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Introduction to Optimal Estimation , ser. Advanced Textbooks in Control and Signal Processing. Springer, 1999. [9]K. Reif, S. Günther, E. Yaz, and R. Unbehauen, “Stochastic stability of the discrete-time extended Kalman filter,” IEEE Transactions on Automatic Control , vol. 44, no. 4, pp. 714–728, 1999.[10] K. Xiong, H... | https://arxiv.org/abs/2503.21643v1 |
G. Reinert, and Y . Swan, “Stein’s method for comparison of univariate distributions,” Probability Surveys , vol. 14, pp. 1–52, 2017. [28] T. Cacoullos, “On upper and lower bounds for the variance of a function of a random variable,” The Annals of Probability , vol. 10, no. 3, pp. 799–809, 1982. [29] J. Nash, “Continui... | https://arxiv.org/abs/2503.21643v1 |
Rolled Gaussian process models for curves on manifolds S. P. Preston1,∗, K. Bharath1, P. C. L´ opez-Custodio2, A. Kume3 Abstract Given a planar curve, imagine rolling a sphere along that curve without slipping or twisting, and by this means tracing out a curve on the sphere. It is well known that such a rolling operati... | https://arxiv.org/abs/2503.21980v1 |
tangent space at a single fixed point on M, and using tangent space projections between the tangent space and M, and vice versa, which causes distortions. Our maps provide a natural way to define a covariance, and hence a Gaussian process model, on M, starting with one in Rd. 1.2 Related works, and contributions “Rolli... | https://arxiv.org/abs/2503.21980v1 |
mean of the Gaussian process in Rd; (iii) to prescribe a finite-dimensional parameterisation of the rolled Gaussian process model, with statistical estimators and tests for the model parameters, in the practically useful setting when sample curves are observed at a finite set of times; and (iv) to establish conditions ... | https://arxiv.org/abs/2503.21980v1 |
injectivity radius of pis its distance to C(p), and the injectivity radius i(M) ofMis the infimum of the injectivity radii of points in M. The next two tools are necessary for moving between Mand its tangent spaces. Geodesics onMare used to define the exponential map exp : TM→M, which maps a point ( p, v) in the tangen... | https://arxiv.org/abs/2503.21980v1 |
said to be symmetric ifg(p) =g(σp(p)) for every p∈M. Relatedly, a distribution νonMis said to possess even symmetry about p∈M, when ν= (expp)#λ, the pushforward under the exponential map at pof a mean-zero distribution λonTpMwith Lebesgue density f, when TpMis identified with Rd, satisfies f(v) =f(−v) for every v∈TpM. ... | https://arxiv.org/abs/2503.21980v1 |
the unwrapping of point x, assumed to be outside the cut locus C{γ(t)}ofγ(t), onto Tγ(0)Mdefined as (t, x)7→Pγ 0←tn exp−1 γ(t)(x)o ∈Tγ(0)M. (3.2) The reverse operation, for a point yonTγ(0)Massociated with t∈[0,1] is the wrapping of point y∈Tγ(0)Monto Mwith respect to the curve γ: (t, y)7→expγ(t)(Pγ t←0y)∈M. (3.3) Give... | https://arxiv.org/abs/2503.21980v1 |
x=γin (3.5); in other words, the unwrapping of a curve with respect to itself is its unrolling. Remark 3.2.Provided xin (c) is outside the cut loci of γ, the wrapping map is the unique inverse of the unwrapping map. Provided yin (d) is such that the argument of expγ(t)in (3.6) is outside the tangential cut locus T {γ(t... | https://arxiv.org/abs/2503.21980v1 |
rolled Gaussian process model. Let z∼GP(m, K ) be a time-indexed Gaussian process in Rdwith mean curve t7→m(t) ={m1(t), . . . , md(t)}⊤∈Rdand covariance (t, t′)7→K(t, t′)∈Sym>0(d). The law of zis characterised by the requirement that for every finite rand times t1, . . . , t r, the d×rmatrix Z:={z(t1), . . . , z (tr)}s... | https://arxiv.org/abs/2503.21980v1 |
space TbMis arbitrary. The rolled Gaussian process, however, is equivariant with respect to this choice. Proposition 3.2 (Equivariance of rolled Gaussian process) .Starting from point b∈Mwith basis UforTbM, letx∼ RGP (m, K ;b, U). If one starts instead from b′∈Mwith basis U′forTb′M, then there exists unique m′andK′such... | https://arxiv.org/abs/2503.21980v1 |
spaced, but the former simplifies notation considerably, and both can be easily relaxed. We use upper case for the discrete-time version of the curve denoted in corresponding lower case; for example, Γ := {γ(t1), . . . , γ (tr)} ∈Mris curve γin discrete time. The four maps in §3.1 have natural analogues for discrete cu... | https://arxiv.org/abs/2503.21980v1 |
basis, Φ. Then there exists a∈Rdand a linear isometry A:Rd→Rdsuch that H′(Xi; Γ) = a1⊤ r+ AH(Xi; Γ)∼ MN M′ wΦ, U′ w,Φ⊤VwΦ , where M′ w=a1⊤ k+AMwandU′ w=AUwA⊤. Under matrix representations of the parallel transport maps, the operator Ais an orthogonal matrix. This Proposition is an analogue of Proposition 3.2, special... | https://arxiv.org/abs/2503.21980v1 |
in the case where the manifold Mhas non-positive sectional curvatures then ˆMMLE w together with ˆUwˆVwin (4.8, 4.9) are asymptotically, for large n, the maximum likelihood estimators, and are exactly so when Γ is the data-generating curve. Another estimator of Γ is based on the discretization of the sample Fr´ echet m... | https://arxiv.org/abs/2503.21980v1 |
the analogue of the Hotelling T2statistic is J= trn ˆV−1 w(ˆM(1) w−ˆM(2) w)⊤ˆU−1 w(ˆM(1) w−ˆM(2) w)o , (4.11) in which ˆUw={(n1−1)ˆU(1) w+ (n2−1)ˆU(2) w}/(n1+n2−2) is a pooled estimate of Uw, and Vw is defined similarly. The null hypothesis is rejected for suitably large Jwith respect to its null distribution, which is... | https://arxiv.org/abs/2503.21980v1 |
respect to Γ onto M. For a curve XonM, the curve’s unwrapping coordinates (4.2), with the unwrapping being with respect to the curve Γ, are H(Xi; Γ) = U⊤n (Xi)↓Γ b−b1⊤ ro . (5.2) 6 Numerical examples 6.1 A prescribed model for heteroscedastic curves on S2 Here we give an example for M=S2, of the steps described in the ... | https://arxiv.org/abs/2503.21980v1 |
data {Xi}as in §6.1. Table 1 shows numerical results from simulating the data then fitting the model to the simulated data, to investigate convergence of estimators ˆMw=ˆMFre wfrom (4.10) and ˆUwand ˆVwfrom (4.8) and (4.9) to their data-generating values. Results in the table show each estimator, with respect to the in... | https://arxiv.org/abs/2503.21980v1 |
of values of test statistic Jin (4.11) simulated under the null hypothesis of equal means, using the permutation procedure described in §4.3 with R= 200 resamples, and the red line shows the observed value of the statistic J= 22.17 computed from the data. This observed value is extreme compared with the null distributi... | https://arxiv.org/abs/2503.21980v1 |
Σ if for every x∈Xthe real-valued random variable ⟨x, Z⟩has a Gaussian distribution onRwith mean ⟨x, µ⟩Xand variance ⟨x,Σx⟩X. With respect to (Ω ,F,P) suppose X1andX2are two random vectors on ( X1,⟨·,·⟩X1) and (X2,⟨·,·⟩X2), respectively, with covariances Σ 1and Σ 2. The random vector ( X1, X2) then assumes values in th... | https://arxiv.org/abs/2503.21980v1 |
discretized curve, Γ, is P˜γ t←t0=P˜γ t←tj···P˜γ t2←t1P˜γ t1←t0, here for tsatisfying tj≤t≤tj+1; in other words, the parallel transport is a composition of parallel transports each along a geodesic, enabling simple computation. To compute the discretized unrolling, Γ↓={γ↓(t1), . . . , γ↓(tm)}, of Γ, from (3.1) and with... | https://arxiv.org/abs/2503.21980v1 |
to show that γ(t) is the stationary point of Fin order for it to be the Fr´ echet mean. Assume otherwise, and let γ(t) be a stationary point of Fwhose unique minimizer is qdistinct from γ(t). Since Fis symmetric about γ(t), we have then that σγ(t)(q) must also minimizer F, which contradicts the uniqueness assumption of... | https://arxiv.org/abs/2503.21980v1 |
desired result; a similar argument applies for k′as well. The chosen m′andk′are unique, since the parallel vector fields along ηdetermined byUandU′that generate the parallel transport maps, respectively, from Tb′MtoTbMand vice versa are unique. If ηis chosen to be the piecewise curve consisting of a segment connecting ... | https://arxiv.org/abs/2503.21980v1 |
γlies in a single chart. Then, since γnare converging to γ, for a large enough n, we may assume that γn, for all n, and γlie in the same chart. Using the reasoning above, we note the solutions may be written in the integral form Vn(t) =Vn(0)−Zt 0A(γn(s))Vn(s)ds, V (t) =V(0)−Zt 0A(γ(s))V(s)ds. Then, the deviation En(t) ... | https://arxiv.org/abs/2503.21980v1 |
Springer Science & Business Media, 2012. Michael Hutchinson, Alexander Terenin, Viacheslav Borovitskiy, So Takao, Yee Teh, and Marc Deisenroth. Vector-valued gaussian processes on Riemannian manifolds via gauge independent projected kernels. Advances in Neural Information Processing Systems , 34:17160–17169, 2021. Pete... | https://arxiv.org/abs/2503.21980v1 |
An Improved Satterthwaite Effective Degrees of Freedom Correction for Weighted Syntheses of Variance Matthias von Davier∗ April 22, 2025 Abstract This article presents an improved approximation for the effective degrees of freedom in the Satterthwaite (1941, 1946) method which estimates the distribution of a weighted c... | https://arxiv.org/abs/2503.22080v4 |
use some well known identities to arrive at the main result. First, recall that V ar(X) =E(X2)−E(X)2and that V ar(aX) =a2V ar(X)and that for independent X1, ..., X Kwe have V ar(P kwkXk) =P kw2 iV ar(Xk). The following equivalencies hold ν2 ∗V ar(S2 ∗) (σ2 ∗)2= 2ν∗↔ν∗= 2(σ2 ∗)2 V ar(S2 ∗) and V ar S2 ∗ =KX k=1w2 kV a... | https://arxiv.org/abs/2503.22080v4 |
K) with ν=KX k=1νk K being the arithmentic mean of the νk.The derivations by von Davier (2025) 7 provide the following expression for the unweighted case: ν∗≈PK k=1S2 k2 1 +K K−12P kνk" PK k=1(S2 k)2 νk+2#. 4 An Improved Effective d.f. Estimator for the Weighted Case Denoting the weighted average of the component ... | https://arxiv.org/abs/2503.22080v4 |
When looking at the unadjusted estimator, a simulation of 4 million 11 replications gives an observed average value of E [Z2 1+Z2 2]2 Z4 1+Z4 2! ≈1 4,000,0004,000,000X i=1[Z2 1+Z2 2]2 Z4 1+Z4 2= 1.41425 . This could be used to find a correctution term Cfor which E(ν∗) = 2 = 3 1 +C 2! 1.41425 this leads to C=6−2×1.41425... | https://arxiv.org/abs/2503.22080v4 |
be made based on an optimality criterion X2 C=X K∈ΩKX ν∈Ων[M(ˆν∗,C)−Kν]2 Kν 14 where the range of components Kand component degrees of freedom νis defined as K∈ΩK={2,3, . . . , K max}andν∈Ων={1,2, . . . , ν max}. The quantity X2 Ccan be understood as a pseudo χ2measure of deviation, which is minimized by finding Copt= ... | https://arxiv.org/abs/2503.22080v4 |
from (5,5)to(100,100). The way the optimal value was determined as Copt≈2.69can be explained based on table 1. The growth of the values of CoptasKmax, νmaxgrow is negatively accelerated and for the difference between (40,40)and(50,50) we see that Coptonly increases by 0.49from 2.6524to2.6655.The percentage increase see... | https://arxiv.org/abs/2503.22080v4 |
true value, that is, ˆν∗,Satter << K ×ν. For example, the true value for K= 6andν= 3is E(ν∗) = 18while the observed average is M(ˆν∗) = 12 .45.Even for ν= 9the estimator does not closely track the true value, for K= 4we find M(ˆν∗) = 31 .37<36.Maybe surprisingly, even for ν= 30 , K= 160we obtain M(ˆν∗) = 4502 .64<4800 ... | https://arxiv.org/abs/2503.22080v4 |
0.06412 Optimal (Kmax, νmax)Adjustment Copt2.69 K 0.03632 Table 6: Pseudo X2values for the original Satterthwaite estimator, and the three adjustments under examination. than the one using C= 2.25. The overall quality of the approximations using different adjustments can be evaluated using the pseudo χ2statistics X2cal... | https://arxiv.org/abs/2503.22080v4 |
provided in this article generalize and further improve the adjusted Satterthwaite formula derived by von Davier (2025) for the unweighted case. It is proposed to estimate the 25 effective degrees of freedom for the general case of K≥2andνk≥1using equation 2 References von Davier, M. (2025). An Improved Satterthwaite (... | https://arxiv.org/abs/2503.22080v4 |
Asymptotic Behavior of Principal Component Projections for Multivariate Extremes Holger Drees University of Hamburg, Department of Mathematics, Germany e-mail: holger.drees@uni-hamburg.de Abstract: The extremal dependence structure of a regularly varying d-dimensional random vector can be described by its angular measu... | https://arxiv.org/abs/2503.22296v1 |
angular measure, which can be any probability measure onSd−1. Reduction of complexity Therefore, multivariate extreme value theory is inherently non-parametric and thus prone to the curse of dimensionality. To render standard methods of mul- tivariate extreme value theory practically feasible in settings of high (and o... | https://arxiv.org/abs/2503.22296v1 |
The goal, though, is very different, namely not to estimate the an- gular measure accurately, but to detect clusters in which the angular measure is approximately concentrated, while any mass of the angular measure in other re- gions (which is considered noise) should be pushed to these clusters. Since these alternativ... | https://arxiv.org/abs/2503.22296v1 |
which will usually not be concentrated on the same set as the limit measure. H. Drees/Asymptotics for PCA for Extremes, March 31, 2025 4 Notation All random variables are defined on some probability space (X,A,P). The ex- pectationwithrespectto Pisdenotedby E,andthelawofsomerandomelement YbyL(Y). For an Rd-valued rando... | https://arxiv.org/abs/2503.22296v1 |
combination of the Cramér-Wold device with the central limit theorem by Lindeberg and Feller shows that k−1/2k/summationdisplay i=1/parenleftig Θ(n,s) i(Θ(n,s) i)⊤−Estn,k(ΘΘ⊤)/parenrightig →Uweakly. Thus, Condition (B) yields k−1/2k/summationdisplay i=1/parenleftbig Θ(n,s) i(Θ(n,s) i)⊤−Σn,k/parenrightbig →Uweakly for... | https://arxiv.org/abs/2503.22296v1 |
ˆΣn,k. According to Theorem 5.1.3 of [10], one has ΠˆW(n) j−ΠW(n) j=−1 2πi/contintegraldisplay ΓˆRn(z)−Rn(z)dz, (3.2) where ˆRn(z) =/parenleftbig ∆n+ (Rn(z))−1/parenrightbig−1=Rn(z)/parenleftbig ∆nRn(z) +I/parenrightbig−1.(3.3) Since∥Rn(z)∥op= 1/min1≤m≤d|z−λ(n) m|<4/(3η)forz∈Im(Γ)(see [10, (5.3)]), we have ∥∆nRn(z)∥op≤... | https://arxiv.org/abs/2503.22296v1 |
[14], we have k/parenleftbig Mn,k(ΠV∗ n,k)−Mn,k(ΠˆVn,k)/parenrightbig =p/summationdisplay i=1d/summationdisplay ℓ=p+1λ(n) i (λ(n) i−ˆλ(n) ℓ)2/vextenddouble/vextenddouble/vextenddoubleΠ(n) i√ k∆nˆΠ(n) ℓ/vextenddouble/vextenddouble/vextenddouble2 HS −d/summationdisplay ℓ=p+1p/summationdisplay i=1λ(n) ℓ (λ(n) ℓ−ˆλ(n) i)2/... | https://arxiv.org/abs/2503.22296v1 |
the proof of Corollary 4.2, by combining Lemma 2.5 and Lemma 3.1 of [14] we obtain with probability of at least 1−ε sup V∈VpMn,k(ΠV∗ n,k)−Mn,k(ΠˆVn,k) ≤in−1/summationdisplay i=1(λ(n) i−λ(n) p)d/summationdisplay j=p+1/vextenddouble/vextenddouble/vextenddoubleΠ(n) i∆nˆΠ(n) j/vextenddouble/vextenddouble/vextenddouble2 HS ... | https://arxiv.org/abs/2503.22296v1 |
the optimal projection. Theorem 5.1. If Condition (B) holds and there is a unique optimal projection in the limit, then k/parenleftbig Mn,k(˜Πn,A)−Mn,k(ΠV∗ n,k)/parenrightbig −→/angbracketleftbig Σ,A2(Π∗−Π⊥)/angbracketrightbig HS(5.2) k/parenleftbigˆMn,k(˜Πn,A)−ˆMn,k(ΠV∗ n,k)/parenrightbig −→ 2⟨U,Π∗A⟩HS+/angbracketleft... | https://arxiv.org/abs/2503.22296v1 |
any given projection matrix, because one would need to know ΠV∗ n,kin order to calculate the pertaining local parameter A. To avoid this problem, one might also think of a different local parametrization centered at the known (random) PCA projection: ˆΠn,B:= e−k−1/2BΠˆVn,kek−1/2B, B∈˜MS. Arguments completely analogous ... | https://arxiv.org/abs/2503.22296v1 |
variable is centered normal with variance σ2 p=p/summationdisplay i=1p/summationdisplay j=1Cov∞(v⊤ iΘΘ⊤vi,v⊤ jΘΘ⊤vj) =Var∞/parenleftigp/summationdisplay i=1(v⊤ iΘ)2/parenrightig =Var∞(∥Π(∗p)Θ∥2). A natural estimator for σ2 pis the empirical variance of the squared norm of ΠˆVp n,kΘ, that is ˆσ2 p:=1 k−1n/summationdis... | https://arxiv.org/abs/2503.22296v1 |
ˆHPCA n,k,˜k:=1 kn/summationdisplay i=1δθtn,k(ΠˆV˜p n,˜kXi). (6.3) In most applications, estimating the angular measure is not the main goal of the statistical analysis, but it is just an important step in order to estimate some other quantities. To evaluate the performance of the proposed PCA procedure, forX= (X1,...,... | https://arxiv.org/abs/2503.22296v1 |
2 or 3 values. Since for large H. Drees/Asymptotics for PCA for Extremes, March 31, 2025 21 0 50 100 150 20000.20.40.6(i) 0 50 100 150 20000.10.20.30.4(ii) 0 50 100 150 20000.050.10.15(iii) 0 50 100 150 20000.050.10.150.2(iv) Fig 1. RMSE of the estimators of the probabilities (i)–(iv) based on ˆHn,k(black, solid), ˆHPC... | https://arxiv.org/abs/2503.22296v1 |
based on (6.3) one should choose ˜ksomewhat larger, because more observations are needed to estimate a subspace of higher dimension accurately. Here we show the results for ˜k= 15. Figure 5 shows the RMSE of the estimator of the four parameters for the Dirichlet model (with t(i)= 0.4and true values 0.5727, 0.1766, 0.57... | https://arxiv.org/abs/2503.22296v1 |
the bias and the variance are substantially increased compared with the estimator with PCA projection on a five dimensional subspace. In contrast, the standard PCA based estimators, which uses the same number of exceedancesineachstepoftheprocedures,nowperformsbetterifthedimension is chosen using the data for all parame... | https://arxiv.org/abs/2503.22296v1 |
Conditional Extreme Value Estimation for Dependent Time Series MARTIN BLADT1,a, LAURITS GLARGAARD1,band THEODOR HENNINGSEN1,c 1Department of Mathematical Sciences, University of Copenhagen, Denmark, amartinbladt@math.ku.dk,blauritsglargaard@mail.dk,cth@math.ku.dk We study the consistency and weak convergence of the con... | https://arxiv.org/abs/2503.22366v1 |
order behaviour and specification of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.2 Asymptotic normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2503.22366v1 |
. . . . . . . . . . . . 16 Funding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 A Proofs of main results from ... | https://arxiv.org/abs/2503.22366v1 |
. . 44 C.1 Conditional Fréchet and Pareto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 C.2 Conditional Max-Stable process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 D Kernel density estimation for mixing sequences . . . . . . . . . ... | https://arxiv.org/abs/2503.22366v1 |
for the conditional tail process. Through simulation studies, we investigate our method across different scenarios. Notably, a distinguishing advantage of our estimator is its simple (and easy to plug-in) variance structure, in contrast to the often intricate variance expressions encountered in the unconditional case. ... | https://arxiv.org/abs/2503.22366v1 |
density, denoted a Kernel, supported on [−1,1]and denote by{ℎ𝑛,𝑛∈N}the corresponding bandwidth (scale) parameter. Let 𝑢𝑥 𝑛,𝑛∈N be a scaling sequence diverging to infinity. Henceforth, we omit the dependence on 𝑥and simply write{𝑢𝑛}. The conditional survival function 𝐹𝑥 is estimated with the Nadaraya–Watson ... | https://arxiv.org/abs/2503.22366v1 |
OVX). The effective sample is calculated using a uniform kernel centred at 𝑥=0.05,0.50,0.95 (highlighted in red, green and blue, respectively) and with ℎ𝑛=0.05. Middle panel: target process 𝑌𝑗 given by the absolute value of the negative log-returns of the West Texas Intermediate (WTI) grade crude oil spot prices w... | https://arxiv.org/abs/2503.22366v1 |
an infinite sum, e.g. in Kulik and Soulier (2020, Eq. (9.5.4)). For𝑠0∈(0,1), define 𝐵𝑥 𝑛(𝑠0)=sup 𝑡≥𝑠0 Eh e𝑉𝑥 𝑛(𝑡)i −𝑡−𝛼𝑥𝑔(𝑥) . Condition 2. The conditions for normality are as follows. 2.1 lim𝑛→∞𝑟𝑛ℎ𝑛𝐹𝑥(𝑢𝑛)=0, 2.2For𝑠,𝑡> 0, lim𝑚→∞lim sup 𝑛→∞𝑟𝑛∑︁ 𝑗=𝑚Eh 𝐾 𝑥−𝑋0 ℎ𝑛 1{𝑌0>𝑠𝑢𝑛}𝐾𝑥−𝑋... | https://arxiv.org/abs/2503.22366v1 |
lim𝑛→∞𝐹𝑥(𝑢𝑛)+ℎ𝑐𝑥𝜂−2 𝑛 𝑛ℎ𝑐𝑥𝑛(𝐹𝑥(𝑢𝑛))2=0, (2). Notice that, in the case of 𝑚-dependence, only the first two conditions are required. In any case, the conditions are directly on the conditional distribution, so the results below do not make use of the time- series structure of the data, but only on the (... | https://arxiv.org/abs/2503.22366v1 |
conditions for consistency, all the above conditions for normality and anti-clustering conditions 2.2 and 2.6 are satisfied. Theorem 9. Assume (1)and that (𝑋𝑗,𝑌𝑗), 𝑗∈Z is a stationary, 𝛽-mixing time series, with 𝛼-mixing coefficients𝛼(𝑗)=O(𝑗−𝜂), for𝜂>2,𝛽-mixing coefficients 𝛽(𝑗)=O(𝑗−𝜈)for𝜈>0, and tha... | https://arxiv.org/abs/2503.22366v1 |
simulated from an auto- regressive (AR) process of order 1 (with coefficient 0 .1 and 0.9 in the low and high dependence cases respectively) transformed to have uniform marginals, similarly for the uniform process 𝑈𝑗 that is then transformed into 𝑌𝑗 via the inverse Fréchet distribution function using the chosen𝛾... | https://arxiv.org/abs/2503.22366v1 |
Ibragimov (1962)). When considering the unconditional version of the CSGMS-process, it is possible to show 𝛽-mixing decay of exponential order (see Kulik and Soulier (2020), Example 13.5.4). In (15) we extend the construction by bundling with the covariate sequence, which itself is i.i.d and in particular of exponenti... | https://arxiv.org/abs/2503.22366v1 |
given the Crude Oil Volatility Index (CBOE OVX) and, if this is the case, how the tail 14 0.00.10.20.30.40.50.00.10.20.30.40.5−0.040.000.04 knnBias 0.00.10.20.30.40.50.00.10.20.30.40.50.000.010.020.03 knnMean squared error 0.0 0.1 0.2 0.3 0.4 0.50.0 0.1 0.2 0.3 0.4 0.5−0.050.00 knnBias 0.0 0.1 0.2 0.3 0.4 0.50.0 0.1 0.... | https://arxiv.org/abs/2503.22366v1 |
both positive and negative values. By construction, the conditional Hill estimator is applicable only for positive 𝑌𝑗 . Splitting the bivariate time series 𝑋𝑗,𝑌𝑗 according to where 𝑌𝑗 is positive and negative respectively, and switching the sign of the negative parts, results in two bivariate time series wit... | https://arxiv.org/abs/2503.22366v1 |
consider extensions in several directions. First, having an estimate of the entire tail at our disposal our results can be used to construct and statistically analyse conditional risk measures such as conditional expected shortfall or conditional tail moments. Secondly, a promising direction is to consider specific mod... | https://arxiv.org/abs/2503.22366v1 |
DREES , H. (2000). Weighted approximations of tail processes for 𝛽-mixing random variables. The Annals of Applied Probability 101274 – 1301. https://doi.org/10.1214/aoap/1019487617 GANNOUN , A., G IRARD , S., G UINOT , C. and S ARACCO , J. (2002). Reference curves based on non-parametric quantile regression. Statistic... | https://arxiv.org/abs/2503.22366v1 |
https://doi.org/10.1007/978-3-031-29040-4 VOLKONSKII , V. A. and R OZANOV , Y. A. (1959). Some Limit Theorems for Random Functions. I. Theory of Probability &; Its Applications 4178–197. https://doi.org/10.1137/1104015 WANG, H. and T SAI, C.-L. (2009). Tail Index Regression. Journal of the American Statistical Associat... | https://arxiv.org/abs/2503.22366v1 |
ℎ𝑛 1{𝑌0>𝑠𝑢𝑛}𝐾𝑥−𝑋𝑗 ℎ𝑛 1{𝑌𝑗>𝑠𝑢𝑛}! ≤1 (𝑛ℎ𝑛𝐹𝑥(𝑢𝑛))2 𝑛∑︁ 𝑗=1E 𝐾2𝑥−𝑋𝑗 ℎ𝑛 1{𝑌𝑗>𝑠𝑢𝑛} +2𝑛𝑛∑︁ 𝑗=1cov 𝐾𝑥−𝑋0 ℎ𝑛 1{𝑌0>𝑠𝑢𝑛},𝐾𝑥−𝑋𝑗 ℎ𝑛 1{𝑌𝑗>𝑠𝑢𝑛}! , where the first term vanishes due to Lemma 5 and condition 1.4. The leading term of the covariance behaves as 1 ℎ𝑛𝐹𝑥(... | https://arxiv.org/abs/2503.22366v1 |
𝑛and relate it to the original eV𝑥 𝑛and in turn eT𝑥 𝑛. The first step is to establish finite dimensional convergence for eV†,𝑥 𝑛. Lemma 6. Assume conditions 1.1-1.4, and 2.1-2.2. Then for all 𝑠,𝑡> 0 lim𝑛→∞cov eV†,𝑥 𝑛(𝑠),eV†,𝑥 𝑛(𝑡) =(𝑠∨𝑡)−𝛼𝑥𝑔(𝑥)∫ 𝐾2(𝑢)d𝑢. (16) Proof. Independence between the b... | https://arxiv.org/abs/2503.22366v1 |
𝑟𝑛 E" 𝐾𝑥−𝑋𝑗 ℎ𝑛2 𝐾𝑥−𝑋0 ℎ𝑛 1{𝑌𝑗>𝑠𝑢𝑛}1{𝑌0>𝑠𝑢𝑛}# +E" 𝐾𝑥−𝑋0 ℎ𝑛2 𝐾𝑥−𝑋𝑗 ℎ𝑛 1{𝑌0>𝑠𝑢𝑛}1{𝑌𝑗>𝑠𝑢𝑛}#! , where the first term is order 𝑜 𝑟𝑛ℎ𝑛𝐹𝑥(𝑢𝑛) as above using condition 2.1 and Lemma 5. The leading term in the sum is dealt with as in the proof of Lemma 6, from which we i... | https://arxiv.org/abs/2503.22366v1 |
𝑛is asymptotically equicontinuous w.r.t. the uniform norm on each interval[𝑎,𝑏]with 0<𝑎<𝑏 , i.e if for all 𝜀>0 lim 𝛿→0lim sup 𝑛→∞Pr 𝑤 eV𝑥 𝑛,𝛿,[𝑎,𝑏] >𝜀 =0. To show this limit, we relate the asymptotic equicontinuity of eV𝑥 𝑛to the pseudo-sample version eV†,𝑥 𝑛as follows. Define the processes eV𝑥 ... | https://arxiv.org/abs/2503.22366v1 |
𝑠,𝑡∈[𝑎,𝑏]|E𝑠(𝒚,𝒙)|=sup 𝑠,𝑡∈[𝑎,𝑏] ∑︁ 𝑗∈Z1{𝑦𝑗>𝑠}𝐾(𝑥𝑗) =∑︁ 𝑗∈Z1{𝑦𝑗>𝑎}𝐾(𝑥𝑗). Lemma 16 implies that lim sup 𝑛→∞𝑚𝑛 𝑛ℎ𝑛𝐹𝑥(𝑢𝑛)Eh 𝑮2 𝑢−1 𝑛𝒀† 1,𝑟𝑛,ℎ−1 𝑛 𝒙−𝑿† 1,𝑟𝑛i =lim sup 𝑛→∞𝑚𝑛 𝑛ℎ𝑛𝐹𝑥(𝑢𝑛)E© «𝑟𝑛∑︁ 𝑗=1𝐾𝑥−𝑋𝑗 ℎ𝑛 1{𝑌𝑗>𝑎𝑢𝑛}ª® ¬2 =lim sup 𝑛→∞var eV... | https://arxiv.org/abs/2503.22366v1 |
𝑘𝑛ℎ𝑛1 𝑔(𝑥){𝑔(𝑥)−𝑔𝑛(𝑥)}∫∞ 1e𝑇𝑥 𝑛(𝑠) 𝑠d𝑠+1 𝑔(𝑥)∫∞ 1eV𝑥 𝑛(𝑠) 𝑠d𝑠 +√︁ 𝑘𝑛ℎ𝑛1 𝑔(𝑥)∫∞ 1Eh e𝑉𝑥 𝑛(𝑠)i −𝑔(𝑥)𝑠−𝛼𝑥 𝑠d𝑠. The last term is negligible due to condition 2.7. In the proof of Theorem 3 we showed∫∞ 1e𝑇𝑥 𝑛(𝑠)/𝑠d𝑠Pr→ 𝛾(𝑥). Combined with condition 2.4 we have that the first ter... | https://arxiv.org/abs/2503.22366v1 |
of results from Section 4 Proof of Theorem 6. Notice that by regular variation it holds that log(𝑢𝑛)=1 𝛼𝑥log(ℓ𝑥(𝑢𝑛))−1 𝛼𝑥log𝑘𝑛 𝑛 . Consequently ℎ𝑛log(𝑢𝑛)=ℎ𝑛1 𝛼𝑥log(ℓ𝑥(𝑢𝑛))−1 𝛼𝑥log𝑘𝑛 𝑛 =(𝐹𝑥(𝑢𝑛))𝑒1 𝛼𝑥log(ℓ𝑥(𝑢𝑛))−1 𝛼𝑥log 𝑛−(1−𝛿) =𝛼−1 𝑥𝑢−𝛼𝑥𝑒 𝑛ℓ𝑥(𝑢𝑛)𝑒log(ℓ𝑥(𝑢𝑛)... | https://arxiv.org/abs/2503.22366v1 |
(𝑡𝑢𝑛)−𝛽𝑥+𝑡−𝛼𝑥𝑢−𝛽𝑥𝑛+𝑐𝑥log(𝑡)(𝑢𝑛𝑡)−𝛼𝑥 ≤cst𝑛 𝑘𝑛𝑡−𝛼𝑥𝑢−𝛼𝑥𝑛(1+log(𝑡)). Equation (9) yields that 𝑢−𝛼𝑥𝑛log(𝑢𝑛)=𝑘𝑛 𝑛+O 𝑢−𝛽𝑥𝑛 , and consequently √︁ 𝑘𝑛ℎ𝑛 𝐹𝑥(𝑡𝑢𝑛) 𝐹𝑥(𝑢𝑛)−𝑡−𝛼𝑥 ≤cst𝑡−𝛼𝑥(1+log(𝑡))√︁ 𝑘𝑛ℎ𝑛𝑛 𝑘𝑛𝑢−𝛼𝑥𝑛log(𝑢𝑛) log(𝑢𝑛) =cst𝑡−𝛼𝑥(1+log(𝑡))√︁ 𝑘... | https://arxiv.org/abs/2503.22366v1 |
since𝛽𝑥/𝛼𝑥>1, by assumption. It then holds that √︁ ℎ𝑛𝑘𝛽𝑥 𝛼𝑥−1 2 𝑛𝑛1−𝛽𝑥 𝛼𝑥≍√︁ 𝑛−𝑒(1−𝛿)𝑛𝛿(𝛽𝑥 𝛼𝑥−1 2)+1−𝛽𝑥 𝛼𝑥=𝑛𝛿(𝑒 2+𝛽𝑥 𝛼𝑥−1 2)− 𝑒 2+𝛽𝑥 𝛼𝑥−1 →0, Conditional Extreme Value Estimation for Dependent Time Series 43 which concludes the last two of the above five conditions. According ... | https://arxiv.org/abs/2503.22366v1 |
log2 + 𝑌0 𝑢𝑛𝐴 |𝑋0=𝑥+ℎ𝑛𝑤1i 𝐹𝑥(𝑢𝑛)ª®® ¬1/2 © «Eh log2 + 𝑌0 𝑢𝑛𝐴 |𝑋0=𝑥+ℎ𝑛𝑤2i 𝐹𝑥(𝑢𝑛)ª®® ¬1/2 →∫∞ 𝐴log2(𝑡)𝛼𝑥𝑡−𝛼𝑥−1d𝑡, uniformly in 𝑤1,𝑤2∈[−1,1]as shown in the proof of Theorem 5. We conclude that also (11) holds. C.2. Conditional Max-Stable process The construction from (15), can be su... | https://arxiv.org/abs/2503.22366v1 |
each 𝑋𝑗admits twice continuously differentiable density 𝑔, and the joint density 𝑔𝑋𝑖,𝑋𝑗of (𝑋𝑖,𝑋𝑗)exists, and is bounded for all 𝑖,𝑗∈Z. Then for any 𝑥∈R, it holds that 𝑔𝑛(𝑥)Pr→𝑔(𝑥). Proof. Fix𝑥∈R. The bias is negligible since E[𝑔𝑛(𝑥)]−𝑔(𝑥)=∫1 ℎ𝑛𝐾𝑥−𝑦 ℎ𝑛 {𝑔(𝑦)−𝑔(𝑥)}d𝑦 =∫ 𝐾(𝑢){𝑔′(𝑥... | https://arxiv.org/abs/2503.22366v1 |
= 𝑛−𝑛 𝑚 𝑚 𝑛2ℎ𝑛Eh 𝐾2 𝑥−𝑋0 ℎ𝑛i ℎ𝑛 1+2 𝑛−j𝑛 𝑚k 𝑚 →0, where we used the Cauchy-Schwarz inequality. The above calculations imply that the remainder is negligible which finishes the proof. Lemma 15. Assume that that ℎ𝑛→0,𝑢𝑛→∞ ,𝑛ℎ𝑛→∞ , that the time series 𝑋𝑗,𝑗∈Z is sta- tionary, each 𝑋�... | https://arxiv.org/abs/2503.22366v1 |
integers. Let(Ω,F,Pr)be a probability space on which the fol- lowing sequences are defined, (i) a stationary R𝑑-valued sequence 𝑋𝑗,𝑗∈Z with𝛽-mixing coef- ficients𝛽𝑗, (ii) a sequence 𝑋𝑗,𝑗∈Z such that the vectors 𝑋† (𝑖−1)𝑟+1,...,𝑋† 𝑖𝑟 ,𝑖∈Zare mutu- ally independent and each have the same distribution... | https://arxiv.org/abs/2503.22366v1 |
Theorem 19. Consider the sequence of processes Z𝑛(𝑓)=𝑚𝑛∑︁ 𝑖=1 𝑍𝑛,𝑖(𝑓)−E 𝑍𝑛,𝑖(𝑓) , where for each 𝑛∈N, 𝑍𝑛,𝑖,𝑖=1,...,𝑚𝑛 ,are i.i.d. separable stochastic processes and assume that the space(G,𝜌)is totally bounded. Suppose that 54 1.For all𝜂>0, lim𝑛→∞𝑚𝑛Eh 𝑍𝑛,1 2 G1n 𝑍𝑛,1 2 G>𝜂oi =0. 2.For ... | https://arxiv.org/abs/2503.22366v1 |
Inference on effect size after multiple hypothesis testing∗ Andreas Dzemski†, Ryo Okui‡, and Wenjie Wang§ March 28, 2025 Significant treatment effects are often emphasized when interpreting and summarizing empirical findings in studies that estimate multiple, possibly many, treatment effects. Under this kind of selecti... | https://arxiv.org/abs/2503.22369v1 |
range of multiple testing procedures, including both step-down and step-up testing techniques. These procedures encompass most methods relevant for empirical applications, such as Bonferroni-corrected t-tests, Holm (1979), Benjamini and Hochberg (1995), Benjamini and Yekutieli (2001), Romano and Wolf (2005), and List, ... | https://arxiv.org/abs/2503.22369v1 |
introduction of a paper tend to focus on the significant results.2Moreover, both the observed significance and insignificance can shape the way research results are framed. For instance, if some effects are insignificant, this can alter the conclusions drawn about the mechanisms behind the significant effects and 2For ... | https://arxiv.org/abs/2503.22369v1 |
contains more than 370 funds, demonstrating that our algorithm can effectively manage conditioning sets based on numerous hypotheses. Funds with significantly positive alphas are generally viewed as outperforming the market. We employ our conditional inference procedures to quantify the excess returns specific to these... | https://arxiv.org/abs/2503.22369v1 |
computational algorithm that makes our methods scalable. Section 6 discusses some properties of our procedures and extends our procedures to more general settings. 7 Section 7 provides an asymptotic justification of our procedures. Sections 8 and 9 provide our empirical applications. Section 10 concludes. All proofs ar... | https://arxiv.org/abs/2503.22369v1 |
of the Gaussian distribution, Z(s)is independent of Xs, implying that the distribution of Xsgiven Z(s) depends only on θsand not on the other elements of θ. A selection event can be represented as a subset of the support of the t-statistics X. Specifically, we derive the set X(S)⊂Rmsuch that ˆS=Sif and only if X∈ X(S).... | https://arxiv.org/abs/2503.22369v1 |
X1∈ X({1}) where X({1}) ={x∈R:x≥Φ−1(1−β)}for one-sided testing and X({1}) ={x∈R:|x| ≥Φ−1(1−β/2)}for two-sided testing. Here, βdenotes the nominal size of the test. Figure 1 shows that the distribution of X1is a truncated Gaussian and typically asymmetric. This asymmetry implies that ˆθ1is conditionally biased for the t... | https://arxiv.org/abs/2503.22369v1 |
values and adjusting the threshold function ¯x. For the rules in Table B.3, this adjustment amounts to replacing αby half its value. We denote the values of the test statistics Xfor which Algorithm 1 selects significant effects SbyX(S)⊆Rm. For the case of two effects, Figure 2 illustrates the region X(S) forS={1}(only ... | https://arxiv.org/abs/2503.22369v1 |
or by negative infinity and the leftmost intersection point, or by the rightmost intersection point and positive infinity. (C) For each interval I: iLet˜xdenote a value in the interior of Iand find the permutation σ∗ Ithat orders the components of x(˜x) in descending order. ii Compute ℓ(I) = max {h∈S,Ωh,s>0}∪{h/∈S,Ωh,s... | https://arxiv.org/abs/2503.22369v1 |
dH(A, B)denote the Hausdorff distance between sets AandB. Suppose that one of the following conditions holds: 1.The selection rule is a 1-sided step-down rule. For all hwithlim supn→∞|Vn,h,s|>0 we have either h∈Sandµn,h=θn,h/√vn,h→ ∞ orh /∈Sandµn,h→ −∞ . 2.The selection rule is a 2-sided step-down rule. For all hwithli... | https://arxiv.org/abs/2503.22369v1 |
relevance in most empirical applications. In addition, this focus allows for a clear presentation of our procedures and results. We are also able to derive sharper theoretical results for this special case than would be possible within a more general framework (see, e.g., Theorem 4). However, alternative selection even... | https://arxiv.org/abs/2503.22369v1 |
P n, there is θ(P)∈RmandV(P)∈ V msuch that forv(P) = ( V1,1(P), . . . , V m,m(P)),Ω = diag−1/2(v(P))V(P)diag−1/2(v(P))and ξΩ(P)∼N(0,Ω(P)), we have lim n→∞sup P∈Pnsup f∈BL1 EPf diag−1/2(v(P)) ˆθ−θ(P) −Ef(ξΩ(P)) = 0. 2. (Consistent variance estimation) For all ϵ >0, we have lim n→∞sup P∈Pnmax j=1,...,mP(|ˆvj/vj(P)−1|>... | https://arxiv.org/abs/2503.22369v1 |
the different grants. We analyze different subgroups as an example of uncorrelated treatment effects estimators and multiple outcomes as an example of correlated treatment effects estimators.5 The details of the estimation procedures are given in Online Appendix A.1. To detect significant effects, we use three FWER-con... | https://arxiv.org/abs/2503.22369v1 |
gains through conditional inference. These gains affect mainly the upper bound, leaving 26 the lower bound virtually unchanged. This asymmetric adjustment reflects the positive bias in positively selected significant effects. Multiple outcomes List, Shaikh, and Xu (2019) also consider a specification without subgroups ... | https://arxiv.org/abs/2503.22369v1 |
The downward corrections behave differently from the corresponding correction in a univariate setting where a more powerful test is associated with a smaller positive bias. In our multivariate setting, we observe a steeper downward correction for the Holm procedure than for the Bonferroni procedure, even though the for... | https://arxiv.org/abs/2503.22369v1 |
and BY (FDR control) procedures, setting the nominal FWER and FDR levels to 10%. Both procedures detect five funds with significantly positive alphas (see Table 3). We now quantify the margin by which they outperform the market. Given that we select a few outperformers (5) from a large pool of mutual funds (371), there... | https://arxiv.org/abs/2503.22369v1 |
the effects of one or more placebo outcomes are found to be insignificant. This type of test is commonly used, for example, when assessing parallel trends in a difference-in-differences design (Roth 2022). In the online appendix, we provide another example where property crime serves as a placebo outcome to test an emp... | https://arxiv.org/abs/2503.22369v1 |
model”. In: Journal of Financial Economics 116.1, pp. 1–22. Fithian, William, Dennis Sun, and Jonathan Taylor (2017). Optimal Inference After Model Selection . arXiv:1410.2597 [math, stat]. Giglio, Stefano, Yuan Liao, and Dacheng Xiu (2021). “Thousands of alpha tests”. In: The Review of Financial Studies 34.7, pp. 3456... | https://arxiv.org/abs/2503.22369v1 |
99–125. 35 Tian, Xiaoying and Jonathan Taylor (2018). “Selective Inference with a Randomized Response”. In:Annals of Statistics 46.2, pp. 679–710. Tibshirani, Ryan J., Jonathan Taylor, Richard Lockhart, and Robert Tibshirani (2016). “Exact Post-Selection Inference for Sequential Regression Procedures”. In: Journal of t... | https://arxiv.org/abs/2503.22369v1 |
into six sections. Appendix A presents an additional empirical example that examines the impact of exposure to conflict on violent crime. This section also includes further analyses related to the example in Section 8. Appendix B supplements the discussion of multiple testing protocols. In particular, Appendix B.1 give... | https://arxiv.org/abs/2503.22369v1 |
valid estimate of V. A.2. Additional empirical application: Exposure to conflict and violent crime We consider an additional empirical application in which considering both insignificant and significant results is crucial in interpreting the results. This application is based on Couttenier, Petrencu, Rohner, and Thoeni... | https://arxiv.org/abs/2503.22369v1 |
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