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classes of the underlying graph, meaning that the set of maximal classes contains all the information of the underlying graph that we can reconstruct from the support of Σ. Additionally, by the definition of maximal classes, we are able to reconstruct a list of possible graphs admitting this set of maximal classes. The... | https://arxiv.org/abs/2504.03466v2 |
l∈[n], λ[k] liλ[k] lsλsj= 0, Lets=j, since λjj̸= 0, ∀k∈N, l∈[n], λ[k] liλ[k] ls= 0. By Lemma 5, it implies that iandjdo not have the same ancestor, and they are not connected by directed paths, i.e.iandjdo not belong to the same maximal class. ■ The following proposition highlights the link between maximal classes and ... | https://arxiv.org/abs/2504.03466v2 |
aiλ[k] aj= 0. It means that there is no directed path from atoiandjrespectively (here amay equal to iorj), which contradicts the assumption. Therefore, J[ω,σij] G ̸= 0, and moreover, J[σij] G̸=0. ■ Proposition 9 shows that two nodes that do not belong to the same maximal class will result in a null column in the Jacobi... | https://arxiv.org/abs/2504.03466v2 |
( JG1)≥n≥2 (see Proposition 10). Addition- ally, since J[σab] G2=0, we have J[σab] G2̸⊥ ⊥J[σa′b′] G2. Combining the results above, we have found a pair of columns of the Jacobian matrix corresponding to σaband σa′b′such that they are linearly independent in M1, and not in M2. Therefore, J(M1)̸=J(M2), and hence by Propo... | https://arxiv.org/abs/2504.03466v2 |
do not apply anymore. The following assumption excludes ”multi-edges”, which is needed in the study of the rank of the Jacobian matrix. In particular, it ensures the existence of ”triplets” in Lemma 10, which is crucial in proving that the Jacobian matrix is of full rank in Theorem 5. Assumption 1. For all graphs G, we... | https://arxiv.org/abs/2504.03466v2 |
are at least two maximal classes, and the undirected subgraph of each maximal class is complete. Next, we introduce a new matrix BGderived from the Jacobian matrix. And it will be shown in Lemma 9 that the Jacobian matrix is full rank if the matrix BGis full rank. 19 Definition 11. LetMGbe a stationary VAR(1) model tha... | https://arxiv.org/abs/2504.03466v2 |
a subset MB GofMG: MB G:={Λ∈MG|det (BG) = 0}, then µG MB G = 0, where µGis the same measure defined in Proposition 5. 20 Proof. First, we prove that the function fG:MG×R+→R (Λ, ω)7→det (BG) is rational. From Appendix A, we know that the parametrization map ϕG:MG×R+→Mn(R) (Λ, ω)7→Σ,s.t. vec (Σ) = In−ΛT⊗ΛT−1 vec (ωIn... | https://arxiv.org/abs/2504.03466v2 |
l}}, it’s clear that B[ω,σij] G0= (Σ 0Λ0)ij(Σ0Λ0)−1 ii+ (Σ 0Λ0)ji(Σ0Λ0)−1 jj= 0. And B[λka,σij] G0=δaih (Σ0Λ0)kj(Σ0Λ0)aa−(Σ0Λ0)ka(Σ0Λ0)aji +δaj (Σ0Λ0)ki(Σ0Λ0)aa−(Σ0Λ0)ka(Σ0Λ0)ai . When a=i, we know that j̸=a, k, l , so (Σ 0Λ0)kj= (Σ 0Λ0)aj= 0, thus δaih (Σ0Λ0)kj(Σ0Λ0)aa−(Σ0Λ0)ka(Σ0Λ0)aji = 0. Similarly, δaj (Σ0Λ0)ki... | https://arxiv.org/abs/2504.03466v2 |
criteria or not are listed below: Here, MCiis the set of maximal classes of the model Mi. Note that although we present cases where the identifiability criteria used in this paper are not satisfied, it does not necessarily mean that the models are not identifiable. These are open cases left for future work. We end this... | https://arxiv.org/abs/2504.03466v2 |
and 2 will be affected, while in B, all the other nodes will be affected. Acknowledgments I would like to express my deepest gratitude to my PhD supervisors: St´ ephane Robin, Viet Chi Tran and Elisa Th´ ebault. I am especially thankful to St´ ephane Robin and Viet Chi Tran, for their unwavering support, guidance, and ... | https://arxiv.org/abs/2504.03466v2 |
Proof of the fact that the parametrization map is rational Proof. Recall that the parametrization map is: ϕG:MG×R+→Mn(R) (Λ, ω)7→Σ,s.t. vec (Σ) = In−ΛT⊗ΛT−1 vec (ωIn). It’s sufficient to prove that for all i∈ n2 , the ithcoordinate function of ϕG: ϕi:MB G×R+→Mn(R) (Λ, ω)7→vec (Σ)i is a quotient of polynomials (QOPs... | https://arxiv.org/abs/2504.03466v2 |
3 Proof. Denote the interaction matrix and covariance matrix for complete graphs with nnodes as Λ cand Σ c respectively. We know from Lemma 1 that the extended Jacobian matrix for the complete graph satisfies: J=J(Λc⊗Λc) +B, (7) where B=(ΣcΛc⊗In) (In2+P) vec (In2)T. . Apply the projection ψGto both sides of the equat... | https://arxiv.org/abs/2504.03466v2 |
Λ′∈MGs.t.λ′ st>0 for all ( s, t)∈EG, then by definition, hij|k0,a0>0. Therefore, hij|k0,a0does not constantly equal zero on MG. In this case, µG({Λ∈MG|hij|k0,a0= 0}) = 0 , because the set of roots of a non-zero real analytic function has Lebesgue measure zero ([5, Section 3.1]). This implies that µGn Λ∈MG|λ[k0] a0iλ[k... | https://arxiv.org/abs/2504.03466v2 |
5 By Lemma 8, we know that under Assumption 1, the condition nr> n′ cis satisfied if and only if the set Cmc Gis empty, i.e. Cmc G= 0, meaning that for all i, j∈[n], either ( i, j) or ( j, i)∈EGori, jdo not belong to the same maximal class. In addition, n′ c=EG. Using similar arguments as before, we define another matr... | https://arxiv.org/abs/2504.03466v2 |
Stochastic Optimization with Optimal Importance Sampling Liviu Aolaritei1, Bart P.G. Van Parys2, Henry Lam3, and Michael I. Jordan1 1Department of Electrical Engineering and Computer Sciences, UC Berkeley, USA liviu.aolaritei@berkeley.edu, jordan@cs.berkeley.edu 2CWI Amsterdam, the Netherlands bart.van.parys@cwi.nl 3In... | https://arxiv.org/abs/2504.03560v1 |
problems of the form min𝜃∈ΘE𝑋∼P[𝐹(𝜃, 𝑋)], where 𝜃is a decision variable in a feasible region Θand𝑋is a random variable drawn from a distribution P. In settings where the loss function 𝐹(𝜃, 𝑋)involves, for example, indicators of rare events, standard sampling-based optimization algorithms become inefficient du... | https://arxiv.org/abs/2504.03560v1 |
earlier. The core idea is to jointly treat the decision variable and the IS distribution, updating both simultaneously using stochastic gradient descent or stochastic approximation (SA), with a diminishing step size. Our scheme applies to general convex objectives 𝑓and to common IS families where the mapping from deci... | https://arxiv.org/abs/2504.03560v1 |
we now discuss in more detail. He et al. [18] consider adaptive IS for stochastic root-finding problems and demonstrate that a stochasticapproximationmethod, coupledwithanISschemeadaptedtothecurrentiterate, canachieve the same asymptotic variance as one adapted to the optimal solution. They refer to this property as mi... | https://arxiv.org/abs/2504.03560v1 |
a random vector distributed according to the probability distribution Pon𝒳⊆R𝑟. A standing assumption in the paper is that the distribution Pis known, or more accurately, that we can generate samples from Pefficiently. We denote by 𝜃⋆an optimal solution of (SO). Stochastic optimization problems of the form (SO) are u... | https://arxiv.org/abs/2504.03560v1 |
(4)), under Assumption 2.1(iv), requires taking 𝐾=∇2𝑓(𝜃⋆)−1, which, in turn, requires knowledge of the unknown optimal solution 𝜃⋆. Moreover, it has been observed that the RM-SA method is sensitive to the choice of the step-sizes, often resulting in poor practical performance (see the discussion in [42, Section 4.5... | https://arxiv.org/abs/2504.03560v1 |
constraints {𝜃∈R𝑠:𝐴⋆ 𝑎𝜃= 0}, where (𝐴⋆ 𝑎𝐴⋆ 𝑎⊤)†denotes the Moore-Penrose inverse of the matrix 𝐴⋆ 𝑎𝐴⋆ 𝑎⊤. Given the partition into active and inactive constraints, the optimality conditions (6) imply the following fact. Fact 2.2 (Optimality conditions) .We have 𝐴⋆ 𝑎𝜃⋆−𝑏⋆ 𝑎= 0,P𝐴⋆𝑎∇𝑓(𝜃⋆) = 0 . (7) ... | https://arxiv.org/abs/2504.03560v1 |
those of the PR-SA method in the affine subspace {𝜃:𝐴⋆ 𝑎𝜃−𝑏⋆ 𝑎= 0}for all 𝑛≥𝑁. Remark 2.3 (Projected SA) .Perhaps surprisingly, the standard projected versions of the RM-SA or PR-SA iterations 𝜃𝑛+1= arg min 𝜃∈Θ{︂ ⟨𝛼𝑛+1𝐺(𝜃𝑛, 𝑋𝑛+1), 𝜃⟩+1 2‖𝜃−𝜃𝑛‖2}︂ (with subsequent averaging for PR-SA) fails to iden... | https://arxiv.org/abs/2504.03560v1 |
with the active constraints, i.e., P 𝐴⋆𝑎∇𝑓(¯𝜃𝑛), may decay at order 1/√𝑛. Consequently, an optimizer aiming for iterates with small (asymptotic) gradient norm should prefer stochastic gradients for which the trace of the projected variance Tr(︀Var𝑋∼P[︀P𝐴⋆𝑎𝐺(𝜃⋆, 𝑋)]︀)︀= Tr(︀P𝐴⋆𝑎Var𝑋∼P[𝐺(𝜃⋆, 𝑋)]P𝐴⋆𝑎)︀... | https://arxiv.org/abs/2504.03560v1 |
𝜇which lives in a closed and convex set ℳ:={𝜇∈R𝑚:𝐶𝜇≤𝑑}, for some technology matrix 𝐶and budget vector 𝑑. We denote the associated likelihood ratio by ℓ(𝑥, 𝜇) := dP/dP𝜇(𝑥), and stochastic gradients by 𝐺𝜇(𝜃, 𝑥) :=ℓ(𝑥, 𝜇)𝐺(𝜃, 𝑥). Moreover, for the IS class to be well behaved, we impose the following a... | https://arxiv.org/abs/2504.03560v1 |
an exponential tilting with naturalparametrization. Indeed, asthenaturalsufficientstatistic 𝑆(𝑥) =𝑥isconsidered, Equation(15) reduces to ℓET(𝑥, 𝜇) = exp(−𝜇𝑥+𝜑(𝜇))with 𝜑(𝜇) = log E𝑋∼P[exp( 𝜇⊤𝑋)] =𝜇2/2. The variance of the stochastic gradients can be bounded by Var𝑋(𝜇)∼P𝜇[︁ 𝐺𝜇(𝜃⋆, 𝑋(𝜇))]︁ =E𝑋(𝜇)∼... | https://arxiv.org/abs/2504.03560v1 |
that ℓMT(𝑥, 𝜇) = dP/dP𝜇(𝑥) = exp(−(|𝑥|−|𝑥−𝜇|)). The variance of the stochastic gradients can be bounded by Var𝑋(𝜇)∼P𝜇[︁ 𝐺𝜇(𝜃⋆, 𝑋(𝜇))]︁ =E𝑋(𝜇)∼P𝜇[︁ 𝐺𝜇(𝜃⋆, 𝑋(𝜇))2]︁ −∇𝑓(𝜃⋆)2=E𝑋∼P[︁ 𝐺(𝜃⋆, 𝑋)2ℓ(𝑥, 𝜇)]︁ =E𝑋∼P[1{𝑋≥𝜃⋆}ℓ(𝑥, 𝜇)] =∫︁∞ 𝜃⋆exp(−2|𝑥|+|𝑥−𝜇|)/2d𝑥 =∫︁𝜇 𝜃⋆exp(−3𝑥+𝜇)/2d𝑥+∫︁∞ ... | https://arxiv.org/abs/2504.03560v1 |
main challenges: Challenge I. The first major challenge is the fact that the optimization problem (IS) which charac- terizes the optimal importance sampler is itself a stochastic optimization problem. Akintothestructuralassumptionsimposedonproblem(SO),wewillimposethefollowingstructural assumptions on the (IS) problem a... | https://arxiv.org/abs/2504.03560v1 |
arg min (𝜃,𝜇)∈Θ×ℳ⎧ ⎪⎨ ⎪⎩⟨𝑛∑︁ 𝑘=0𝛼𝑘+1⎡ ⎣𝐺𝜇𝑘(𝜃𝑘, 𝑋(𝜇𝑘) 𝑘+1) 𝐻(𝜃𝑘, 𝜇𝑘, 𝑋𝑘+1)⎤ ⎦,⎡ ⎣𝜃 𝜇⎤ ⎦⟩+1 2⃦⃦⃦⃦⃦⃦⎡ ⎣𝜃−𝜃0 𝜇−𝜇0⎤ ⎦⃦⃦⃦⃦⃦⃦2⎫ ⎪⎬ ⎪⎭ ¯𝜃𝑛=1 𝑛𝑛−1∑︁ 𝑖=0𝜃𝑖,¯𝜇𝑛=1 𝑛𝑛−1∑︁ 𝑖=0𝜇𝑖,(19) with step size 𝛼𝑛=𝛼/𝑛𝛾, for 𝛾∈(1/2,1)and some constant 𝛼 > 0. In Section 5 we will study the converge... | https://arxiv.org/abs/2504.03560v1 |
may recursively apply the approach developed in this paper as many times as necessary, ultimately terminating with one of the heuristic procedures described in point (i) or (ii). Secondary importance sampling lies beyond the scope of this paper. While we do not pursue this direction further, the theoretical results dev... | https://arxiv.org/abs/2504.03560v1 |
(19), its first-order optimality condition guarantees that ⟨𝑛∑︁ 𝑘=0𝛼𝑘+1⎡ ⎣𝐺𝑘 𝐻𝑘⎤ ⎦+⎡ ⎣𝜃𝑛+1 𝜇𝑛+1⎤ ⎦,⎡ ⎣𝜃−𝜃𝑛+1 𝜇−𝜇𝑛+1⎤ ⎦⟩≥0, for all 𝜃∈Θand𝜇∈ℳ. In particular, this holds true for 𝜃⋆∈Θand𝜇⋆∈ℳ, showing that 𝑅𝑛+1≥0. Standard algebraic manipulations show that 𝑅𝑛+1can be rewritten as 𝑅𝑛+1=⟨𝑛∑︁ 𝑘=... | https://arxiv.org/abs/2504.03560v1 |
Hölder’s inequality, which allows us to use Assumption 4.1(iii) to recover the final bound. Moreover, on the event 𝒜:={𝐴⋆ 𝑎𝜃𝑛=𝑏⋆ 𝑎, 𝐴⋆ 𝑖𝜃𝑛< 𝑏𝑖}, Assumption 5.1(iii) guarantees that the upper bound‖∇𝜇𝑣(𝜃𝑛, 𝜇𝑛)−∇ 𝜇𝑣(𝜃⋆, 𝜇𝑛)‖≤ 𝑐3‖𝜃𝑛−𝜃⋆‖2holds. Putting everything together, the second term in (26... | https://arxiv.org/abs/2504.03560v1 |
𝐻𝑘−∇ 𝜇𝑣(𝜃𝑘, 𝜇𝑘)⎤ ⎦⃦⃦⃦⃦⃦⃦√︁ 𝑏𝑛𝑖+1⃦⃦⃦⃦⃦⃦⎡ ⎣𝜃⋆−𝜃𝑛𝑖+1 𝜇⋆−𝜇𝑛𝑖+1⎤ ⎦⃦⃦⃦⃦⃦⃦a.s.→0, using Lemma B.2 and (32). Moreover, ⃦⃦⃦⃦⃦⃦𝑛𝑖∑︁ 𝑘=0𝛼𝑘+1⎡ ⎣∇𝑓(𝜃𝑘)−∇𝑓(𝜃⋆) ∇𝜇𝑣(𝜃𝑘, 𝜇𝑘)−∇ 𝜇𝑣(𝜃⋆, 𝜇⋆)⎤ ⎦⃦⃦⃦⃦⃦⃦⃦⃦⃦⃦⃦⃦⎡ ⎣𝜃⋆−𝜃𝑛𝑖+1 𝜇⋆−𝜇𝑛𝑖+1⎤ ⎦⃦⃦⃦⃦⃦⃦≤√ 𝐶√︁ 𝑏𝑛𝑖+1⃦⃦⃦⃦⃦⃦⎡ ⎣𝜃⋆−𝜃𝑛𝑖+1 𝜇⋆−𝜇𝑛𝑖+1⎤ ⎦⃦⃦⃦⃦⃦⃦... | https://arxiv.org/abs/2504.03560v1 |
𝜆1∈R𝑝1 ++. (ii)−∇𝜇𝑣(𝜃⋆, 𝜇⋆) =𝐶⋆ 𝑎⊤𝜆2, for some 𝜆2∈R𝑝2 ++. Assumptions 5.4(i)-(ii) are relatively standard constraint qualifications for constrained optimization problems, with clear geometric meaning. Specifically, Assumptions 5.4(i) requires that −∇𝑓(𝜃⋆)should belong to the relative interior of the normal... | https://arxiv.org/abs/2504.03560v1 |
asymptotic negligibility of the martingale difference process. For this, we introduce the following notation. As in the proof of Theorem 5.2, we consider the filtration ℱ𝑛:=𝜎(𝑋(𝜇𝑘−1) 𝑘, 𝑋𝑘|𝑘≤𝑛). Moreover, for ease of notation, we define the noise vector 𝜉𝑘:=⎡ ⎣𝐺𝑘 𝐻𝑘⎤ ⎦−⎡ ⎣∇𝑓(𝜃𝑘) ∇𝜇𝑣(𝜃𝑘, 𝜇𝑘)⎤ ⎦.... | https://arxiv.org/abs/2504.03560v1 |
0P𝐶⋆𝑎⎤ ⎦ 𝜖𝑛:=⎡ ⎣P𝐴⋆𝑎𝐴⋆⊤ 𝑖(𝜆𝐴⋆ 𝑖,𝑛−1−𝜆𝐴⋆ 𝑖,𝑛) P𝐶⋆𝑎𝐶⋆⊤ 𝑖(𝜆𝐶⋆ 𝑖,𝑛−1−𝜆𝐶⋆ 𝑖,𝑛)⎤ ⎦−𝛼𝑛+1⎡ ⎣P𝐴⋆𝑎∇2𝑓(𝜃⋆)(I−P𝐴⋆𝑎)(𝜃𝑛−𝜃⋆) P𝐶⋆𝑎∇2 𝜇𝑣(𝜃⋆, 𝜇⋆)(I−P𝐶⋆𝑎)(𝜇𝑛−𝜇⋆)⎤ ⎦. Using again the facts P 𝐴⋆𝑎𝐴⋆ 𝑎⊤= 0, P𝐶⋆𝑎𝐶⋆ 𝑎⊤= 0and the optimality conditions P 𝐴⋆𝑎∇𝑓(𝜃⋆) = 0and P𝐶⋆𝑎∇𝜇𝑣(... | https://arxiv.org/abs/2504.03560v1 |
implies the marginal convergence in distribution, we immediately recover the CLT √𝑛(︁¯𝜃𝑛−𝜃⋆)︁𝑑→𝒩(︁ 0,Q†Var𝑋(𝜇⋆)∼P𝜇⋆[︁ 𝐺𝜇⋆(𝜃⋆, 𝑋(𝜇⋆))]︁ Q†)︁ , with Q :=P𝐴⋆𝑎∇2𝑓(𝜃⋆)P𝐴⋆𝑎. Finally, invoking Assumption 2.1(iii), the delta method yields the projected gradient CLT as stated below. Corollary 5.10 (Asymptoti... | https://arxiv.org/abs/2504.03560v1 |
Operations Research , 2023. [11] A. Deo and K. Murthy. Importance sampling for minimization of tail risks: A tutorial. In 2024 Winter Simulation Conference , pages 1353–1367, 2024. [12] J. C. Duchi and F. Ruan. Asymptotic optimality in stochastic optimization. The Annals of Statistics , 49(1):21–48, 2021. [13] D. Eglof... | https://arxiv.org/abs/2504.03560v1 |
Efficient estimators from a slowly convergent robbins-monro procedure. School of Oper. Res. and Ind. Eng., Cornell Univ., Ithaca, NY, Tech. Rep , 781, 1988. [37] J. S. Sadowsky and J. A. Bucklew. On large deviations theory and asymptotically efficient monte carlo estimation. IEEE transactions on Information Theory , 36... | https://arxiv.org/abs/2504.03560v1 |
all 𝑛∈N. The asymptotic normality result requires the following two assumptions. Assumption A.5 (Generic Asymptotic Normality I) . (i) There exists 𝑐 >0such that for all 𝑤∈𝒯we have 𝑤⊤𝐻𝑤≥𝑐‖𝑤‖2. (ii) There exists 𝐶 <∞such that E[︀‖𝜉𝑛‖2|ℱ𝑛−1]︀≤𝐶. Moreover, for some Σ≥0, 1√𝑛𝑛−1∑︁ 𝑘=0𝜉𝑘𝑑→𝒩 (0,Σ). We wou... | https://arxiv.org/abs/2504.03560v1 |
𝑛+1 2‖𝐺𝑛‖2. Since 𝑅𝑛is adapted to the filtration ℱ𝑛=𝜎(𝑋(𝜇𝑘−1) 𝑘|𝑘≤𝑛), we can take the conditional expectation E[·|ℱ𝑛]on both sides and obtain E[𝑅𝑛+1|ℱ𝑛]≤𝑅𝑛−𝛼𝑛+1⟨∇𝑓(𝜃𝑛), 𝜃𝑛−𝜃⋆⟩+𝛼2 𝑛+1 2E[‖𝐺𝑛‖2|ℱ𝑛], (46) where we have used the fact that E[𝐺𝑛|ℱ𝑛] =E𝑋∼P𝜇𝑛[𝐺𝜇𝑛(𝜃𝑛, 𝑋)ℓ(𝑋, 𝜇 𝑛)] ... | https://arxiv.org/abs/2504.03560v1 |
𝐴⋆ 𝑖𝜃⋆< 𝑏⋆ 𝑖, there exists some random 𝑁 <∞such that 𝐴⋆ 𝑖𝜃𝑛< 𝑏⋆ 𝑖, for all 𝑛≥𝑁. Therefore, 𝐴𝜃𝑛 𝑖=𝐴⋆ 𝑖, for all 𝑛≥𝑁. We will now show that there exists some random 𝑁 <∞such that 𝐴𝜃𝑛𝑎=𝐴⋆ 𝑎, for all 𝑛≥𝑁. We start by rewriting iteration (45) as 𝜃𝑛+1= arg min 𝜃∈Θ{︂ ⟨𝑔, 𝜃⟩+⟨𝑣𝑛, 𝜃⟩+1 2𝑏... | https://arxiv.org/abs/2504.03560v1 |
𝑏𝑛+1=∑︀𝑛 𝑘=0𝛼𝑘+1. Then, 1 𝑏𝑛+1𝑛∑︁ 𝑘=0𝛼𝑘+1⎡ ⎣𝐺𝑘 𝐻𝑘⎤ ⎦a.s.→⎡ ⎣∇𝑓(𝜃⋆) ∇𝑣(𝜃⋆, 𝜇⋆)⎤ ⎦. Proof.We start by upper-bounding ⃦⃦⃦⃦⃦⃦1 𝑏𝑛+1𝑛∑︁ 𝑘=0𝛼𝑘+1⎡ ⎣𝐺𝑘 𝐻𝑘⎤ ⎦−⎡ ⎣∇𝑓(𝜃⋆) ∇𝑣(𝜃⋆, 𝜇⋆)⎤ ⎦⃦⃦⃦⃦⃦⃦ by ⃦⃦⃦⃦⃦⃦1 𝑏𝑛+1𝑛∑︁ 𝑘=0𝛼𝑘+1⎛ ⎝⎡ ⎣𝐺𝑘 𝐻𝑘⎤ ⎦−⎡ ⎣∇𝑓(𝜃𝑘) ∇𝜇𝑣(𝜃𝑘, 𝜇𝑘)⎤ ⎦⎞ ⎠⃦⃦⃦⃦⃦⃦+⃦⃦⃦⃦⃦⃦1 𝑏... | https://arxiv.org/abs/2504.03560v1 |
Common Drivers in Sparsely Interacting Hawkes Processes Alexander Kreiss Leipzig University, Institute of Mathematics and Enno Mammen Heidelberg University, Institute of Applied Mathematics and Wolfgang Polonik UC Davis, Department of Statistics April 8, 2025 Abstract We study a multivariate Hawkes process as a model f... | https://arxiv.org/abs/2504.03916v1 |
in a practical appli- cation. For instance, when analyzing social media data, one might wonder about whether the age impacts the behavior of the users. However, since it is not uncommon that related actors in a social network are of similar age and that their behavior is influenced by their neighbors in the network, we... | https://arxiv.org/abs/2504.03916v1 |
our model is that the larger λn,i(t), the higher is the probability of an event happening in a small neighborhood around time t. Hence, an event in process Nn,jat a given time t(corresponding to an action of actor j) will increase the probability of an action of actor iat time tinstantaneously by an amount depending on... | https://arxiv.org/abs/2504.03916v1 |
al. (2017) considered high-dimensional processes with a constant baseline intensity, and Cai et al. (2022) studied multivariate Hawkes process models in a high- dimensional setting, where the transferring functions are estimated non-parametrically using B-spline approximations along with a group penalty. In these paper... | https://arxiv.org/abs/2504.03916v1 |
but are only reacting to others. This could be reflected by promoting sparsity in the estimation of α∗ n, e.g., by introducing an L1-penalty also on the estimates of α∗ n. We will mention below the optimization problems (4)and(5)how they have to be changed in order to achieve sparsity inα∗ n. We will, however, not purs... | https://arxiv.org/abs/2504.03916v1 |
( C, α) can be performed for each iseparately because LS i(C, α, θ ) is in fact only a function of Ci·andαi. More precisely, we denote for any fixed θ pCn,i ·(θ),pαn,i(θ) := argmin c∈[0,∞)n,a≥01 TLSi(c, a, θ ) + 2ωi∥c∥1, (5) 5 where the minimum is taken over all vectors cand numbers athat may appear as i-th row of m... | https://arxiv.org/abs/2504.03916v1 |
will define shortly. In the proof of the below Theorem 3.8, we show that the motivation from van de Geer et al. (2014) essentially transfers to our setting. We have left to define the matrix Θ n. The idea is that Θ napproximates 6 Σn(qCn,qαn,qθn)−1(even though the latter might not exist). We use a procedure inspired by... | https://arxiv.org/abs/2504.03916v1 |
TLSi(Ci·, αi, θ) + 2ωi∥Ci·∥1 . For formulating our main results, we need to introduce some notation and assumptions. Assumption (PE1): There are constants Kα, KC>0and bounded, convex, open sets Kβ⊆Rp andKγ⊆Rsuch that, for all n∈N,(β∗ n, γ∗ n)∈Θ := Kβ×Kγ,α∗ n∈(0, Kα)n,C∗ n∈[0, KC)n×n, and(C∗ n, α∗ n, θ∗ n)∈ H n. We wil... | https://arxiv.org/abs/2504.03916v1 |
have under (PE1) on the event Tn(an, bn, dn, en)∩ΩRCC,n(L,eHn) that 1 nT Ψn(·;qCn,qαn,qθn)−λn 2 T+2 nnX i=1ωi∥qCn,i·−C∗ n,i·∥1+an n∥qαn−α∗ n∥1+ebn∥qθn−θ∗ n∥1 ≤L(C∗ n, α∗ n)2 4ϕcomp(S1(C∗n), ..., S n(C∗n);L;eHn)2. Theorem 3.2 is a classical result from the LASSO literature. For our purposes, the most inter- esting part ... | https://arxiv.org/abs/2504.03916v1 |
j∈{1,...,n} γ∈Kγ16µRT 0Rt− 0g(t−r;γ)dNn,j(r)2 dNn,i(t) (µ−ϕ(µ))T2 +567g2N2 0log2(nT)·(logn+ log( nT) +α3logT) (µ−ϕ(µ))T2, pVµ e,T:= sup γ∈Kγµ µ−ϕ(µ) ×nX i=1ZT 0 nX j=1C∗ n,ij4t−R 01R 0d dγg(t−r;sγ+ (1−s)γ∗ n)dsdN n,j(r) nT!2 dNn,i(t) +576L2 gN2 0(1 +α4) max i=1,...,n∥C∗ n,i·∥2 1log3(nT) (µ−ϕ(µ))n2T2. Recall the defin... | https://arxiv.org/abs/2504.03916v1 |
n∥qαn−α∗ n∥1+1 n qCn−C∗ n 1=OP(rnsn), where snis defined as in Lemma 3.3. Recall also the definitions of σjandτjfrom 2.1. We make the following assumption on the rates. Assumption (D2) It holds that n3 2√ Tlog2(nT)qSn∥Θθ,n∥∞r2 ns2 n=oP(1), max j=1,...,p+11 τj=OP(1),and n3 2√ Trnsnmax j=1,...,p+1σj=o(1). Assumption (D3)... | https://arxiv.org/abs/2504.03916v1 |
part of Assumption (D4) implicitly requires the covariates the covariates Xn,i(t) to stabilize. Thus, Xn,i(t) should fulfill some weak stationarity condition. ButXn,i(t) being independent of iis not violating this assumptions. Theorem 3.8. Suppose that Assumptions (D1)-(D4) hold. Then, √ nT Θ0,θ,nVnΘT 0,θ,n−1 2 θn−θ... | https://arxiv.org/abs/2504.03916v1 |
The above definition can be used to formulate a restricted eigenvalue condition similar to the one that can be found in Chapter 6.2 in B¨ uhlmann and van de Geer (2011) or Bickel et al. (2009) and thus falls in the general class of compatibility conditions. We define Ei(C, α, θ ) := LS i(Ci·, αi, θ) + 2ZT 0Ψn,i(t;Ci·, ... | https://arxiv.org/abs/2504.03916v1 |
of radius rcentred around θ∗ n. Suppose furthremore that sup i=1,...,n(1 +|Si(C∗ n)|)(1 +∥C∗ n,i ·∥2 1+∥C∗ n,i ·(θn)∥2 1)2log2(nT) n=OP(1). (12) Then, 1 T Ψn,i(·;pCn,i ·,pαn,i,θn)−λn,i 2 2+ 2ωi∥pCn,i ·−C∗ n,i ·∥1+ 2an|pαn,i−α∗ n,i| =OP(1)· (1 +|Si(C∗ n)|)log4(nT) T , ∥pCn,i ·−C∗ n,i ·∥1=OP(1)· (1 +|Si(C∗ n)|)log2(nT... | https://arxiv.org/abs/2504.03916v1 |
particularly challenging as ω∈[0,∞)nis in fact a collection of tuning parameters. But we note that in (5), for any given θ, it is possible to choose ωiindependently of each other. While this is true for (5), it does not hold for (4), where we also optimize over θ. Nevertheless, we take the independence in (5) as motiva... | https://arxiv.org/abs/2504.03916v1 |
end for return ω 4.3 Simulation Study 4.3.1 Data Generating Process We consider n= 10 many vertices which are observed over T= 34 days. The baseline intensities depend on q= 1 covariate, which we update hourly, that is, Xn,i(t) is piecewise constant on intervals that have length 1 /24. The values of the covariates is t... | https://arxiv.org/abs/2504.03916v1 |
in each realization: The full model as described in Section 4.3.1 and a slim-oracle model that contains no covariates but assumes the true value of γ0to be known. To select the tuning parameter, for computational reasons, we use the cross-validation from Section 4.2 only for one data set in the full model and use the s... | https://arxiv.org/abs/2504.03916v1 |
common drivers. To investigate the situation further, we show in Figure 5 the percentage of non-zero estimations per edge in each model before and after the de-biasing step. In the full model, we see that the networks after de-biasing of θseem to be generally less dense compared to the first stage estimation (cf. Figur... | https://arxiv.org/abs/2504.03916v1 |
we see that the difference is not so systematic. Finally, Figure 10 in Appendix H shows the root mean squared error of the estimation of all of Cn. After de-biasing, the RMSE is a bit higher. This might be due to the higher bias induced by the higher sparsity. In Figure 4, the size of the vertices is proportional to αn... | https://arxiv.org/abs/2504.03916v1 |
the de-biasing is possible under some assumptions. These assumptions, unfortunately, restrict the level of sparsity. If one is only interested in the slower convergence rate, the sparsity is allowed to be much lower. An interesting future research direction would therefore be to understand if this restriction in the sp... | https://arxiv.org/abs/2504.03916v1 |
θ) =αTVn(β)α+ tr CnΓn(γ)CT n + 2αTdiag CnGn(θ)T −2αTvn(β)−2tr CnAn(γ)T , (14) where tr( A) denotes the trace of the matrix Aand diag( A) the diagonal of the matrix A written as column vector. The following result serves as a definition of Γ−1 nin case Γ nis not invertible. 26 Lemma A.1. The matrix Γn(γ)is positiv... | https://arxiv.org/abs/2504.03916v1 |
=2Vn(β)α+ 2diag( CnGn(θ)T)−2vn(β) + 2 vn(β)−diag( CnGn(θ)T)−Vn(β)α = 0. This completes the proof. Equation (15) shows that optimization in ccan be understood as a pure least squares linear model with δ= 0. Equation (16) shows the same about optimization in α. Moreover, we can see from (16) that finding the baseline p... | https://arxiv.org/abs/2504.03916v1 |
l=k+ 1, the statement remains true because then the columns kandlcover different blocks. Similarly, if kis odd and l=k−1, the statement remains true. Finally, it also holds if l=mork=m, when mis odd. So we have left to check the situation that k < m is odd and l=k+ 1 (and the case that k≤mis even and l=k−1 but this wor... | https://arxiv.org/abs/2504.03916v1 |
now use the cluster representation to prove Lemma B.1. 30 Proof of Lemma B.1. Theorem 7 in Br´ emaud and Massouli´ e (1996), states that there is a unique multivariate Hawkes process N(with finite expectations) with intensity given by (1), where ν0(Xn,i(t);β) is replaced by νiif the largest eigenvalue of the matrix A:=... | https://arxiv.org/abs/2504.03916v1 |
function as in (1)where ν0(Xn,i(t);β∗ n)is replaced by νiand(C∗ n, α∗ n, γ∗ n) = (C, α, γ )arbitrary. Suppose that a0:= sup i∈{1,...,n}∥Ci·∥1ZT 0g(t;γ)dt < 1. Since a0<1, there are ε∈(0,1)andr >0such that |ex−1| ≤ε a0|x|for all |x| ≤r. Then, for alls∈[0,∞)nwith∥s∥1≤r(1−ε)it holds that nX l=1 logE esTWl ≤r, (20) ∞X k=... | https://arxiv.org/abs/2504.03916v1 |
fact that Nb,l[−(k+ 1)A,−kA) is Poisson distributed with rate Aνiαi logE eaP∞ k=1Pn l=1P|Nb,l[−(k+1)A,−kA)| m=1Nl i,m(k) =∞X k=1nX l=1logE E eaNl i,m(k)|Nb,l[−(k+1)A,−kA)| =∞X k=1nX l=1Aνiαi E eaNl i,m(k) −1 ≤A∥ν∥∞∥α∥∞rε2 a0(1−ε), by (21) of Lemma B.3, which we apply here with s∈[0,∞)nwith si=a=r(1−ε) and sj=... | https://arxiv.org/abs/2504.03916v1 |
of Hansen et al. (2015) for a uni-variate process, i.e., in the notation of Hansen et al. (2015), M= 1. We keep the notation as in the theorem for the convenience of the reader. We have H(t) =4 Tν0(Xn,i(t);β), which is uniformly bounded in absolute value by B:=4νi T. The integral conditions are hence true and we consid... | https://arxiv.org/abs/2504.03916v1 |
since P(ΩN)→1 by Lemma B.2. Proof of Lemma 3.3. In the situation of Corollary 3.7, the requirements of Theorem 3.2 are fulfilled if we choose for a suitable Keb>0 ebn=Keblog2(nT)r max 1,1 nPn i=1∥C∗ n,i·∥2 1 √ T ≥max Kblog(nT), Kelog2(nT)q 1 nPn i=1∥C∗ n,i·∥2 1 √ nT, ωi=Kdlog2(nT)√ T. Moreover, using these definiti... | https://arxiv.org/abs/2504.03916v1 |
0p+1 0n 2ω11n ... 2ωn1n + 0p+1 −µα −µC , (29) where 0 pand 1 pare vectors of zeros or ones of size p, and µC∈Rn2is the vector (µC,11, ..., µ C,1n, µC,21, ..., µ 2n, ..., µ C,n1, ..., µ C,nn). Moreover, µC,ijqCn,ij= 0 and µα,iqαn,i= 0 for all i, j= 1, ..., n . We may rearrange (29) to obtain 0p+... | https://arxiv.org/abs/2504.03916v1 |
the last step follows by Assumption (D2). Plugging (34) and (33) in (32) yields √ nT θn−θ∗ n =−√ nTΘθ,nWn+oP(1). (35) We hence need to study the asymptotic behaviour of −√ nTΘθ,nWn. Recall that Mn,iare the counting process martingales. We have by definition of WnandMn,i −√ nTΘθ,nWn=−Θθ,n1√ nTnX i=1 ∂θ ∂α ∂C LSi(C... | https://arxiv.org/abs/2504.03916v1 |
row of C(i)and the i-th entry of α(i), we can merge all these minimizers into a single ( C∗ n(θ), α∗ n(θ)) at which the minimum is attained for all isimultaneously. Before proving Corollary 3.13, we present a technical lemma, which we prove in Appendix G. Lemma C.3. Let Assumptions (A0), (A1), and (PE1)-(PE3) hold. We ... | https://arxiv.org/abs/2504.03916v1 |
n,i(θn), α∗ n,i(θn),θn −Ψn,i ·;C∗ n,i(θ∗ n), α∗ n,i(θ∗ n), θ∗ n 2 T =1 T vi(t;θn)Tα∗ n,i(θn) C∗ n,i ·(θn) −vi(t;θ∗ n)Tα∗ n,i(θ∗ n) C∗ n,i ·(θ∗ n) 2 T =1 T vi(t;θn)−vi(t;θ∗ n)Tα∗ n,i(θn) C∗ n,i ·(θn) −vi(t;θ∗ n)Tα∗ n,i(θ∗ n)−α∗ n,i(θn) C∗ n,i ·(θ∗ n)−C∗ n,i ·(θn) 2 T ≤2 T vi(t;θn)−vi(t;θ∗ n)Tα∗ n,i(θn)... | https://arxiv.org/abs/2504.03916v1 |
Kl 1(k), ..., Kl n(k)) and define the logarithm of the Laplace transform of Kl(1) by ϕn,l(s) := log E esTKl(1) . Let finally ϕn:= (ϕn,1, ..., ϕ n,n). Denote a:=R∞ 0g(t;γ)dt. We note that Kl j(1) for j= 1, ..., n equals the number of events of a counting process with intensity function Cjlg(t;γ) and hence Kl j(1) is P... | https://arxiv.org/abs/2504.03916v1 |
=E e−sT(Wl(k)−Wl(p−2))+g(1)(s)TKl(p−1) =E eg(p−k−1)(s)TKl(k+1) . By applying (45) repeatedly, we can continue =E E eg(p−k−1)(s)TKl(k+1) Kl(k), ..., Kl(1) =E eϕn(g(p−k−1)(s))TKl(k) =...=E eϕ◦(k+1) n (g(p−k−1)(s))TKl(0) =eh ϕ◦(k+1) n (g(p−k−1)(s))i l, where ϕ◦(k) n:=ϕn◦···◦ ϕnk-times. By manually checking the... | https://arxiv.org/abs/2504.03916v1 |
|qγn−γ∗ n| ≤an n∥qαn−α∗ n∥1+bn qβn−β∗ n 1+1 nnX i=1dn,i qCn,i·−C∗ n,i· 1+en|qγn−γ∗ n| =an n∥qαn−α∗ n∥1+1 nnX i=1dn,i qCn,i·−C∗ n,i· 1+ebn∥qθn−θ∗ n∥1, whereebn≥max( bn, en). Using this in (49) yields on the event Tn(an, bn, dn, en) the following basic inequality: 1 nTE(qCn,qαn,qθn) + 2 qPn≤1 nTE(C∗ n, α∗ n, θ∗ n) + 2P∗ ... | https://arxiv.org/abs/2504.03916v1 |
we define ηat this point without motivation. Choose first c′′ 2such that Kα N(T+A) AT+Kα∥ν∥∞+ sup i=1,..,n∥C∗ n,i·∥1gN!Dν 2+Lν ≤c′′ 2max 1,sup i=1,...,n∥C∗ n,i·∥1! log(nT). Letη:=4KαLν(2p+α2) c′′ 2√ µ−ϕ(µ) max(1,supi=1,...,n∥C∗ n,i·∥1)nT. We conclude that P sup β∈Kβ 2 nTnX i=1α∗ n,iZT 0ν0(Xn,i(t);β)−ν0(Xn,i(t);β∗ n) ... | https://arxiv.org/abs/2504.03916v1 |
2max 1,sup i=1,...,n∥C∗ n,i·∥1! log(nT)η, by choice of c′′ 2, (25), and the fact that the number of jumps of each Nn,ion [0 , T] is on Ω N bounded by N(T+A)/A. By choice of η, we have that c′′ 2max 1,sup i=1,...,n∥C∗ n,i·∥1! log(nT)η≤√wx≤q pVµ bx≤bn 2. Hence, (53) = 0. Part involving dn,i:LetKγ,n,ηbe a finite, discrete... | https://arxiv.org/abs/2504.03916v1 |
≤c′′ 4log2(nT) n ∥C∗ n∥1+nX i=1∥C∗ n,i·∥2 1! . We define then η:=24LgN0(1 +α4) max i=1,...,n∥C∗ n,i·∥1p µ−ϕ(µ)c′′ 4(∥C∗n∥1+Pn i=1∥C∗ n,i·∥2 1)T. We compute using the fundamental theorem of calculus P sup γ∈Kγ 2 nTnX i,j=1C∗ n,ijZT 0Zt− 0g(t−r;γ)−g(t−r;γ∗ n) |γ−γ∗n|dNn,j(r)dMn,i(t) > en 56 =P sup γ∈Kγ nX i,j=12C∗ n... | https://arxiv.org/abs/2504.03916v1 |
Lemma C.1. Since, Rn,ab= Σ n,ab(Ca, αa, θa) for some intermediate parameters that are the same within each row of Rn, we may ignore their dependence on the row and may simply study the row-wise difference between Σ n(C1, α1, θ1) and Σ n(C, α, θ ) for a parameter (C, α, θ ) that lies between ( C1, α1, θ1) and ( C2, α2, ... | https://arxiv.org/abs/2504.03916v1 |
0α1,iν0,r(Xn,i(t);β1)nX j=1C1,ijZt− 0g′(t−r;γ1)dNn,j(r) 61 −αiν0,r(Xn,i(t);β)nX j=1CijZt− 0g′(t−r;γ)dNn,j(r)dt ! ≤K nmax i=1,...,n∥C1,i·∥1N∥α1−α∥1+K∥C1∥1 nN∥β1−β∥1+KN n∥C1−C∥1 +K∥C1∥1 nN|γ1−γ| Leta=γandb=γ: |Rn,ab−Σn,ab(C1, α1, θ1)| = 2 nTnX i=1 ZT 0 nX j=1C1,ijZt− 0g′(t−r;γ1)dNn,j(r) 2 − nX j=1CijZt− 0g′(t−r;γ)d... | https://arxiv.org/abs/2504.03916v1 |
argument for (63) goes similarly: Define Hi(t) :=1√ nT∂αkΨn,i(t;C∗ n,i ·, α∗ n,i, θ∗ n) =1√ nTν0(Xn,i(t);β∗ n) 1(i=k). Thus, only Hkis different from zero and we may again use M= 1 in Theorem 3 of Hansen et al. (2015). It holds that |Hk(t)| ≤K√ nT:=Bfor some constant Kfor all t∈[0, T]. The integral condition of Theorem... | https://arxiv.org/abs/2504.03916v1 |
the proof of the lemma. 65 G Proofs of Section 3.3 Proof of Theorem 3.11. The proof runs exactly along the lines of the proof of Theorem 6.2 in B¨ uhlmann and van de Geer (2011). However, for completeness we repeat it here in our setting. We have 1 TEi(pCn(θ),pαn(θ), θ) + 2ωi∥pCn,i ·(θ)∥1 =1 TLSi(pCn,i ·(θ),pαn,i(θ), θ... | https://arxiv.org/abs/2504.03916v1 |
than the inequality we want to prove. Case II : Suppose that 1 T Ψn,i(·;C∗ n,i ·(θ), α∗ n,i(θ), θ)−λn,i 2 T <14 3ωi∥pCn,iSi(C∗n)(θ)−C∗ n,iSi(C∗n)(θ)∥1+ 2an|pαn,i(θ)−α∗ n,i(θ)|. 67 In this case, we obtain from (67) 2 T Ψn,i(·;pCn,i ·(θ),pαn,i(θ), θ)−λn,i 2 T+ 2ωi∥pCn,iSc i(C∗n)(θ)∥1 ≤14ωi pCn,iSi(C∗(θ))(θ)−C∗ n,iSi(C∗n)... | https://arxiv.org/abs/2504.03916v1 |
every entry of the difference vi(t;θ)−vi(t;θ∗ n) can be uniformly (in tand i) be bounded by log( nT)· ∥θ−θ∗ n∥2times a constant. Let us assume for the remainder of the proof that we are on the event Ω N. Let now θ∈Θ be arbitrary, and let ( c, a) be such that cisi-th row of a matrix C∈[0,∞)n×n, and ais the i-th entry of... | https://arxiv.org/abs/2504.03916v1 |
·(θn)∥1 . The other case, θn−θ∗ n 2≥ α∗ n,i(θn)−α∗ n,i C∗ n,i ·(θn)− C∗ n,i ·T! 1, is stronger than what we wanted to prove. Therefore, the proof is complete because P(ΩN)→ 1. H Presentation of additional simulation results In this section, we collect further results of the simulation study presented in Section 4.3.... | https://arxiv.org/abs/2504.03916v1 |
Ann. Statist. , 32 (2):407–499, 2004. G. Fang, G. Xu, H. Xu, X. Zhu, and Y. Guan. Group network hawkes process. Journal of the American Statistical Association , 119:2328 – 2344, 2023. URL https://doi.org/10.1080/ 01621459.2022.2102019 . N. R. Hansen, P. Reynaud-Bouret, and V. Rivoirard. Lasso and probabilistic inequal... | https://arxiv.org/abs/2504.03916v1 |
Regression Discontinuity Design with Distribution-Valued Outcomes David Van Dijcke∗ Department of Economics University of Michigan, Ann Arbor April 8, 2025 Abstract This article introduces Regression Discontinuity Design (RDD) with Distribution- Valued Outcomes (R3D), extending the standard RDD framework to settings wh... | https://arxiv.org/abs/2504.03992v1 |
minimum wage is implemented along a state border, the outcome of interest could be the distribution of goods prices in each establishment, rather than a single average price. These settings are marked by two layers of randomness: one across units (districts, establishments), and one within (students within districts, g... | https://arxiv.org/abs/2504.03992v1 |
estimate is a conditional Wasserstein barycenter (Fr´ echet mean), which has the important property of being the central tendency of the observed quantile functions in probability space (Agueh and Carlier, 2011; Fan and M¨ uller, 2024). Importantly, this Fr´ echet (second) estimator is closely linked to the local polyn... | https://arxiv.org/abs/2504.03992v1 |
setting, and its identifying smoothness assumption highly restric- tive. This can be seen in Figure 1. Because of random sampling, the observed distributions (rainbow) exhibit discontinuous changes, violating the Q-RD assumption. The average dis- tributions (gray) do evolve smoothly, however. Of course, when treatment ... | https://arxiv.org/abs/2504.03992v1 |
Panaretos, 2022; Zhou and M¨ uller, 2024) and for local Fr´ echet regression (Chen and M¨ uller, 2022; Iao et al., 2024; Qiu et al., 2024). As noted above, I contribute to this literature by deriving uniform confidence bands for local Fr´ echet regression in Wasserstein space, complementing related results for global F... | https://arxiv.org/abs/2504.03992v1 |
version of the canonical regression discon- tinuity design. First, I formally introduce the setting, before providing several concrete examples from the literature. Then, I introduce a new definition of “local average quantile treatment effects” (LAQTE) appropriate for this setting, where the average is over random qua... | https://arxiv.org/abs/2504.03992v1 |
districts, or government agencies. Example 2 (Institutions) .Clark (2009) considers a British reform allowing public high schools to become autonomous (directly funded by the central instead of the local government) if a majority of parents vote in favor. The paper finds large increases in examination pass rates at sch... | https://arxiv.org/abs/2504.03992v1 |
local treatment effect is (Hahn et al., 2001) E[Z1−Z0|X= 0], the conditional expectation of the jump in the outcome variable at the threshold. In the R3D setting, YTis a full distribution function. An intuitive generalization of the classical average treatment effect to settings with distribution-valued outcomes is giv... | https://arxiv.org/abs/2504.03992v1 |
that even this simple collection of random Gaussian distributions satisfies the weaker continuity assumption in I1 but still fails continuity in quantiles. The following example establishes this formally. I include the proof here for intuition. Example 4. Suppose (X, Y)∈(R,Y)andY|X=x∼N(N(g(x),1),1)for some continuous f... | https://arxiv.org/abs/2504.03992v1 |
with the following local linear estimators, R3D :1 nnX i=1s±,i(h)QYi(q) Q-RDD :1 nnX i=1s±,i(h)1(Zi≤z). As can be seen, the R3D approach firstestimates quantiles and only then runs a local linear regression. This properly accounts for the two-level randomness intrinsic to the R3D setting. Distribution estimation at a g... | https://arxiv.org/abs/2504.03992v1 |
to Company B, then even if all untreated companies have identical distributions in the absence of treatment (an unrealistically strong assumption), the outcome distributions of those companies under treatment will still differ. 2.5 Estimators To estimate the local average quantile treatment effects introduced above, I ... | https://arxiv.org/abs/2504.03992v1 |
p≥1. Following Chiang et al. (2019), I build bias correction into the estimator by leveraging Remark 7 in (Calonico et al., 2014), which establishes an equivalence between explicitly bias-corrected estimators and estimators where the MSE-optimal bandwidth is chosen based on a pilot estimator of lower order – which is t... | https://arxiv.org/abs/2504.03992v1 |
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