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A Fourier-based inference method for learning interaction kernels in particle systems Grigorios A. Pavliotis∗Andrea Zanoni† Abstract We consider the problem of inferring the interaction kernel of stochastic interacting particle systems from observations of a single particle. We adopt a semi-parametric approach and repr... | https://arxiv.org/abs/2505.05207v1 |
term in the drift with respect to the (unique) invariant measure of the process. A detailed analysis of this approach can be found in [41]. Furthermore, a fully nonparametric estimation of the interaction potential has been studied using different methodologies based on deconvolution [2], data-driven kernel estimators ... | https://arxiv.org/abs/2505.05207v1 |
that the analysis can be extended to the setting considered in [34]. This is confirmed in the numerical experiments that are presented in Section 4. The analysis of our inference methodology in the presence of phase transitions will be presented elsewhere. In this work, we consider the problem of learning the interacti... | https://arxiv.org/abs/2505.05207v1 |
this paper. •We extend our previous work on the application of the method of moments to inference problems for stochastic interacting particle systems. In particular, we introduce a semi-parametric methodology for learning the interaction kernel from the observation of a single-particle trajectory. •For the semi-parame... | https://arxiv.org/abs/2505.05207v1 |
Since for the construction of ψkwe only need the moments {M(r)}2k−1 r=0, it is sufficient to estimate them using the empirical moments /tildewiderM(r) T,N=1 T/integraldisplayT 0Yr tdt,/tildewiderM(r) T,I=1 II/summationdisplay i=1/tildewideYr i, (2.5) depending on whether we have continuous-time or discrete-time observa... | https://arxiv.org/abs/2505.05207v1 |
uniform boundedness of the moments and the estimate (2.6), following the proof of [40, Lemma 4.5(i)] we obtain the desired result. We finally remark that the dependence of Conjis not explicitly emphasized since j≤k. Lemma 2.3. Under Assumption 1.1, for all k≥0and for all q∈[1,2)there exists a constantC=C(k)>0, independ... | https://arxiv.org/abs/2505.05207v1 |
which implies /vextenddouble/vextenddouble/vextenddoubleψk−/tildewideψk/vextenddouble/vextenddouble/vextenddouble2 L2(ρ)≤C1 /tildewidec2 k k/summationdisplay j=0/vextendsingle/vextendsingle/vextendsingleλkj−/tildewideλkj/vextendsingle/vextendsingle/vextendsingle 2 +C/parenleftbigg1 ck−1 /tildewideck/parenrightbigg2... | https://arxiv.org/abs/2505.05207v1 |
It now remains to approximate the entries of the matrix and the right-hand side in the system (3.3), since they cannot be computed exactly. Due to ergodicity and propagation of chaos, using the empirical moments, and neglecting the reminder e(K), we define the linear system whose solution/tildewideβ(K) T,Nis the estima... | https://arxiv.org/abs/2505.05207v1 |
underdetermined system. This can also be seen formally from equation (3.1), where we have “infinitely many” linear equations for “ 2×infinitely many” unknowns. Therefore, our methodology does not allow us to simultaneously infer both the drift term and the interaction kernel. We remark that this property is common to a... | https://arxiv.org/abs/2505.05207v1 |
By definition of /tildewideγ(K) T,Nandγ(K), we have E/bracketleftig/vextenddouble/vextenddouble/vextenddouble/tildewideγ(K) T,N−γ(K)/vextenddouble/vextenddouble/vextenddouble/bracketrightig =E /radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtK/summationdisplay i=0 1 /tildewidecii/summationdisplay j... | https://arxiv.org/abs/2505.05207v1 |
tion 2.4, and Lemmas 2.2 and 2.6, we obtain (iii), which concludes the proof. The second approximation we make by passing from the exact system (3.3)to the final system(3.4)is to remove the term e(K). In the next result, we justify this step by showing thate(K)vanishes for large values of K. Lemma 3.6. Under Assumption... | https://arxiv.org/abs/2505.05207v1 |
3.8 must disappear as Kincreases. First, notice that ϵ(K)→0asK→∞because W′∈L2(ρ). In principle, we should be able to relate the decay of the generalized Fourier coefficients of the interaction kernel (or its derivative) with the regularity of W, using results from approximation theory such as Jackson’s inequality, as w... | https://arxiv.org/abs/2505.05207v1 |
the constant function) exact ( ψk) and approximated (/tildewideψk) orthogonal polynomials with respect to the invariant measure ρof the mean-field Ornstein–Uhlenbeck process. Top: we fix T= 10 000 and varyN= 5,50,500. Bottom: we fix N= 500and varyT= 100,1 000,10 000. Notice that, in this case, the orthogonal polynomial... | https://arxiv.org/abs/2505.05207v1 |
we fix the quadratic confining potential V(x) =x2/2and set the diffusion coefficient σ= 1. In the numerical experiments we will consider the following interaction potentials: W1(x) =x4 4−x2 2, W2(x) = cosh(x), W3(x) =D/parenleftig 1−e−a(x2−r2)/parenrightig2, W4(x) =−A√ 2πe−x2 2.(4.2) All of these interaction potentia... | https://arxiv.org/abs/2505.05207v1 |
since it only requires the approximation of the moments of the invariant measure and the solution of a low-dimensional linear system, and computationally cheap. Moreover, because of its versatility, it can infer complex interaction kernels. On the other hand, its main limitation, illustrated in the convergence analysis... | https://arxiv.org/abs/2505.05207v1 |
2024. [3]C. Amorino, A. Heidari, V. Pilipauskait ˙e, and M. Podolskij ,Parameter estimation of discretely observed interacting particle systems , Stochastic Process. Appl., 163 (2023), pp. 350–386. [4]C. Amorino and V. Pilipauskait ˙e,Kinetic interacting particle system: parameter estimation from complete and partial d... | https://arxiv.org/abs/2505.05207v1 |
1–144. [23]S. N. Gomes, G. A. Pavliotis, and U. Vaes ,Mean Field Limits for Interacting Diffusions with Colored Noise: Phase Transitions and Spectral Numerical Methods , Multiscale Model. Simul., 18 (2020), pp. 1343–1370. [24]S. N. Gomes, A. M. Stuart, and M.-T. Wolfram ,Parameter estimation for macroscopic pedestrian ... | https://arxiv.org/abs/2505.05207v1 |
Theory, 120 (2003), pp. 185–216. [44]L. Sharrock, N. Kantas, P. Parpas, and G. A. Pavliotis ,Online parameter estimation for the McKean-Vlasov stochastic differential equation , Stochastic Process. Appl., 162 (2023), pp. 481–546. [45]J. Sirignano and K. Spiliopoulos ,Mean field analysis of neural networks: a law of lar... | https://arxiv.org/abs/2505.05207v1 |
arXiv:2505.05670v1 [econ.EM] 8 May 2025Estimation and Inference in Boundary Discontinuity Design s∗ Matias D. Cattaneo†Rocio Titiunik‡Ruiqi (Rae) Yu§ May 22, 2025 Abstract Boundary Discontinuity Designs are used to learn about treatment effec ts along a continu- ous boundary that splits units into control and treatment ... | https://arxiv.org/abs/2505.05670v1 |
At. The observed response variable is Yi=1(Xi∈A0)·Yi(0) + 1(Xi∈A1)·Yi(1). Without loss of generality, we assume that the boundary belongs to the treatment group, t hat is,bd(A1)¢A1and B∩A0=∅. Boundarydiscontinuitydesignsarecommonlyusedinquantitativesoc ial,behavioral,andbiomed- ical sciences. For example, consider the ... | https://arxiv.org/abs/2505.05670v1 |
for students at the margin of program eligibility, as determined by their bivariate score Xi= (SABER11 i,SISBEN i)¦∈B. The parameter τ(x) captures policy-relevant heterogeneous treatment effects along the boundary B: for example, in Figure 1a,τ(b1) corresponds to a (local) average treatment effect for students with high ... | https://arxiv.org/abs/2505.05670v1 |
The estimator based on distance exhibits a not iceable higher bias near (but not at) the boundary kink point x=b21, relative to the bivariate local polynomial estimator based on the bivariate score. A contribution of our paper is to provide a t heory-based explanation of this empirical phenomena, which turns out to be ... | https://arxiv.org/abs/2505.05670v1 |
3below). On the other hand, the uniform inference results require some additional technical care, and are new to the lit erature. The main potential issue is that a uniform distributional approximation is established over the lower-dimensional man- ifoldB, and thus its shape can affect the validity of the results. Secti... | https://arxiv.org/abs/2505.05670v1 |
the class of k-times continuously differentiable functions from X toY, andCk(X) is a shorthand for Ck(X,R). For a Borel set S¦X, the De Giorgi perimeter of S isperim(S) = supg∈D2(X)/integraltext R21(x∈S)divg(x)dx/∥g∥∞, where div is the divergence operator, and D2(X) denotes the space of C∞functions with values in R2and ... | https://arxiv.org/abs/2505.05670v1 |
For each treatment group t∈ {0,1}, boundary point x∈B, and distance score Di(x), the uni- variate distance-based local polynomial estimator /hatwideγt(x) is the sample analog of the coefficients associated with the best (weighted, local) mean square approximation of the conditional expecta- tionE[Yi|Di(x)] based on rp(Di... | https://arxiv.org/abs/2505.05670v1 |
left derivative is constant, but the right derivative is equal to infinity. The supplemental 10 (a) Distance to b∈B. (b) Distance-based Conditional Expectation. Figure 2: Lack of Smoothness of Distance-Based Conditional Expectation near a Kink. Note: Analytic example of θ1,b(r) =E[Y(1)|Di(b) =r],rg0, for distance transf... | https://arxiv.org/abs/2505.05670v1 |
that |Bn(x)| fChκ'(p+1)for allh∈[0,ε]andx∈B. (ii)Suppose B=γ([0,L])whereγis a one-to-one function in Cι+2([0,L],R2)for some L >0. Suppose there exists δ,ε >0such that for all x∈Bo=γ([δ,L−δ]),r∈[0,ε], andt= 0,1, the set{u∈R2:d(u,x) =r}intersects with bd(At)at only two points in B. Then, for all 12 x∈γ([δ,L−δ]),θ0,x(·)an... | https://arxiv.org/abs/2505.05670v1 |
1/hatwideΨ−1 t,x1/hatwideΥt,x1,x2/hatwideΨ−1 t,x2e1, and /hatwideΥt,x1,x2=h2En/bracketleftBig rp/parenleftBigDi(x1) h/parenrightBig kh(Di(x1))/parenleftbig Yi−rp(Di(x1))¦/hatwideγt(x1)/parenrightbig 1(Di(x1)∈It) ×rp/parenleftBigDi(x2) h/parenrightBig¦ kh(Di(x2))/parenleftbig Yi−rp(Di(x2))¦/hatwideγt(x2)/parenrightbig 1... | https://arxiv.org/abs/2505.05670v1 |
distance variable as the univariate covariate, and thus ignoring its intrinsic bivariate dimen- sion, implies that the resulting bandwidth choice will be smaller n−1/(3+2p)< n−1/(4+2p), thereby undersmoothing the point estimator (relative to the correctoptimal MSE bandwidth choice). As a consequence, the point estimato... | https://arxiv.org/abs/2505.05670v1 |
as another simple rule-of-thumb bandwidth implementation. Furt her implementation details and simulation evidence are given in Cattaneo et al. (2025). 4 Analysis based on Bivariate Location The previous section demonstrated the potential detrimental aspect s of employing distance-based local polynomial regression to an... | https://arxiv.org/abs/2505.05670v1 |
but rather due to the choice of distance-based (univariate) local polynomial approximation: e.g., rp(∥u∥) (in/hatwideτdis(x)) vs.Rp(u) (in/hatwideτ(x)). In other words, Theorem 3demonstrates that a bivariate local polynomial approximation does not suffer from the bias documented in in Lemma 2even when the radial kernel ... | https://arxiv.org/abs/2505.05670v1 |
boundary designs that would lead to invalid stati stical inference. 4.3 Implementation The bivariate local polynomial estimator /hatwideτ(x) and associated t-statistic /hatwideT(x) is fully adaptive to kinksandotherirregularitiesoftheboundary B. Therefore, itisstraightforwardtoimplementlocal and global bandwidth select... | https://arxiv.org/abs/2505.05670v1 |
college attendance remains constant as students becom e wealthier ( /hatwideτ(x)≈0.3 forx∈ {b1,...,b21}), but the treatment effects appear to decrease as students exhibit higher 23 (a) Point Estimation (b) Confidence Interval and Bands (c) Treatment Effects Heatmap (d) p-values Heatmap Figure 3: Boundary Treatment Effect E... | https://arxiv.org/abs/2505.05670v1 |
∥x1−x2∥q−+q,fL. (iv)σ2(x) =V[Yi|Xi=x]is continuous on XwithL−1finfx∈Xσ2(x)fsupx∈Xσ2(x)fL. In addition, let Tbe the class of all distance-based estimators Tn(Un(x))withUn(x) = ((Yi,∥Xi−x∥)¦: 1fifn)for each x∈X. Then, liminf n→∞n1/4inf Tn∈Tsup P∈PNPEP/bracketleftBig sup x∈B/vextendsingle/vextendsingleTn(Un(x))−µ(x)/vexte... | https://arxiv.org/abs/2505.05670v1 |
Persistent Effects of Peru’s Mining Mita,” Econometrica , 78, 1863–1903. Diaz, J. D., and Zubizarreta, J. R. (2023), “Complex Discontinuity Design s Using Covariates for Policy Impact Evaluation,” Annals of Applied Statistics , 17, 67–88. Federer, H. (2014), Geometric measure theory , Springer. Galiani,S.,McEwan,P.J.,an... | https://arxiv.org/abs/2505.05670v1 |
. . . . . . . . . . . . . . . 3 SA-1.2 Mapping between Main Paper and Supplement . . . . . . . . . . . . . . . . . . . . . . . . 4 SA-2 Location-Based Methods 5 SA-2.1 Preliminary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 SA-2.2 Point Estimation . . . . . . . . . . . . . . . .... | https://arxiv.org/abs/2505.05670v1 |
. . . . . 19 SA-5.2 Proof of Lemma SA-2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 SA-5.3 Proof of Lemma SA-2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 SA-5.4 Proof of Lemma SA-2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2505.05670v1 |
40 SA-6 Proofs for Section SA-3 41 SA-6.1 Proof of Lemma SA-3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 SA-6.2 Proof of Lemma SA-3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 SA-6.3 Proof of Lemma SA-3.3 . . . . . . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2505.05670v1 |
. . . . . . . . . . . . . . . 50 SA-7.2 Proof of Lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 SA-7.3 Proof of Theorem 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 SA-8 Proofs for Section SA-4 61 SA-8.1 Proof of Lemma SA-4.1 . . . . . . ... | https://arxiv.org/abs/2505.05670v1 |
a function fon a metric space ( S,d),∥f∥∞=supx∈X|f|, ∥f∥Lip,∞=supx,x′∈S|x−x′| d(x,x′). For a probability measure Qon (S,S) andpg1, define ∥f∥Q,p= (/integraltext S|f|pdQ)1/p. For a set E¦Rd, denote by m(E) the Lebesgue measure of E. (iii)Empirical Process . We use sta ndard empirical process notations: En[g(vi)] =1 n/sum... | https://arxiv.org/abs/2505.05670v1 |
assumption on d-variate kernel function and assignment boundary. Assumption SA–2 (Kernel Function and Bandwidth) Lett∈ {0,1}. (i)K:Rd→[0,∞)is compact supported and Lipschitz continuous, or K(u) = 1(u∈[−1,1]d). (ii) liminf h³0infx∈B/integraltext AtKh(u−x)du≳1. Under the assumptions imposed, for t∈ {0,1}, we have /hatwid... | https://arxiv.org/abs/2505.05670v1 |
SA-2.3 (Asymptotic Normality) Suppose Assumptions SA–1andSA–2hold. Ifnhd→ ∞andnhdh2(p+1)→0, then sup u∈R/vextendsingle/vextendsingle/vextendsingle/vextendsingleP/parenleftbig/hatwideT(ν)(x)fu/parenrightbig −Φ(u)/vextendsingle/vextendsingle/vextendsingle/vextendsingle=o(1),x∈B. For any 0 < ³ <1, define the confidence inte... | https://arxiv.org/abs/2505.05670v1 |
(Regularity Conditions for Distance) d:Rd×Rd→Ris a metric on Rdequivalent to the Euclidean distance, that is, there exists positive constants CuandClsuch that Cl∥x−x′∥ fd(x,x′)fCu∥x−x′∥for allx,x′∈X. Assumption SA–4 (Kernel Function) Lett∈ {0,1}. 10 (i)k:R→[0,∞)is compact supported and Lipschitz continuous, or k(u) = 1... | https://arxiv.org/abs/2505.05670v1 |
=/parenleftbigg /hatwideÄdis(x)−c³/radicalBig /hatwideΞx,x,/hatwideÄdis(x)+c³/radicalBig /hatwideΞx,x/parenrightbigg . Then, P/parenleftBig Ä(x)∈/hatwideIdis(x,³)/parenrightBig →1−³,x∈B. SA-3.3 Uniform Inference Theorem SA-3.3 (Uniform Convergence Rate) Suppose Assumptions SA–1,SA–3andSA–4hold. Ifnhd log(1/h)→ ∞, then ... | https://arxiv.org/abs/2505.05670v1 |
allu∈Rq. (ii)LetSupp(F) =∪f∈FSupp(f). A probability measure QFon (Rq,B(Rq)) is a surrogate measure for P with respect to Fif (i)QFagrees with Pon Supp( P)∩Supp(F). (ii)QF(Supp(F)\Supp(P)) = 0. LetQF= Supp(QF). (iii) For q= 1 and an interval I¦R, the pointwise total variation of FoverIis pTVF,I= sup f∈Fsup Pg1sup PP∈IP−... | https://arxiv.org/abs/2505.05670v1 |
constant, c=cG+cR,Y+k,d=dGdR,Yk, and rn= min/braceleftBig(cd 1Md+1 GTVdEG)1/(2d+2) n1/(2d+2),(cd/2 1cd/2 2MGTVd/2EGLd/2)1/(d+2) n1/(d+2)/bracerightBig , c1=dsup x∈QGd−1/productdisplay j=1Ãj(∇ϕG(x)),c2= sup x∈QG1 Ãd(∇ϕG(x)). SA-4.3 Multiplicative-Separable Empirical Process The following Lemma SA-4.2generalizes Cattaneo... | https://arxiv.org/abs/2505.05670v1 |
greater than 2 d(van der Vaart and Wellner , 1996, Example 2.6.1), and by van der Vaart and Wellner (1996, Theorem 2.6.4), for any discrete measure Q, we have N(G,∥·∥Q,2,ε)f2d(4e)2dε−4d,0< εf1. It then follows that for any discrete measure Q, N(F,∥·∥Q,2,ε∥H∥Q,2)≲N(H,∥·∥Q,2,ε/2∥H∥Q,2)+N(G,∥·∥Q,2,ε/2)≲2dh−dε−d+2d(32e)dε−... | https://arxiv.org/abs/2505.05670v1 |
follows that supQN(F,∥·∥Q,2,ϵ∥F∥Q,2)≲(diam(X) ϵh))d, wheresupis over all finite probability distribution on (X×R,B(X)¹B(R)). Denote A1=diam(X) h,A2=d. We have sup QN(F,∥·∥Q,2,ϵ∥F∥Q,2)≲(A1/ϵ)A2, ϵ∈(0,1]. Case 2:Kis the uniform kernel. Consider hn(À,x) =/parenleftbiggÀ−x h/parenrightbiggv1 hd1(À∈At), À,x∈X, andH={(À,u)∈X×... | https://arxiv.org/abs/2505.05670v1 |
x,y∈B∥M3,x,y∥≲P/radicalbigg log(1/h) nhd+log(1/h) nv 2+vhd. Term M 4,x,y. Notice that {gn(·;x,y)Ã2 t(·) :x,y∈B}is a VC-type of class with constant envelope function Ch−dfor some suitable Cand sup x,y∈Bsup À∈X/vextendsingle/vextendsinglegn(À;x,y)Ã2(À)/vextendsingle/vextendsingle≲h−d, sup x,y∈BE/bracketleftbig gn(Xi;x,y)... | https://arxiv.org/abs/2505.05670v1 |
1(Xi∈At)ui. Then,T(ν)(x) =/summationtextn i=1ZiandE[Zi] = 0 and V[Zi] =n−1. By Berry-Essen Theorem, sup u∈R/vextendsingle/vextendsingle/vextendsingleP/parenleftBig T(ν)(x)fu/parenrightBig −Φ(u)/vextendsingle/vextendsingle/vextendsingle≲B−1 nn/summationdisplay i=1E[|Zi|3], whereBn=/summationtextn i=1V[Zi] = 1. Moreover,... | https://arxiv.org/abs/2505.05670v1 |
SA-5.11 Proof of Theorem SA-2.7 First, we consider the class of functions Ft={K(ν) t(·;x) :x∈B},t∈ {0,1}. W.l.o.g., we can assume X= [0,1]d, andQFt=PXis a valid surrogate measure for PXwith respect to Ft, andϕFt=Idis a valid normalizing transformation (as in Lemma SA-4.1). This implies the constants c1andc2from Lemma S... | https://arxiv.org/abs/2505.05670v1 |
t(u;x)̸= 0 implies Et,x=u∈ {y∈At: (y−x)/h∈Supp(K)}, and 1(u∈At)K(ν) t(u;x) = 1(u∈Et,x)K(ν) t(u;x),∀u∈X. By the assumption that the De Giorgi perimeter of Et,xsatisfies L(Et,x)fChd−1and using TV{gh}f M{g}TV{h}+M{h}TV{g}, we have TVG= sup x∈BTV{gx}fsup x∈B/summationdisplay t∈{0,1}TV{ 1AtK(ν) t(·;x)}fsup x∈B/summationdispl... | https://arxiv.org/abs/2505.05670v1 |
in this lemma, sup u∈R/vextendsingle/vextendsingle/vextendsingle/vextendsingleP/parenleftbigg sup x∈B/vextendsingle/vextendsingle/vextendsingle/vextendsingle/hatwideT(ν)(x)/vextendsingle/vextendsingle/vextendsingle/vextendsinglefu/parenrightbigg −P/parenleftbigg sup x∈B/vextendsingle/vextendsingle/vextendsingleT(ν)(x)/... | https://arxiv.org/abs/2505.05670v1 |
2. ■ SA-5.12 Proof of Theorem SA-2.8 The result follows from Lemma SA-5.3, Lemma SA-5.4and Lemma SA-5.5. ■ SA-5.13 Proof of Theorem SA-2.9 LemmaSA-2.8and Dominated Convergence Theorem give sup u∈R/vextendsingle/vextendsingle/vextendsingle/vextendsingleP/parenleftbigg sup x∈B/vextendsingle/vextendsingle/vextendsingle/ve... | https://arxiv.org/abs/2505.05670v1 |
Ã= supf∈FE[f(Xi,Yi)2]1/2. Then, Ã2≲sup x∈BE[∥e¦ ¿gx∥2 ∞(|Yi|+∥e¦ ¿hx∥∞)21(kh(Di(x))̸= 0)]≲h−d. To check for the covering number of F, notice that compare to the proof of Lemma SA-2.1, we have one more terme¦ νgxhx=rp/parenleftBig d(z,x) h/parenrightBig kh(d(z,x))γ∗ t(x)¦rp(d(z,x)). All terms except for γ∗ t(x) can be h... | https://arxiv.org/abs/2505.05670v1 |
/vextendsingle/vextendsingle/vextendsinglee¦ 1(/hatwideΨ−1 t,x−Ψ−1 t,x)Ot,x/vextendsingle/vextendsingle/vextendsingle≲P/radicalbigg 1 nhd/parenleftBigg/radicalbigg 1 nhd+1 n1+v 2+vhd/parenrightBigg . The decomposition Equation ( SA-3.1) then gives the result. ■ SA-6.7 Proof of Theorem SA-3.2 DefineTdis(x) = Ξ−1/2 x,xe¦ ... | https://arxiv.org/abs/2505.05670v1 |
to Cattaneo et al. (2024, Lemma 7), there exists a constant c′only depending on canddthat for any 0 fεf1, sup QN/parenleftBig hd/2Ht,∥·∥Q,1,(2c+1)d+1ε/parenrightBig fc′ε−d−1+1, 48 where supremum is taken over all finite discrete measures. Taking a constant envelope function MHt= (2c+1)d+1h−d/2, we have for any 0 < εf1, ... | https://arxiv.org/abs/2505.05670v1 |
Ch/parenrightbiggl k/parenleftbigg∥(Cu−Cs,Cv)∥ Ch/parenrightbigg µ1(C(u−s,v))C2 4dudv =/integraldisplay2/C 0/integraldisplay2/C 0/parenleftbigg1 h/parenrightbigg2/parenleftbigg∥(u−s,v)∥ h/parenrightbiggl k/parenleftbigg∥(u−s,v)∥ h/parenrightbigg Cµ1((u−s,v))1 4dudv (2)=/integraldisplay2 0/integraldisplay2 0/parenleftbi... | https://arxiv.org/abs/2505.05670v1 |
= (1,0). LetT: [0,∞)→[0,∞) to be a continuous increasing function that satisfies ∥µ◦T(r)∥2=r2,∀r∈[0,h]. Initial Case: l= 1,2,3. We will show that TisClon (0,h). For notational simplicity, define another function ϕ: [0,∞)→[0,∞) byϕ(t) =∥µ(t)∥2. Using implicit derivations iteratively, ϕ◦T(r) =r2, ϕ′(T(r))T′(r) = 2r, ϕ′′(T(... | https://arxiv.org/abs/2505.05670v1 |
finite limits at 0. Continue this argument, we see thatµ2(·) ∥µ(·)∥isCp+1on (0,∞) and lim r³0dv dtv(µ2(t)/∥µ(t)∥) exist and are uniformly bounded for all x∈Band for all 0 fvfp+1. SincearcsinisCp+1with bounded (higher order derivatives) on [ −1/2,1/2],AisCp+1on (0,¶) and for all 0fvfp+1, lim r³0A(v)(t) exist and are unif... | https://arxiv.org/abs/2505.05670v1 |
For each r∈R, define ˜r(y) =r(y) 1(|y| fÄn), y∈R, and define the class ˜R={˜r:r∈R}. For an overview of our argument, suppose ZR nis a mean-zero Gaussian process indexed by G×R∪GטR, whose existence will be shown below, then we can decompose by: Rn(g,r)−ZR n(g,r) =/bracketleftbig Rn(g,˜r)−ZR n(g,˜r)/bracketrightbig +/brac... | https://arxiv.org/abs/2505.05670v1 |
three parts that if we chooseÄnsuch that rnÄn≍√MGEGÄ−v/2 n, then the approximation error can be bounded by E/bracketleftbig ∥Rn−ZR n∥G×R/bracketrightbig ≲(dlog(cn))3/2rv v+2n(√MGEG)2 v+2+dlog(cn)MGn−v/2 2+v +dlog(cn)MGn−1/2/parenleftBig√MGEG rn/parenrightBig2 v+2, whered=dG+dR,Y+k, andc=cGcR,Yk. ■ SA-8.2 Proof of Lemma... | https://arxiv.org/abs/2505.05670v1 |
simplicity, we denote Π1(AM,n(P,1)) byΠ1instead. Now define the projected empirical process Π1An(g,h,r,s) =Π1Mn(g,r)+Π1Mn(h,s), g∈G,h∈H,r∈R,s∈S, 64 whereΠ1Mn(g,r) andΠ1Mn(h,s) are given in Equation ( SA-10) inCattaneo and Yu (2025), that is, Π1Mn(g,r) =1√nn/summationdisplay i=1/parenleftbig Π1[g,r](xi,yi)−E[Π1[g,r](xi,y... | https://arxiv.org/abs/2505.05670v1 |
result is derived under the assumption thatsupx∈XE[exp(|yi|)|xi=x]<∞. For the result under the condition supx∈XE[|yi|2+v|xi=x]<∞, we can use the same truncation argument as in Section SA-8.1(proof of Lemma SA-4.1) and the VC-type conditions for G,H,R,Sto get the stated conclusions. ■ 67 References Cattaneo, M. D., Chan... | https://arxiv.org/abs/2505.05670v1 |
arXiv:2505.06088v1 [math.PR] 9 May 2025Approximations for the number of maxima and near-maxima in independent data Fraser Daly∗ May 12, 2025 Abstract In the setting where we have nindependent observations of a random variable X, we derive explicit error bounds in total variation distance when approximating the number o... | https://arxiv.org/abs/2505.06088v1 |
Steutel [11] establish a limiting (shifted) mixed Poisson distribution for this quantity as n→ ∞ , and note that this mixed Poisson simplifies to a geometric distribution in the case where Xis in the maximum domain of attraction of a Gumbel distribution. This geometric limit for observations close to the maximum is gen... | https://arxiv.org/abs/2505.06088v1 |
2 below, while part (b) is proved in Section 3. Before moving on to state our Poisson approximation results, we illustrate the upper bounds of Theorem 1 in our motivating example of geometrically distributed data. Example 2. Suppose that P(X=j) = p(1−p)j−1forj= 1,2, . . .and some p∈(0,1), so that X∼Geom( p) has a geome... | https://arxiv.org/abs/2505.06088v1 |
— 0.251 0.073 0.039 Table 1: The upper bound of Theorem 3 in the case where X∼Geom(1 −µ/n) for selected values of µandn. The symbol ‘–’ indicates that the upper bound is larger than 1, and therefore uninformative within a certain distance of one of the order statistics of our sample X1, . . . , X n. To that end, we let... | https://arxiv.org/abs/2505.06088v1 |
∞ , and also gives reasonable numerical bounds on dTV(Kn(a,1)−1, Z), even for moderate values of nanda. This is illustrated in Figure 2, which gives the upper bound (5) in the cases n= 20 and n= 100 for various values of a. Example 7. LetX∼U(0, b) have a uniform distribution on the interval (0 , b), so that f(x) = 1 /b... | https://arxiv.org/abs/2505.06088v1 |
= (1 −α)∞X k=0αkf(k) = (1 −α)f(0)−(1−α) log(1 −α)∞X k=1kf(k)P(L=k) = (1−α)f(0) +α E[L]∞X k=1kf(k)P(L=k) = (1 −α)f(0) + αE[f(L⋆)]. That is, L⋆−1 is equal in distribution to IαL⋆, where Iαhas a Bernoulli distribution with mean α and is independent of L⋆. This leads us to define the following Stein equation for a logarith... | https://arxiv.org/abs/2505.06088v1 |
2, K⋆= 1|eK⋆>1) =P(eK⋆= 2|eK⋆>1)P(K⋆= 1) =1−α αP(K⋆= 2) =2(1−α) αE[K]P(K= 2), (12) so that P(eK⋆−1̸=IαK⋆)≤α 1−2(1−α) αE[K]P(K= 2) . (13) Our result then follows on combining (11) and (13). Remark 10.When bounding the probability P(eK⋆−1 =K⋆|eK⋆>1) in (12) we could, of course, include further terms of the form P(eK⋆=j... | https://arxiv.org/abs/2505.06088v1 |
= 1 + Iβ/γW⋆+ (1−Iβ/γ)W , 11 where Iβ/γis a Bernoulli random variable with mean β/γwhich is independent of all else. It then follows that we may write W′= 1 + 1−IγIβ/γ W+IγIβ/γW⋆= 1 + IγIβ/γ+B+C , where B∼MBin( n−ℓ−1, Q+), Q+= 1−IγIβ/γ Q+IγIβ/γQ⋆, andC∼MBin(1 ,(1−IγIβ/γ)Q), and where, conditional on Q,Q⋆and the Ber... | https://arxiv.org/abs/2505.06088v1 |
. . . , n −1}:X′ i=M}|. Using an argument analogous to that of Lemma 2.1 of Brands et al.[3], we then have that P(K⋆ n=k|M=m) =n−1 k−1P(X=m)k−1P(X≤m−1)n−k P(X≤m)n−1, so that K⋆ n−1|M∼Bin(n−1, q(M)), (19) where q(m) =P(X=m) P(X≤m). We will return to this mixed binomial representation of K⋆ n−1 in the proof of Theorem ... | https://arxiv.org/abs/2505.06088v1 |
that E[Y†] =E[Y] =λ, from Theorem 1.C(ii) of [2] we have that dTV(Y†, Y)≤Var(( n−1)q(M)) λ=(n−1)2Var(q(M)) λ. Hence, dTV(K⋆ n−1, Y)≤E[q(M)] +(n−1)2Var(q(M)) λ=(n−1)E[(Kn)3] (n−2)E[(Kn)2]−(n−2)E[(Kn)2] (n−1)E[Kn],(24) where the final equality follows from the expressions (20) for the first two moments of q(M). The concl... | https://arxiv.org/abs/2505.06088v1 |
arXiv:2505.06190v1 [econ.EM] 9 May 2025Beyond the Mean: Limit Theory and Tests for Infinite-Mean Autoregressive Conditional Durations Giuseppe Cavalierea, Thomas Mikoschb,Anders Rahbekc and Frederik Vilandtc May 9, 2025 Abstract Integrated autoregressive conditional duration (ACD) models serve as natural counterparts t... | https://arxiv.org/abs/2505.06190v1 |
of large numbers and central limit theorems, cannot be applied directly, or even applied at all, as they rely on the assumption of a deterministically increasing number of observations. A main implication of the randomness of n(t) is that a crucial role is played by the tail index κ∈(0,∞) of the marginal distribution o... | https://arxiv.org/abs/2505.06190v1 |
in combination with testing infinite mean against finite mean. The latter may be stated as test- ingα+β≥1, against the alternative α+β <1 (κ≤1 against κ >1). Notice that this idea is reminiscent of testing for finite variance in strictly stationary GARCH processes as in Francq and Zako¨ ıan (2022), in double-autoregres... | https://arxiv.org/abs/2505.06190v1 |
if 0 < α 0+β0<1,xiis stationary and ergodic with tail index κ0>1,such that xihas finite mean, E[xi] =µ0=ω0(1−(α0+β0))−1<∞. On the other hand, if α0+β0>1 andE[log (α0εi+β0)]<0,xiis stationary and ergodic but with tail index κ0<1, and hence infinite mean, E[xi] =∞. Our focus here is on the yet unexplored case of integrat... | https://arxiv.org/abs/2505.06190v1 |
we discuss (Q)ML estimation under the IACD restriction α0+β0= 1 and discuss t- and LR-based test statistics for this restriction. Finally, in Section 3.3 we revisit these tests and show how to implement tests of infinite expectation of durations. 3.1 Asymptotics for the unrestricted QMLE The main result about QMLE of t... | https://arxiv.org/abs/2505.06190v1 |
of Theorem 3.1, as t→ ∞ ,τt→dN (0,1). 6 3.2 Asymptotics for the restricted QMLE Next, turn to QML estimation under the restriction α+β= 1, corresponding to the IACD model. Introduce the parameter vector ϕ∈R2, where ϕ= (ϕ1, ϕ2)′= (ω, α)′∈Φ⊂[0,∞)2, such that θ=θ(ϕ) = (ω, α, 1−α)′∈Θ for ϕ∈Φ. The restricted QML estimator ˜... | https://arxiv.org/abs/2505.06190v1 |
level η. These different testing scenarios are investigated using Monte Carlo simulation in Section 4, and applied to cryptocurrency data in the Section 5. 4 Simulations Using Monte Carlo simulation, in this section we assess the finite sample performance and accuracy of the asymptotic properties of testing based on th... | https://arxiv.org/abs/2505.06190v1 |
0 .056 0 .059 0 .057 0 .059 62500 0 .063 0 .065 0 .054 0 .056 0 .051 0 .053 1.0 100 0 .145 0 .340 0 .063 0 .103 0 .056 0 .070 500 0 .130 0 .165 0 .044 0 .052 0 .046 0 .059 2500 0 .061 0 .064 0 .051 0 .059 0 .056 0 .061 12500 0 .059 0 .062 0 .057 0 .060 0 .056 0 .058 62500 0 .060 0 .063 0 .054 0 .056 0 .050 0 .052 2.0 1... | https://arxiv.org/abs/2505.06190v1 |
the shape parameter ν. Some size distortions are present for shorter samples, especially in the case of over-dispersion (σ2 ε(v) = 2 >1) and for stronger persistence (e.g., α0= 0.15), where the QLRt-based test is oversized, in particular relatively to τt. Nonetheless, both tests demonstrate good finite-sample size cont... | https://arxiv.org/abs/2505.06190v1 |
.049 0 .054 Notes: The table reports empirical rejection probabilities for the one-sided t-tests based on τt, with q= 1.64. See also Table 1. large samples. A noticeable difference between the two tests is that the (left sided) test for the infinite expected duration hypothesis, H∞:α+β≥1, the ERPs are above the nominal... | https://arxiv.org/abs/2505.06190v1 |
following facts. First, as expected, the ERPs increase as t(hence, med {n(t)}) gets larger. Second, the ERPs increase monotonically as c <0 moves away from the null of c= 0. Apart from the fact that the distance from the null increases in −c, this also reflects that by eq.(8) in CMRV, n(t)/t→a.s.1/E[xi] = (1 −(α+β))/ω0... | https://arxiv.org/abs/2505.06190v1 |
Bitcoin Mini Trust (ticker: BTC), Grayscale Ethereum Mini Trust (ETH), Grayscale Bitcoin Trust (GBTC), Grayscale Ethereum Trust (ETHE) and Bitwise Bitcoin (BITB). Intra-day durations for the observed ETFs are measured in seconds (with decimal precision down to nano-seconds) and are obtained from the limit order book re... | https://arxiv.org/abs/2505.06190v1 |
the empirical distribution of the ˆ εi’s does not appear to be exponential, again consistent with findings commonly reported in the financial durations literature; see, for example, Section 5.3.1 of Hautsch (2012) and the references therein. Returning to the results in Table 3, we first test the null hypothesis of infi... | https://arxiv.org/abs/2505.06190v1 |
large samples are required for some parameter config- urations. Finally, we have applied our results to recent high-frequency trading data on various 16 cryptocurrency ETFs. The empirical evidence indicates heavy-tailed duration distributions, and in most cases, the integrated ACD hypothesis is not rejected in favor of... | https://arxiv.org/abs/2505.06190v1 |
their Applications , 131, 151-171. Jensen, Søren T. and Anders Rahbek (2004) “Asymptotic Inference for Nonstationary GARCH, ” Econometric Theory , 20(6), 1203–1226 . Lee, SanG-Won and Bruce E. Hansen (1994) “Asymptotic Theory for the GARCH(1, 1) Quasi-Maximum Likelihood Estimator”, Econometric Theory, 10, 29–52. Ling, ... | https://arxiv.org/abs/2505.06190v1 |
(ω, α, β )′, conditions (C.1)-(C.3) in CMRV hold by the proof of Theorem 2 there .To see this, for ndeterministic, define Ln(θ) by replacing n(t) bynin (2.4), and define Sn(θ) and In(θ) similarly. Then, by CMRV, as n→ ∞ , 1√nS[n·](θ0)→wΩ1/2 SB(·),1 nIn(θ0)→a.sΩI, where B(·) is a three dimensional Brownian motion on [0 ... | https://arxiv.org/abs/2505.06190v1 |
arXiv:2505.06405v1 [math.CO] 9 May 2025Mixing and Merging Metric Spaces using Directed Graphs Mahir Bilen Can1and Shantanu Chakrabartty2 1Department of Mathematics, Tulane University, USA, mahirbilencan@gmail.com 2Department of Electrical and Systems Engineering, Washington University in St. Louis, USA, shantanu@wustl.... | https://arxiv.org/abs/2505.06405v1 |
dense graphs. Introduced in the foundational work by Lov´ asz and Szegedy [10], they provide a powerful framework for studying the properties of large graphs and their limiting behavior, analogous to how probability distributions describe the limits of sequences of random variables [1]. Corollary 1.2. In the limit N→ ∞... | https://arxiv.org/abs/2505.06405v1 |
the edge ( u1, v1) inG1(resp. ( u2, v2) inG2). If (( u1, u2),(v1, v2)) is an edge of G1□G2, then we define p(u1,u2),(v1,v2):=( pu1,v1ifu2=v2, pu2,v2ifu1=v1. The last corollary uses dX,G,Pto understand the nature of a product metric space. Let X1, . . . , X Nbe metric spaces. Let N1∈ {1, . . . , N −1}and let N2:=N−N1. F... | https://arxiv.org/abs/2505.06405v1 |
j=k. Adirected acyclic graph (DAG) is a directed graph that contains no cycles. Here, a cycle is a path that starts and ends at the same vertex. 4 LetG1= (V1, E1) and G2= (V2, E2) be directed graphs, where E1⊆V1×V1and E2⊆V2×V2. The Cartesian product of directed graphs G1□G2is a directed graph with the following propert... | https://arxiv.org/abs/2505.06405v1 |
all points a, b∈X, the distance between aandbinXis equal to the distance between their images f(a) and f(b) inY. In other words: dY(f(a), f(b)) =dX(a, b). (2.1) Notice that an isometry is necessarily injective. Unless otherwise noted, we assume that isometries are surjective as well. When we write ‘ fis an isometry of ... | https://arxiv.org/abs/2505.06405v1 |
field Fq. In [2], Brualdi, Gravis, and Lawrance introduced metrics on the Fq-vector space KN(which is isomorphic toFN q) defined by posets. Let Pbe a poset on the set [ N] ={1, . . . , N }. The P-weight of v= (v1, . . . , v N)∈FN qis defined by ωP(v) =|{i∈[N]|i⪯jfor some j∈supp( v)}|. Brualdi, Gravis, and Lawrance show... | https://arxiv.org/abs/2505.06405v1 |
following lemma will be useful for our next theorem. Lemma 3.2. Leta:X×X→[0,1)be a distance function on a set X. Then d:X×X→R defined by d(x, y) =−log(1−a(x, y)) where x, y∈Xis also a distance function. Proof. The non-negativity, identity of indiscernibles, and the symmetry properties of dare straightforward to verify.... | https://arxiv.org/abs/2505.06405v1 |
zi)≤NX i=1Dai(xi, yi) +NX i=1Dai(yi, zi). Using (3.5), this inequality becomes Da1,...,aN(x,z)≤Da1,...,aN(x,y) + Da1,...,aN(y,z), where x= (x1, . . . , x N),y= (y1, . . . , y N), and z= (z1, . . . , z N). This shows that Da1,...,aN satisfies the triangle inequality. The other metric properties of Da1,...,aN(non-negativ... | https://arxiv.org/abs/2505.06405v1 |
the proof of the base case of our induction. We now assume that our claim holds for N−1, and proceed to prove it for N. Notice the decomposition da1,...,aN(x,y) = 1−(1−d1(x1, y1))a1)| {z } α(x,y)+ (1−d1(x1, y1))a1)| {z } β(x,y) 1−NY i=2(1−di(xi, yi))ai! | {z } γ(x,y), (3.8) where α(x,y),β(x,y), and γ(x,y) are nonnegati... | https://arxiv.org/abs/2505.06405v1 |
i∈[N](1−di(xi, yi))1/pji , (4.1) where x= (x1, . . . , x N) and y= (y1, . . . , y N) are from X. 13 We are now ready to prove our Corollary 1.1. Let us recall its statement for convenience. We assume that the underlying sets in the main result, X1, . . . , X Nare finite and that the corresponding metrics d1, . . . , ... | https://arxiv.org/abs/2505.06405v1 |
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