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Inference for spherical location under high concentration

Davy Paindaveine, Thomas Verdebout

arXiv:1901.00359v2math.STstat.TH

TL;DR

Prior high-concentration spherical-location results were limited in their asymptotic regime and distributional scope. This paper studies a broader rotationally symmetric model as concentration grows with sample size, establishing nuisance-dependent rates and optimality results for standard procedures.

  • Problem

    Prior results for high-concentration spherical-location inference used κ→∞ with n fixed, focused on FvML distributions, and rarely addressed local-alternative power or optimality.

    Source: §1 · p. 5

  • Method

    The paper studies rotationally symmetric distributions indexed by location θ, concentration κ, and angular function f, with κn diverging at an arbitrary rate as n grows.

    Source: §1 · pp. 2, 5

  • Results

    The spherical mean’s consistency rate depends on f, and under mild assumptions the paper establishes Le Cam optimality for it and the Watson and Wald tests.

    Source: §1 · p. 5

  • Takeaways & Limitations

    High-concentration inference can be studied beyond FvML distributions, with the spherical mean and Watson and Wald tests having the stated optimality properties under mild assumptions.

    Source: §1 · p. 5

    The confidence zones described include a disconnected form and a form that is not practically usable when f is an unspecified nuisance.

    Source: §3 · p. 10

Abstract

from arXiv · show

Motivated by the fact that circular or spherical data are often much concentrated around a location $\pmb\theta$, we consider inference about $\pmb\theta$ under "high concentration" asymptotic scenarios for which the probability of any fixed spherical cap centered at $\pmb\theta$ converges to one as the sample size $n$ diverges to infinity. Rather than restricting to Fisher-von Mises-Langevin distributions, we consider a much broader, semiparametric, class of rotationally symmetric distributions indexed by the location parameter $\pmb\theta$, a scalar concentration parameter $\kappa$ and a functional nuisance $f$. We determine the class of distributions for which high concentration is obtained as $\kappa$ diverges to infinity. For such distributions, we then consider inference (point estimation, confidence zone estimation, hypothesis testing) on $\pmb\theta$ in asymptotic scenarios where $\kappa_n$ diverges to infinity at an arbitrary rate with the sample size $n$. Our asymptotic investigation reveals that, interestingly, optimal inference procedures on $\pmb\theta$ show consistency rates that depend on $f$. Using asymptotics "\`a la Le Cam", we show that the spherical mean is, at any $f$, a parametrically super-efficient estimator of $\pmb\theta$ and that the Watson and Wald tests for $\mathcal{H}_0:{\pmb\theta}={\pmb\theta}_0$ enjoy similar, non-standard, optimality properties. We illustrate our results through simulations and treat a real data example. On a technical point of view, our asymptotic derivations require challenging expansions of rotationally symmetric functionals for large arguments of $f$.

1. Introduction.

The paper extends high-concentration spherical-location inference beyond the parametric FvML family to rotationally symmetric distributions with a functional nuisance. It studies a double asymptotic regime and establishes inference and optimality results whose behavior can depend on that nuisance.

1. Introduction.

  • The paper studies rotationally symmetric distributions with location θ, concentration κ, and an angular function f, including FvML distributions when f(z)=exp(z).The density is proportional to f(κx′θ).Source: §1 · p. 2
  • High-concentration inference is motivated by the possibility that fixed-κ asymptotics may not adequately describe finite-sample behavior when concentration is large.The introduction contrasts this with the increasing difficulty of inference as κ decreases toward zero.Source: §1 · pp. 3–4
  • Earlier high-concentration asymptotics fixed n, focused on FvML distributions, and rarely addressed local-alternative power or optimality.The paper aims to remove these restrictions while allowing κn to diverge at an arbitrary rate with n.Source: §1 · p. 5
  • The paper studies inference as κn diverges at an arbitrary rate, finding that the spherical mean’s consistency rate depends on f and establishing Le Cam optimality for the mean and Watson and Wald tests under mild assumptions.Confidence zones centered at the spherical mean are also considered.Source: §1 · p. 5

2. High concentration.

This section defines high concentration through concentration of probability mass near the location and characterizes angular functions that produce it. It gives exponential-power examples and develops moment asymptotics under the resulting function class.

2. High concentration.

  • The authors leave κn and f unidentified because they are nuisance parameters and they wish to include functions that may not be differentiable at zero.They do not impose the normalizations f(0)=f′(0)=1.Source: §2 · p. 6
  • High concentration means that the probability mass in every fixed spherical cap centered at θn converges to one as n diverges.The property is determined by the angular function f.Source: §2 · p. 7
  • Under the stated regularity conditions, κϕf(κ) tending to a finite positive constant or to zero does not yield high concentration.The section characterizes high concentration as requiring f to diverge at least exponentially fast at infinity.Source: §2 · pp. 7–8
  • Every exponential-power function f(z)=exp(z^b), for b>0, belongs to the section’s class F of angular functions.This family includes the FvML angular function.Source: §2 · p. 8
  • For f∈F and κn diverging, the section studies asymptotic moments and states that the first cosine moment satisfies en1=1+o(1).The section does not provide the expansion of 1−en1 because it is not needed for the paper’s purposes.Source: §2 · p. 9

3. Point estimation.

Under high concentration, the spherical mean converges faster than the usual root-n rate, with its rate depending on the angular function f. The section develops confidence caps centered at the mean and examines their feasibility and finite-sample behavior.

3. Point estimation.

  • For f(z)=exp(z^b), b>0, the spherical mean rate is n^((b+1)/2), allowing arbitrarily fast polynomial convergence as b increases.For small b, the rate can be arbitrarily close to the usual root-n rate.Source: §3 · p. 10
  • A second-order result yields confidence zones in the form of connected spherical caps centered at the spherical mean.The first-order zone is problematic because it consists of two antipodal caps and is not connected.Source: §3 · pp. 10–11
  • Replacing the unknown κ_nϕ_f(κ_n) with (p−1)/(1−e_n²) has no asymptotic impact and makes feasible confidence caps possible.The feasible zones retain the connected spherical-cap form centered at the spherical mean.Source: §3 · p. 11
  • In Monte Carlo simulations with M = 10,000 samples of size n = 100 across stated a,b combinations, Figure 1 supports theoretical results, with possibly one exception involving T Feasible.Source: §3 · p. 12
  • At n=10,000, Figure 2 agrees with theory for a=1, while fit remains imperfect for a=0.5 because that distribution is not highly concentrated.The authors note that this does not contradict the theory, which applies at higher concentrations.Source: §3 · p. 12

4. Hypothesis testing.

The section studies Watson and Wald tests of H0: θ=θ0 under high concentration. Their statistics remain asymptotically equivalent, with size α and non-trivial power at local alternatives calibrated to the concentration-dependent rate.

4. Hypothesis testing.

  • Under high concentration, Watson and Wald statistics are asymptotically equivalent in probability under the null and considered local alternatives, with asymptotic size α regardless of f or κ_n's divergence rate.The alternatives are parameterized using θ_n=θ_0+ν_nτ_n, with τ_n converging to τ.Source: §4 · pp. 13–14
  • The tests have non-trivial asymptotic power at local alternatives scaled by ν_n, which is the consistency rate for testing under high concentration.The excerpt characterizes these powers as lying between α and 1.Source: §4 · p. 14
  • Compared with fixed-concentration root-n alternatives, high concentration permits detection of less severe alternatives; the testing rate is tied to ν_n.The spherical mean's consistency rate depends on f, as noted in the point-estimation results.Source: §4 · p. 14
  • The simulations compare rejection frequencies with limiting powers across alternatives of increasing severity, with ℓ=0 denoting the null.Tests were run at asymptotic level α=5%.Source: §4 · p. 15
  • In simulations, Watson–Wald asymptotic equivalence appeared by n=100 except in the lowest-concentration case.Agreement with limiting powers improved as sample size increased.Source: §4 · p. 15

5. Local asymptotic normality.

The section uses local asymptotic normality to connect high-concentration inference rates with contiguity. It establishes optimality properties for the spherical mean and the Watson and Wald tests, while showing that a tangent-space interpretation also holds in the original spherical problem.

5. Local asymptotic normality.

  • The LAN analysis requires strengthened assumptions on f, which most angular functions satisfy, sometimes with a mild restriction on κn’s growth.For f_b(z)=exp(z^b) with b≥1, the stated FLAN condition holds regardless of the sequences κn and tn.Source: §5 · pp. 15–16
  • The spherical mean’s f-dependent convergence rate is the contiguity rate, making it rate-consistent and asymptotically optimal under the convolution theorem.The optimality result is tied to the LAN framework and its assumptions on f.Source: §5 · p. 17
  • Watson and Wald tests have non-trivial local power under contiguous alternatives, with Wald inheriting Watson’s result through their asymptotic equivalence.The equivalence under the null extends to the contiguous alternatives.Source: §5 · pp. 17–18
  • Under the LAN assumptions, the Watson and Wald tests are Le Cam optimal, and the optimal test is locally asymptotically maximin at level α.The Watson test’s optimality holds at any f meeting the assumptions; the Wald result follows from asymptotic equivalence and contiguity.Source: §5 · pp. 18–19
  • The Watson test is a spherical Hotelling test on tangent-space projections, and the LAN result proves its optimality in the original curved experiments.The projected data are spherically symmetric under the null, with a standardization matrix proportional to the identity.Source: §5 · p. 19

6. Real data illustration.

The real-data illustration applies spherical-location inference to 62 magnetic remanence directions. The data appear highly concentrated, and the reported tests do not reject rotational symmetry.

6. Real data illustration.

  • The 62 rock-specimen directions are highly concentrated, with an FvML concentration estimate of 76.12, comparable in order to n=62.The dataset is on S2 and had previously been used for inference on spherical location.Source: §6 · pp. 19–20
  • The three rotational-symmetry tests returned p-values .844, .305, and .607, with none rejecting at usual nominal levels.The same tests also did not reject on any of 62 leave-one-out subsamples.Source: §6 · p. 20
  • The 95% confidence cap is centered at the spherical mean (.210, .104, .972)′ and has very small size.The authors present it as an illustration of the high-concentration setting.Source: §6 · pp. 20–21

7. Wrap up.

The paper studies spherical-location inference when concentration diverges at any rate with sample size, beyond the parametric and fixed-sample high-concentration settings of earlier work. The procedures retain validity and gain f-dependent rates, with Le Cam optimality under stated conditions.

7. Wrap up.

  • The analysis covers rotationally symmetric distributions with κn diverging at an arbitrary rate, extending beyond FvML and asymptotics with n fixed.The earlier high-concentration literature described here was restricted to the parametric FvML class and κ diverging with n fixed.Source: §7 · p. 21
  • Under high concentration, the spherical mean remains consistent and asymptotically normal, while Watson and Wald tests continue to meet the asymptotic level constraint.The procedures also have faster consistency rates than in the fixed-κ scenario, and those rates depend on f.Source: §7 · p. 21
  • Under mild assumptions on f, the inference procedures have Le Cam-type optimality properties, though some angular functions require κn to grow sufficiently quickly.For f_b(z)=exp(z^b), the stated growth restriction is at least (log n)^2 for some b, while b≥1 requires no such restriction; this includes FvML.Source: §7 · p. 21

A.1. Proof of Theorem 2.1.

The proof reduces high concentration to divergence of a ratio of integrals. It rules out cases where κϕf(κ) approaches a positive constant or decreases to zero, and uses eventual convexity or concavity properties of f.

A.1. Proof of Theorem 2.1.

  • When zϕf(z) is eventually nondecreasing with values at least 1, f is eventually convex; the proof uses an analogous argument for the nonincreasing case with values in [0,1].The convexity or concavity conditions are imposed on a sufficiently large tail of the domain.Source: §A.1 · p. 22
  • High concentration is equivalent to Aκ/Bκ diverging to infinity.The proof expresses this through the limiting fraction Aκ/(Aκ+Bκ) tending to one.Source: §A.1 · p. 22
  • If κϕf(κ) approaches a positive constant or decreases to zero, the proof concludes that f does not provide high concentration.The two cases are treated separately in the proof.Source: §A.1 · p. 24

A.2. Proof of Proposition 2.1.

The proof establishes Proposition 2.1 for f_b(z) = exp(z^b) by verifying Condition (2.5) for every b > 0. Its bounds handle the cases b < 1 and b ≥ 1 separately.

A.2. Proof of Proposition 2.1.

  • For b ∈ (0, 1), the function (1 − r)^b − 1 + br is nonpositive on [0, 2], by monotonicity from zero.For all b > 0, its absolute value is also bounded by a constant times r^2 on [0, 2].Source: §A.2 · p. 25
  • For f_b(z) = exp(z^b), the proof reduces Proposition 2.1 to verifying Condition (2.5) for any b > 0.The other conditions are stated to be trivially fulfilled.Source: §A.2 · p. 26

A.3. Proof of Theorem 2.2.

The proof of Theorem 2.2 uses Lemma A.4 together with relations (A.4)–(A.5). Lemma A.4 controls variance terms through a U-statistic formulation and integral bounds.

A.3. Proof of Theorem 2.2.

  • Lemma A.4 bounds variance terms using a U-statistic formulation and separates contributions involving s and its approximation.The proof also bounds squared differences by terms involving (1 − s)^2 and (1 − s̃)^2.Source: §A.3 · pp. 28–30
  • The integral bounds apply Condition (2.5) after the substitution z = (1 − s)κ_nϕ_f(κ_n) in two integrals.This is one step in the proof of Lemma A.4.Source: §A.3 · p. 30
  • Theorem 2.2 follows by combining Lemma A.4 with relations (A.4)–(A.5).The proof first invokes Lemma A.4 and then applies it alongside those relations.Source: §A.3 · p. 31

A.4. Proofs of Theorems 3.1, 3.2 and 3.3.

The appendix proves the asymptotic results through delta-method and limit arguments, a reduction of the Wald statistic, and bounds on likelihood terms. The proofs use tangent-normal decompositions and conditions on f where indicated.

A.4. Proofs of Theorems 3.1, 3.2 and 3.3.

  • A uniform second-order delta method supports the proofs of the spherical-mean limits, including the normalized-mean transformation.The proof of Theorem 3.2 uses the Hessian of the coordinate map x_j/∥x∥.Source: §A.4 · pp. 31, 32, 34
  • For Theorem 3.1, the proof establishes asymptotic normality through Lindeberg checks and derives the corresponding weak limit.The argument first considers projections along unit vectors.Source: §A.4 · pp. 33–34
  • Lemma A.6 controls tangent-normal decomposition terms when κ_n diverges, providing a preliminary result for Theorem 3.3.Its proof shows the listed remainder terms converge to zero in probability.Source: §A.4 · pp. 35–36

A.5. Proofs of Lemma 4.1 and Theorem 4.1.

  • The Wald statistic is reduced to W_n + o_P(1), establishing the asymptotic equivalence used in Theorem 4.1.The proof applies Lemma 4.1 in this reduction.Source: §A.5 · p. 39

A.6. Proofs of Proposition 5.1 and Theorem 5.1.

  • The proof of Theorem 5.1 decomposes its remaining terms and uses f ∈ F_LAN(p, κ_n, ∥τ_n∥) to bound them.The stated condition is used to conclude that the relevant remainder terms are o_P(1).Source: §A.6 · pp. 44–46
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