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inliers is highly limited. To the best of our knowledge, this specific area remains largely unexplored, indicating a significant opportunity for further research. One potential extension of the extreme value inlier mixture model is to address scenarios where inliers are concentrated around a specific point rather than ... | https://arxiv.org/abs/2502.19793v1 |
Journal of Hydrology , 18(3-4):257–271. Do Nascimento, F. F., Gamerman, D., and Lopes, H. F. (2012). A semiparametric bayesian approach to extreme value estimation. Statistics and Computing , 22:661–675. Fischer, S. (2018). A seasonal mixed-pot model to estimate high flood quantiles from different event types and seaso... | https://arxiv.org/abs/2502.19793v1 |
) 92.2 u 10.9861 11.3489 ( 11.4183 ) 1.1752 (8.6897, 13.4214) 1.8504 ( 4.3832 ) 0.3628 ( 0.4322 ) 94.3 ξ 0.2 0.1496 ( 0.1442 ) 0.3339 (-0.6739, 0.6399) 0.1585 ( 0.3001 ) -0.0504 ( -0.0558 ) 93.7 σ 5 5.6882 ( 6.0615 ) 2.8003 (2.2598, 13.1484) 7.7169 ( 14.235 ) 0.6882 ( 1.0615 ) 95.1 400α 0.1 0.1007 (NA) 0.015 (0.0692, 0... | https://arxiv.org/abs/2502.19793v1 |
error (MSE), bias, and convergence probability (CP). 30 Sample Size Parameters True Value Sample Mean BSE BCI MSE Bias CP 300α 0.3 0.2997 (NA) 0.0273 (0.2681, 0.3749) 7e-04 (NA) -3e-04 (NA) 96.4 η 1 1.022 ( 1.0169 ) 0.1214 (0.9976, 1.4702) 0.0098 ( 0.0096 ) 0.022 ( 0.0169 ) 97.1 β 5 4.908 ( 4.9528 ) 0.5401 (2.9843, 5.1... | https://arxiv.org/abs/2502.19793v1 |
σ 5 5.0135 ( 5.7996 ) 0.9093 (3.5399, 7.0413) 0.9672 ( 3.7267 ) 0.0135 ( 0.7996 ) 94.1 Table 10: Simulation results based on N= 2000 replications for sample sizes n= 300 ,400,500,750,and 1000, with an inlier proportion of 30% and heavy-tailed generalized Pareto distributions (GPD). The table presents the sample mean of... | https://arxiv.org/abs/2502.19793v1 |
0.0494 (0.9377, 1.1335) 0.0025 ( 0.0021 ) 0.0142 ( 0.0048 ) 95.6 β 5 4.9227 ( 4.9833 ) 0.2973 (4.1721, 5.3589) 0.1048 ( 0.1054 ) -0.0773 ( -0.0167 ) 90.3 u 10.9861 11.5441 ( 11.4225 ) 1.0475 (9.212, 13.2607) 1.5883 ( 1.5418 ) 0.558 ( 0.4363 ) 92.6 ξ -0.2 -0.2184 ( -0.2264 ) 0.1528 (-0.5135, 0.0846) 0.0249 ( 0.0223 ) -0... | https://arxiv.org/abs/2502.19793v1 |
0.6848 (7.9644, 10.6759) 0.5769 ( 4.4615 ) -0.0538 ( 1.565 ) 92.3 ξ -0.2 -0.2251 ( -0.2335 ) 0.1326 (-0.5607, -0.0453) 0.0201 ( 0.0483 ) -0.0251 ( -0.0335 ) 93.0 σ 5 5.083 ( 4.8666 ) 0.9574 (3.325, 7.0527) 1.0449 ( 2.4929 ) 0.083 ( -0.1334 ) 93.4 1000α 0.3 0.2993 (NA) 0.0153 (0.2772, 0.337) 2e-04 (NA) -7e-04 (NA) 96.9 ... | https://arxiv.org/abs/2502.19793v1 |
1.4156 (1.9517, 7.1467) 3.4103 ( 6.0285 ) 0.2854 ( 0.5703 ) 91.1 750α 0.1 0.0997 (NA) 0.0111 (0.0734, 0.1166) 1e-04 (NA) -3e-04 (NA) 97.8 η 1 1.0172 ( 1.0052 ) 0.0563 (0.9318, 1.1507) 0.0033 ( 0.0028 ) 0.0172 ( 0.0052 ) 96.0 β 5 4.9135 ( 4.9875 ) 0.338 (3.938, 5.2803) 0.1387 ( 0.1401 ) -0.0865 ( -0.0125 ) 91.9 u 10.986... | https://arxiv.org/abs/2502.19793v1 |
) 0.0129 ( 0.0089 ) 96.2 β 5 4.9415 ( 4.9773 ) 0.471 (3.8489, 5.6909) 0.2583 ( 0.2703 ) -0.0585 ( -0.0227 ) 92.7 u 9.7295 9.57 ( 11.3408 ) 0.7616 (7.8512, 10.8358) 0.6861 ( 6.6199 ) -0.1595 ( 1.6112 ) 94.1 ξ 0 -0.0398 ( -0.0706 ) 0.1897 (-0.445, 0.3035) 0.0388 ( 0.1425 ) -0.0398 ( -0.0706 ) 95.1 σ 5 5.1865 ( 5.6851 ) 1... | https://arxiv.org/abs/2502.19793v1 |
Stein’s unbiased risk estimate and Hyv¨ arinen’s score matching Sulagna Ghosh Nikolaos Ignatiadis sulagnag@uchicago.edu ignat@uchicago.edu Frederic Koehler Amber Lee fkoehler@uchicago.edu amberlee0516@uchicago.edu Draft Manuscript, February 2025 Abstract We study two G-modeling strategies for estimating the signal dist... | https://arxiv.org/abs/2502.20123v1 |
distributions that may (or may not) contain G⋆. Finally, we estimate µivia ˆµi=EbG[µi|Zi]. The connection to Hyv¨ arinen’s score matching is as follows. Via the Eddington/Tweedie formula [Dyson, 1926, Efron, 2011], the Bayes denoiser may be expressed in terms of the score sG(z), that is, the derivative of the log-margi... | https://arxiv.org/abs/2502.20123v1 |
The key ingredients of the proof include showing that when the Fisher divergence between fG⋆andfGis small, their score derivatives are also close (Theorem 8). This latter result builds on both an induction argument from Jiang and Zhang [2009] (presented in Lemma 9) and powerful results on functional log-Sobolev inequal... | https://arxiv.org/abs/2502.20123v1 |
that we introduce in (7) of Section 2. The focus therein is on arguing consistency (that is, that the empirical Bayes regret converges to zero), however, the proof technique used can only establish a rate of O(n−1/2) (up to log factors), rather than the rate O(n−1) in (6) (formally stated in Theorem 7). Further importa... | https://arxiv.org/abs/2502.20123v1 |
understanding finite-sample properties of Hyv¨ arinen’s score matching. For instance, as far as we know, the only result on minimax rate optimality of Hyv¨ arinen’s score matching in a nonparametric setting is available for certain infinite dimensional kernel exponential families studied by Sriperumbudur et al. [2017].... | https://arxiv.org/abs/2502.20123v1 |
(equivalently, SURE), not maximum likelihood. This message (in the case of denoising) was also emphasized by Hyv¨ arinen [2008] and Xie et al. [2012]. 2 The general setting We start by extending the scope of (1). We allow for side information encoded via covariates Xitaking values in a generic space X, and heteroscedas... | https://arxiv.org/abs/2502.20123v1 |
Xiincludes σ2 i, we see that σ2 iinfluences the marginal density in two ways: first, it determines the noise level (variance, σ2 i) of the Gaussian kernel used in the convolution, and second, it may influence the distribution of µivia the conditional distribution G(· |Xi). Other covariates in Xionly influence the margi... | https://arxiv.org/abs/2502.20123v1 |
present the arguments underlying SURE and Score Matching in a unified way. In each case we start with the risk of interest, e.g., the MSE, E[{µi−(Zi+σ2 is(Wi))}2], for SURE and Fisher divergence, E[{s⋆(Wi)− s(Wi)}2], for SM. Moreover, in each case we subtract a constant term that does not depend ons(·), namely E[(µi−Zi... | https://arxiv.org/abs/2502.20123v1 |
(An alternative expression for SURE) .A second order generalization of the Eddington/Tweedie formula yields Var G[µi|Wi] =σ2 i+σ2 i∂ ∂zs(Wi), see e.g., Efron [2011, Equation (2.8)]. Using this formula allows us to rewrite SURE in (15) as: SURE( G) =1 nnX i=1n −σ2 i+ (Zi−EG[µi|Wi])2+ 2 Var G[µi|Wi]o . The interpretation... | https://arxiv.org/abs/2502.20123v1 |
:G∈ G) , as long as the following centered class has low Rademacher complexity: M◦:= hG(·) :hG(w, µ) =−σ2(z−µ) (sG◦(w)−sG(w)) +σ4∂ ∂z{sG◦(w)−sG(w)}, G∈ G . Proposition 4 (Compound risk control) .In the above setting, it holds that, Eµ" 1 nnX i=1 Zi+σ2 isbG(Wi)−µi 2−1 nnX i=1 Zi+σ2 isG◦(Wi)−µi 2# ≤4Rµ n(M◦). 10 In a... | https://arxiv.org/abs/2502.20123v1 |
however, their result only applies to a finite collection of linear smoothers. 11 and define the corresponding greatest (post)fixed point7by ˆr:= sup {r≥0 :r2≤cW(r)}. Then 1 nnX i=1σ4 i sG⋆(Wi)−sbG(Wi) 2≤ˆr2. To turn Lemma 6 into sharp convergence guarantees, we require an implication of the following form: 1 nnX i=1σ... | https://arxiv.org/abs/2502.20123v1 |
on the regret. The NPMLE is free of any tuning parameters. Moreover, computing bGNPMLEis computationally streamlined with the (by now) standard proposal of Koenker and Mizera [2014]: consider a discretization of priors Gthat may be represented as G=PK j=1πjδuj where δujdenotes a Dirac point mass at uj,u1, . . . , u Kis... | https://arxiv.org/abs/2502.20123v1 |
be polynomial inn−1. Proving Theorem 8 requires powerful machinery. To state this machinery and its impli- cations, it will be convenient to consider in addition to the Fisher divergence F(fG⋆||fG) between fG⋆andfG, defined in (5), their Kullback-Leibler divergence as well as their squared Hellinger distance: KL (fG⋆||... | https://arxiv.org/abs/2502.20123v1 |
of Xi. Variants of model (20) appear throughout the literature; see Section 4.1 below for several examples. If we marginalize over µiin (20), we recover the following standard heteroscedastic regression model: Zi|Xi∼N m⋆(Xi), A⋆(Xi) +σ2 i , i = 1, . . . , n. (21) The posterior mean in model (20) is, EG⋆[µ|Z=z, x=x] =... | https://arxiv.org/abs/2502.20123v1 |
Ziand assume that for K > 0, Zi=µi+ξi,E[ξi] = 0,Var [ξi] =σ2 i,E[exp( tξi)]≤exp K2σ2 it2/2 for all t,(29) that is ξ:=Zi−µiisKσi-sub-Gaussian. The reason we can relax Gaussianity is that as noted e.g., by Kou and Yang [2017], Ignatiadis and Wager [2019], the conditionally linear nature of the shrinkage rules in (26) i... | https://arxiv.org/abs/2502.20123v1 |
inequalities for nonconvex classes (for example, in the nonconvex case the oracle estimate may not be uniquely defined, see e.g., Lee et al. [1996]). 4.1 Examples We consider a few different settings to illustrate the result of Theorem 13. Pure regression setting. Suppose we take L:={1}, where our notation here identif... | https://arxiv.org/abs/2502.20123v1 |
µi=Zi; (ii.) lim supn→∞n−1Pn i=1E (µi−ˆµGL i)2 ≤infλ,b E {µi−b(σ2 i)−(1−λ(σ2 i))Zi}2 if σ2 i7→E[µi|σ2 i] and σ2 i7→Var[µi|σ2 i] are uniformly continuous (and further technical conditions); (iii.) n−1Pn i=1E (µi−ˆµGL i)2|σ2 i =O(n−2/3) when Var[ µi|σ2 i] = 0 and σ2 i7→E µi|σ2 i isL-Lipschitz. Our SURE-tuning ap... | https://arxiv.org/abs/2502.20123v1 |
i=1n λ◦(Xi)(λ◦(Xi)−ˆλ(Xi))o (ξ2 i−σ2 i). The challenge associated with turning the inequality of Proposition 14 into the fast rates of Theorem 13 is conceptually related to our discussion following equation (19). Our argument would be relatively standard, if we could show an implication of the following form: 1 nnX i=1... | https://arxiv.org/abs/2502.20123v1 |
the conditional distribution G(· |Xi). This second method is called SURE-THING (T his H elps I n Neural G -modeling). All computations are done using PyTorch. SURE-PM. Consider first the class of K-atomic distributions for SURE-PM: GPM:= G:G=KX j=1πjδuj, u∈RK, π∈∆K−1 , where ∆K−1denotes the probability simplex.... | https://arxiv.org/abs/2502.20123v1 |
. , n , and returns estimates ˆ µ(b) i,i= 1, . . . , n . Our reported metric (for that estima- tor) is: [MSE :=1 BBX b=11 nnX i=1 µ(b) i−ˆµ(b) i2 . 22 Table 1: In-sample MSE of the estimators over 50 simulations in the homoscedastic problem without side-information and normal prior as in (33). σ2 ⋆ 0.1 1 5 SURE-PM 0.... | https://arxiv.org/abs/2502.20123v1 |
by Zhao [2021, Section 6.1], also see our discussion in Section 1.1 on related work. The main difference in implementation lies in the optimization method we use for computing bG.12Moreover, our simulation is accompanied by the sharp rate in Theorem 7, while Zhao [2021] does not provide a theoretical analysis of this e... | https://arxiv.org/abs/2502.20123v1 |
though in the DGPs µistrongly depends on σ2 iand so G⋆/∈ G. By contrast, SURE-THING is well-specified in all settings except the uniform likelihood setting. 25 The results of the simulation are shown in Figure 1. We summarize some key observa- tions: SURE-THING outperforms all other estimators (beyond, of course, the o... | https://arxiv.org/abs/2502.20123v1 |
places substantial mass therein. How is it possible that the in-sample MSE of SURE-PM is so much smaller than of the NPMLE? One explanation is provided by Figure 3. We first focus on its first column which pertains to the low variance component ( σ2 i= 0.1). The bottom row plots the marginal density of the low variance... | https://arxiv.org/abs/2502.20123v1 |
Data Science Institute cluster. References A. Abadie and M. Kasy. Choosing among regularized estimators in empirical economics: The risk of machine learning. The Review of Economics and Statistics , 101(5):743–762, 2019. M. Arbel and A. Gretton. Kernel conditional exponential family. In A. Storkey and F. Perez- Cruz, e... | https://arxiv.org/abs/2502.20123v1 |
(FOCS) , pages 684–695. IEEE, 2022. N. Cohen, E. Greenshtein, and Y. Ritov. Empirical Bayes in the presence of explanatory variables. Statistica Sinica , pages 333–357, 2013. 29 D. D. Cox. A penalty method for nonparametric estimation of the logarithmic derivative of a density function. Annals of the Institute of Stati... | https://arxiv.org/abs/2502.20123v1 |
, volume 32, 2019. N. Ignatiadis, S. Saha, D. L. Sun, and O. Muralidharan. Empirical Bayes mean estimation with nonparametric errors via order statistic regression on replicated data. Journal of the American Statistical Association , 118(542):987–999, 2023. G. M. James, P. Radchenko, and B. Rava. Irrational exuberance:... | https://arxiv.org/abs/2502.20123v1 |
the Ninth Annual Conference on Computational Learning Theory , pages 140–146, 1996. J. Leiner, B. Duan, L. Wasserman, and A. Ramdas. Data fission: Splitting a single data point. Journal of the American Statistical Association , pages 1–12, 2023. J. Li, S. S. Gupta, and F. Liese. Convergence rates of empirical Bayes est... | https://arxiv.org/abs/2502.20123v1 |
approach to statistics. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics , pages 157–163. The Regents of the University of California, 1956. E. T. Rosenman, G. Basse, A. B. Owen, and M. Baiocchi. Combining observational and ex... | https://arxiv.org/abs/2502.20123v1 |
Media, 2006. 34 R. Vershynin. High-dimensional probability: An introduction with applications in data sci- ence, volume 47. Cambridge university press, 2018. P. Vincent. A connection between score matching and denoising autoencoders. Neural Computation , 23(7):1661–1674, 2011. M. Wainwright. High-Dimensional Statistics... | https://arxiv.org/abs/2502.20123v1 |
and σq=∥X∥Lq<∞for some q≥2. Then |E[X1(|X|< s|)]≤2s(σq/s)q. Lemma 18 (Section 2.5 and 2.7 of Vershynin [2018]) .IfXis a mean-zero random variable which is 1-subgaussian, then ∥X∥Lq≲√q. If instead Xis 1-subexponential, then ∥X∥Lq≲ q. Lemma 19. Suppose that Z= sup a∈A nX i=1aiξi where ξiis independent, mean zero, 1-sub-G... | https://arxiv.org/abs/2502.20123v1 |
take expectation overXin EBRegret( G⋆,bG), then given fixed Xi’s, E[hG(W)] =1 nnX i=1E[hG(Zi, Xi)] =1 nnX i=1E[hG(Wi)]. Then, using (38) and Lemma 21, we have: Eh EBRegret( G⋆,bG)i =E σ4sbG(W)2+ 2σ4∂ ∂zsbG(W) −E σ4sG⋆(W)2+ 2σ4∂ ∂zsG⋆(W) ≤2E" sup G∈G E[hG(W)]−1 nnX i=1hG(Wi) # ≤4E" sup G∈G 1 nnX i=1εihG(Wi) # = 4Rn... | https://arxiv.org/abs/2502.20123v1 |
max {a, b}fora, b≥0 and h′(z) =∂h(z)/∂zdenotes the derivative of h(·) with respect to z. Notice that when fG(z)≥ρ, then sρ G(z) =sG(z) (and in particular, for ρ= 0,sρ G(z) =sG(z) for all z). We will need the following function: ˜L(ρ) :=p −log(2πρ2). (42) Now let us give some properties of the regularized score and also... | https://arxiv.org/abs/2502.20123v1 |
η < ρ and˜L2(η)≥˜L2(ρ)≥2, since ˜L2(y) is decreasing in y2, where ˜L(y) =p −log (2 πy2). 42 Now, if fG(x)≥ρ, where 0 < ρ≤(2πe2)−1 2, we have ˜L2(ρ)≥˜L2(fG(x)) and ˜L2(ρ)≥˜L2 (2πe2)−1 2 = 2, since ˜L2(y) is decreasing in y2. From Lemma A.1 in Jiang and Zhang [2009], 0≤f′′ G(x) fG(x)+ 1≤˜L2(fG(x)), =⇒ − 1≤f′′ G(x) fG(x... | https://arxiv.org/abs/2502.20123v1 |
define, similar to Jiang and Zhang [2009], Gmwith at most N≤D·log(1 ερ) points supported on [ −M, M ],Gm,ηwith at most Npoints supported on 0,±η2,±2η2, . . ., which are at most at a distance of η2from any point under Gm, and finally ˜Gm,ηsupported on 0 ,±η2,±2η2, . . .with at most Npoints and the weights coming from an... | https://arxiv.org/abs/2502.20123v1 |
the first claim. Using the consequence of the Cauchy-Schwarz inequality that ( a−b)2= (a−c+c−b)2≤2(a−c)2+ 2(c−b)2, we have that Z f′ G⋆(z)−f′ G(z) 2 fG⋆(z)∨ρ+fG(z)∨ρdz =Zf′ G⋆(z) fG⋆(z)−f′ G(z) fG⋆(z)2fG⋆(z)2 fG⋆(z)∨ρ+fG(z)∨ρdz ≤2Zf′ G⋆(z) fG⋆(z)−f′ G(z) fG(z)2fG⋆(z)2 fG⋆(z)∨ρ+fG(z)∨ρdz + 2Zf′ G(z) fG(z)−f′ G(z) ... | https://arxiv.org/abs/2502.20123v1 |
the symmetric bound for f⋆. C.2.3 Proof of Proposition 11 As mentioned in the main text, the argument for this proposition appears in Koehler et al. [2023] (see there for further related references). Koehler et al. [2023] states the result for the Kullback-Leibler divergence, however, the first inequality between squar... | https://arxiv.org/abs/2502.20123v1 |
in (48) (which depends on C, M > 0). Then it holds that: Rn(δ; Star( Sρ∗ c))≲C,Mδ√n(log(1 /δ) + log( n)) +√logn n(log(1 /δ) + log( n))2.(52) 4. Furthermore, there exists a constant C′=C′(C, M )>0 such that δ=C′log3/2n√n, satisfies Rn(δ; Star( Sρ∗ c))≤δ2 2˜L(ρ∗). 53 Proof. Part 1: This result follows from Lemma 22. Part... | https://arxiv.org/abs/2502.20123v1 |
the complexity of the following class. Mρ c= hG(·) =sρ ⋆(·){sρ ⋆(·)−sρ G(·)}+∂ ∂z{sρ ⋆(·)−sρ G(·)}:G∈ P(M) . (53) Lemma 27 (Complexity of centered regularized noise process class) .We have the following results for the centered noise process class and its star hull. 1. For 0 < ρ < (2πe3)−1/2, we have that sup h∈Mρ c∥... | https://arxiv.org/abs/2502.20123v1 |
[2019, Theorem 14.1], which we apply for the class Star( Sρ∗ c) studied in Lemma 26. In particular, by the former lemma, we may take t=δ, which we derive probability bounds on A′ n. Theorem 14.1 of Wainwright [2019] then yields that on the event A′ nthe following holds for any h∈Star(Sρ∗ c): E h(Zi)2 −bE h2 ≤1 2E ... | https://arxiv.org/abs/2502.20123v1 |
G) −E sρ∗ ⋆n sρ∗ G−sρ∗ ⋆o +∂ ∂z(sρ∗ ⋆−sρ∗ G) by Zr:= sup G∈P(M):Eh (sρ∗ ⋆{sρ∗ G−sρ∗ ⋆}+∂ ∂z(sρ∗ ⋆−sρ∗ G))2i ≲(logn)4r2( bE sρ∗ ⋆n sρ∗ G−sρ∗ ⋆o +∂ ∂z(sρ∗ ⋆−sρ∗ G) −E sρ∗ ⋆n sρ∗ G−sρ∗ ⋆o +∂ ∂z(sρ∗ ⋆−sρ∗ G)) . By Theorem 16 (and the boundedness statement from Lemma 22), we have Zr≲E[Zr] +rlog2nr log(2/τ) n+˜L(ρ∗) ... | https://arxiv.org/abs/2502.20123v1 |
L with λ◦∈Rnandb◦(·)∈ Bwith b◦∈Rn. 14Where, as explained at the beginning of the proof, we combine the inequalities (ˆ r)2≤cW(ˆr)≤cW(ri)≤ r2 i+1to inductively prove that ˆ r≤ristarting from the base case ˆ r≤r0. 60 •We often identify Lwith its projection onto Rn(that is, we interpret Las a subset of Rn) and analogously... | https://arxiv.org/abs/2502.20123v1 |
(ci+fiξi)2 −(ci+fiξi)2 . 62 We observe that 1 nX i∈A(ci+fiξi)2=1 nX i∈AE (ci+fiξi)2 +1 nX i∈A (ci+fiξi)2−E (ci+fiξi)2 ≥1 nX i∈AE (ci+fiξi)2 −1 nZ =1 nX i∈AE c2 i+ 2cifiξi+f2 iξ2 i −1 nZ ≥1 2nX i∈A(c2 i+f2 i)−1 nZ, where the last step is by the assumptions and the AM-GM inequality. Observe that, using the fact... | https://arxiv.org/abs/2502.20123v1 |
the diagonal matrix D with Dii=Kσ2 i, G1(r) :=E" sup b:∥b−b◦∥2≤r1 nnX i=1λ◦ i(b◦ i−bi)ζi# ,where ζ∼N(0, D), G2(r) :=E" sup λ:∥λ−λ◦∥2≤r1 nnX i=1(2λ◦ iµi−b◦ i)(λ◦ i−λi)ζi# ,where ζ∼N(0, D), T3(r) :=E" sup λ:∥λ−λ◦∥2≤r1 nnX i=1λ◦ i(λ◦ i−λi)(ξ2 i−σ2 i)# and Rn(r1, r2) :=E sup (b,λ)∈Kr1,r2 1 nnX i=1εi((λi−λ◦ i)µi−(bi−b◦ i)) ... | https://arxiv.org/abs/2502.20123v1 |
(b,λ)∈Kr1,r2 1 nnX i=1εi((λi−λ◦ i)µi−(bi−b◦ i) +σmaxp log(4n/δ) sup (b,λ)∈Kr1,r2 1 nnX i=1εi(λ◦ i−λi) # . 66 Definition and lower bound on H.Define H(r1, r2) :=Z1(r1) +Z2(r2) +Z3(r2) + 52Kσmaxp log(4n/δ)Rn(r1, r2) + 26K2σ2 maxr2p log(4n/δ)r log(2/δ) n+ 360K2σ2 maxlog(4n/δ) log(2 /δ) n. Then we have shown that H(r1, r2)... | https://arxiv.org/abs/2502.20123v1 |
arXiv:2502.20206v7 [math.ST] 30 Apr 2025ON THE GLIVENKO-CANTELLI THEOREM FOR REAL-VALUED EMPIRICA L FUNCTIONS OF STATIONARY α-MIXING AND β-MIXING SEQUENCES OUSMANE COULIBALY1AND HAROUNA SANGAR ´E2 A/b.sc/s.sc/t.sc/r.sc/a.sc/c.sc/t.sc. In this paper we extend the classical Glivenko-Cantelli theor em to real-valued empir... | https://arxiv.org/abs/2502.20206v7 |
Such a result, also known as the fundamental theorem of stati stics, is the frequentist paradigm (as opposed to the Bayesian paradigm) in statistic s. In the form of ( 1.3), the Glivenko-Cantelli law has gone through a large number of stu dies for a variety of types GLIVENKO-CANTELLI CLASSES 3 of dependence. It has als... | https://arxiv.org/abs/2502.20206v7 |
give a brief introduction to the no tions of α-mixing and β- mixing. But before that, section 2is devoted to entropy numbers and Vapnik- ˇCervonenkis classes. In section 4, we give our GC class regarding real-valued empirical funct ions for 4 OUSMANE COULIBALY1AND HAROUNA SANGAR ´E2 arbitrary stationary rv’s sequences ... | https://arxiv.org/abs/2502.20206v7 |
by Sangar´ e and Lo (2015 ). Theorem 1. (Sangar´ e and Lo (2015 )). LetX1,X2,...be an arbitrary sequence of rv’s, and let(fi,n)i≥1be a sequence of measurable functions such that Var[fi,n(Xi)]<+∞, fori≥1 andn≥1. If for some δ,0< δ <3 (3.1) C1= sup n≥1sup q≥1Var/parenleftBigg 1 q(3−δ)/4q/summationdisplay i=1fi,n(Xi)/pare... | https://arxiv.org/abs/2502.20206v7 |
we are also going to provide results for sequences veri fyingα-mixing and β-mixing conditions, we make a brief summary of these notions. 3.3.α-mixing. Let us have a brief recall on φ-mixing. Let ( Ω,A,P) be a probability space andA1,A2two sub σ−algebras of A. Theα-mixing coefficient is given by α(A1,A2) = sup{|P(A)P(B)−P... | https://arxiv.org/abs/2502.20206v7 |
defined on a probability space (Ω,A,P). We have sup x∈R|Fn(x)−F(x)| →0,almost surely, as n→+∞ whenever the following general conditions hold : for some δ∈]0;3[ (4.1) sup q≥11 q(3−δ)/2 q/summationdisplay i=1Var/parenleftbig I{Xi≤x}/parenrightbig +q/summationdisplay (i/ne}ationslash=j)=1Cov/parenleftbig I{Xi≤x},I{Xj≤x}/... | https://arxiv.org/abs/2502.20206v7 |
≤2β(|t−s|)/bardblXt/bardbl∞/bardblXs/bardbl∞, where/bardblXt/bardbl∞= supω∈Ω|Xt(ω)|is the supremum norm of Xt. Proof. The covariance of XtandXsis given by Cov(Xt,Xs) =E[XtXs]−E[Xt]E[Xs]. We focus on the first term, E[XtXs]. By introducing the σ-algebra Ft=σ(Xℓ:ℓ≤t), we can write E[XtXs] =E[XtE[Xs|Ft]]. Thus, the covaria... | https://arxiv.org/abs/2502.20206v7 |
–β-Mixing Condition •Result: Studied β-mixing sequences and provided invariance principles for e mpir- ical processes. •Assumption: Required that β-mixing coefficients decay at a rate satisfying:/summationtext∞ n=1β(n)< +∞. •Limitation: This assumption is very strong, as many real-wo rld processes (e.g., financial time se... | https://arxiv.org/abs/2502.20206v7 |
of Theoretical Statistics . 60, pp. 41-62. http://dx.doi.org/10.17654/TS060020041 . Shao, Q. M. (1995). Weak convergence of multidimensional em pirical processes for strong mixing sequences. Chinese Ann. Math. Ser. A 7, 547-552. Shao, Q. M. and Yu, H. (1996). Weak convergence for weighted e mpirical processes of depend... | https://arxiv.org/abs/2502.20206v7 |
Linear type conditional specifications for multivariate count variables Yang Lu1Wei Sun2 Abstract : This paper investigates conditional specifications for multivariate count vari- ables. Recently, the spatial count data literature has proposed several conditional models such that the conditional expectations are linear... | https://arxiv.org/abs/2502.20227v1 |
higher dimensions. In this paper, we fill the gap by studying alternative conditional specifications for multivari- ate count distributions, such that the conditional expectations E[Xj|X1,...,Xj−1,Xj+1,...,Xn], j= 1,...,n , are all linear in the conditioning variables. A starting, toy example is the following condition... | https://arxiv.org/abs/2502.20227v1 |
model to the n−dimensional case and study its solutions. Since the conditional expectations do not completely characterize the distribution, we cannot work out all the solutions to this latter specification. Nevertheless, we show that there are many more models that satisfy the linear conditional expectation assumption... | https://arxiv.org/abs/2502.20227v1 |
conditional distributions of X|YandY|Xare of the form (3) and αand βare related to the Poisson parameters given by: α=λ0 λ0+λ2, β =λ0 λ0+λ1. (9) Note that (9) is compatible with (7). We end this section by an important model that is the solution to the compound autore- gressive model (3). Example 1 (Poisson-gamma conju... | https://arxiv.org/abs/2502.20227v1 |
following two extra constraints: θ4[θ1+θ3(θ1−θ2)] =θ2[θ3+θ1(θ3−θ4)], (13) and (θ1−θ2)(θ3−θ4)<1. 7 In this case, XandYboth have NB marginals with pgf’s: E[uX] =1 /bracketleftbigg 1 +θ1+θ3(θ1−θ2) 1−(θ1−θ2)(θ3−θ4)(1−u)/bracketrightbiggδ,E[vY] =1 /bracketleftbigg 1 +θ3+θ1(θ3−θ4) 1−(θ1−θ2)(θ3−θ4)(1−v)/bracketrightbiggδ, (14... | https://arxiv.org/abs/2502.20227v1 |
Poisson distribution with parameter β4U2. Finally, we assume that the marginal distribution of U1is with shape parameter δand rate parameterβ0<β 1. By the Bayes’ formula, this implies that X|U1is Poisson with parameter (β1−β0)U1. Thus, the conditional distributions alternate between the pair of conditional distribution... | https://arxiv.org/abs/2502.20227v1 |
6ishould not be constant variables such as zero. 11 Proof. We chooseZ= 0in eqs.(18) and (19), and take marginal expectations on all sides. We get: E[uX|Y,Z = 0] =/parenleftbigg E[uW11]/parenrightbiggY E[uϵ1], E[vY|X,Z = 0] =/parenleftbigg E[vW31]/parenrightbiggX E[vϵ2]. Then we can apply Theorem 2 to the joint distribu... | https://arxiv.org/abs/2502.20227v1 |
beta distribution with parameters (α,β 1)and(α,β 2), respectively, and ϵ(resp.,η) follows NB distributions with number of successes parameter β2 (resp.,β1) and common probability parameter θbetween 0 and 1. 13 This compatible model has the stochastic (trivariate NB reduction) representation: X=(d)Z+ϵ, Y (d)=Z+η, whereZ... | https://arxiv.org/abs/2502.20227v1 |
of jointly mix distribution is, for instance, the multinomial distribution. This example, however, involves a multivariate distribution whose support is not Nn 0. More- over, the regression coefficients are all negative and equal to −1. 15 Example 5. Let us assume that (Z1,Z2,Z3)follows the three-dimensional multinomia... | https://arxiv.org/abs/2502.20227v1 |
)satisfying (5)? This remains an open problem but in the remainder of this subsection, we will provide some partial answers. First, most of the bivariate distributions that we have examined in section 3 and 4 that solve the compound autoregressive or the random coefficient model involve some sort of common factor, whic... | https://arxiv.org/abs/2502.20227v1 |
(33)-(35) allows for a solution is relegated to the Appendix. Whether or not Theorem 7 can be extended to include the case where either a > 1or c >1remains an open problem. The main difficulty is that, if say a≥1, then any solution to (5) must satisfy the condition that P(X=N) = 0 . Indeed, P(X=N)>0implies E[Y|X=n] =an... | https://arxiv.org/abs/2502.20227v1 |
implications for the statistical modeling of count data, espe- cially for the recent literature on spatial count data (Glaser et al., 2022; Karlis et al., 2024): i)Many of the recently proposed conditional Poisson model with linear conditional expec- tation, or the integer autoregressive model, may not be compatible pe... | https://arxiv.org/abs/2502.20227v1 |
of conjugate priors for discrete exponential families. Statistica Sinica , 11(2):409–418. Chutoo, A., Karlis, D., Khan, N. M., and Jowaheer, V. (2021). The unilateral spatial autogres- sive process for the regular lattice two-dimensional spatial discrete data. SORT-Statistics and Operations Research Transactions , 45(1... | https://arxiv.org/abs/2502.20227v1 |
Chutoo, A., Mamode Khan, N., and Jowaheer, V. (2024). The multilateral spatial integer-valued process of order 1. Statistica Neerlandica , 78(1):4–24. Lee, L.-F. (2023). Spatial econometrics: Spatial autoregressive models , volume 1. World Scientific. Lu, Y. (2019). The predictive distributions of thinning-based count ... | https://arxiv.org/abs/2502.20227v1 |
0. Similarly, we can change the roles of uandvin eq.(42) and get: ∞/summationdisplay n=11 n!/bracketleftbiggdnD1 dun(c(v))−dnD1 dun(0)/bracketrightbigg un=∞/summationdisplay n=11 n!/bracketleftbiggdnD2 dun(v)−dnD2 dun(0)/bracketrightbigg a(u)n. 26 Similar arguments leads to: γ2/bracketleftbigg β1w2+β2/bracketrightbigg ... | https://arxiv.org/abs/2502.20227v1 |
g(x)=A(1,x) G(1,x)=C=B(1,y) H(1,y)=/parenleftigy i/parenrightig E[p(1−p)y−1] E[(1−p)y]h(y−1) h(y),∀x,y, (51) whereCis a constant. Takingx= 2,y≥2in eq.(47), we get: A(0,2)H(0,y)+A(1,2)H(1,y)+A(2,2)H(2,y) =G(0,2)B(0,y)+G(1,2)B(1,y)+G(2,2)B(2,y). But the first terms on the LHS and RHS are equal, and the second terms on ... | https://arxiv.org/abs/2502.20227v1 |
θ 3=b δ, θ 4=b δ−a. We have: θ2>−1⇔d δ+ 1>c;θ4>−1⇔b δ+ 1>a, and (13) holds⇔/parenleftiggb δ−a/parenrightigg/parenleftiggd δ+b δ·c/parenrightigg =/parenleftiggd δ−c/parenrightigg/parenleftiggb δ+d δ·a/parenrightigg ⇔(b−aδ)(d+bc) = (d−cδ)(b+ad) ⇔[ad(1−c) +bc(a−1)]δ=b2c−ad2. (i) Suppose that a≥1,ac< 1, i.e.,a>1,ac... | https://arxiv.org/abs/2502.20227v1 |
V[ϵ2] =V[Y]−a21Cov(X,Y )−a23Cov(Y,Z), V[ϵ3] =V[Z]−a31Cov(X,Z)−a32Cov(Y,Z). Hence, in matrix form, A:= 1−a12−a13 −a21 1−a23 −a31−a32 1 =DC−1, 35 where C= V[X] Cov(X,Y ) Cov(X,Z) Cov(X,Y )V[Y] Cov(Y,Z) Cov(X,Z) Cov(Y,Z)V[Z] is the covariance matrix of (X,Y,Z )andDis the diagonal matrix with diagonal ... | https://arxiv.org/abs/2502.20227v1 |
arXiv:2502.20368v1 [math.ST] 27 Feb 2025Minimax rate for learning kernels in operators Sichong Zhang1, Xiong Wang2, and Fei Lu2 1Department of Applied Mathematics and Statistics, Johns Ho pkins University, Baltimore, USA. 2Department of Mathematics, Johns Hopkins University, Balt imore, USA. Learning kernels in operato... | https://arxiv.org/abs/2502.20368v1 |
Probability of cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Minimax lower rate 18 4.1 The reduction scheme and innovations . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 A bound for total variation distance . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.3 Proofs of th... | https://arxiv.org/abs/2502.20368v1 |
details. The problem of recovering the kernel φfrom data is at the intersection of statistical learning and inverse problems. In essence, it is a deconvolution from multiple function-valued input- output data pairs. The deconvolution renders it a severely i ll-posed inverse problem, while the randomness of the data end... | https://arxiv.org/abs/2502.20368v1 |
these studies employ different settings and method s, their minimax rates coincide because, in each case, the inverse problem in the large sampl e limit involves a compact normal operator (see Section 1.2for detailed comparisons). In contrast, when the spectral d ecay is exponential, to the best of our knowledge, our wo... | https://arxiv.org/abs/2502.20368v1 |
form Yipxq “αpxq `ş φpx,yqXipyqdy`εipxq. However, we note that predictor error is for forward estimation, which contrasts with the inverse problem of learning kernels in operators. Inverse problems and their statistical variants. Classical ill-posed inverse problem solves φin the model Aφpxiq `εi“fpxiqfrom discrete dat... | https://arxiv.org/abs/2502.20368v1 |
Table 1: Comparison of well-posed and ill-posed statistica l learning problems: the normal oper- ator in the large sample limit, the Sobolev spaces, the domin ating order of the bias and variance terms. Here, Hβ“ tφpxq “ř8 k“1ck? 2sinpnxq:ř8 k“1c2 kk2βă 8,xP r0,πsuis the classical spectral Sobolev space of the operator... | https://arxiv.org/abs/2502.20368v1 |
2}v}2, (2.4) for a constant τą0that is unform for all NandvPRN. 6 Condition pB1qis used for the minimax upper rate, and Condition pB2qis used for the minimax lower rate. The noise can be either Gaussian or non-Gaussian, and the spa ceYcan be either finite or infinite-dimensional. When Yis finite-dimensional, the linear ma... | https://arxiv.org/abs/2502.20368v1 |
X|gruspx,sq|2νpdxq . (2.6) With the above exploration measure ρ, the forward operator of Model ( 1.1) defined a square- integrable Y-valued random variable when the volume of Sis finite. That is, for any φPL2 ρpSq, we have by Cauchy-Schwartz inequality, E“ }Rφrus}2 Y‰ “E«ż Xˆż Sφpsqgruspx,sqds˙2 νpdxqff ďE„ż Xż Sφ2psqg2r... | https://arxiv.org/abs/2502.20368v1 |
“ tfpmqPL2pr0,1sq:ż1 0fptqdt“0, fpkqp0q “fpkqp1q,1ďkďmu, andGis the Green’s function for the problemdm dsmφ“ψinW0 m,per; see e.g., [Wah90 , Section 2.1] . 9 Example 2.6 Consider the nonlocal operator with radial interaction kernel: Rφruspxq “ż |y|ďδφp|y|qrupx`yq ´upxqsdy“ż r0,δsφpsqrupx`sq `upx´sq ´2upxqsds(2.8) forxPX... | https://arxiv.org/abs/2502.20368v1 |
gression [ GKKW06 ,CS02,Tsy08 ]. Classical spaces provide a universal quantification of the s moothness independent of the model and measure ρ, making them suitable for problems for classical regressio n problems that estimate fpxqin the model Y“RfpXq`εwithRfpxq “fpxqfrom data tpXm,Ymqu. For these problems, the normal o... | https://arxiv.org/abs/2502.20368v1 |
ill-conditioned with the smallest eigenvalue oscillating near λn, the eigenvalue of An,8, as shown in Lemma 3.2below. In contrast, the tLSE is stable by using the LSE only when the eigenvalues of An,Maren’t too small, and it is zero otherwise. The next lemma shows that in the large sample limit, the tamed LSE recovers ... | https://arxiv.org/abs/2502.20368v1 |
term is bounded by Opn1`2r Mqfor polynomial spectral decay and Opern Mqfor exponential spectral decay, paralleling the Opn Mqrate in classical regression. This variance is further decomposed into a sampling error component and a negligible term arising from the event Ac(the cutoff event for small eigenvalues), with each... | https://arxiv.org/abs/2502.20368v1 |
2βr`2r`1 `op1qff M´2βr 2βr`2r`1 “«ˆ2σ2 a˙2βr 2βr`2r`1 pbβL2q2r`1 2βr`2r`1hˆβr 1`2r˙ `op1qff M´2βr 2βr`2r`1 ď pCβ,r,L,a,b,σ `op1qqM´2βr 2βr`2r`1, wherehpxq “2x1 2x`1`x´2x 2x`1andCβ,r,L,a,b,σ “3´ 2σ2 a¯2βr 2βr`2r`1pbβL2q2r`1 2βr`2r`1by the fact that supxą0hpxq “hp1q “3sinceh1pxq “ ´2logx p2x`1q2hpxq# ą0,if0ăxă1; ă0,ifxą1. ... | https://arxiv.org/abs/2502.20368v1 |
supvPSn´1ş ΘZpθqπv,γpdθqby´infvPSn´1xAn,Mv,vy and the trace TrpAn,Mq. The infimum term corresponds to the smallest eigenvalue, gi ving a PAC bound for the left-tail probability. The trace term is ty pically controlled by truncation (see [Mou22 ]), and we only need a rough bound P"TrpAn,Mq něλ1`1* ďP# TrpAn,Mq ěnÿ k“1λk`... | https://arxiv.org/abs/2502.20368v1 |
writingpφ“řnM`LM´1 k“nMpθkψkandφ“řnM`LM´1 k“nMθkpφqψk, we bound the supreme of the expectations from below by the average test error (see Secti on4.3for its proof): inf pφPΦMsup φPΦMEφr}pφ´φ}2 L2ρs “inf pφPΦMsup φPΦMnM`LM´1ÿ k“nMEφ„ˇˇˇpθk´θkpφqˇˇˇ2 ě` L´1 ML2nM`LM´1ÿ k“nMλβ k˘ 2´LMinf pφPΦMmin kÿ φPΦMPφ`pθk‰θkpφq˘ .(4... | https://arxiv.org/abs/2502.20368v1 |
( 4.3) (which follows from Lemma 4.2), (4.5), and Lemma 4.3, we obtain inf pφPL2ρsup φPHβ ρpLqEφr}pφ´φ}2 L2ρs ě1 4inf pφPΦMsup φPΦMEφr}pφ´φ}2 L2ρs “1 4inf pφPΦMsup φPΦMnM`LM´1ÿ k“nMEφ„ˇˇˇpθk´θkpφqˇˇˇ2 ě1 4` L´1 ML2nM`LM´1ÿ k“nMλβ k˘ 2´LMinf pφPΦMmin kÿ φPΦMPφ`pθk‰θkpφq˘ ě2´3` L´1 ML2nM`LM´1ÿ k“nMλβ k˘ˆ 1´1 2b τML2L´1 ... | https://arxiv.org/abs/2502.20368v1 |
in minimax lower rate Proof of (4.5).Since the set ΦMhas2LMelements, we reduce the supremum to the average, sup φPΦMEφr}pφ´φ}2 L2ρs ě2´LMÿ φPΦMEφr}pφ´φ}2 L2ρs. Meanwhile, by orthogonality of tψkuand definition of the set ΦM, we get Eφr}pφ´φ}2 L2ρs “nM`LM´1ÿ k“nMEφ„ˇˇˇpθk´θkpφqˇˇˇ2 “nM`LM´1ÿ k“nML´1 ML2λβ kPφ´ pθk‰θkpφq... | https://arxiv.org/abs/2502.20368v1 |
Thus, we only need to show the following bounds for the first term: Er}A´1 n,M¯cn,M}21As ď16κL2 Mλβ´1 nnÿ k“1λk. (A.2) 24 Since }A´1 n,M} ď4λ´1 nfor either case of A, we have Er}A´1 n,M¯cn,M}21As ď16λ´2 nEr}¯cn,M}2s “16λ´2 nnÿ k“1Er|¯cn,Mpkq|2s. By sample independence and that E” xRφHKnrums,RψkrumsyYı “ xLGφHKn,ψkyL2ρ“0... | https://arxiv.org/abs/2502.20368v1 |
k“1E” λ´2 kpAn,Mq1AEr`˜U¯dn,M˘2 k|An,Msı . To compute Er`˜U¯dn,M˘2 k|An,Ms, note that `˜U¯dn,M˘ k“nÿ l“1˜ukl¯dn,Mplq “nÿ l“1˜ukl1 MMÿ m“1xεm,Rψlrumsy “1 MMÿ m“1xεm,nÿ l“1˜uklRψlrumsy. Then, since εmis independent of um, Assumption 2.2pB1qimplies Er`˜U¯dn,M˘2 k|An,Ms “1 M2Mÿ m“1E« xεm,nÿ l“1˜uklRψlrumsy2|An,Mff ďσ21 M2Mÿ... | https://arxiv.org/abs/2502.20368v1 |
Chebyshev’s equaility and the fact that ErTrpAn,Mqs “řn k“1λkimply P"TrpAn,Mq něλ1`1* ďP# TrpAn,Mq ěnÿ k“1λk`n+ ďVarpTrpAn,Mqq n2ďκ´1 nMnÿ k“1λ2 kďκλ2 1 M. The next lemma, from [ Mou22 , Section 2.3] (see also in [ WSL23 ] for a constructive proof), controls the approximate term in the application of the PAC- Bayesian ... | https://arxiv.org/abs/2502.20368v1 |
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