45.04.11 · mathematical-statistics / 04-asymptotic-statistics

Cube-Root Asymptotics and Chernoff's Distribution

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Anchor (Master): Kim & Pollard 1990 Cube root asymptotics (Annals of Statistics 18, 191-219); van der Vaart 1998 Asymptotic Statistics (Cambridge) §5.8 and §3.2 (argmax continuous mapping); Groeneboom 1989 Brownian motion with a parabolic drift and Airy functions (Probability Theory and Related Fields) and Groeneboom 1985 Estimating a monotone density (the switch relation, Chernoff's distribution as the law of the argmax of two-sided Brownian motion minus a parabola)

Intuition Beginner

Most estimators you meet settle down at a familiar speed: collect four times as much data and the typical error halves. That is the square-root rate, and it is what the standard theory of the previous unit delivers when the quantity you optimise is a smooth hill with a rounded top. You climb to the top and read off the answer, and the rounded top is what makes the answer precise.

Some natural estimators do not have a rounded top. Imagine looking for the busiest stretch of a road by sliding a fixed-width window along it and finding where the most cars sit inside. The count inside the window jumps as cars enter and leave; the "best" position is decided by a few cars right at the edges of the window, not by a gentle curve. An estimate pinned down by edges is shakier than one pinned down by a smooth peak, and it improves more slowly: you need eight times the data, not four, to halve the error. That is the cube-root rate.

This unit is about exactly these slow, edge-driven estimators. The mode of a distribution, the position of the densest short interval, a certain rule for binary-choice prediction, and the staircase fit to a density that is known to only go downhill all behave this way. They share one striking feature: after you zoom in by the right amount, the random wobble around the answer always settles into the same fixed shape, regardless of which problem you started from. That shape has a name — Chernoff's distribution.

The one-sentence takeaway: when an estimator is found at a sharp edge rather than a smooth peak, it converges at the cube-root rate, and its rescaled error follows a single universal law, Chernoff's distribution.

Visual Beginner

Picture two ways the thing you are maximising can look near its best point. On the left, a smooth rounded hill: the height changes gently, and the top is easy to locate sharply because small sideways moves cost height steadily. On the right, a jagged ridge sitting on top of a gentle downhill bowl: the gentle bowl says roughly where to go, but the jagged top means the exact best point is decided by a few random spikes near the summit. The smooth hill gives the square-root rate; the jagged ridge gives the cube-root rate.

criterion near its best point	who decides the answer	rate	data to halve the error
smooth rounded hill	the gentle curvature of the peak	square-root	four times
jagged ridge on a gentle bowl	a few random edge effects	cube-root	eight times

The takeaway: a rounded peak locates the answer fast, while an answer decided at a sharp edge improves slowly, and the rescaled wobble of that edge-decided answer is always the same fixed shape.

Worked example Beginner

We compare how much data two estimators need, using concrete numbers, to make the cube-root penalty plain. Suppose a smooth-peak estimator has a typical error of $0.10$ when you have $n = 1000$ data points, and it converges at the square-root rate. Suppose an edge-decided estimator also has a typical error of $0.10$ at $n = 1000$ , but it converges at the cube-root rate.

Step 1. The square-root rule. At the square-root rate the error scales like $1$ divided by the square root of the count. To halve the error from $0.10$ to $0.05$ , you must quadruple the data: $1000 \times 4 = 4000$ points.

Step 2. The cube-root rule. At the cube-root rate the error scales like $1$ divided by the cube root of the count. To halve the error you must multiply the data by $2^{3} = 8$ : from $1000$ to $8000$ points.

Step 3. Push further. To cut the error to one-tenth, $0.10$ down to $0.01$ : the square-root estimator needs $1 0^{2} = 100$ times the data, $100, 000$ points; the cube-root estimator needs $1 0^{3} = 1000$ times the data, one million points.

Step 4. What this tells us. The cube-root estimator is not broken — it still homes in on the truth — but it is hungry. Each extra digit of precision costs ten times more data than for a square-root estimator at the cheaper rate. This is the price of an answer decided by a few random edge effects rather than by a smooth curve, and it is why people work hard to smooth these estimators when they can.

Check your understanding Beginner

Formal definition Intermediate+

Let $X_{1}, \dots, X_{n}$ be i.i.d. from a law $P$ on $X$ , with empirical measure $P_{n} = n^{- 1} \sum_{i} δ_{X_{i}}$ and notation $P_{n} f, P f$ as in 45.04.04. An M-estimator $\hat{θ}_{n}$ maximises $M_{n} (θ) = P_{n} m_{θ}$ over $θ \in Θ \subseteq R^{k}$ , with population criterion $M (θ) = P m_{θ}$ and maximiser $θ_{0}$ . Modes of convergence $d, P$ and the $o_{P} / O_{P}$ calculus are those of 45.04.01; the empirical process $G_{n} f = n (P_{n} - P) f$ and Donsker / manageable classes are those of 45.05.03.

Definition (the standard versus the cube-root regime). The smooth M-estimator theory of 45.04.04 applies when the map $θ \mapsto m_{θ}$ is differentiable in quadratic mean and the criterion increments contract quadratically, $P (m_{θ} - m_{θ_{0}})^{2} = O (∣ θ - θ_{0} ∣^{2})$ ; this yields the $n$ rate. The cube-root regime is the case in which $θ \mapsto m_{θ}$ is too rough for that contraction — typically $m_{θ}$ involves an indicator or a hard threshold in $θ$ — so that the increments contract only linearly,

P (m_{θ} - m_{θ_{0}})^{2} ≍ ∣ θ - θ_{0} ∣ (θ \to θ_{0}),

while the population criterion still has a non-degenerate quadratic maximum.

Definition (the sharp-edge / Kim-Pollard structure). The criterion has the sharp-edge structure at $θ_{0}$ if:

(K1) quadratic population maximum: $M (θ) = M (θ_{0}) - \frac{1}{2} (θ - θ_{0})^{T} V (θ - θ_{0}) + o (∣ θ - θ_{0} ∣^{2})$ for a positive-definite $V$ ;

(K2) linear envelope contraction: there is a neighbourhood on which $P (m_{θ} - m_{θ_{0}})^{2} \leq C ∣ θ - θ_{0} ∣$ and the class ${m_{θ} - m_{θ_{0}} : ∣ θ - θ_{0} ∣ < δ}$ is manageable (a uniform-entropy / Donsker-type control, 45.05.03);

(K3) local covariance: the localized increments have a limiting covariance kernel linear in the separation, $Cov (m_{θ_{0} + g} - m_{θ_{0}}, m_{θ_{0} + h} - m_{θ_{0}})$ rescaled gives a kernel $H (g, h)$ with $H (h, h)$ of order $∣ h ∣$ .

Definition (localized criterion). Fix the cube-root rescaling $θ = θ_{0} + h n^{- 1/3}$ . The localized criterion process is

Z_{n} (h) = n^{2/3} [M_{n} (θ_{0} + h n^{- 1/3}) - M_{n} (θ_{0})], h \in R^{k} .

The rescaled estimator $\hat{h}_{n} = n^{1/3} (\hat{θ}_{n} - θ_{0})$ is the argmax of $Z_{n}$ . The power $n^{2/3}$ is the balance point: under (K1) the deterministic part $n^{2/3} [M (θ_{0} + h n^{- 1/3}) - M (θ_{0})] \to - \frac{1}{2} h^{T} V h$ stays $O (1)$ , and under (K2)-(K3) the stochastic part $n^{2/3} (P_{n} - P) (m_{θ_{0} + h n^{- 1/3}} - m_{θ_{0}}) = n^{1/6} G_{n} (m_{θ_{0} + h n^{- 1/3}} - m_{θ_{0}})$ has variance $n^{1/3} \cdot P (m_{θ_{0} + h n^{- 1/3}} - m_{θ_{0}})^{2} ≍ n^{1/3} \cdot ∣ h ∣ n^{- 1/3} = O (∣ h ∣)$ .

Definition (Chernoff's distribution). Let $W$ be a two-sided standard Brownian motion on $R$ (independent standard Brownian motions on $[0, \infty)$ and $(- \infty, 0]$ , $W (0) = 0$ ), as constructed from 02.15.01. Chernoff's distribution is the law of

Z = argmax_{t \in R} (W (t) - t^{2}),

the location of the maximum of two-sided Brownian motion minus a parabola. It is symmetric about $0$ , has a density expressible through Airy functions, and is the canonical limit law of cube-root asymptotics.

Counterexamples to common slips Intermediate+

The rate is $n^{1/3}$ , not $n^{1/2}$ , because of (K2), not because the criterion is non-smooth in $x$ . A criterion can be discontinuous in the data $x$ yet smooth in $θ$ (the median's $sign (x - θ)$ has a smooth population map $θ \mapsto P sign (X - θ)$ , giving the $n$ rate of 45.04.04). The slow rate needs roughness in $θ$ , registered as the linear — not quadratic — envelope contraction.
The limit is an argmax, not a Gaussian. At the smooth rate the limit is $N (0, V^{- 1} Σ V^{- 1})$ , a linear image of a Gaussian fluctuation. At the cube-root rate the argmax operation is genuinely nonlinear: $n^{1/3} (\hat{θ}_{n} - θ_{0}) d argmax (G - \frac{1}{2} h^{T} V h)$ , which is not Gaussian and has no closed-form mean-plus-noise structure.
No plug-in variance exists. Because the limit is the argmax of a process, the asymptotic "variance" is a fixed numerical constant scaling Chernoff's law, not a functional of $P$ you can estimate by differentiating the criterion. The naive sandwich of 45.04.04 is undefined here — there is no $A$ matrix, since the relevant derivative does not exist.
Two-sided Brownian motion, not one-sided. The limit process runs over all of $R$ ; the argmax can fall on either side of $0$ . Restricting to $[0, \infty)$ would change the law. The symmetry of Chernoff's distribution is a consequence of the symmetric two-sided construction.

Key theorem with proof Intermediate+

The signature result is the Kim-Pollard cube-root theorem: under the sharp-edge structure the rescaled M-estimator converges in distribution to the argmax of a Gaussian process with parabolic drift, and in the one-dimensional canonical case this argmax is Chernoff's distribution. The proof has two parts — the localized criterion converges as a process to a Gaussian-minus-parabola, and the argmax continuous-mapping theorem transfers that to the argmax — mirroring the Taylor-plus-Slutsky mechanism of 45.04.04 but with the linearisation replaced by an argmax.

Theorem (Kim-Pollard cube-root theorem). Let $\hat{θ}_{n}$ maximise $M_{n} = P_{n} m_{θ}$ with $\hat{θ}_{n} P θ_{0}$ , $θ_{0}$ interior to $Θ \subseteq R^{k}$ , and suppose the sharp-edge structure (K1)-(K3) holds. Then

n^{1/3} (\hat{θ}_{n} - θ_{0}) d argmax_{h \in R^{k}} (G (h) - \frac{1}{2} h^{T} V h),

where $G$ is a centred Gaussian process with continuous sample paths, $G (0) = 0$ , and covariance $Cov (G (g), G (h)) = H (g, h)$ from (K3), provided the argmax on the right exists and is a.s. unique. In the scalar canonical case $H (g, h) = ∣ g ∣ + ∣ h ∣ - ∣ g - h ∣$ (so $Var G (h) = 2∣ h ∣$ , the covariance of two-sided Brownian motion), the limit equals $(4/ V)^{1/3} Z$ with $Z$ Chernoff's distribution.

Proof. Write the localized criterion $Z_{n} (h) = n^{2/3} [M_{n} (θ_{0} + h n^{- 1/3}) - M_{n} (θ_{0})]$ and split it into a deterministic and a stochastic part,

Z_{n} (h) = D_{n} (h) n^{2/3} [M (θ_{0} + h n^{- 1/3}) - M (θ_{0})] + S_{n} (h) n^{1/6} G_{n} (m_{θ_{0} + h n^{- 1/3}} - m_{θ_{0}}),

using $M_{n} - M = (P_{n} - P) = n^{- 1/2} G_{n}$ and $n^{2/3} n^{- 1/2} = n^{1/6}$ .

The drift converges. By (K1) the second-order Taylor expansion gives $D_{n} (h) = n^{2/3} [- \frac{1}{2} (h n^{- 1/3})^{T} V (h n^{- 1/3}) + o (n^{- 2/3} ∣ h ∣^{2})] \to - \frac{1}{2} h^{T} V h$ for each fixed $h$ , uniformly on compacts.

The fluctuation converges to a Gaussian process. The summands of $S_{n} (h)$ are the localized increments $f_{n, h} = m_{θ_{0} + h n^{- 1/3}} - m_{θ_{0}}$ . Their covariance is $n^{1/3} Cov_{P} (f_{n, g}, f_{n, h})$ , which converges to $H (g, h)$ by (K3); finite-dimensional convergence to a Gaussian vector follows from the multivariate central limit theorem 37.03.02 (the triangular-array Lindeberg condition holds because each $f_{n, h}$ is bounded and its variance is $O (n^{- 1/3})$ ). Tightness of $h \mapsto S_{n} (h)$ on compacts follows from the manageability hypothesis in (K2), which gives a maximal inequality controlling the modulus of continuity (45.05.03); thus $S_{n} ⇝ G$ as processes in the topology of uniform convergence on compacts.

Assemble. Adding the deterministic drift, $Z_{n} ⇝ Z$ where $Z (h) = G (h) - \frac{1}{2} h^{T} V h$ , a Gaussian process with parabolic drift. Because $V ≻ 0$ the parabola dominates $G$ at infinity (a centred Gaussian process with $Var G (h) = O (∣ h ∣)$ grows like $∣ h ∣ lo g ∣ h ∣$ by the law of the iterated logarithm, slower than $∣ h ∣^{2}$ ), so $Z$ attains its maximum at a finite, a.s. unique $h^{*} = argmax Z$ .

Argmax continuous mapping. The functional $argmax$ is continuous (in the uniform-on-compacts topology) at sample paths possessing a unique well-separated maximum with at-infinity decay; since $Z$ has such paths a.s. and $\hat{h}_{n} = n^{1/3} (\hat{θ}_{n} - θ_{0}) = argmax Z_{n}$ is tight (the consistency $\hat{θ}_{n} P θ_{0}$ plus a rate bound from (K1)-(K2) confine $\hat{h}_{n}$ to compacts with high probability), the argmax continuous-mapping theorem of 45.04.04 in its process form (45.05.03) gives $\hat{h}_{n} d h^{*} = argmax_{h} (G (h) - \frac{1}{2} h^{T} V h)$ .

The scalar canonical reduction. With $k = 1$ , $H (g, h) = ∣ g ∣ + ∣ h ∣ - ∣ g - h ∣$ is the covariance of two-sided Brownian motion $W$ , so $G = d 2 W$ after the standardisation, and $Z (h) = 2 W (h) - \frac{1}{2} V h^{2}$ . The Brownian scaling identity (self-similarity) of 02.15.01, $W (c h) = d c W (h)$ , lets one rescale $h = a t$ to convert $2 W (a t) - \frac{1}{2} V a^{2} t^{2}$ into a constant multiple of $W (t) - t^{2}$ : choosing $a$ so the two coefficients align gives $argmax_{h} Z = (4/ V)^{1/3} argmax_{t} (W (t) - t^{2}) = (4/ V)^{1/3} Z$ . $□$

Bridge. This theorem builds toward the asymptotic theory of every shape-constrained and threshold estimator — the Grenander monotone-density NPMLE, the maximum-score estimator, current-status and interval-censoring isotonic estimators, the least-median-of-squares regression — and it appears again in the inference theory of 45.04.10, where the failure of the ordinary bootstrap at this rate forces the m-out-of-n and smoothing repairs. The foundational reason the rate slows to $n^{1/3}$ is the balance struck in the localized criterion: a deterministic parabola of order $n^{2/3} ∣ h n^{- 1/3} ∣^{2}$ must be matched against a fluctuation of order $n^{1/6} ∣ h n^{- 1/3} ∣$ , and equating the two exponents forces the $n^{- 1/3}$ localisation — the same exponent-balancing that in 45.04.04 produced $n^{- 1/2}$ from a quadratic fluctuation. This is exactly the argmax theorem of 45.04.04 run at the level of localized processes rather than at the level of the criterion: there consistency was the argmax continuous-mapping applied to deterministic limits, and here asymptotic distribution is the same operation applied to a Gaussian-process limit. Putting these together, the central insight is that the universal limit law $Z$ is the argmax of two-sided Brownian motion minus a parabola, so every cube-root estimator inherits one shape from the single structural fact that its localized criterion is Brownian plus a quadratic drift; the bridge is the recognition that the smooth regime's linear response (a derivative, hence a Gaussian) is replaced here by a nonlinear response (an argmax, hence Chernoff's law) precisely because the criterion has no derivative in $θ$ to linearise.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Derive the cube-root localisation exponent from scratch. Suppose the deterministic drift of the criterion at displacement $δ$ is $- c_{1} δ^{2}$ and the standard deviation of the empirical fluctuation of the increment is $c_{2} δ / n$ . Find the displacement scale $δ_{n}$ at which drift and fluctuation balance, and confirm it is $n^{- 1/3}$ .

Hint

Set the magnitude of the drift equal to the standard deviation of the fluctuation: $c_{1} δ^{2} = c_{2} δ / n$ , and solve for $δ$ as a power of $n$ .

Answer

Balancing $c_{1} δ^{2} = c_{2} δ / n$ gives $δ^{2} = (c_{2} / c_{1}) δ^{1/2} n^{- 1/2}$ , so $δ^{3/2} = (c_{2} / c_{1}) n^{- 1/2}$ and $δ = (c_{2} / c_{1})^{2/3} n^{- 1/3}$ . The displacement scale is $δ_{n} ≍ n^{- 1/3}$ , hence $n^{1/3} (\hat{θ}_{n} - θ_{0}) = O_{P} (1)$ . The linear $δ$ inside the square root (from the linear envelope contraction K2) is what produces $1/3$ rather than the $1/2$ that a $δ^{2}$ inside would give.

Exercise 4 (medium, symbolic).

(Grenander estimator and the switch relation.) Let $\hat{F}_{n}$ be the empirical distribution function and $\hat{f}_{n}$ the Grenander estimator of a decreasing density — the left derivative of the least concave majorant of $\hat{F}_{n}$ . State the switch relation expressing the event ${\hat{f}_{n} (x_{0}) \leq a}$ as an argmax event for the process $s \mapsto \hat{F}_{n} (s) - a s$ , and explain why this puts the Grenander estimator into the cube-root framework.

Hint

The least concave majorant's slope drops below $a$ exactly where the process $\hat{F}_{n} (s) - a s$ is maximised; the maximiser is the inverse function of $\hat{f}_{n}$ .

Answer

The switch relation states ${\hat{f}_{n} (x_{0}) \leq a} = {U_{n} (a) \geq x_{0}}$ where $U_{n} (a) = argmax_{s} [\hat{F}_{n} (s) - a s]$ is the location at which the concave majorant has slope $a$ . Thus the value $\hat{f}_{n} (x_{0})$ is read off an argmax of a centred-empirical-process-plus-linear-drift; localising $a$ near $f (x_{0})$ and $s$ near $x_{0}$ , the empirical process $\hat{F}_{n} - F$ contributes a Brownian fluctuation while the smooth curvature $f^{'} (x_{0}) \neq = 0$ supplies the parabolic drift, giving $n^{1/3} (\hat{f}_{n} (x_{0}) - f (x_{0})) d ∣4 f (x_{0}) f^{'} (x_{0}) ∣^{1/3} (2 Z)$ . The argmax-of-Brownian-minus-parabola structure is exactly the canonical case of the theorem.

Exercise 5 (medium, symbolic).

(Maximum-score estimator.) Manski's estimator maximises $M_{n} (θ) = P_{n} [(2 Y - 1) sign (X^{T} θ)]$ over $∣ θ ∣ = 1$ for binary $Y$ . Explain why the indicator/sign in $θ$ makes (K2) hold with linear rather than quadratic contraction, and hence why the estimator is cube-root rate while the population criterion is smooth.

Hint

The increment $m_{θ} - m_{θ_{0}}$ is non-zero only on the thin lens between the hyperplanes ${X^{T} θ = 0}$ and ${X^{T} θ_{0} = 0}$ , whose $P$ -measure is of order $∣ θ - θ_{0} ∣$ .

Answer

The increment $m_{θ} - m_{θ_{0}} = (2 Y - 1) [sign (X^{T} θ) - sign (X^{T} θ_{0})]$ vanishes except where $X$ lies in the wedge between the two hyperplanes, a region of $P$ -measure $≍ ∣ θ - θ_{0} ∣$ when $X$ has a bounded density near the boundary. Hence $P (m_{θ} - m_{θ_{0}})^{2} ≍ ∣ θ - θ_{0} ∣$ — the linear contraction (K2) — even though $M (θ) = P [(2 Y - 1) sign (X^{T} θ)]$ is a smooth integral with a quadratic maximum at $θ_{0}$ (K1) because integrating the indicator over $X$ smooths it. Linear contraction with quadratic drift is precisely the sharp-edge structure, so $n^{1/3} (\hat{θ}_{n} - θ_{0})$ converges to an argmax of a Gaussian-process-minus-parabola.

Exercise 6 (medium, symbolic).

(Bootstrap inconsistency.) Explain heuristically why the ordinary nonparametric bootstrap fails to estimate the law of $n^{1/3} (\hat{θ}_{n} - θ_{0})$ consistently, and state one repair (m-out-of-n or smoothing).

Hint

The limit is an argmax functional, which is not Hadamard-differentiable; bootstrap consistency for the empirical process does not transfer through a non-smooth functional.

Answer

The bootstrapped argmax does not have the right conditional limit law: resampling reproduces the empirical process well, but the argmax map is not (Hadamard-)differentiable, so the delta-method route to bootstrap validity of 45.05.06 breaks, and the bootstrap distribution of $n^{1/3} (\hat{θ}_{n}^{*} - \hat{θ}_{n})$ does not converge to Chernoff's law — it retains a random, data-dependent limit. Repairs: the m-out-of-n bootstrap resamples $m = o (n)$ points so that $m^{1/3} (\hat{θ}_{m}^{*} - \hat{θ}_{n})$ is consistent (the smaller resample restores the asymptotic decoupling), or one smooths the criterion (kernel-smoothed maximum score, smoothed isotonic estimators) to push the estimator back toward a faster, bootstrappable rate. Subsampling is the general-purpose fix because it requires only that a limit law exist.

Exercise 7 (hard, symbolic).

(Chernoff's mode estimator / the shorth.) Chernoff estimates the mode as the midpoint of the shortest interval of fixed length $2 h_{n}$ containing the most data, equivalently the maximiser of $P_{n} [1 {∣ X - θ ∣ \leq h_{n}}]$ . Taking $h_{n}$ fixed for simplicity, show this fits (K1)-(K3) and identify $V$ in terms of the density curvature at the mode.

Hint

The population criterion is $M (θ) = F (θ + h) - F (θ - h)$ ; expand to second order at the maximiser $θ_{0}$ and compute the increment variance from the symmetric-difference set.

Answer

The criterion $m_{θ} (x) = 1 {∣ x - θ ∣ \leq h}$ has population version $M (θ) = F (θ + h) - F (θ - h)$ , maximised at $θ_{0}$ where $f (θ_{0} + h) = f (θ_{0} - h)$ (the symmetric case $θ_{0} =$ mode); the second derivative is $M^{''} (θ_{0}) = f^{'} (θ_{0} + h) - f^{'} (θ_{0} - h) < 0$ , so (K1) holds with $V = - M^{''} (θ_{0})$ . The increment $m_{θ} - m_{θ_{0}}$ is the indicator of the symmetric difference of the two windows, a set of measure $≍ ∣ θ - θ_{0} ∣$ , giving $P (m_{θ} - m_{θ_{0}})^{2} ≍ ∣ θ - θ_{0} ∣$ — (K2) with the indicator class manageable (a VC class, 45.05.04). The localized covariance (K3) is that of two-sided Brownian motion since disjoint window-shifts touch disjoint data. Hence $n^{1/3} (\hat{θ}_{n} - θ_{0}) d (4 a / V^{2})^{1/3} Z$ for a constant $a$ from the local Poisson intensity of points entering and leaving the windows, the original Chernoff (1964) result.

Exercise 8 (hard, symbolic).

(Why two-sided, and the symmetry of $Z$ .) Using the Brownian scaling identity from 02.15.01 and the time-reversal symmetry of Brownian motion, prove that $Z = argmax_{t} (W (t) - t^{2})$ is symmetric about $0$ , i.e. $Z = d - Z$ .

Hint

Consider $W (t) = W (- t)$ , which is again a two-sided Brownian motion, and track what the substitution $t \mapsto - t$ does to the parabola $t^{2}$ .

Answer

Let $W (t) := W (- t)$ . Since two-sided Brownian motion is invariant under $t \mapsto - t$ (the two independent half-line Brownian motions swap roles, each remaining a standard Brownian motion), $W = d W$ as processes on $R$ . Now $W (t) - t^{2} = W (- t) - (- t)^{2}$ , so the map $t \mapsto - t$ carries the path $t \mapsto W (t) - t^{2}$ to the path $s \mapsto W (s) - s^{2}$ (with $s = - t$ ); the parabola is even, so it is unchanged. The argmax of the second path is $- Z$ by the substitution, and since $W = d W$ the second path has the same law as the first, giving $- Z = d Z$ . The two-sidedness is essential: a one-sided $W$ on $[0, \infty)$ would force $Z \geq 0$ and destroy the symmetry.

Advanced results Master

The cube-root theorem extends to nonparametric pointwise problems where the parabolic drift comes from local density curvature, to interval-censoring and current-status models through the same switch relation, and to a sharp characterisation of when the rate degrades further; the inference difficulty it creates is repaired by subsampling and by smoothing, and the limit law is computable through the parabolic-drift Brownian analysis. The smooth M-estimation of 45.04.04 is the rigid centre from which all of these are the non-smooth deformation.

Theorem 1 (Kim-Pollard, the general cube-root M-estimation theorem). Under (K1)-(K3) with the localized increment class manageable for an envelope $F$ with $P F^{2} < \infty$ , the localized criterion $Z_{n} ⇝ Z = G - \frac{1}{2} h^{T} V h$ in the uniform-on-compacts topology, where $G$ is the centred Gaussian process with covariance $H$ of (K3); consequently $n^{1/3} (\hat{θ}_{n} - θ_{0}) d argmax_{h} Z (h)$ . The proof is the localized empirical-process convergence of 45.05.03 composed with the argmax continuous-mapping theorem; the manageability hypothesis replaces the Donsker condition of the $n$ theory and supplies the maximal inequality controlling the modulus of continuity that yields tightness ^{[Kim, J. & Pollard, D. — Cube root asymptotics]}.

Theorem 2 (Grenander estimator and Chernoff's distribution; Groeneboom). For a density $f$ that is decreasing with $f (x_{0}) > 0$ , $f^{'} (x_{0}) < 0$ , the Grenander NPMLE $\hat{f}_{n}$ satisfies

n^{1/3} (\hat{f}_{n} (x_{0}) - f (x_{0})) d \frac{1}{2} f (x_{0}) f^{'} (x_{0})^{1/3} (2 Z),

with $Z = argmax_{t} (W (t) - t^{2})$ Chernoff's distribution. The route is the switch relation, converting the value of the monotone NPMLE into the argmax of $s \mapsto F_{n} (s) - a s$ (a Brownian-bridge-plus-linear-drift after localisation), and then the parabolic-drift analysis. Groeneboom's explicit study of $W (t) - t^{2}$ via Airy functions gives the density of $Z$ , $f_{Z} (z) = \frac{1}{2} g (z) g (- z)$ for a function $g$ solving an Airy boundary-value problem, with $Var Z \approx 0.26$ ^{[Groeneboom, P. — Estimating a monotone density]}.

Theorem 3 (maximum-score estimator; Kim-Pollard). Manski's maximum-score estimator $\hat{θ}_{n} = argmax_{∣ θ ∣ = 1} P_{n} [(2 Y - 1) sign (X^{T} θ)]$ for the binary-choice model with $med (Y ∣ X) = 1 {X^{T} θ_{0} > 0}$ converges at rate $n^{1/3}$ to the argmax of a Gaussian process with parabolic drift over the sphere's tangent space, the multivariate Chernoff-type limit; the indicator criterion supplies the linear contraction (K2) while the smoothing-by-integration of the indicator supplies the quadratic drift (K1). The rate and limit are why inference for maximum score is hard and why Horowitz's smoothed maximum-score estimator, replacing the sign by a smooth kernel, recovers a faster ( $n^{2/5}$ or better) bootstrappable rate at the cost of a bandwidth and a bias ^{[Manski, C. F. — Maximum score estimation of the stochastic utility model of choice]}.

Theorem 4 (bootstrap inconsistency and subsampling repair). The ordinary nonparametric bootstrap is inconsistent for the law of $n^{1/3} (\hat{θ}_{n} - θ_{0})$ in the cube-root problems above: the conditional law of $n^{1/3} (\hat{θ}_{n}^{*} - \hat{θ}_{n})$ given the data converges to a random distribution, not to Chernoff's law, because the argmax functional is not Hadamard-differentiable and the empirical-process bootstrap of 45.05.06 does not pass through it. The m-out-of-n bootstrap with $m \to \infty$ , $m / n \to 0$ , and subsampling (Politis-Romano) are consistent: they estimate the limit law of any statistic with a non-degenerate limit, requiring only its existence, and so furnish valid confidence intervals at the $n^{1/3}$ rate; cross-referenced with the bootstrap theory of 45.04.10 ^{[Kim, J. & Pollard, D. — Cube root asymptotics]}.

Theorem 5 (Chernoff's mode estimator and the original limit; Chernoff). The estimator of the mode as the centre of the densest interval, $\hat{θ}_{n} = argmax_{θ} P_{n} 1 {∣ X - θ ∣ \leq h_{n}}$ (with $h_{n}$ fixed or shrinking slowly), satisfies $n^{1/3} (\hat{θ}_{n} - θ_{0}) d c Z$ for an explicit constant $c$ built from the density and its curvature at the mode; this is the 1964 calculation in which the limit law $argmax_{t} (W (t) - t^{2})$ first appeared, before its general role was recognised. The same two-sided-Brownian-minus-parabola object recurs because the localized count of points entering and leaving the sliding window is, in the limit, a two-sided Brownian increment carried by a Poisson intensity, against the smooth curvature of the density ^{[Chernoff, H. — Estimation of the mode]}.

Synthesis. The foundational reason a single law governs every sharp-edge estimator is that the localized criterion is always a Gaussian-process-minus-parabola: the population's quadratic maximum (K1) supplies the parabola while the linear envelope contraction (K2) and the local covariance (K3) force the fluctuation to be two-sided Brownian motion, and the argmax of that fixed object is Chernoff's distribution. The central insight is that the smooth M-estimation of 45.04.04 and the cube-root theory are one machine read at two levels — there the argmax continuous-mapping theorem acted on deterministic limits to give consistency and a derivative linearised the criterion into a Gaussian, while here the same argmax theorem acts on a Gaussian-process limit because no derivative exists to linearise; this is exactly the passage from a linear response (a sandwich variance) to a nonlinear one (an argmax law).

The rate exponent is not arbitrary: balancing the parabolic drift $n^{2/3} δ^{2}$ against the Brownian fluctuation $n^{2/3} δ / n$ pins the localisation to $δ ≍ n^{- 1/3}$ , which is exactly the structural fact that the $n$ theory's quadratic envelope contraction has degraded to a linear one. Putting these together, the inference pathology — no plug-in variance, an inconsistent bootstrap, the m-out-of-n and smoothing repairs of 45.04.10 — is the dual face of the same nonlinearity: the argmax functional that produces the universal limit is the very non-differentiability that defeats the delta-method bootstrap of 45.05.06, so the universality of Chernoff's law and the difficulty of using it are two readings of one structure, the absence of a derivative in $θ$ .

Full proof set Master

The process convergence and argmax transfer are proved in the Key theorem; the rate-balancing is Exercise 3 and the symmetry of $Z$ is Exercise 8. The remaining Master claims are recorded here.

Proposition 1 (the parabola dominates the Gaussian; existence and uniqueness of the limiting argmax). Let $G$ be a centred Gaussian process on $R^{k}$ with continuous paths, $G (0) = 0$ , and $Var G (h) = O (∣ h ∣)$ , and let $V ≻ 0$ . Then $Z (h) = G (h) - \frac{1}{2} h^{T} V h$ attains a finite maximum a.s., and the maximiser is a.s. unique.

Proof. For the at-infinity decay, the increments $G (h) - G (g)$ have variance bounded by a constant times $∣ h - g ∣$ on bounded sets (sub-linear growth of the variance plus continuity), so by the Borell-TIS / Dudley entropy bound $sup_{∣ h ∣ \leq R} G (h)$ has Gaussian tails with mean $O (R lo g R)$ . Hence on the shell $∣ h ∣ \in [R, 2 R]$ , $Z (h) \leq sup_{∣ h ∣ \leq 2 R} G (h) - \frac{1}{2} λ_{m i n} (V) R^{2}$ , which $\to - \infty$ a.s. as $R \to \infty$ because $R lo g R = o (R^{2})$ . Therefore $Z (h) \to - \infty$ as $∣ h ∣ \to \infty$ a.s., and a continuous function decaying to $- \infty$ attains its supremum on a compact set. Uniqueness: for a Gaussian process the event that the maximum is attained at two distinct points has probability zero, because at fixed $g \neq = h$ the difference $Z (g) - Z (h)$ has a non-degenerate Gaussian (hence atomless) law, so ${Z (g) = Z (h) = max Z}$ is null; a separability/continuity argument upgrades this to a.s. uniqueness over all pairs. $□$

Proposition 2 (the cube-root rate; tightness of $\hat{h}_{n}$ ). Under (K1)-(K3), $n^{1/3} (\hat{θ}_{n} - θ_{0}) = O_{P} (1)$ .

Proof. The argument is a peeling / chaining bound on the centred criterion. For displacements with $∣ θ - θ_{0} ∣ \in [2^{j - 1} δ, 2^{j} δ]$ the population drift is at most $- c 2^{2 (j - 1)} δ^{2}$ by (K1), while the maximal fluctuation of $(P_{n} - P) (m_{θ} - m_{θ_{0}})$ over that shell is, by the manageability maximal inequality of 45.05.03 and the linear envelope (K2), of order $n^{- 1/2} (2^{j} δ)^{1/2}$ . For $\hat{θ}_{n}$ to lie in the shell the fluctuation must exceed the drift, $n^{- 1/2} (2^{j} δ)^{1/2} ≳ 2^{2 j} δ^{2}$ , which (summing the geometric series over $j$ ) has probability tending to $0$ once $δ = M n^{- 1/3}$ with $M$ large. Hence $P (n^{1/3} ∣ \hat{θ}_{n} - θ_{0} ∣ > M) \to 0$ as $M \to \infty$ , uniformly in $n$ . $□$

Proposition 3 (Chernoff's distribution is well defined and symmetric). The variable $Z = argmax_{t} (W (t) - t^{2})$ , $W$ two-sided Brownian motion, is a.s. finite, a.s. unique, and symmetric about $0$ .

Proof. Finiteness and uniqueness are Proposition 1 with $k = 1$ , $G = W$ (variance $∣ t ∣$ , so $O (∣ t ∣)$ ), $V = 2$ . Symmetry is Exercise 8: $W (t) = W (- t)$ is again two-sided Brownian motion, the parabola is even, and the substitution $t \mapsto - t$ sends $argmax (W (t) - t^{2})$ to $- argmax (W (s) - s^{2}) = d - Z$ , giving $Z = d - Z$ . $□$

Proposition 4 (the scaling reduction to the canonical parabola). If $Z (h) = σ W (h) - \frac{1}{2} V h^{2}$ with $W$ two-sided Brownian, $σ, V > 0$ , then $argmax_{h} Z = (σ^{2} / V^{2})^{1/3} 2^{1/3} Z$ , hence the limit of the M-estimator is a known constant times Chernoff's variable.

Proof. Substitute $h = a t$ . By the Brownian scaling identity of 02.15.01, $W (a t) = d a^{1/2} W (t)$ , so $Z (a t) = d σ a^{1/2} W (t) - \frac{1}{2} V a^{2} t^{2}$ . Factor out $σ a^{1/2}$ : $Z (a t) = d σ a^{1/2} (W (t) - \frac{V a ^{3/2}}{2 σ} t^{2})$ . Choose $a$ so that $V a^{3/2} / (2 σ) = 1$ , i.e. $a = (2 σ / V)^{2/3}$ ; then $Z (a t) = d σ a^{1/2} (W (t) - t^{2})$ , a positive multiple of the canonical process, whose argmax (in $t$ ) is $Z$ . Since $argmax_{h} Z = a argmax_{t} Z (a t) = a Z$ , and $a = (2 σ / V)^{2/3} = 2^{2/3} (σ / V)^{2/3}$ , the maximiser equals $2^{2/3} (σ / V)^{2/3} Z$ . Specialising $σ = 2$ (the standardised two-sided-Brownian fluctuation $Var G (h) = 2∣ h ∣$ ) recovers $(4/ V)^{1/3} Z$ of the Key theorem. $□$

Connections Master

The smooth M- and Z-estimator theory 45.04.04 is the regime this unit departs from: there the increment class contracts quadratically and a derivative linearises the criterion into a Gaussian limit at the $n$ rate with a sandwich variance, whereas here the increment class contracts only linearly, no derivative exists, and the argmax of a Gaussian-process-minus-parabola replaces the linear response at the $n^{1/3}$ rate. The argmax continuous-mapping theorem that delivered consistency there is the same tool that delivers the limit distribution here, applied one level up to processes.

The empirical-process theory of Donsker and manageable classes 45.05.03 supplies the technical engine: the localized fluctuation $n^{1/6} G_{n} (m_{θ_{0} + h n^{- 1/3}} - m_{θ_{0}})$ converges to a Gaussian process precisely because the increment class is manageable, and the maximal inequality that bounds its modulus of continuity is what yields both the tightness in the Key theorem and the rate bound of Proposition 2. The VC-class control of indicator criteria 45.05.04 is what verifies manageability for the maximum-score and shorth examples.

Two-sided Brownian motion and its scaling self-similarity 02.15.01 are the source of the universal limit: Chernoff's distribution is the argmax of $W (t) - t^{2}$ , and the Brownian scaling identity $W (c t) = d c W (t)$ is exactly what reduces every cube-root estimator's limit to a constant times this one object (Proposition 4). The local-time and excursion theory of 37.06.01 governs the fine path structure of the maximiser.

The bootstrap theory 45.04.10 meets its sharpest failure here: the ordinary nonparametric bootstrap is inconsistent for the cube-root limit because the argmax functional is not Hadamard-differentiable, so the functional-delta-method bootstrap of 45.05.06 does not apply, and the m-out-of-n bootstrap and subsampling are the standard repairs. This is the concrete counterexample that motivates the differentiability hypotheses of bootstrap consistency.

The boundary and non-regular failures of the likelihood-ratio trinity 45.04.09 are a parallel breakdown of the standard $n$ / chi-squared machinery — there a parameter on the edge of its cone, here a criterion with no derivative — both reminding that the regularity hypotheses of 45.04.04 and 45.04.03 are load-bearing rather than cosmetic.

Historical & philosophical context Master

The limit law now called Chernoff's distribution appeared first in Herman Chernoff's 1964 study of mode estimation, where he found that the centre of the densest fixed-width interval converges at rate $n^{1/3}$ to the argmax of two-sided Brownian motion minus a parabola ^{[Chernoff 1964]}. The object recurred independently in Charles Manski's 1975 maximum-score estimator for discrete-choice models, whose non-smooth indicator criterion produced the same slow rate, and in the analysis of monotone-density and monotone-regression NPMLEs.

The general structural theorem — that a wide family of M-estimators with a sharp-edge criterion share the $n^{1/3}$ rate and an argmax-of-Gaussian-process-minus-parabola limit — was proved by Jeankyung Kim and David Pollard in 1990, who isolated the manageability and quadratic-mean hypotheses and routed the proof through the argmax continuous-mapping theorem on empirical processes ^{[Kim & Pollard 1990]}. The explicit analysis of the canonical process $W (t) - t^{2}$ — its argmax density via Airy functions and the moments of Chernoff's distribution — is due to Piet Groeneboom, who also formalised the switch relation connecting isotonic NPMLEs to argmax problems and developed the cube-root theory of shape-constrained estimation through the 1980s ^{[Groeneboom 1985]}. The maximum-score rate and its limit were placed in the Kim-Pollard framework, and the subsequent recognition that the ordinary bootstrap fails at this rate, with the m-out-of-n and subsampling repairs, is due to work of Abrevaya, Huang, Léger, MacGibbon, Politis, Romano, and others through the 1990s and 2000s.

Bibliography Master

@article{KimPollard1990,
  author  = {Kim, Jeankyung and Pollard, David},
  title   = {Cube root asymptotics},
  journal = {The Annals of Statistics},
  volume  = {18},
  number  = {1},
  year    = {1990},
  pages   = {191--219}
}

@article{Chernoff1964,
  author  = {Chernoff, Herman},
  title   = {Estimation of the mode},
  journal = {Annals of the Institute of Statistical Mathematics},
  volume  = {16},
  year    = {1964},
  pages   = {31--41}
}

@incollection{Groeneboom1985,
  author    = {Groeneboom, Piet},
  title     = {Estimating a monotone density},
  booktitle = {Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, Vol. II},
  editor    = {Le Cam, Lucien M. and Olshen, Richard A.},
  publisher = {Wadsworth},
  year      = {1985},
  pages     = {539--555}
}

@article{Groeneboom1989,
  author  = {Groeneboom, Piet},
  title   = {Brownian motion with a parabolic drift and Airy functions},
  journal = {Probability Theory and Related Fields},
  volume  = {81},
  number  = {1},
  year    = {1989},
  pages   = {79--109}
}

@article{Manski1975,
  author  = {Manski, Charles F.},
  title   = {Maximum score estimation of the stochastic utility model of choice},
  journal = {Journal of Econometrics},
  volume  = {3},
  number  = {3},
  year    = {1975},
  pages   = {205--228}
}

@book{vanderVaart1998cuberoot,
  author    = {van der Vaart, Aad W.},
  title     = {Asymptotic Statistics},
  publisher = {Cambridge University Press},
  series    = {Cambridge Series in Statistical and Probabilistic Mathematics},
  year      = {1998}
}

@book{GroeneboomJongbloed2014,
  author    = {Groeneboom, Piet and Jongbloed, Geurt},
  title     = {Nonparametric Estimation under Shape Constraints},
  publisher = {Cambridge University Press},
  year      = {2014}
}

Prerequisites

45.04.04

Tier anchors

beginner: van der Vaart 1998 Asymptotic Statistics (Cambridge) §5.8 (the heuristic for the n^{1/3} rate: when the criterion has a sharp ridge rather than a smooth peak, the optimum is found at a slower rate; the mode and the maximum-score examples described informally); Groeneboom & Jongbloed 2014 Nonparametric Estimation under Shape Constraints (Cambridge) Ch. 3 (the Grenander estimator of a monotone density and the picture of the least-concave-majorant / isotonic fit)
intermediate: van der Vaart 1998 Asymptotic Statistics (Cambridge) §5.8 (Theorem 5.52 / 5.53, the cube-root theorem: rescaling by n^{1/3} and convergence of the localized criterion to a Gaussian-process-minus-parabola whose argmax is the limit); Kim & Pollard 1990 (Annals of Statistics) §2-3 (the sharp-edge / quadratic-drift structure and the argmax continuous-mapping route)
master: Kim & Pollard 1990 Cube root asymptotics (Annals of Statistics 18, 191-219); van der Vaart 1998 Asymptotic Statistics (Cambridge) §5.8 and §3.2 (argmax continuous mapping); Groeneboom 1989 Brownian motion with a parabolic drift and Airy functions (Probability Theory and Related Fields) and Groeneboom 1985 Estimating a monotone density (the switch relation, Chernoff's distribution as the law of the argmax of two-sided Brownian motion minus a parabola)

References

van der Vaart — Asymptotic Statistics · Cambridge University Press, Cambridge Series in Statistical and Probabilistic Mathematics, 1998; §5.8 (rate of convergence and the cube-root theorem: when the population criterion behaves quadratically near its maximiser while the empirical fluctuations of the localized criterion are of order n^{-2/3}, the M-estimator converges at rate n^{1/3} to the argmax of a Gaussian process with parabolic drift), §3.2 (the argmax / argmin continuous-mapping theorem for processes with a.s. unique maximisers), §5.7 (the contrast with the n^{1/2} smooth-criterion regime); §19 (empirical processes) for the localized-process convergence)
Kim, J. & Pollard, D. — Cube root asymptotics · Annals of Statistics 18 (1990), 191-219 (the general cube-root M-estimation theorem: a sharp-edge / argmax structure in which the criterion increments have a non-degenerate quadratic mean and an empirical-process covariance scaling like |s-t|, yielding n^{1/3} rate and a limit equal to the argmax of a Gaussian-process-minus-parabola; the maximum-score, shorth/mode, and related examples)
Groeneboom, P. — Estimating a monotone density · Proc. Berkeley Conf. in Honor of Jerzy Neyman and Jack Kiefer (1985), Vol. II, 539-555; and Groeneboom, Brownian motion with a parabolic drift and Airy functions, Probability Theory and Related Fields 81 (1989), 79-109 (the switch relation between the isotonic NPMLE and an argmax problem, and the explicit analysis of Z = argmax_{t}(W(t) - t^2) — Chernoff's distribution — via Airy functions)
Chernoff, H. — Estimation of the mode · Annals of the Institute of Statistical Mathematics 16 (1964), 31-41 (Chernoff's estimator of the mode as the centre of the densest short interval, its n^{1/3} rate, and the limiting law of the argmax of two-sided Brownian motion minus a parabola now bearing his name)
Manski, C. F. — Maximum score estimation of the stochastic utility model of choice · Journal of Econometrics 3 (1975), 205-228; and Kim & Pollard (1990) §6 (the maximum-score estimator for the binary-choice / median-regression model, its non-smooth indicator criterion, the n^{1/3} rate, and the Chernoff-type limit for the rescaled argmax)

Estimated time

beginner: 20m
intermediate: 55m
master: 95m