its reflection cannot be told apart by a symmetric covariate whatever their tails do. What it does not do is make the amount of the copula’s asymmetry cost nothing.

That is a much narrower failure than “the guarantee does not hold”, and it is the one worth carrying: an odd marginal buys invariance to the direction of the dependence’s asymmetry and buys nothing against its size.

What heavy tails are doing #

The mechanism is about where the mass is, and it is the same mechanism as the cancellation the previous essay is about with the direction taken out.

An asymmetric copula concentrates the dependence in one tail of the latent pair. What the mean split’s interaction can leak depends on how much of the covariate’s mass sits where that concentration is. A normal covariate has thin tails, so relatively little mass lies where a Clayton copula’s dependence is strongest; the heavy-tailed one has, by construction, a great deal more.

So the leak is a product of two things: how much the dependence is concentrated, and how much of the covariate is where the concentration is. Parity controls neither. It controls whether the mass is symmetric, which is enough to make the two tails cancel when the dependence is symmetric too, and is exactly nothing when it is not.

What a trial with symmetric covariates should conclude #

The practical reading is narrower than either the original sentence or its qualification.

If the dependence between the covariates is radially symmetric, a symmetric covariate is enough. That is established across six marginals and three copulas at four decimals, and it is not a small class: a Gaussian copula is what a multivariate normal has, a t copula is what heavy-tailed elliptical data has, and a Frank copula is a common flexible symmetric alternative. Most of what anybody fits is in there.

If it is not, symmetry buys the direction and not the level, and the level is worse than a normal’s. A trial cannot tell which case it is in from its covariates’ marginals, because the marginals are exactly what radial symmetry is not about.

And the check that would settle it is a rank statistic nobody computes. The radial gap — how far the copula is from being symmetric under reflection — is 0.0000 for the three symmetric copulas and 0.1359 for the asymmetric pair, and it is estimable from a sample of pairs. Nothing in this collection estimates it, and it is the obvious thing to add to a trial’s covariate screening: a rank correlation says how much dependence there is, and this says whether the guarantees that dependence is being handled by apply at all.

Where the doubling comes from, checked #

The mass argument above predicts something specific and it is worth checking rather than asserting: if the leak under an asymmetric copula rises with how much of the covariate’s mass sits in the tails, then the six marginals should order by their tail weight rather than by their skewness.

They very nearly do. Under the lower-tail copula the leaks are 7.707% for a normal covariate, 14.229% for the heavy-tailed symmetric one, and then 0.002%, 3.431%, 6.162% and 4.543% for the four skewed ones — where the last four are the direction effect and the first two are the size effect with the direction switched off.

Under the reflection the same four skewed marginals run 25.996%, 36.213%, 33.264% and 40.288%, and the ordering among them is not the ordering of their skewnesses either: the covariate skewed at 4.75 leaks less than the one skewed at 2.26. Neither half’s leak is monotone in its own summary statistic, which is the same conclusion the previous essay reaches about additivity, arrived at within a single column.

So the mass argument is consistent with the table and is not established by it. What would establish it is a family of symmetric marginals with tails varying and everything else fixed, and this field has exactly one such marginal.

What the other two rules do in the same row #

The mean split is one of three rules and the row reads differently for the other two, which is worth a paragraph because it says the finding is about the rule rather than about the covariate.

A median split leaks nothing, here as everywhere: under 10⁻¹⁶ at every one of the thirty combinations. Its zero is arithmetic — a centred median split takes the values ±½ so its square is a quarter identically — and neither heavy tails nor an asymmetric copula has anything to break. The last essay of this field is what that is worth.

A threshold at a value leaks less with the heavy-tailed covariate than with a normal one: 14.853% against 16.417% under a Gaussian copula, and 5.417% against 5.331% under a Clayton — barely moving. It never had a zero, so what is measured is a size, and the size is nearly indifferent to the marginal’s tails.

So the doubling belongs to the mean split alone. That is not a coincidence: the mean split is the only one of the three whose guarantee is a symmetry argument, and a symmetry argument is exactly what a second asymmetry can undo. A rule with an arithmetic guarantee has nothing to undo, and a rule with no guarantee has nothing to lose.

What survives of the original sentence #

It is not normality the guarantees needed — and the amended version is worth writing out, because it is shorter than the qualification suggests.

Under a radially symmetric copula, the guarantee needs the marginal to be odd and nothing else, and that is established at four decimals across six marginals whose tails run from a normal’s to an exponential’s. Heavy tails are free there, and a practitioner with a symmetric covariate and no reason to doubt the dependence’s symmetry is exactly as protected as the parity field says.

Under an asymmetric copula the marginal’s parity buys the direction and not the size. The two asymmetric copulas still agree with each other, which they would not if parity bought nothing — a skewed covariate makes them differ by a factor of ten — so the symmetry is doing real work and the work it is doing is smaller than it looked.

And nothing in this collection has a condition on the marginal that is sufficient. The three rules have been varied over six marginals and five copulas, and exactly one of them has a guarantee that survives every combination: the median split, whose zero is arithmetic rather than a symmetry argument, and which is the subject of the last essay of this field.