Why Constants Matter in Distribution Testing:
From Uniformity to Calibration

1. Distribution testing beyond rates

A central lesson from property testing and modern nonparametric statistics is that testing can be much easier than estimation. If \(p\) is an unknown distribution on \(N\) categories, estimating \(p\) in total variation requires order \(N\) samples. By contrast, testing whether \(p\) is uniform can often be done with order \(\sqrt{N}\) samples, up to the dependence on the separation parameter \(\epsilon\).

The emphasis on constants is familiar from other parts of statistics. In regular parametric estimation, the \(n^{-1/2}\) rate is only the first-order message; the Fisher information determines the sharp asymptotic variance and therefore distinguishes between experiments and estimators with the same rate. Similarly, in nonparametric estimation, minimax rates describe how risk scales with sample size, but Pinsker's constant gives the sharp asymptotic benchmark for the leading minimax risk over smoothness classes; see, for example, Nussbaum (1999). These constants are not cosmetic refinements: they are the quantities that turn asymptotic theory into quantitative risk predictions. The same principle applies in distribution testing. Once the detection rate is known, the next question is the sharp constant governing the limiting risk.

The testing-versus-estimation phenomenon was made especially clear by Paninski (2008), who showed that in the sparse regime the number of repeated observations, or collisions, contains enough information to detect nonuniformity even when most categories are unobserved.

This rate-level perspective has been extremely successful. It tells us when testing is possible and when it is impossible. However, it does not fully determine the statistical performance of a concrete test. Two tests may both require

samples and yet have noticeably different error probabilities at the same \(n,N,\epsilon\). In other words, sample-complexity rates can identify the right scale while still hiding the most practically relevant comparison.

A sharper question is therefore:

This question is not only a refinement of theory. It changes the way we compare tests.

2. A simple Gaussian analogy

The importance of constants is easiest to see in the Gaussian testing problem

Here it is natural to decide between \(H_0\) and \(H_1\) based on whether the test statistic \(X\) exceeds a threshold \(t\). The risk, denoted \(R(t)\), is the sum of the Type-I and Type-II errors:

The threshold that minimizes the risk is \(t = u/2\), so the minimal risk is

Thus, the value of the signal-to-noise ratio \(u\) directly determines the risk. For example,

whereas

From a rate perspective, both are just fixed nonzero signal-to-noise ratios. But from a risk perspective, they correspond to very different testing performance.

Sharp-constant distribution testing seeks analogous formulas for high-dimensional discrete testing problems. The goal is not only to say that a problem becomes solvable when a certain normalized separation diverges, but to identify the precise signal-to-noise ratio that determines the limiting risk.

3. Uniformity testing problem

Uniformity goodness-of-fit testing is the cleanest setting in which we ask these questions. The data \((V_1,\ldots,V_n)\) are i.i.d. from a distribution \(\mathbf{q} = (q_1,\ldots,q_N)\) on \(N\) categories, and the null hypothesis is

Under the alternative, \(\mathbf{q}\) is separated from uniformity in some metric, for example total variation, \(\ell_2\), or more generally \(\ell_p\).

Many test statistics have been proposed for this problem. The most common are the collision test, the chi-squared test, and the empirical total variation test. At the rate level, several of these procedures can be optimal or nearly optimal in appropriate regimes. But they are not equivalent at the constant level. Gupta and Price (2022) make this point explicitly in their work on sharp constants for uniformity testing via the Huber statistic: theoretical rate analyses can fail to rank the empirical performance of common uniformity testers, and constant factors explain part of this discrepancy.

This suggests that the right asymptotic object is not only the detection boundary, but the limiting risk curve.

4. Sharp minimax risk

A sharp minimax formulation asks for

where \(\mathcal{A}_\epsilon\) is the alternative class and \(\psi\) ranges over all tests. One may also invert the minimax-risk curve to obtain two familiar quantities: the minimax separation radius and the sample complexity. For a target total risk \(\gamma\in(0,1)\), define the \(\gamma\)-minimax separation radius at sample size \(n\) by

Similarly, for a fixed separation level \(\epsilon\), define the \(\gamma\)-sample complexity by

In large-alphabet uniformity testing against \(\ell_p\)-separated alternatives, Kipnis (2026b) characterizes this minimax risk at the constant level.

In the remainder of this note, we specialize to \(p=1\), both for simplicity and to match the common \(\ell_1\) formulation used in much of the distribution-testing literature. Throughout, \(\epsilon\) denotes \(\ell_1\) distance from uniformity, \(\|\mathbf{q}-\mathbf{u}\|_1=\sum_i|q_i-1/N|\), which is twice the total-variation distance.

In the Poissonized model, in the intermediate regime where the risk converges to a nontrivial constant, the limiting risk takes the Gaussian form

where \(u\) is the effective signal-to-noise ratio.

With the notation and normalization of Kipnis (2026b), this signal-to-noise ratio may be written in the form

In the small-signal regime, using \(\sinh(x)\sim x\), this becomes

Thus the high-dimensional goodness-of-fit problem behaves, at the level of minimax risk, like a one-dimensional Gaussian testing problem.

The analysis also identifies the asymptotically minimax test. The optimal statistic is a linear functional of the occupancy histogram,

for \(m=1,\ldots,n\). In the very sparse regime, the optimal statistic is driven by collisions \(Z_2\). In denser regimes, it behaves more like a chi-squared statistic. This provides a constant-level explanation of why collision and chi-squared tests can each be natural, but in different sampling regimes.

5. What constants reveal that rates hide

Sharp constants add three pieces of information that rates alone do not provide.

First, they distinguish between tests that have the same detection boundary. If two tests both succeed when

rate theory treats them as equivalent. Sharp-risk theory can show that one has a substantially smaller limiting error probability than the other.

Second, constants identify the correct signal-to-noise ratio. In the large-alphabet uniformity problem, the relevant quantity is the effective signal-to-noise ratio \(u_{\epsilon,n,N}\). Once \(u_{\epsilon,n,N}\) is known, a risk target can be translated directly into a sample-size requirement.

Third, constants make asymptotic theory operational. Suppose we want total minimax risk at most \(\gamma\). The Gaussian formula gives the requirement

or equivalently

For a level-\(\alpha\) test with Type-II error at most \(\beta\), the analogous Gaussian approximation gives

where \(z_{1-\alpha}\) is the \(1-\alpha\) quantile of the standard normal distribution (i.e., \(\Phi(z_t)=t\)).

6. From uniformity testing to calibration testing

Calibration assessment in machine learning provides a natural setting in which sharp constants become operational. Consider a predictive model that outputs a conditional distribution function \(\hat{F}(y\mid x)\). Given a test observation \((X,Y)\), define the probability integral transform (PIT)

Under perfect distributional calibration, the PIT is uniform:

Thus testing calibration of a continuous predictive model can be reduced to a uniformity testing problem for PIT values.

Let \(Q\) denote the distribution function of \(V\), and suppose it has density \(q\). The Expected Calibration Error (ECE) is a natural population-level calibration distance:

For \(p=1\), this is the area between the PIT density and the uniform density.

In practice, however, calibration is rarely tested directly in the continuum. The PIT values are binned. If \(N\) equal-width bins are used, with edges \(0=t_0<t_1<\cdots<t_N=1\), then the bin counts

are compared to the uniform baseline \(n/N\). Writing

the binned calibration problem becomes the multinomial goodness-of-fit problem

The corresponding discretized calibration error is

This formulation is developed in Kipnis (2026a), which uses sharp minimax uniformity testing to derive optimal binning and minimax calibration tests for continuous predictive models.

7. Binning creates two competing errors

The number of bins \(N\) is not merely a plotting choice. It is a statistical resolution parameter. Choosing \(N\) too small creates discretization bias: localized or oscillatory departures from calibration may cancel inside wide bins. Choosing \(N\) too large creates statistical noise: the \(n\) test samples are spread too thinly across bins, and the goodness-of-fit test loses power.

The smoothing effect of binning is visible from the identity

By the triangle inequality, we have

Thus,

Binning can only reduce the apparent calibration error. If \(q(v)-1\) oscillates within a bin, positive and negative deviations may cancel, making a miscalibrated predictive model appear nearly calibrated at that binning resolution.

This is the first side of the trade-off. The second side is statistical. For a fixed binned separation level \(\epsilon\), the relevant signal-to-noise ratio for minimax calibration testing is, in the small-signal regime,

Thus, once \(N\) is large enough to resolve the miscalibration, increasing \(N\) further reduces the signal-to-noise ratio by spreading the same sample size over more bins.

8. Sharp constants give an optimal binning rule

The sharp minimax theory gives a direct way to translate a desired testing guarantee into a maximum number of bins. For the binned calibration problem, consider the minimax risk

In the asymptotic regime where

the minimax risk satisfies

Moreover, this risk is attained by the statistic

This statistic is closely related to chi-squared, but the subtraction of \(O_j\) stabilizes the variance in sparse large-\(N\) regimes. In particular, the variance of \(T_N\) remains constant even if \(n\) is random with a Poisson distribution, so \(T_N\) is robust to sample-size fluctuations.

Given a target total risk \(R\), the condition

is equivalent to

Solving this inequality gives the maximum number of bins

Equivalently, if one wants a level-\(\alpha\) test with worst-case Type-II error at most \(\beta\), the Gaussian approximation gives the condition

and hence

This formula makes precise the design principle:

Using fewer bins may hide continuous miscalibration through bin averaging. Using more than \(N_{\max}\) may resolve the miscalibration visually but make it statistically undetectable.

9. An oscillatory calibration example

The binning trade-off is especially clear for oscillatory miscalibration. Suppose a miscalibrated predictive CDF perturbs the true conditional CDF in probability space according to

where \(k\) is the number of oscillation cycles and \(a\in(0,1)\) is the amplitude. This is a valid CDF perturbation and corresponds to a miscalibrated quantile map. The PIT density is

with CDF \(Q(v)=v+\tfrac{a}{2\pi k}\sin(2\pi k v)\). This density oscillates around the uniform density. Its continuous \(\ell_1\)-calibration error is

However, after binning into \(N\) equal-width bins, the apparent calibration error is attenuated:

Thus, when \(N\) is too small relative to the oscillation frequency \(k\), the miscalibration cancels inside bins and the binned ECE is close to zero. When \(N\) is very large, the oscillation is resolved, but the expected count per bin \(n/N\) becomes small and sampling noise dominates.

For example, Kipnis (2026a) considers

for which

For \(p=1\) and target minimax risk \(R=0.1\), the sharp-constant formula gives

The resulting risk curve is illustrated in Figure 1.

With too few bins, bin averaging hides the oscillatory miscalibration. Around the intermediate resolution, the oscillation is resolved and the test has power. With too many bins, sampling noise dominates and power is lost again.

This example illustrates why constants matter. A rate-level statement says that testing is possible at the right asymptotic scale. The sharp-constant formula says how fine the binning can be while still achieving a desired risk level.

10. A broader lesson

Uniformity testing is often considered a solved problem because the sample-complexity rate is known. Calibration testing is sometimes treated as an engineering problem because binning appears to be a practical detail. Both views miss the same point: constants matter when theory is used to choose a procedure.

Sharp constants do not merely refine asymptotic rates. They identify the effective Gaussian experiment hidden inside a high-dimensional testing problem. They explain why different statistics are optimal in different regimes. And they turn qualitative detectability statements into quantitative design rules.

From this perspective, uniformity testing is not only a classical benchmark problem. It is a laboratory for understanding how to design reliable tests in modern data-analysis pipelines.

Calibration binning is one such pipeline. The number of bins determines the resolution at which miscalibration can be seen, but also the statistical power with which it can be detected. Sharp constants provide a principled way to navigate this trade-off.

11. Takeaway

In distribution testing, this distinction matters because many natural procedures share the same rate but differ in risk. In calibration testing, it matters because binning choices determine whether a finite validation set can actually detect the miscalibration one hopes to measure.

Sharp constants therefore provide a bridge from asymptotic theory to statistical practice. They turn large-alphabet uniformity testing from a purely theoretical problem into a guide for choosing the resolution of calibration diagnostics.