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˜A=n/intersectiondisplay i=1{/tildewideXi=Xiand (/tildewideVi=Vi)I(i > K)},˜Ac=n/uniondisplay i=1{/tildewideXi/ne}ationslash=Xior (/tildewideVi/ne}ationslash=Vi)I(i > K)}. L2risk on the event ˜A If the event ˜Aoccurs, then ˆfK(ω) coincides with the untruncated estimator ˜fK(ω), i.e., ˜fK(ω) =1 2π(n−K)n/summationdisplay... | https://arxiv.org/abs/2504.00919v1 |
(/tildewideVi/ne}ationslash=Vi)·I(i > K)}]. From (24), we know/summationtextn i=1Pr[/tildewideXi/ne}ationslash=Xi] =O(n−3) forτ2 n≥8log1+δ(n). Next, we need to bound from above, for i > K: Pr[{/tildewideXi=Xiand/tildewideVi/ne}ationslash=Vi}] ≤Pr /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/... | https://arxiv.org/abs/2504.00919v1 |
=E[ˇZi|/tildewiderWi] =E[E[ˇZi|/tildewideYi]|/tildewiderWi]. By the same arguments as in the proof of Proposition 3.2. of Butuce a et al. (2023), one can show that almost surely E[ˇZi|/tildewideYi] =/tildewideYi. Furthermore, for ever j= 0,..,K, E[/tildewideYi,j|/tildewiderWi,j=wj] = ˜τn/parenleftbigg1 2+wj 2˜τn/parenr... | https://arxiv.org/abs/2504.00919v1 |
obtain the following bound for the bias: if τ2 n= 8log1+δ(n) and ˜τn= 16log1+δ(n)τ2 n for some δ >0, then {E[ˇfK(ω)]−f(ω)}2/lessorsimilarK n+/braceleftBigg log(K)K−2s, f∈Ws,∞(L0,L), K−2s+1, f ∈Ws,2(L), where the constant C >0 depends on s, L, iff∈Ws,2(L), and additionally on L0, iff∈ Ws,∞(L0,L).Next, by the law of tota... | https://arxiv.org/abs/2504.00919v1 |
Fori≥2, we prove (11), that is I(Xi|Z1:(i−1))≤(1+9 4(eα−1)2)/parenleftbig I(Xi,Xi−1|Z1:(i−2))+I(Xi|Z1:(i−2))/parenrightbig . Let us go back to (38). We decompose1 2(eα+1) =1 2(eα−1)+1 and use that: /vextendsingle/vextendsingle/vextendsingle˙LXi|Z1:(i−2) θ(xi)/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vex... | https://arxiv.org/abs/2504.00919v1 |
have I(Xi,Xi−K,...,X i−ℓK) =1 2tr/bracketleftBigg/parenleftbigg Σ(θ)−1∂ ∂θΣ(θ)/parenrightbigg2/bracketrightBigg , where the covariance matrix is given by the tridiagonal symmetric ma trix: Σ(θ) =σ0·Iℓ+1+σK(θ)·M1, Mi,j=Ind(|i−j|= 1), whereIℓ+1denotes the identity matrix of size ℓ+1 and Inddenotes the indicator function ... | https://arxiv.org/abs/2504.00919v1 |
Mastorakis, S., Jan, M. A., Alalmaie, A. Z., and Na nda, P. (2023). Privacy- enhanced living: A local differential privacy approach to secure sma rt home data. In 2023 IEEE International Conference on Omni-layer Intelligent Syste ms (COINS) , pages 1–6. IEEE. Wang, D. and Xu, J. (2020). Tight lower bound of sparse covar... | https://arxiv.org/abs/2504.00919v1 |
Causal Models for Growing Networks Gecia Bravo-Hermsdorff1,aLee M. Gunderson2,bKayvan Sadeghi2,c 1School of Informatics, University of Edinburgh, Edinburgh, Scotland, UK 2Department of Statistical Science, University College London, London, England, UK a,b,c{gecia.bravo@gmail.com, l.gunderson@ucl.ac.uk, k.sadeghi@ucl.a... | https://arxiv.org/abs/2504.01012v1 |
the causal mechanisms generating the edges (with respect to node deletion and marginalization). By systematically enumerating causal models with these properties, we find statistically-streamlined models for growing networks that exhibit emergent features charac- teristic of real-world networks, and offer a baseline fr... | https://arxiv.org/abs/2504.01012v1 |
ural numbers Nwith the standard ≤ordering. Intuitively, we can think of the nodes as “arriving” in that order, and then deciding with other nodes to connect to. We refer to specific node indices with lower-case letters (sometimes iandj, and sometimes a,b,c,d), with ordering implied lexicographically. Unspecified nodes ... | https://arxiv.org/abs/2504.01012v1 |
a parent or as a child) are also deleted.And the relabeling maps the remaining causal arrows: φ (ij) (kl) = φ (ij) φ (kl) A meta-DAG is a set of causal arrows between dyad vari- ables. We want to classify all meta-DAGs that are invariant to this action of node deletion and relabeling. That is, what sets of caus... | https://arxiv.org/abs/2504.01012v1 |
(ij). and the child dyad (ij). The middle column are the names we have given to each type of causal arrow. The right column shows the causal arrows from the parent dyad to the child dyad in terms of lexicographic node indices a < b < c < d . See Figs. 2 and 3 for the rep- resentation of the causal arrows in the growing... | https://arxiv.org/abs/2504.01012v1 |
de- scribe a way to include both OLDandNEWcausal arrows by using structural equations that can be unrolled into an asynchronous generative process. 2.4.2 Arrows between Dyads Not Sharing a Node The next three types of causal arrows in Table 1 are from a parent dyad to a child dyad with nonodes in common. Fig. 3 display... | https://arxiv.org/abs/2504.01012v1 |
are not observed cannot simply be deleted from a causal model; in order to preserve the causal and probabilistic relationships between the remaining variables, they must be marginal- ized. For a causal DAG, marginalizing a variable is a two-step process: first add arrows from all of its parents to all of its children, ... | https://arxiv.org/abs/2504.01012v1 |
for all cases in which the child dyad of the first causal arrow is the parent dyad of the next causal arrow, which types of causal arrows could point from the parent dyad of the first arrow to the child dyad of the second? For example:PATH PATH={FAR}, since the effect of traversing two PATH causal arrows is always equi... | https://arxiv.org/abs/2504.01012v1 |
[1999]. The simplest statement of their model [Pósfai and Barabási, 2016] has a single parameter, m. Initialize the network with clique of mnodes. At each iteration, select mnodes proportional to their current degree, and add a new node connected to each of these selected nodes. Many extensions to this model have been ... | https://arxiv.org/abs/2504.01012v1 |
avg degree θin+θout = 1⟨d⟩ → 2αln( n) + Cpoly avgdegree ⟨d⟩ →C×nρ ρ=θin+θout−1 γ=2−θout 1−θout const avgdegree ⟨d⟩ →2α 1− θin+θout p d ∝d−γγ=1 +θin θin Figure 4: Sparsity and power-laws in the DAPA model. 3.2.1 Three Sparsity Regimes Sparsity can be characterized in terms of the average de- gree as a function of th... | https://arxiv.org/abs/2504.01012v1 |
current citation network. 6Estimating the power-law exponent of a degree distribution is notoriously tricky [Clauset et al., 2009]. No finite network is truly scale-free; even if there is an obvious power-law that fits the majority of the degree distribution, there are necessarily deviations at the extremities.By appro... | https://arxiv.org/abs/2504.01012v1 |
sort of causal structure, with dyads containing older nodes influencing dyads containing newer nodes. Conversely, the causal model with HUBandNEWdepends on the other “quadrant” of dyads, and instead has a sort of “bottom-up” sort of causal structure (see its causal meta-DAG at the top- right of Fig. 7). That is, the dy... | https://arxiv.org/abs/2504.01012v1 |
14(1): 26569, 2024. Chen Avin, Hadassa Daltrophe, Barbara Keller, Zvi Lotker, Claire Mathieu, David Peleg, and Yvonne-Anne Pigno- let. Mixed preferential attachment model: Homophily and minorities in social networks. Physica A:Statistical Mechanics anditsApplications, 555:124723, 2020. Albert-László Barabási and Réka A... | https://arxiv.org/abs/2504.01012v1 |
Daniel M Roy. Bayesian models of graphs, arrays and other exchangeable random struc- tures. IEEE Transactions onPattern Analysis and Machine Intelligence, 37(2):437–461, 2014. Judea Pearl. A probabilistic calculus of actions. In Uncertainty inArtificial Intelligence , pages 454–462. Elsevier, 1994. Judea Pearl. Causali... | https://arxiv.org/abs/2504.01012v1 |
( bottom-right ). X12 X13 X23 X14 X24 X34 X15 X25 X35 X45X12 X13 X23 X14 X24 X34 X15 X25 X35 X45 X12 X13 X23 X14 X24 X34 X15 X25 X35 X45X12 X13 X23 X14 X24 X34 X15 X25 X35 X45 Figure 7: Causal graphs with multiple types of causal arrows between dyads. Causal meta-DAGs between dyads of a growing network with 5nodes that... | https://arxiv.org/abs/2504.01012v1 |
at each step is E(n+ 1)−E(n) =nX i=1α+θindin i+θoutdout i n+α+β−1 =αn+ θin+θout E(n) n+α+β−1(12) We make an ansatz of constant average degree E(n) =C1×n+g(n) (13) where g(n) =o n is subdominant. We will first solve for C1to obtain the asymptotic average degree ⟨d⟩= 2C1, then we will verify our assumption that g(n) ... | https://arxiv.org/abs/2504.01012v1 |
exponent ρ=θin+θout−1. Note that this does not fix C1. To see why, let us attempt to verify our ansatz g(n) =o nlogn . We again equate the remaining lower-order terms: g(n+ 1)−g(n) = 1 +ρg(n) n+O nρ−1 (28) In this case, it seems as though our ansatz is not verified, with g(n)being the same order as the “dominant”... | https://arxiv.org/abs/2504.01012v1 |
(36) We can extract the power law of the degree distribution from the dependence of the expected degree on the node index j. Since dj is monotonically decreasing in j, the probability density will be proportional to the reciprocal of the magnitude of the derivative with respect to j: p d ∝ d dj dj −1 (37) For a power... | https://arxiv.org/abs/2504.01012v1 |
Confidence Bands for Multiparameter Persistence Landscapes Inés García-Redondo∗Anthea Monod†Qiquan Wang‡ April 3, 2025 Abstract Multiparameter persistent homology is a generalization of classi- cal persistent homology, a central and widely-used methodology from topological data analysis, which takes into account densit... | https://arxiv.org/abs/2504.01113v1 |
extension of the classical persistence landscape. As such, the known functional and statistical prop- erties of persistence landscapes carry over to the multiparameter setting [Bub15, Vip20]. This paper extends the statistical framework of confidence bands from single-parameter [CFL+14a, CFL+14b] to multiparameter pers... | https://arxiv.org/abs/2504.01113v1 |
Banach space, given by central limit theorems. We also outline the construction of confidence bands—the dual construction to hypothesis testing—using bootstrap methods. CentralLimitTheorems(CLT) holdforbothsingle-andmultiparameter persistence landscapes [Bub15, Vip20]: Suppose Xis a Borel measurable r.v.onsomeprobabili... | https://arxiv.org/abs/2504.01113v1 |
it to define a distance L2(Q)over the class of functions Fas∥f−g∥2 Q,2=R |f−g|2dQ. Let N(F, L2(Q), ϵ)be the covering number of F, i.e., the size of the smallest ϵ-net in this metric. Lastly, recall that given a family of measurable functions F={f: Ω→R}, ameasurable envelope of this collection is the “smallest” measurab... | https://arxiv.org/abs/2504.01113v1 |
three distinct topological 7 Table 1:Mean accuracies after 5-fold cross validation for the MBD classifier using the standard or multiplier bootstrap in single (SPH) or multiparameter (MPH) persistence. Models trained over nsubsamples of eachclassofshapes: spheres, toriiandKleinbottles, asexplainedinFigure1. SPH Standar... | https://arxiv.org/abs/2504.01113v1 |
of Intelligence: An “Erlangen Programme” for AI. Q.W. is funded by a CRUK–Imperial College London Convergence Science PhD studentship (2021 cohort, PIs Monod/Williams), which is supported by Cancer Research UK under grant reference [CANTAC721 \10021]. References [Bub15] Peter Bubenik. Statistical Topological Data Analy... | https://arxiv.org/abs/2504.01113v1 |
On spectral gap decomposition for Markov chains Qian Qin School of Statistics University of Minnesota Abstract Multiple works regarding convergence analysis of Markov chains have led to spectral gap decomposition formulas of the form Gap( S)≥c0h inf zGap( Qz)i Gap( ¯S), where c0is a constant, Gap denotes the right spec... | https://arxiv.org/abs/2504.01247v1 |
¯S characterizing the chain’s transition between the subsets, and Qzdepicting the chain’s movement within a subset Xz. This type of relation has been applied to Metropolis-Hastings algorithms (Guan and Krone, 2007), tempering algorithms (Woodard et al., 2009) and the reversible jump algorithm (Qin, 2025+). See Madras a... | https://arxiv.org/abs/2504.01247v1 |
¯Sis its ideal- ized counterpart. 2. Spectral gap decomposition holds when Sis a hybrid data augmentation algorithm with two intractable conditional distributions, as opposed to one in the study of Andrieu et al. (2018b). Additionally, a unified framework allows one to extend spectral gap decomposition in a generic set... | https://arxiv.org/abs/2504.01247v1 |
J∗g⟩ρforf∈L2 0(ρ),g∈L2 0(ρ′). A linear operator J:L2 0(ρ)→L2 0(ρ) is self-adjoint if J=J∗. It is positive semi-definite if it is self adjoint and ⟨f, Jf⟩ρ≥0. Assume that J:L2 0(ρ)→L2 0(ρ) is self-adjoint, and its operator norm ∥J∥ρ≤1. For f∈L2 0(ρ), define the Dirichlet form EJ(f) =∥f∥2 ρ− ⟨f, Jf⟩ρ. Define the right sp... | https://arxiv.org/abs/2504.01247v1 |
dynamic of a Markov chain into global and local components, based on a finite partition or covering of the state space. 5 Let (X,A, π) be a probability space. Suppose that Xhas a partition X=Sk z=1Xz, where kis a positive integer, and the Xz’s are non-overlapping measurable subsets such that π(Xz)>0. Let M andNbe Mtks ... | https://arxiv.org/abs/2504.01247v1 |
be probability spaces. Let ˜ π:A×C → [0,1] be a probability measure such that ˜ π(A×Z) =π(A) forA∈ A and ˜π(X×C) =ϖ(C) forC∈ C. Assume that ˜ πhas the decomposition ˜π(d(x, z)) =π(dx)πx(dz) =ϖ(dz)ϖz(dx). This means that, if ( X, Z)∼˜π, then X∼π,Y∼ϖ,Z|X=x∼πx, and X|Z=z∼ϖz. An ideal data augmentation algorithm (Tanner an... | https://arxiv.org/abs/2504.01247v1 |
iaccording to the probability vector ( p1, . . . , p k), draw x′ ifrom Hi,x−i(xi,·), and set x′ −i=x−i. The following result was proved in Qin et al. (2025+). Proposition 4. (Qin et al., 2025+) In the context of this subsection, Gap( S)≥ min iess inf uGap( Hi,u) Gap( ¯S), where the essential infimum is taken with res... | https://arxiv.org/abs/2504.01247v1 |
for some t∈Z+, there exist positive constants κ1, . . . , κ tsuch that, for s∈ {0, . . . , t −1}, the following “approximate conservation of variance” holds almost surely: For f∈L2 0(νs), E[var νs+1(f)|(Wi)s i=0]≥κs+1varνs(f). (4) Then Gap( Kt)≥tY s=1κs (5) Proposition 5 allows one to bound the spectral gap of an Mtk K... | https://arxiv.org/abs/2504.01247v1 |
as πrestricted to the hyperplane Lw,θ(x,w). An ideal ℓ-dimensional hit-and-run algorithm targeting πsimulates a Markov chain through the following procedure: Given the current state x∈X, an orthonormal basis ( w1, . . . , w ℓ) is drawn from the distribution ν; then the new state is drawn from ϖw,θ(x,w). The correspondi... | https://arxiv.org/abs/2504.01247v1 |
H2,x1) Gap( ¯S),Gap( ˆS2)≥ 1−ess sup x1∥H2,x1∥2 φ2,x1 Gap( ¯S). Here, we have used the fact that H2,x1is reversible, which implies that Gap( H2 2,x1) = 1−∥H2,x1∥2 φ2,x1. In Appendix‘C.5, we establish the following proposition relating Tto the other Mtks. Proposition 7. In the context of this subsection, 1− ∥T∥2 φ≥ ... | https://arxiv.org/abs/2504.01247v1 |
operator P∗E∗EP. This operator has kernel form P∗E∗EP(x,dx′) =Z Y×Z×YP∗(x,d(y, z))ϖz(dy′)P((y′, z),dx′) ifPandP∗can be represented as integral kernels. We shall compare Dirichlet forms associated with P∗ˆQPto those associated with the operator P∗E∗EP. A spectral decomposition formula of the form (1) with ¯S=P∗E∗EPwould... | https://arxiv.org/abs/2504.01247v1 |
condition will be imposed: (C1) One can find (i) a linear transformation P:L2 0(π)→L2 0(˜π) satisfying (H2), (ii) a collection of Mtks {Qz}z∈Zsatisfying (H3), (iii) an Mtk Q: (Y×Z)×(B × C )→[0,1] reversible with respect to ˜ πsatisfying (H4), and (iv) a linear transformation R:L2 0(˜π)→L2 0(π) such that Q=R∗Rand that T... | https://arxiv.org/abs/2504.01247v1 |
•˜π:A ×2[k]→[0,1] is such that ˜π(A× {z}) =π(A∩Xz) =Z Aπ(dx)1Xz(x). In other words, ˜ πis the distribution of an X×[k]-valued random element ( X, Z) where X∼π andZsatisfies X∈XZ. •Forz∈[k] and A∈ A,ϖz(A) =ωz(A∩Xz) =π(A∩Xz)/π(Xz). Note that, if ( X, Z)∼˜π, then Z∼ϖ, and X|Z=z∼ϖzas required. •S=M1/2NM1/2. •Forf∈L2 0(π) a... | https://arxiv.org/abs/2504.01247v1 |
that S(j)can be viewed as a version of P∗ˆQPby taking ˆQ=Q(j). In hybrid Gibbs-like algorithms (e.g., data augmentation, random-scan Gibbs, and hit-and-run algo- rithms), having two choices of Qzcorresponds to having two choices of Markovian approximations to approximate conditional distributions. See Table 1. For two ... | https://arxiv.org/abs/2504.01247v1 |
it does not admit a positive right spectral gap. Moreover, a weak Poincar´ e inequality leads to a quantitative convergence bound, often of subgeometric nature. We refer readers to Andrieu et al. (2022) and Power et al. (2024). The following two result, established in Appendix B, can be regarded as analogues of Theorem... | https://arxiv.org/abs/2504.01247v1 |
are positive constants c1(w)andc2(w)such that, for x∈Rkandu∈R,c1(w)≤ −w⊤U(x+uw)w≤c2(w). Then, if σ2 w= 1/c2(w)in the Metropolis- within-hit-and-run algorithm, it holds that Gap( Hw,θ(x,w))≥C∗[c1(w)/c2(w)], and C∗ inf wc1(w) c2(w) Gap( ¯S)≤Gap( S)≤Gap( ¯S), where C∗is some universal constant. Proof. It follows from Pr... | https://arxiv.org/abs/2504.01247v1 |
on a data set with ξ≈226. Right: τ(Sλ) var Sλ(fi)/∥fi−πfi∥2 πplotted against τ(S) var S(fi)/∥fi−πfi∥2 π. The dashed line goes through the origin and has slope 1. 26 be uniform. We then use the output of each chain to estimate var S(fi)/var¯S(fi) for i= 1, . . . , 30, where fi(β1, . . . , β 30) =βi. This is accomplished... | https://arxiv.org/abs/2504.01247v1 |
on showing that the norms and spectral gaps of the two operators are equal. Define the map U:L2 0(ρ)→L2 0(ρ0) as follows: Uf(w) =f(w) forw∈Ω0. Then Uis invertible, andU−1g(w) =g(w)1Ω0(w) for g∈L2 0(ρ0) and w∈Ω. (Note that two functions in L2 0(ρ) are equal as long as they coincide ρ0-a.e. on Ω 0.) In fact, Uis unitary.... | https://arxiv.org/abs/2504.01247v1 |
osc ≥ EP∗E∗EP(f)−¯α(s)∥f∥2 osc. This establishes the desired result. B.4 Proof of Proposition 22 Proof. By Popoviciu’s inequality (Popoviciu, 1935), when s <1, for f∈L2 0(π), ∥f∥2 π≤1 4∥f∥2 osc≤sES(f) +˜β(s)∥f∥2 osc. Assume that s≥1. Then, by Lemma 20 and the weak Poincar´ e inequality for P∗E∗EP, for f∈L2 0(π),s1>0, a... | https://arxiv.org/abs/2504.01247v1 |
π0=Z Xπ0(dx) [f(x)−π0f]2 ≤Z Xπ0(dx) [f(x)−πf]2 ≤∆∥f−πf∥2 π. As a consequence, Gap( N) = inf fEN(f−π0f) ∥f−π0f∥2π0≥1 Θinf fES(f−πf) ∥f−πf∥2π=1 ΘGap( S). (9) Proposition 2 then follows from (7), (8), and (9). 32 C.2 Random-scan hybrid Gibbs samplers Recall the setting of Section 3.4: X=X1× ··· × Xk, where each Xiis a Pol... | https://arxiv.org/abs/2504.01247v1 |
demonstrate that Proposition 5 can be derived using Proposition 3, which is encompassed by Corollary 13. Fixt∈Nands∈ {0, . . . , t −1}. To utilize Proposition 3, we shall demonstrate that Ks corresponds to a data augmentation algorithm, while Ks+1is associated with a particular hybrid data augmentation algorithm, as de... | https://arxiv.org/abs/2504.01247v1 |
x, w, u )7→ Hw,θ(x,w)(u,B) is measurable for every measurable B⊂Rℓ. Our goal is to prove Proposition 6. To this end, we show that ¯Sdefines a data augmentation algorithm, and Sdefines a hybrid version of that algorithm, as described in Sections 3.3 and 5.3. The elements in Sections 3.3 and 5.3 are identified below. •Z=... | https://arxiv.org/abs/2504.01247v1 |
that, for h∈L2 0(˜π) and x2∈X2, Eh(x2) =Z X1h(x1, x2)φ1,x2(dx1). •Forg∈L2 0(φ2) and ( x1, x2)∈X,E∗g(x1, x2) =g(x2). One may then verify the following: •Pis an Mtk that is reversible with respect to ˜ π, so∥P∗P∥π=∥P∥2 ˜π≤1. Thus, Psatisfies (H2). •Qx2is reversible with respect to φ1,x2forx2∈X2, and (H3) is satisfied by ... | https://arxiv.org/abs/2504.01247v1 |
andZhang, X. (2024). Rapid mixing of Glauber dynamics via spectral independence for all degrees. SIAM Journal on Computing FOCS21–224. Chen, Y. andEldan, R. (2022). Localization schemes: A framework for proving mixing bounds for Markov chains. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science . IEEE... | https://arxiv.org/abs/2504.01247v1 |
rate of the Gibbs sampler with various scans. Journal of the Royal Statistical Society, Series B 57157–169. Liu, K. (2023). Spectral Independence: A New Tool to Analyze Markov Chains . University of Washington. Lov´asz, L. (1999). Hit-and-run mixes fast. Mathematical Programming 86443–461. Lov´asz, L. andVempala, S. (2... | https://arxiv.org/abs/2504.01247v1 |
arXiv:2504.01318v2 [math.ST] 21 Apr 2025Tail Bounds for Canonical U-Statistics and U-Processes with Unbounded Kernels∗ Abhishek Chakrabortty†and Arun Kumar Kuchibhotla‡ April 22, 2025 Abstract In this paper, we prove exponential tail bounds for canonical (or degenerate) U- statistics and U-processes under exponential-t... | https://arxiv.org/abs/2504.01318v2 |
functionals encountered in missing data or causa l inference problems ( Robins et al. ,1994;Bang and Robins ,2005), as well as in the literature on adaptive estimation of functionals based on so-called higher order influence functions (Robins et al. ,2008,2017;Liu et al. ,2021). Apart from the nonparametric and semipara... | https://arxiv.org/abs/2504.01318v2 |
the paper and derive non-asymptotic moment as well as tail bounds when the non-degenerate U-statistics is of the form ( 1). Our main tool is the decoupling inequality proved in de la Pe˜ na (1992). We refer to de la Pe˜ na and Gin´ e (1999, Chapter 3) for more details regarding decoupling in U-statistics. After derivin... | https://arxiv.org/abs/2504.01318v2 |
convergence as can be obtained from the results of Gin´ e et al. (2000). This is because the bound of Kolesko and Lata/suppress la (2015) does not depend on the variance. We are not aware of any tail b ounds in the literature that implies the correct rate of convergence as w ell as the optimal tail behavior. We also no... | https://arxiv.org/abs/2504.01318v2 |
UD n:=/summationdisplay 1≤i/ne}ationslash=j≤nfi,j(Zi,Zj), whereZ1,...,Znare independent random variables and {fi,j(·,·) : 1≤i/\e}atio\slash=j≤n}is a collection of degenerate (or canonical) kernels, i.e., E[fi,j(Zi,Zj)|Zi] = 0 =E[fi,j(Zi,Zj)|Zj]. We assume the following on the degenerate kernel fi,j: (A1) For 1≤i/\e}ati... | https://arxiv.org/abs/2504.01318v2 |
i=1E[ξ2 i]/parenrightiggp/2 +KpppE/bracketleftbigg max 1≤i≤n|ξi|p/bracketrightbigg ,forp≥2, (6) for independent mean-zero random variables ξ1,...,ξn. If|ξi| ≤Balmost surely, then this inequality is equivalent to Bernstein’s inequali ty. However, for unbounded random variables ξi(for examples, those satisfying only a s... | https://arxiv.org/abs/2504.01318v2 |
a moment and tail bound for degenerate U-statistics. The appearance of the constants Λ αand Λβmight make this result difficult to apply in some applications. For this reason, we provide ou r second result assuming a little more structure on the kernel. Suppose we have nindependent random variables Z1= (X1,Y1),Z2= (X2,Y2)... | https://arxiv.org/abs/2504.01318v2 |
i,2and Ψ′ j,2, are respectively non-zero, which can only happen with only a small probability under Assumption (B1) . Finally, the fourth term can be non-zero only if both Φ i,2and Ψ′ j,2are non-zero which can happen with even smaller probability. These four terms leads to four different degenerate U-statistics that wil... | https://arxiv.org/abs/2504.01318v2 |
rate of convergence. These results parallel the Glivenko-Cantelli theorems well-known for empirical processes. Functional l imit theorems were established inNolan et al. (1988). Exponential tail bounds that parallel the classical Bern stein’s in- equality for non-degenerate and degenerate U-statistics were given in Arc... | https://arxiv.org/abs/2504.01318v2 |
j=1,j/ne}ationslash=iE/bracketleftbig σ2 j,ψ(Xj)w2 i,j(x,Xj)/bracketrightbig 1/2 , /bardbl(φwψ)W/bardbl2→2:= sup w∈Wsup {qi}sup {pj}/summationdisplay 1≤i/ne}ationslash=j≤nE/bracketleftbig qi(Xi)σi,φ(Xi)wi,j(Xi,X′ j)σj,ψ(X′ j)pj(X′ j)/bracketrightbig . Here in the definitions, the supremum over {qi}(or{pj}) represents ... | https://arxiv.org/abs/2504.01318v2 |
and Wellner (2011). Theorem 4. Suppose Frepresent a class of real-valued functions f:χ×χ→R uniformly bounded by Rwith the envelope function F. Then there exists a universal constantC > 0such that E/bracketleftigg sup f∈F/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationtext 1≤i/ne}ationsl... | https://arxiv.org/abs/2504.01318v2 |
(γ1,...,γn). Define Wj(γ) =/summationdisplay 1≤i≤n, i/ne}ationslash=jE[fi,j(Zi,Z′ j)γi(Zi)|Z′ j]. Degeneracy of {fi,j}implies that Wj’s are mean zero independent random variables. Hence, by Theorem B.1 of Kuchibhotla and Chakrabortty (2022), we get E sup γ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vex... | https://arxiv.org/abs/2504.01318v2 |
i(εi,Zi)≤1,En/summationdisplay j=1ζ2 i(ε′ i,Z′ i)≤1 , Cp:=E max 1≤i≤nE n/summationdisplay j=1Φ2 i,1w2 i,j(Xi,X′ j)/parenleftbig Ψ′ i,1/parenrightbig2/vextendsingle/vextendsingleXi,Yi p/2 +E/parenleftigg max 1≤j≤nE/bracketleftiggn/summationdisplay i=1Φ2 i,1w2 i,j(Xi,X′ j)/parenleftbig Ψ′ i,1/parenrightbig... | https://arxiv.org/abs/2504.01318v2 |
for p≥1, E/bracketleftig |U(3) n|p/bracketrightig ≤Kp βCp ψ(logn)p/βpp/β∗/bracketleftig (logn)p/2Υp φ+ (BwTφ)p(logn)p/bracketrightig +Kp βCp ψ(logn)p/βpp/β∗/bracketleftig pp/2Υp φ+pp(BwTφ)p/bracketrightig ,(E.11) 26 where Υ2 φ:= max xn/summationdisplay i=1,i/ne}ationslash=jE[w2 i,j(Xi,x)σ2 i,φ(Xi)]. To control U(... | https://arxiv.org/abs/2504.01318v2 |
. (E.13) Observe that n/summationdisplay i=1E[|ξi|p] =n/summationdisplay i=1E/bracketleftbig |ξi|p/BD{|ξi|>δ0}/bracketrightbig +n/summationdisplay i=1E/bracketleftbig |ξi|p/BD{|ξi|≤δ0}/bracketrightbig (a) ≤2E/bracketleftbigg max 1≤i≤n|ξi|p/bracketrightbigg +n/summationdisplay i=1E/bracketleftbig |ξi|p/BD{|ξi|≤δ0}/brack... | https://arxiv.org/abs/2504.01318v2 |
p2 j(Xj)/bracketrightbig ≤1,n/summationdisplay j=1E/bracketleftbig q2 i(Xi)/bracketrightbig ≤1 . Thus the result follows. 32 S.2 Proofs of Results in Section 3 S.2.1 Proof of Theorem 3 Similar to U(ℓ) n,1≤ℓ≤4 defined in the proof of Theorem 2, we define U(1) n(W) := sup w∈W/vextendsingle/vextendsingle/vextendsingle/v... | https://arxiv.org/abs/2504.01318v2 |
n,2(W) +pp/2Σp/2 n,2(W) +ppΛp 2(W)/bracketrightig +Kppp/α∗/bracketleftig (logn)p/2Σp/2 n,2(W) + (logn)pΛp 2(W)/bracketrightig . By a similar calculation, we get E/bracketleftig |U(3) n(W)|p/bracketrightig ≤Kppp/β∗/bracketleftig Ep n,1(W) +pp/2Σp/2 n,1(W) +ppΛp 2(W)/bracketrightig +Kppp/β∗/bracketleftig (logn)p/... | https://arxiv.org/abs/2504.01318v2 |
sequence ( p1,...,pn) satisfying/summationtextn j=1/integraltext p2 j(x)PXj(dx)≤1. Substituting ( E.22) and ( E.21) in ( E.20), we get II≤Kppp/2 E sup {qi}sup w∈W/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay j=1ε′ jΨ′ j,1ℓj(X′ j;{qi},w)/vextendsingle/vextends... | https://arxiv.org/abs/2504.01318v2 |
of the Maximal Inequality (Theorem 4) The following moment bound of Rademacher chaos is used in the proof. See corollary 3.2.6 of de la Pe˜ na and Gin´ e (1999) and inequalities leading to (4.1.20) on page 167 of de la Pe˜ na and Gin´ e (1999). 41 Lemma 9. LetZbe a homogeneous Rademacher chaos of degree 2, that is, Z:=... | https://arxiv.org/abs/2504.01318v2 |
Adamczak, R. (2006). Moment inequalities for U-statistics. Ann. Probab. , 34(6):2288– 2314. 44 Adamczak, R. and Kutek, D. (2023). On orlicz spaces satisfyi ng the hoffmann-j {\o} rgensen inequality. arXiv preprint arXiv:2310.04163 . Arcones, M. A. and Gin´ e, E. (1993). Limit theorems for U-processes. Ann. Probab. , 21(... | https://arxiv.org/abs/2504.01318v2 |
Methods for Comparing Distributions . PhD thesis, Carnegie Mellon University. Kolesko, K. and Lata/suppress la, R. (2015). Moment estimates for cha oses generated by symmetric random variables with logarithmically convex tails. Statist. Probab. Lett. , 107:210–214. Kuchibhotla, A. K. and Chakrabortty, A. (2022). Moving... | https://arxiv.org/abs/2504.01318v2 |
On Robust Empirical Likelihood for Nonparametric Regression with Application to Regression Discontinuity Designs Qin Fang1, Shaojun Guo2, Yang Hong3, and Xinghao Qiao4 1University of Sydney Business School, Sydney, Australia 2Institute of Statistics and Big Data, Renmin University of China, China 3School of Mathematics... | https://arxiv.org/abs/2504.01535v1 |
nonparametric likelihood-based inference method that has seen a wave of advancements in its applications to parametric, semiparametric, and nonparametric models; see, e.g., Kitamura et al. (2004); Xue and Zhu (2007b); Chen and Van Keilegom (2009); Bravo et al. (2020); Matsushita and Otsu (2021); Xue (2023), and Yu and ... | https://arxiv.org/abs/2504.01535v1 |
and no satisfactory solution has yet been established in the literature. To tackle this challenge, we propose a novel bias-corrected EL framework, called robust EL, that delivers valid confidence intervals across a wider range of bandwidth selections for nonparametric regression and RDD. Unlike existing methods, which ... | https://arxiv.org/abs/2504.01535v1 |
including those considered in Xu (2013, 2020), Chiang et al. (2019), Qu and Yoon (2019), and Dong et al. (2023). It is worth noting that Calonico et al. (2014, 2018) made significant contributions to robust inference in RDD analysis. Specifically, they developed a robust bias-corrected inference procedure based on norm... | https://arxiv.org/abs/2504.01535v1 |
local linear estimator of mpxqwith bandwidth hthen simplifies to pm1,hpxq“1 nnÿ i“1Wi,0,1,hpxqYi“řn i“1Wi,hpxqYiřn i“1Wi,hpxq. Note that pm1,hpxqsatisfies the weighted moment equationřn i“1Wi,hpxqpYi´θq“0.Inspired 7 by this, Chen and Qin (2000) (referred to as CQ) introduced the original empirical log- likelihood ratio... | https://arxiv.org/abs/2504.01535v1 |
presenting the theoretical results, we introduce a refined asymptotic framework that characterizes a more flexible and practically reasonable relationship between the smooth- ing bandwidth hand the pilot bandwidth b. For local polynomial estimation, balancing bias and variance typically results in the optimal bandwidth... | https://arxiv.org/abs/2504.01535v1 |
capture the variability associated with the bias estimators in a simple and effective way. 3 Methodology In this section, we develop a new EL framework in the nonparametric regression setting, referred to as robust EL. We first propose two sets of robust weights in Section 3.1 and then construct the corresponding robus... | https://arxiv.org/abs/2504.01535v1 |
log-likelihood ratios for mpxqin model (2). To be specific, theTaylor-expansion-based Robust EL ratio and Difference-based Robust EL ratio are 14 respectively given by lTRpθq“´ 2 max"nÿ i“1logpnpiqˇˇˇˇpiě0,nÿ i“1pi“1,nÿ i“1piW‹ i,h,bpxqpYi´θq“0* , lDRpθq“´ 2 max"nÿ i“1logpnpiqˇˇˇˇpiě0,nÿ i“1pi“1,nÿ i“1piW˛ i,h,bpxqpYi´... | https://arxiv.org/abs/2504.01535v1 |
h˙j , W´ i,h“p1´TiqKhpXiqˆ S´ 2,h´S´ 1,hXi h˙ , S´ j,h“1 nnÿ i“1p1´TiqKhpXiqˆXi h˙j . LetZipθ, aq“` W` i,hpYi´θ´aq, W´ i,hpYi´aq˘T.The original empirical log-likelihood func- tion, introduced by Otsu et al. (2015), is given by lSpθ, aq“´ 2 max"nÿ i“1logpnpiqˇˇˇˇpiě0,nÿ i“1pi“1,nÿ i“1piZipθ, aq“0* . and the correspondin... | https://arxiv.org/abs/2504.01535v1 |
is nonparametrically identified as τF“limxÑ0`EpYi|Xi“xq´limxÑ0´EpYi|Xi“xq limxÑ0`EpTi|Xi“xq´limxÑ0´EpTi|Xi“xq”µY`´µY´ µT`´µT´. The original empirical log-likelihood ratio, proposed by Otsu et al. (2015), for τFcan then be formulated as lFpθq“ inf pa,b`,b´qPAˆr0,1sˆr0,1slFpθ, a, b`, b´q, where lFpθ, a, b`, b´qis defined... | https://arxiv.org/abs/2504.01535v1 |
of the estimators in both settings. Fur- thermore, Conditions 6(iv) and 8(ii) place standard constraints on the conditional variance of the observed outcome and treatment, respectively, allowing for potential heterogeneity across the threshold. Theorem 3. Assume that Conditions 3–7hold. Then, as nÑ8, lS,TRpτSqDÑχ2 1,an... | https://arxiv.org/abs/2504.01535v1 |
approaches and their robustness to both handb. Second, all other competing methods suffer from significant size and coverage distortions. As expected, the Orig method is highly sensitive to bandwidth selection, with its performance deteriorating severely when hbecomes slightly larger. Meanwhile, both conventional metho... | https://arxiv.org/abs/2504.01535v1 |
(0.899) (0.892) (0.888) (0.911) (0.914) DB 0.071 0.061 0.055 0.050 0.046 0.061 0.050 0.043 0.039 0.035 (0.927) (0.934) (0.931) (0.928) (0.928) (0.931) (0.924) (0.933) (0.939) (0.934) TR 0.085 0.074 0.066 0.061 0.057 0.074 0.060 0.052 0.047 0.043 (0.944) (0.944) (0.954) (0.953) (0.952) (0.948) (0.945) (0.953) (0.955) (0... | https://arxiv.org/abs/2504.01535v1 |
0.24 0.27Orig TB DB TR DR CCT 0.800.850.900.951.00 BandwidthEmpirical coverage 0.15 0.18 0.21 0.24 0.27(a)n“500, b“1.2h 0.000.050.100.150.20 BandwidthEmpirical size 0.15 0.18 0.21 0.24 0.27 0.800.850.900.951.00 BandwidthEmpirical coverage 0.15 0.18 0.21 0.24 0.27 (b)n“500, b“1.5h 0.000.050.100.150.20 BandwidthEmpirical... | https://arxiv.org/abs/2504.01535v1 |
Table 3: Comparison of average and standard deviation (in parentheses) of bandwidths h andb, empirical sizes, empirical coverages and average interval lengths over 10000 simulation runs for Model 3. The best performances, with empirical sizes close to 5% and empirical coverages close to 95%, are in bold font. Methodn“5... | https://arxiv.org/abs/2504.01535v1 |
2000 Turkish Population Census, as ana- lyzed in Meyersson (2014). The goal is to examine the effect of Islamic political representation in the 1994 municipal elections on high school attainment for women whose education could have been influenced between 1994 and 2000. The matched dataset includes n“2629 mu- nicipalit... | https://arxiv.org/abs/2504.01535v1 |
nonparametric quantile regression and quantile RDD; see, e.g., Qu and Yoon (2019) and Xu (2020). A key challenge in this context is how to conduct uniform inference across various quantile levels within our proposed robust EL framework. These topics fall beyond the scope of the current paper and will be pursued elsewhe... | https://arxiv.org/abs/2504.01535v1 |
Biometrika 108: 661–674. Meyersson, E. (2014). Islamic rule and the empowerment of the poor and pious, Econometrica 82: 229–269. 34 Otsu, T. (2012). Empirical likelihood for nonparametric additive models, Journal of the Japan Statistical Society 41: 159–186. Otsu, T., Xu, K.-L. and Matsushita, Y. (2015). Empirical like... | https://arxiv.org/abs/2504.01535v1 |
i“1Zipxq´λ2nÿ i“1Z2 ipxq␣ 1`oPp1q( “σ2 1,κpxq σ2 0pxq#? nhU 1pxq σ1,κpxq+2␣ 1`oPp1q( , implying that lTBtmpxquDÑγ1,κχ2 1, where γ1,κ“σ2 1,κpxq{σ2 0pxq. We now provide the proofs for equations (S.2)-(S.5). To begin, we introduce some nota- tion. Define Sas the matrix pµj`k´2q1ďj,kď3, and let µj“ş1 ´1ujKpuqdu, ν j“ş1 ´1u... | https://arxiv.org/abs/2504.01535v1 |
rU1pxq, which corresponds to (S.6). First, it follows from Lemma 1 of Fan and Zhang (1999) that pm2,bpxq´mpxq“1 nfpxqnÿ i“1˘wi,bpxqεi`OP˜ b3`b2log1{2n n1{2b1{2`logn nb¸ , holds uniformly over xPr0,1s, where ˘ wi,bpxq“eT 1S´1t1,pXi´xq{b,pXi´xq2{b2uTKbpXi´xq. 6 Given that b2lognÑ0 and nh3b4Ñ0,for each Xiandx, pm2,bpXiq´p... | https://arxiv.org/abs/2504.01535v1 |
1`oPp1q( , where ˘Ki,bpxqis defined in Section A.1.2. Then, hU˛ 21pxq“h nnÿ i“1␣ Z˛ i,1pxq(2 “h nnÿ i“1␣ wi,hpxq´µ2˘Ki,bpxq`µ2˘wi,bpxq(2ε2 if2pxq␣ 1`oPp1q( “σ2 2,κpxqt1`oPp1qu. Since ripxq“mpXiq´mpxq“Op|Xi´x|qaround x, we have hU˛ 22pxq“h nnÿ i“1␣ Z˛ i,2pxq(2“h nnÿ i“1␣ wi,hpxq(2r2 ipXiqf2pxq␣ 1`oPp1q( “OP` h2˘ , hU˛ 2... | https://arxiv.org/abs/2504.01535v1 |
that pλ0“pV´1 nUn`oPpn´1{2h1{2qand max 1ďiďn|pλT 0Z‹ ipτS, µ´q| “ oPp1q. Hence, we have lS,TRpτS, µ´q“nUT npV´1 nUn␣ 1`oPp1q( . Analogously, lS,TRpτS, µ´`Cpnhq´1{2q “ 2 logt1`pλT rZ‹ ipτS, µ´`Cpnhq´1{2qu, wherepλr 14 satisfies: nÿ i“1ZipτS, µ´`Cpnhq´1{2q 1`pλT rZ‹ ipτS, µ´`Cpnhq´1{2q“0. Furthermore, pλr“pV´1 ntUn´Cpnhq... | https://arxiv.org/abs/2504.01535v1 |
(0.916) (0.844) (0.674) Setting: b“1.2h TB 0.071 0.061 0.055 0.050 0.046 0.061 0.050 0.043 0.039 0.035 (0.868) (0.863) (0.869) (0.864) (0.869) (0.872) (0.859) (0.861) (0.863) (0.872) DB 0.071 0.061 0.055 0.050 0.046 0.061 0.050 0.043 0.039 0.035 (0.910) (0.917) (0.917) (0.919) (0.899) (0.910) (0.904) (0.912) (0.929) (0... | https://arxiv.org/abs/2504.01535v1 |
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