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neral assumption of independent (but not necessarily identically distributed) /u1D44B/u1D456. /square Proof of Theorem 7. (=⇒) Assume the Pufferfish condition is satisfied, and fix P/u1D703∈Θsuch that /u1D437∼P/u1D703. Without loss of generality, consider the test for /u1D43B0:/u1D4600vs./u1D43B1:/u1D4601. By the post-pro... | https://arxiv.org/abs/2504.12520v1 |
{/u1D456, /u1D457}. We similarly rewrite: P/u1D703(M(/u1D43A)=/u1D714| {/u1D456, /u1D457}∉/u1D438)=1 2/summationdisplay.1 /u1D454∈G/u1D45BP(M(/u1D454\ {/u1D456, /u1D457})=/u1D714)P/u1D703(/u1D43A=/u1D454\ {/u1D456, /u1D457} | {/u1D456, /u1D457}∉/u1D438). For any fixed /u1D454, sinceMis/u1D700–edge DP, we know that P(M(/... | https://arxiv.org/abs/2504.12520v1 |
Proceedings of the 17th ACM SIGKDD international conferenc e on Knowledge dis- covery and data mining , pp. 1253–1261. Cuff, Paul and Lanqing Yu (2016). “Differential privacy as a mutual information constraint”. In: Proceedings of the 2016 ACM SIGSAC Conference on Computer an d Communications Security , pp. 43–54. Cumm... | https://arxiv.org/abs/2504.12520v1 |
analysis : A survey”. In: IEEE Transactions on Knowledge and Data Engineering 35.1, pp. 108–127. Jorgensen, Zach, Ting Yu, and Graham Cormode (2016). “Publi shing attributed social graphs with formal privacy guarantees”. In: Proceedings of the 2016 international conference on manage - ment of data , pp. 107–122. Kairou... | https://arxiv.org/abs/2504.12520v1 |
In: State of the Art Applications of Social Network Analysis , pp. 139–161. Tschantz, Michael Carl, Shayak Sen, and Anupam Datta (2020) . “SoK: Differential privacy as a causal property”. In: 2020 IEEE Symposium on Security and Privacy (SP) . IEEE, pp. 354–371. Wasserman, Larry and Shuheng Zhou (2010). “A statistical f... | https://arxiv.org/abs/2504.12520v1 |
Shrinkage priors for circulant correlation structure models Michiko Okudo1and Tomonari Sei1 1Department of Mathematical Informatics Graduate School of Information Science and Technology The University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, JAPAN, Email: okudo@mist.i.u-tokyo.ac.jp; sei@mist.i.u-tokyo.ac.jp Abs... | https://arxiv.org/abs/2504.12615v1 |
remove a multiplicative redundancy between αandλ, we assume pY i=1λi= 1 (6) without loss of generality. Lemma 1.1. Under the restriction (6), the parameters αandλare identifiable from Σ. 2 Proof. We see from the expression (3) that all the diagonal elements of Rhave the same value. This means that Ris a constant multip... | https://arxiv.org/abs/2504.12615v1 |
and the proposed prior are also given. 2 Construction of shrinkage priors In this section, we assume that λ(θ) is log-linear, that is, log λ(θ) = (log λk(θ))p k=1is linear in θ= (θ1, . . . , θ d). We also assume that vectors ( ∂/∂θ i) logλ(i= 1, . . . , d ) are linearly independent. For example, the full model for λtha... | https://arxiv.org/abs/2504.12615v1 |
is constant. Let µm=X mod ( k+l−1)=m1 pλkλ−1 l, then |I+R◦R−1|=|I+Qdiag( µ1, . . . , µ m)Q∗| =pY m=1(1 +µm) =pY m=1 1 +X mod ( k+l−1)=m1 pλkλ−1 l , (8) where A◦Bis the Hadamard product of AandB. We consider a new parametrization βi= log αi(i= 1, . . . , p ) 6 so that gβiβj= (δij+rijrij).Then, from (8), |gβ|=|I+R◦R−... | https://arxiv.org/abs/2504.12615v1 |
and Komaki (2022), where finite-sample properties of πSare studied. The prior πShas an optimal property regarding the asymptotic KL risk in a class of priors πa(θ, β)∝ πJ(θ, β)a(a∈R). Theorem 3.1. Suppose that λ(θ)is the full model (12). Let πa(θ, β)∝πJ(θ, β)a(a∈R).Regarding the asymptotic KL risk, when θ1, . . . , θ d... | https://arxiv.org/abs/2504.12615v1 |
2 sinhpθ 22 α−1(α−1)⊤+ 1 p 2 sinhpθ 22 + 2! diag( α−2 1, . . . , α−2 p), gθθ=p(p−1)/2, respectively. The Jeffreys prior is πJ(θ, α)∝ 1 p 2 sinhpθ 22 + 2!p−1 2 pY i=1α−1 i! . We have some theorems about shrinkage priors in the settings. Recall that βi= log αi. 11 Proposition 3.1. Letπc(θ, β)∝πJ(θ, β)c(c∈R). The Bay... | https://arxiv.org/abs/2504.12615v1 |
spiked covariance model. Annals of Statistics , 46:1742–1778, 2018. E. I. George, F. Liang, and X. Xu. From minimax shrinkage estimation to minimax shrinkage prediction. Statistical Science , 27:92–94, 2012. J. Jiang and Y. V. Hui. Spectral density estimation with amplitude modulation and outlier detection. Annals of t... | https://arxiv.org/abs/2504.12615v1 |
On perfect sampling: ROCFTP with Metropolis-multishift coupler Nabipoor, Majid nabipoor@ualberta.ca University Health Network, Toronto, Canada April 18, 2025 Abstract ROCFTP is a perfect sampling algorithm that employs various random opera- tions, and requiring a specific Markov chain construction for each target. To o... | https://arxiv.org/abs/2504.12872v1 |
Section 2 delves into the fundamentals and definitions of random operations, the normal multishift coupler, CFTP, and ROCFTP, while also addressing the determination of the block length Tin ROCFTP. Section 3 further explores the Metropolis-multishift coupler, including its func- tionality, convergence properties, the i... | https://arxiv.org/abs/2504.12872v1 |
that if a Markov chain moves backward in time, it will pointwise converge to the stationary distribution. Aldous (1990) investigated a Markov chain with random walk stationary, and through a reversed time transition matrix, proved that the chain converges to uniform distribution. Johnson (1996) proposed monotone coupli... | https://arxiv.org/abs/2504.12872v1 |
& Rhee (2014) proposed a new approach for unbiased estimation of Markov chain stationary expectations. They defined Z:=PN K=0∆k P(N≥k)as an estimator of E[f(X)] where ∆ k=f(Xk)−f(Xk−1), and Nis aZ+-valued random variable independent of ∆k. They established a coupling between XkandXk−1to ensure ∆ k−→0 ask−→ ∞ . 5 They d... | https://arxiv.org/abs/2504.12872v1 |
Therefore L=−f−1(U) and R=f−1(U), where f−1is the inverse function of f. Take a random point within the rectangle X∼Uniform (L, R), and apply mapping ML,R,X(s) = ⌊s+R−X R−L⌋(R−L) +X. This method is illustrated in Figure 1. In this algorithm, σ= 1, and it can be changed for any σ. Note that sis the initial value or the ... | https://arxiv.org/abs/2504.12872v1 |
the start block in Figure A.2, increasing it by 2nwhere n= 2,3,···in the next iterations. The second block contains 22new vectors, shown as a white block in the top right of the start block. It is necessary to continue this block to time t= 0. Note that this sampling is simultaneous, so we have to store the randomness ... | https://arxiv.org/abs/2504.12872v1 |
the algorithm 100 Ktimes for starting ranges of (−100,100), ( −50,50), (−10,10), with various selections for block time T. It is evident that as Tincreases, palso increases, and the total time τto generate a sample decreases up to a certain range, beyond which τstarts to increase. For more details, refer to Table 1. (ˆ... | https://arxiv.org/abs/2504.12872v1 |
chain typically demonstrates be- havior closely resembling monotonicity. We will begin by developing and applying the Metropolis-multishift algorithm to various target distributions before integrating it as a random operation in ROCFTP. 13 3.1 Metropolis sampler The Metropolis algorithm stands as one of the most influe... | https://arxiv.org/abs/2504.12872v1 |
+ 0 .1Beta (500,500) The Metropolis-multishift coupler implemented simultaneously over the range (-10,10) for the first case of N(0,1). Figure A.4 depicts the initial partitioning of the starting range into intervals or sources by this coupler, emitting one or several signals from each source. Subsequently, these signa... | https://arxiv.org/abs/2504.12872v1 |
illustrated in Figures A.9 and A.10, where the mode is discernible in both cases. In the sixth case, we increased the weight of the uniform distribution up to 0.9. While the mode is detectable in the figure, it becomes more ambiguous for larger weights. Despite the increased computational cost of the Metropolis-multish... | https://arxiv.org/abs/2504.12872v1 |
sup A∈F L(X∈A)− L(ˆX∈A) . Suppose T∗ ˆ0denotes the coupling time of Markov chains {Xt},{ˆ0t}; and T∗ ˆ1represents the coupling time of Markov chains {Xt},{ˆ1t}in a successful block with length T. The coupling event inequality (Lindvall (1992), Thorisson (2000)) implies that XT∗ ˆ0=ˆ0T∗ ˆ0=⇒ ∥L (Xt)− L(ˆ0t)∥ ≤2P(T∗ ˆ0... | https://arxiv.org/abs/2504.12872v1 |
uniformly selected from the interval [ L, R]. Once Xis generated, the algorithm creates an interval around it. Similar intervals are then constructed on the left and right sides of this interval, effectively dividing the starting range into intervals. The mapping then proposes a unique value for each of these intervals... | https://arxiv.org/abs/2504.12872v1 |
is N(30,1); the first case selects a start range of [ −100,−95], resulting in coalescence around -70 after 100 steps. Although, in the second case, the start range is [−200,−190], leading to coalescence around -100 after 200 steps! When dealing with mixture distributions featuring multiple modes, selecting the wrong st... | https://arxiv.org/abs/2504.12872v1 |
of a range around the mode(s). While identifying the MIR in complex posteriors is challenging, keeping it in mind proves invaluable. Now, the question arises: Can the algorithm generate a sample from the unlikely range? If so, what is the probability? Suppose ϕ:X −→ X represents the random operation of the Metropolis-m... | https://arxiv.org/abs/2504.12872v1 |
100 paths encompasses the experiment with 10 paths. In fact, these experiments run jointly using the same randomness. 26 Table 2 displays the mean time of coalescence for the selected four targets, calculated from 1000 replications. It shows minimal differences in the mean coalescence time between two paths and more pa... | https://arxiv.org/abs/2504.12872v1 |
the primary chain for sampling. The two auxiliary chains {ˆ0t}and{ˆ1t}function as a diagnostic mechanism to select an independent sample from the stationary distribution. The convergence of the chain to the stationary distribution πis established by the first coupling event. We are prepared to sample using ROCFTP with ... | https://arxiv.org/abs/2504.12872v1 |
Journal of Computation and Mathematics 13, 246–259. Besag, J., Green, P., Higdon, D. & Mengersen, K. (1995), ‘Bayesian computation and stochastic systems’, Statistical science pp. 3–41. Besag, J. & Green, P. J. (1993), ‘Spatial statistics and bayesian computation’, Journal of the Royal Statistical Society: Series B (Me... | https://arxiv.org/abs/2504.12872v1 |
Biometrika 60(3), 607–612. Propp, J. G. & Wilson, D. B. (1996), ‘Exact sampling with coupled markov chains and applications to statistical mechanics’, Random structures and Algorithms 9(1-2), 223– 252. Sokal, A. D. & Thomas, L. E. (1988), ‘Absence of mass gap for a class of stochastic contour models’, Journal of Statis... | https://arxiv.org/abs/2504.12872v1 |
described in the article. The package also contains examples provided in the article. https://cran.r-project.org/web/packages/ROCFTP.MMS/index.html A.3 CFTP R code ev<- function(vec) { even<- NULL for(i in (1: floor(length(vec)/2))) { even<- c(even, 2*i) } return(even) } L.R<- function() { z<- rnorm(1,0,1) u<- runif(1,... | https://arxiv.org/abs/2504.12872v1 |
Query Complexity of Classical and Quantum Channel Discrimination Theshani Nuradha1,∗and Mark M. Wilde1,† 1School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14850, USA Quantum channel discrimination has been studied from an information-theoretic perspective, wherein one is interested in... | https://arxiv.org/abs/2504.12989v1 |
Here, the task is to use the unknown quantum channel several times ( ntimes) to determine which channel was applied. Quantum channel discrimination in the asymptotic regime (i.e., n→ ∞ ) has been studied in [12–20]. Also Ref. [21] studied some settings in which a finite number of channel uses are sufficient to distingu... | https://arxiv.org/abs/2504.12989v1 |
the number of distinct channels to be discriminated. Note: After the completion of our work, we came across the independent study [39], which also studies the non-asymptotic regime of quantum channel discrimination. II. PRELIMINARIES AND NOTATIONS In this section, we establish some notation and recall various quantitie... | https://arxiv.org/abs/2504.12989v1 |
3.20]) and the reasoning used to establish [55, Proposition 55]. Precisely, the SDP is given by h bF(N,M)i1/2 = sup λ≥0,WRB∈Herm λ:λIR≤TrB[WRB], ΓN RBWRB WRBΓM RB ≥0 , (20) where ΓN RBand ΓM RBare the Choi operators of NandM, respectively. We make use of the following inequalities throughout our work: Let NA→BandMA... | https://arxiv.org/abs/2504.12989v1 |
this experiment is as follows: pe,{Q,A},ρR1A1(p,N, q,M, n):=pTr[(IRnBn−QRnBn)ρRnBn] +qTr[QRnBnτRnBn].(32) Given p, the distinguisher can minimize the error-probability expression in (32) over all adaptive strategies {Q,A}and the initial state ρR1A1. The optimal error probability pe(p,N, q,M, n) of hypothesis testing is... | https://arxiv.org/abs/2504.12989v1 |
ε, δ):= inf n∈N:βε N(n)∥M(n) ≤δ . (40) Definition 4 (Query Complexity of M-ary Channel Discrimination) .Letε∈[0,1], and letS:={(pm,Nm)}M m=1be an ensemble of Mchannels. The query complexity n∗(S, ε)of M-ary channel discrimination is defined as follows: n∗(S, ε):= inf{n∈N:pe(S, n)≤ε}. (41) Remark 1 (Equivalent Expres... | https://arxiv.org/abs/2504.12989v1 |
channels, then we have that n∗(p,N, q,M, ε) = Θ ln 1 ε −lnF(N,M)! . (53) Proof. Proof follows by adapting Corollary 7 and identifying that, for classical channels with classical inputs, the following equality holds: F(N,M) =bF(N,M) =FH(N,M). Remark 3 (Special Classes of Channels) .For some special classes of channels,... | https://arxiv.org/abs/2504.12989v1 |
two channels by adaptive methods and its application to quan- tum system, IEEE Transactions on Information Theory 55, 3807 (2009). [13] T. Cooney, M. Mosonyi, and M. M. Wilde, Strong converse exponents for a quantum channel discrimination problem and quantum-feedback-assisted communication, Communications in Mathematic... | https://arxiv.org/abs/2504.12989v1 |
Nuradha, and M. M. Wilde, Quantum doeblin coefficients: Interpre- tations and applications (2025), arXiv:2503.22823. [34] X. Huang and L. Li, Query complexity of unitary operation discrimination, Physica A: Sta- tistical Mechanics and its Applications 604, 127863 (2022). [35] A. Kawachi, K. Kawano, F. Le Gall, and S. T... | https://arxiv.org/abs/2504.12989v1 |
quantum channel estimation and discrimination, Quantum Information Processing 20, 78 (2021). [56] K. Matsumoto, Quantum fidelities, their duals, and convex analysis (2014), arXiv:1408.3462 [quant-ph]. [57] S. Cree and J. Sikora, A fidelity measure for quantum states based on the matrix geometric mean (2020), arXiv:2006... | https://arxiv.org/abs/2504.12989v1 |
n) = inf {Q,A},ρR1A11 2(1− ∥pρRnBn−qτRnBn∥1). (B8) Choosing ncopies of the input state and passing it through channel NorMin its n uses, the final state before the measurement is applied evaluates to be either of these states (N⊗n(ρ⊗n RA) orM⊗n(ρ⊗n RA)). This is one strategy that is included in the set of all adaptive ... | https://arxiv.org/abs/2504.12989v1 |
pqh bdB(N,M)i2, (B41) which concludes the proof of the lower bound. 19 Appendix C: Proof of Theorem 9 Lower bound: Letα∈(1,2]. Then, by the data-processing inequality for the geometric R´ enyi divergence under the measurement channel (comprised of the POVM elements Q, I−Q) bDα(ρ∥τ)≥1 α−1ln Tr[Qρ]αTr[Qτ]1−α+ Tr[( I−Q)ρ... | https://arxiv.org/abs/2504.12989v1 |
inf {Q,A},ρR1A1MX m=1pmTr (IRnBn−Qm RnBn)ρm RnBn , (D1) where Qdenotes a POVM ( Q1 RnBn, . . . , QM RnBn) satisfying Qm RnBn≥0 for all m∈ {1, . . . , M } andPM m=1Qm RnBn=IRnBn. For 1 ≤m̸= ˜m≤M, choose LRnBnandTRnRnto be positive semi-definite operators satisfying LRnBn+TRnBn=IRnBn−Qm RnBn−Q˜m RnBn. With that, define... | https://arxiv.org/abs/2504.12989v1 |
Spatial Confidence Regions for Excursion Sets with False Discovery Rate Control Howon Ryu1, Thomas Maullin-Sapey2,3, Armin Schwartzman1,4, and Samuel Davenport1 1Division of Biostatistics and Bioinformatics, University of California, San Diego, San Diego, CA, USA 2Big Data Institute, Nuffield Department of Population H... | https://arxiv.org/abs/2504.13124v1 |
uncertainty for the excursion set ,Ac= {v∈ V:µ(v)> c}, corresponding to the locations where µ(v), the true mean at each element vin the image space V, is greater than the threshold c∈R. We will refer to the elements of Vas locations. For images defined on a 2D/3D lattice, the locations are pixels/voxels respectively an... | https://arxiv.org/abs/2504.13124v1 |
[23], and [51, 8] for multiple testing confidence intervals. Different correctional methods for di- rectional FDR control are presented in [52] for neuroimaging and [32] for genetics data applications, while [31] presents error control under different correlational structures, or under independence [36, 38]. (a) Separa... | https://arxiv.org/abs/2504.13124v1 |
c (1) at each location v∈ Vwith corresponding null set (Ac)C⊆ V, the complement of Ac. To do so, let t(v) =ˆµ(v)−c ˆσ(v)/√n be the t-statistic for testing against the level cas in (1) where ˆµ(v)andˆσ(v)are the point estimates of the true mean and standard deviation at each location v∈ V respectively from ni.i.d. sampl... | https://arxiv.org/abs/2504.13124v1 |
cwe consider testing against the negative one-sided null hypothesis at level c, defined as HL 0(v) :µ(v)≥cvs. HL A(v) :µ(v)< c. (4) Construction of the lower confidence region is then formally given by Algorithm 2. This differs from Algorithm 1 in that it incorporates a two-stage adaptive procedure, as explained below.... | https://arxiv.org/abs/2504.13124v1 |
the rejection status (negative direction, one-sided). The color of the text in each cell corresponds to the color of the area in Figure 3 (b) and (c). Figure 3: Lower confidence region schematic: (a) Example signal and the projection of the excursion set at and above c(Ac). (b) A hypothetical lower confidence region ˆA... | https://arxiv.org/abs/2504.13124v1 |
lower confidence regions which have joint (instead of separate) error rate control. To do so, we shall combine the directional p-values from the upper and lower direction tests into a single BH algorithm. 9 3.1 Hypothesis Testing We now propose a confidence region construction procedure for jointly testing two null hyp... | https://arxiv.org/abs/2504.13124v1 |
in the image are tested twice with each location being the true non-null hypothesis for at least one of the directions considered. This means that the FDR is effectively controlled at a levelm 2m·α. We can take advantage of this and use a nominal level of 2αinstead of αwhile still providing FDR control at a level α, le... | https://arxiv.org/abs/2504.13124v1 |
it is defined as linearly gradual increase or decrease across the grid. The circle signal is generated by smoothing (FWHM 8 pixels) over a circle of radius 12 pixels which has magnitude 1 on the background of field of value -1. The noise field is obtained from smoothing the Gaussian random field N(0,1.52)by a kernel of... | https://arxiv.org/abs/2504.13124v1 |
considered for the ramp signal. Among noise smoothing settings, only FWHM = 5 is reported, as the other two settings showed similar results. The signals used for error simulation are identical to the signals in confidence region illustration (Figure 5) where the signal generation process is explained in detail. There a... | https://arxiv.org/abs/2504.13124v1 |
lower (adaptive). For the ramp signal, the signal smoothing is non-existent, thus showing the same result for three different levels of signal smoothing. The simulation results suggest that in general, the empirical FDR is effectively 16 controlled under the 0.05 level across four methods and for all values of c. This ... | https://arxiv.org/abs/2504.13124v1 |
like the FDR simulation, the asymmetric pattern in the circle signal for joint method stems from the asymmetry in the number of voxels in the circle signal and the background. Otherwise, the pattern of empirical FNDR demonstrated by the step and circle signals are similar. Generally, lower BH shows higher FNDR than low... | https://arxiv.org/abs/2504.13124v1 |
c, the yellow area including the red area denotes the excursion set ˆAc, and the blue area including the yellow area denotes the lower confidence region ˆA− c. The rows differ in the threshold level cby which the confidence regions are constructed. The columns denote different methods. Overall, FDR controlling hypothes... | https://arxiv.org/abs/2504.13124v1 |
when more voxels are thought to be rejected. In the context of negative one-sided testing, this is equivalent to when there are less number of voxels above cthan below c. 6 Conclusion and Discussion This paper has developed upper and lower confidence regions for the excursion set Ac, with spatial FDR control, based on ... | https://arxiv.org/abs/2504.13124v1 |
c, the lower confidence region with two-stage adaptive procedure provides better results (as signified by the circle signal simulation; c.f. Figure 7 third row). Most fMRI data would fall under this case. A further observation of note is the presence of the peaks in the empirical FDR plots atc=−1,1for the circle and st... | https://arxiv.org/abs/2504.13124v1 |
false discovery rate”. In: Biometrika 93.3 (2006), pp. 491–507. [8] YoavBenjaminiandDanielYekutieli.“Falsediscoveryrate–adjustedmultiplecon- fidence intervals for selected parameters”. In: Journal of the American Statistical Association 100.469 (2005), pp. 71–81. [9] Yoav Benjamini and Daniel Yekutieli. “The control of... | https://arxiv.org/abs/2504.13124v1 |
al. “The minimal preprocessing pipelines for the Human Connectome Project”. In: Neuroimage 80 (2013), pp. 105–124. [30] Javier Gonzalez-Castillo et al. “Whole-brain, time-locked activation with simple tasks revealed using massive averaging and model-free analysis”. In: Proceedings of the National Academy of Sciences 10... | https://arxiv.org/abs/2504.13124v1 |
Van Essen et al. “The Human Connectome Project: a data acquisition perspective”. In: Neuroimage 62.4 (2012), pp. 2222–2231. [51] Asaf Weinstein, William Fithian, and Yoav Benjamini. “Selection adjusted confi- dence intervals with more power to determine the sign”. In: Journal of the Amer- ican Statistical Association 1... | https://arxiv.org/abs/2504.13124v1 |
denotes the empirical FNDR. The red line denotes the joint method, the blue line the separate upper (BH), the yellow the separate lower (BH), and the green the separate lower (adaptive). 34 Figure17: FNDRsimulationresultforthecirclesignal. Therowsdifferinnoisesmooth- ing (FWHM 0, 5, and 10) and the columns differ in si... | https://arxiv.org/abs/2504.13124v1 |
How Much Weak Overlap Can Doubly Robust T-Statistics Handle? Jacob Dorn∗ April 23, 2025 Abstract In the presence of sufficiently weak overlap, it is known that no regular root-n-consistent estimators exist and standard estimators may fail to be asymptotically normal. This paper shows that a thresholded version of the s... | https://arxiv.org/abs/2504.13273v2 |
known that the semiparametric bound for estimation of ψ0is finite and can be achieved by the doubly robust Augmented Inverse Propensity Weighted (AIPW) estimator with known conditional mean outcome and propensity functions. I show that semi- parametric efficiency holds even when using nonparametric estimates of the two... | https://arxiv.org/abs/2504.13273v2 |
threshold bn=n−βe/(2βe+d)log(n)(3βe+d)/(2βe+d)suffices, regardless of the weak overlap parameter γ0. When the outcome and propensity smoothness orders are the same β >0, then thresholded AIPW can handle weak overlap of order γ0, so long as: γ0>max2β2+ 2βd+d2 β(2β+d),4β2 4β2−d2 . Under Lipschitz continuity of both nui... | https://arxiv.org/abs/2504.13273v2 |
from Crump et al. (2009). Other proposals targeting new samples include reweighting towards higher-precision populations (Yang and Ding, 2018; Li et al., 2018) or clipping strategies that Winsorize weights above (Lee et al., 2011; Ionides, 2008).1D’Amour et al. (2021) argue that weak overlap is likely to be prevalent i... | https://arxiv.org/abs/2504.13273v2 |
is new (Stone, 1982; Hall et al., 1997; Ga¨ ıffas, 2005; Mou et al., 2023). The plan of the paper is as follows. Section 2 presents the setting and main theoretical results. Section 3 interprets these results as minimal black-box consistency rates and as minimal smoothness rates. Section 4 considers implications for pa... | https://arxiv.org/abs/2504.13273v2 |
indicate max {a, b}. I define H¨ older smoothness using a multivariate version of the notation of Tsybakov (2009): a function fis in the H¨ older smoothness class Σ( β, L) if the ⌊β⌋-order multivariate derivatives Dαf=∂∥α∥ ∂xα1 1∂xα2 2...fsatisfy ∥Dαf(x)−Dαf(x′)∥ ≤L∥x−x′∥β−⌊β⌋, where I write Dαf(x) for Dαfevaluated at ... | https://arxiv.org/abs/2504.13273v2 |
is finite for all P∈P. (ii) Suppose Assumption 1 holds for some γ0∈(1,2), and there is a P∈PandC′>0 such that P(e(X)≤π)≥C′πγ0−1for all π∈(0,1]. Then the semiparametric bound is infinite for P. I will require certain rates on the nuisance functions e(X) and µ(X). I write the worst-case rates as re,n andrµ,n. Assumption ... | https://arxiv.org/abs/2504.13273v2 |
It shows that under suitable rate restrictions, the clipped AIPW estimator is first-order equivalent to an oracle clipped AIPW estimator, both estimators are consistent and asymptotically normal, and simple Wald confidence intervals are well- calibrated. A common strategy for deriving confidence intervals for unthresho... | https://arxiv.org/abs/2504.13273v2 |
assumption. Assumption 4 (Nongeneracy or faster rates) .One of the following two conditions hold: 10 (i)Nondegenerate overlap . There exists some ρ >0 such that for all P∈Pandπ∈[0,1],P(e(X)≤ π/2)≤(1−ρ)P(e(X)≤π). (ii)Faster rates .rµ,nb(γ0−1)2/γ0n ≪n−1/2. Assumption 4(i) is a uniform version of the requirement that P(e(... | https://arxiv.org/abs/2504.13273v2 |
with bn= 0 is semiparametrically efficient, and the associated Wald confidence interval ˆCn(α)satisfies lim sup n→∞sup P∈P P(ψ(P)∈ˆCn(α))−(1−α) = 0. The logic of Corollary 2 is to show that any sequence of bn→0 has a second-order effect on estimation, so that unthresholded and thresholded AIPW are first-order equivalen... | https://arxiv.org/abs/2504.13273v2 |
slower consis- tency rates. Conditional on P, larger values of bncorrespond to a smaller value of EP[D/max{e(X), bn}2], faster oracle consistency, and greater asymptotic power. Proposition 3 implies a worst-case consistency rate over distributions in P. I focus on the case of very weak overlap, because Corollary 2 show... | https://arxiv.org/abs/2504.13273v2 |
there is some η >0 such that rµ,nre,nlog(1/re,n)≪n−1/2. Then there exists a bn→0 such that clipped AIPW t-statistics are asymptotically well-calibrated. Example 3 (Shared rates, very weak overlap) .Suppose Assumption 5 holds for some γ0>1 and rµ,n, re,n≪ n−1/3. Then there exists a bn→0 such that clipped AIPW t-statisti... | https://arxiv.org/abs/2504.13273v2 |
achievable global (and pointwise) rate . Suppose Psatisfies Assumptions 6 and 7. Then there exists an estimator ˆµ(x)such that for all ϵ >0, there is a finite c(ϵ)such that lim sup n→∞sup P∈PP(∥ˆµ−µ∥∞> c(ϵ)ψn)≤ϵ. 16 Further, the estimator can be computed without knowledge of the overlap bound γ0. Recall that under stri... | https://arxiv.org/abs/2504.13273v2 |
the propensity function is infinitely-differentiable, thresholded AIPW with a Lipschitz-continuous conditional outcome mean can handle weak overlap of order γ0>2(d+1) d+2. More generally, under very weak overlap, if βµ, βe>d√ γ2 0+4γ0−4+2−γ0 4(γ0−1), then it is feasible to achieve standard inference with thresholded ... | https://arxiv.org/abs/2504.13273v2 |
3, which has P(e(X)≤π)∼πγ0−1for all πsmall enough. The bias in the thresholded region with an inconsistent outcome regression estimate is generally on the order of P(e(X)≤bn)∼bγ0−1 n. However, by Corollary 3, the oracle AIPW (and oracle IPW) standard deviation is on the order of n−1/2bγ0/2−1 n ≪bγ0−1 n. This heuristic ... | https://arxiv.org/abs/2504.13273v2 |
propensity estimate, but also often corresponds to a relatively slow consistency rate with relatively little power. Three alternative rules of thumb target faster consistency rates and laxer propensity requirements. While the main text focuses on black-box requirements on consistency rates in terms γ0, the proof goes t... | https://arxiv.org/abs/2504.13273v2 |
([0,1]) i.i.d. and setting e(X) =X1/(γ0−1). I present results for 5,000 simulations of increasingly large samples. I estimate both the propensity and outcome regressions with five-fold cross-fitting. I use shrinkage cubic splines and REML estimation, as implemented by the mgcv package in R. In this setting, Theorem 2 e... | https://arxiv.org/abs/2504.13273v2 |
this setting when the propensity score is known, and both the asymptotic normality and the clipped IPW estimator’s first-order bias are visible to the naked eye, although the clipped IPW estimator also exhibits visible skew in small samples. I test for t-statistic normality using a Shapiro-Wilk test. The test rejects n... | https://arxiv.org/abs/2504.13273v2 |
small values of e(X). As a result, when such observations are treated, a small number of observations can receive substantial leverage in outcome regression, and the predictions of E[Y|X= 0, D= 1] can be driven by a small number of observations. In Appendix B (Figures 9 through 11), I conduct the same experiments, but ... | https://arxiv.org/abs/2504.13273v2 |
There is a meaningful density of units with estimated propensities near zero, suggesting weak overlap. This pattern is similar to the findings of Crump et al., although there are slight differences, presumably due to my use of cross-fitting. I compare AIPW estimators for various trimmed subsamples to the clipped AIPW e... | https://arxiv.org/abs/2504.13273v2 |
of zero is omitted from the graph because the resulting confidence interval of [-568.8, 581.3] would make the graph difficult to read. simulation results of Section 5.1 suggest that trimmed AIPW may slightly undercover. Taken together, these results illustrate that under weak overlap, targeting the causal effect within... | https://arxiv.org/abs/2504.13273v2 |
the mortality of older americans. American Economic Review , 105(3):1067–1104. Bruns-Smith, D., Dukes, O., Feller, A., and Ogburn, E. L. (2024). Augmented balancing weights as linear regression. Callaway, B. and Sant’Anna, P. H. (2021). Difference-in-differences with multiple time periods. Journal of Econometrics , 225... | https://arxiv.org/abs/2504.13273v2 |
(2011). Weight trimming and propensity score weighting. PLoS ONE , 6(3). Lei, L., D’Amour, A., Ding, P., Feller, A., and Sekhon, J. (2021). Distribution-free assessment of population overlap in observational studies. Li, F., Morgan, K. L., and Zaslavsky, A. M. (2018). Balancing covariates via propensity score weighting... | https://arxiv.org/abs/2504.13273v2 |
standard deviation defined in Proposition 4. Next, I describe the key new results for nonparametric regression. This result shows that in nonparametric regression, if the propensity function is sufficiently smooth, then nature cannot severely concentrate treated 33 observations within a given bandwidth of any point. Th... | https://arxiv.org/abs/2504.13273v2 |
p = 2.43e−08 −4−202 −4−202 −6−4−202 −2.5 0.02.5−4−202 −202 −6−4−202 −4−202−4−202−4−202 −4−202−4−2020.00.10.20.30.4 0.00.10.20.30.4 0.00.10.20.30.40.00.10.20.30.4 0.00.10.20.30.4 0.00.10.20.30.40.00.20.40.6 0.00.30.60.9 0.000.250.500.751.000.00.20.40.60.8 0.00.51.0 0.00.51.01.5 T StatisticDensity Figure 10: Histograms o... | https://arxiv.org/abs/2504.13273v2 |
for various sample sizes, but with trimming instead of clipping. 37 C Proofs The proofs, including proofs of the claims in Appendix A, are split into sections showing asymptotic properties of oracle clipped AIPW (Appendix C.1), consistency of estimated clipped AIPW (Appendix C.2), and black-box consistency rates (Appen... | https://arxiv.org/abs/2504.13273v2 |
0P(e(X)≤t)dt≥Z1 0Ctγ0−1dt=C γ0. Lemma 3. Assume bn→0. Then for all large n, the following inequalities hold throughout P: (i)P(e(X)> πmin/2)≥πmin/2 (ii)E[e(X)/{e(X)∨bn}2]≥πmin/2 (iii)E[|ϕn−EP(n)[ϕn]|q]≤(4M)qE[e(X)/{e(X)∨bn}2]/bq−2 n (iv)E[|ϕn|q]≤(8M)qE[e(X)/{e(X)∨bn}2]/bq−2 n Proof of Lemma 3. I take these proofs one a... | https://arxiv.org/abs/2504.13273v2 |
ρ(bn, P(n)) σ(bn, P(n))3√n≤8EP(n)h D|Y−µ(X)|3 max{e(X),bn}3+|µ(X)−EP(n)[µ(X)]|3i σ(bn, P(n))3√n ≤O(Mq)EP(n)h D max{e(X),bn}2i bnσ(bn, P(n))3√n+O(Mq) σ(bn, P(n))3√n =O(Mq, σ2 min)EP(n)D max{e(X), bn}2−1/2 (b2 nn)−1/2+o(1) = o(1). 42 Proof of Proposition 4. LetP(n) be a sequence of distributions in P. By Lemma 7 and th... | https://arxiv.org/abs/2504.13273v2 |
2γ0−1C+γ0−1 2−γ0C bγ0−2 n+ 2Crγ0−1 e,nb−1 n =O bγ0−2 n 1 +δ1−γ0 n =O(bγ0−2 n), 44 so that re,nEP(n)h 1{e(X)>bn+re,n} e(X)−re,ni =o(1). Note that this bound may be lax. Whether the propensity rate requirement could be weakened is an open question for future work. Regardless, in this remaining case under the propens... | https://arxiv.org/abs/2504.13273v2 |
Asymptotic Normality and Rates Lemma 9. Suppose the conditions of Proposition 4 hold and let P(n)be a sequence of distributions in P. Then, EP(n)h D max{e(X),bn}2i =O 1 +bγ0−2 nlog(1/bn)1{γ0−2} , with a constant that only depends on Cand γ0. Proof of Lemma 9. LetP(n) be given. Then: EP(n)D max{e(X), bn}2 =EP(n)e(X... | https://arxiv.org/abs/2504.13273v2 |
holds. Lemma 11. Suppose the requirements of Theorem 1’ hold. Then by implication, rµ,nP(n)(e(X)≤bn)≪ n−1/2r EP(n)h D max{e,bn}2i . Proof of Lemma 11. rµ,nP(n)(e(X)≤bn) = rµ,nP(n)(e(X)≤bn)r EP(n)h D max{e,bn}2i s EP(n)D max{e, bn}2 ≪n−1/2s EP(n)D max{e, bn}2 . (A3’(c)) Lemma 12 (Oracle consistency) .Ifn−1/2... | https://arxiv.org/abs/2504.13273v2 |
14 hold and rµ,n→0. Let P(n)be a sequence of distributions inP. Recall the definitions of ϕnas the oracle clipped influence function and ˆϕnthe estimated influence function. Then EP(n)[ϕ2 n] =V ar P(n)(ϕn) +O(1)andP(n)nh ˆϕ2 n−ϕ2 ni =oP(n) V ar P(n)(ϕn) . Proof of Lemma 16. First, note that: EP(n)[ϕ2 n] =V ar P(n)(ϕn... | https://arxiv.org/abs/2504.13273v2 |
clip (bn) =1 nnX i=1ϕ(Zi|bn,ˆη(−k)) =X knk n1 nkX i:k(i)=kϕ(Zi|bn,ˆη(−k)) | {z } “ˆψAIPW, (k) clip(bn)”. I write ˆ rk≡σ−1 n ˜ψAIPW, (k) clip(bn)−ˆψAIPW, (k) clip(bn) . I wish to show thatP knk nˆrk=oP(n)(1). I consider an arbitrary kand quantify the bias and variance of ˆ rkgiven the data and nuisance estimates from ... | https://arxiv.org/abs/2504.13273v2 |
= 0. (Theorem 1) Proof of Corollary 2. By construction, there is a sequence of bn→0 such that re,n≪bn,re,n≪bn≪ n−1/2/rµ,n2∗min{1/γ0,γ0/(4(γ0−1))}. For such a bn, Assumption 2(b) holds by rµ,nre,n≪n−1/2andγ0>2. Further, Assumption 4(ii) holds because ( γ0−1)2/γ0>1. Recall the definition of the oracle clipped AIPW est... | https://arxiv.org/abs/2504.13273v2 |
applies, ˆσ2 n/V ar (ˆψAIPW clip (bn))→P1 and Eh ˆψAIPW clip (bn)−ψ(P)i →P∞. As a result,EP[ˆψAIPW clip (bn)−ψ(P)] ˆσn→P∞ and for any fixed α∈(0,1/2), P(ψ(P)∈ˆCn) =P ˆψAIPW clip (bn)−ψ(P) ˆσn∈[zα/2, z1−α/2]! =P ˆψAIPW clip (bn)−Eh ˆψAIPW clip (bn)i σn+oP(σn)∈ Bn+zα/2+oP(1), Bn+z1−α/2+oP(1) =POP(σn) σn+oP(σn)∈ ... | https://arxiv.org/abs/2504.13273v2 |
htends to zero from above, the left-hand side is of a lower order than the right-hand side. By continuity, h∗ n∈(0,1] and satisfies this weak inequality. Take c∗= min0≤j≤α(Mou )C1 1−γ0 ⌈α(Mou )+2⌉(h∗)d γ0−1−j. I claim that for this c∗, the the content of Lemma 20 holds. Proof by contradiction. Suppose, not, and there i... | https://arxiv.org/abs/2504.13273v2 |
∥x−x0∥∥x−x0∥ hnj∗ +o(hj∗ n). Letn≥n′imply that the o(hj∗ n) term is at most half as large as supx∈Anhj∗ ncj∗ x−x0 ∥x−x0∥ ∥x−x0∥ hnj∗ , as well as to imply that x0+hnv∈[−1,1]dif and only if there is an h >0 such that x0+hv∈[−1,1]d. Then: P(n) e(X)≥sup x∈Ane(x)|D= 1, X∈An ≥P(n) hj∗ ncj∗X−x0 ∥X−x0∥∥X−x0∥ hnj∗... | https://arxiv.org/abs/2504.13273v2 |
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