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significant part CSS T. Since we impose assumptions on the expected value of CT, i.e. matrix C∞, we use concentration inequalities that directly follow from Assumption ( C). These inequalities allow us to show that with high probability as T→ ∞ the matrices CTandC∞become close to each other in the following sense. Lemm... | https://arxiv.org/abs/2501.16703v1 |
Bound for the martingale term Another crucial step in our analysis involves deriving a bound for the martingale compo- nent ϵT. The result is stated as follows: Proposition 6.4. Under Assumptions (A1)-(A3)and(C), the inequalities P |ϵj T|> x ≤2 exp −Tx2 2(√ K+M) + 6 exp −T 36 . (6.7) and P(∥ϵT,S∥2> x)≤2sexp −Tx2... | https://arxiv.org/abs/2501.16703v1 |
that P(B3)→0and P(B4)→0if (√ K+M)(M2,T+1 rT)2(τmin√s+ 2L)2 λ2+τ2 min s+Ks! slogp τ2 minT→0. (6.12) Proof. Due to Assumption ( B1), we have that for all j /∈ S 1 wj=|eθj| ≤ M 2,T+OP1 rT and since by definition B3= ∃j /∈ S: CSCS T(CSS T)−1ϵT,Sj−s ≥1−κ−ε 2λwj , and B4= ∃j /∈ S: ϵj T ≥1−κ−ε 2λwj , for large consta... | https://arxiv.org/abs/2501.16703v1 |
by using (6.3) and (6.4) as (p−s) max j /∈SP CSCS ∞(CSS T)−1−CSCS ∞(CSS ∞)−1 j−s 2≥ε 4M1,T ≤(p−s) max j /∈SP CSCS ∞ j−s 2∥(CSS T)−1∥op∥CSS T−CSS ∞∥op∥(CSS ∞)−1∥op≥ε 4M1,T ≤(p−s)P ∥CSS T−CSS ∞∥op≥ετ2 min 8LM1,T + (p−s)P ∥(CSS T)−1∥op>2 τmin ≤6s2(p−s) exp −Tε2τ4 min 2304KL2s2M2 1,T! + 12s2(p−s) exp −Tτ2 mi... | https://arxiv.org/abs/2501.16703v1 |
in- equalities and applications to random dynamical systems and diffusions. Ann. Probab. 32, 3B (2004), 2702–2732. [19]Fan, J. Comments on ≪Wavelets in statistics: A review ≫by A. Antoniadis. Statis- tical Methods & Applications 6 , 2 (1997), 131–138. [20]Fan, J., and Li, R. Variable selection via nonconcave penalized ... | https://arxiv.org/abs/2501.16703v1 |
the mckean–vlasov stochastic differential equation. Stochastic Processes and their Applications 162 (2023), 481–546. [42]Sur, P., and Cand `es, E. J. A modern maximum-likelihood theory for high- dimensional logistic regression. Proceedings of the National Academy of Sciences 116, 29 (2019), 14516–14525. [43]Trottner, L... | https://arxiv.org/abs/2501.16703v1 |
Rethinking the Win Ratio: A Causal Framework for Hierarchical Outcome Analysis Mathieu Even Theremia and PreMeDICaL Inria-Inserm, University of Montpellier, France and Julie Josse PreMeDICaL Inria-Inserm, University of Montpellier, France April 24, 2025 Abstract Forhierarchical multivarariates outcomes, the FDA recomme... | https://arxiv.org/abs/2501.16933v3 |
and a treated patient. Each pair is then considered as a “Win” if the outcome of the treated patient is considered as more favorable than the control one, as a “Loss” if it is considered as less favorable, and as a “Null” if the two patient outcomes are comparable. We here illustrate the hierarchical comparison process... | https://arxiv.org/abs/2501.16933v3 |
for all pi,jqPCNN,iis the treated patient with features Xithat are the closest features amongst all controlled patients to the features Xjof the controlled patient j. Nearest Neighbor pairings are in fact reminiscent of stratified pairings, since they can be seen as the extreme limit of stratification. 4 0.40.50.60.7 2... | https://arxiv.org/abs/2501.16933v3 |
introduction for readability purposes is wpy, y1q“1tyąy1u. 6 defined. Our proxy is a better proxy than the population-level estimand, as it compares the outcome of an individual being treated, with the outcome of a controlled patient that has the exact same features , rather than any random controlled patient. Informal... | https://arxiv.org/abs/2501.16933v3 |
Their analysis demonstrated how comparisons of all patient pairs contributed to "wins" for empagliflozin and placebo across four levels of the outcome hierarchy, resulting in an unstratified Win Ratio of 1.38, with accompanying confidence intervals and related metrics. They also discussed appro- priate and inappropriat... | https://arxiv.org/abs/2501.16933v3 |
the proposed method in the presence of dependent subjects [Zhang et al., 2022] or clusters [Zhang and Jeong, 2021]. Causal effect measures to assess treatment effects. One of the key contribution of our paper is to design an adequate estimand for pairwise comparisons, that Win Ratio and Generalized Pairwise Comparisons... | https://arxiv.org/abs/2501.16933v3 |
concept of an intervention by positing the existence of two valuesYip0qandYip1qfor the outcomes of interest, for the two situations where the patient has been exposed to treatment or not. These values are called potential outcomes , and they lie in some outcomes space Y. The following assumption, often stated as the St... | https://arxiv.org/abs/2501.16933v3 |
as pointed out by Ajufo et al. [2023]. Notations and terminology. For a sequence pZnqně0of random variables with values in a metric space pZ,dqandZa random variable in Z, we say that Znconverges in probability towardsZif for anyεą0we have that PpdpZn,ZqąεqÑ0. We say that a measurable event Eis almost sure if its probab... | https://arxiv.org/abs/2501.16933v3 |
fewer dimensions, improving NN’s effectiveness. This approached is detailed in Section 6 when studying the CRASH-3 data. As highlighted in Example 1, using different pairings leads to different Win Proportions (and thus to different Win Ratios and Net Benefits), to the point where treatment recommendations may even dif... | https://arxiv.org/abs/2501.16933v3 |
causal estimands that are functions of tPpYip0qq,PpYip1qqu. What makes it possible to estimate ErwpYip1q|Yip0qqswith the Risk Difference is the fact that thanks 3IfYip0q, Yip1q „ Bernoullip1{2q, the ATE with the risk difference ErYip1q´Yip0qsis identifiable (via e.g. taking the mean on test and control groups, in a RCT... | https://arxiv.org/abs/2501.16933v3 |
E“ wpYpXiqp1q,Yiq|Ti“0‰ , or, equivalently: E“ wpYi,YpXiqp0qq|Ti“1‰ . Under Assumption 1 (SUTVA), we have that these statistical estimands are equal to their associated causal estimands. 20 From the estimands τ“τ‹orτ“τpop, the Win Ratio estimand related to the estimator ˆRWRdefined in Equation (5) writes as: RWRdef“τ 1... | https://arxiv.org/abs/2501.16933v3 |
if the marginals are absolutely continuous with respect to the Lebesgue measure, and where dℓ2 is theℓ2distance between densities. Finally, next proposition formalizes the excess risk when usingτ‹orτpopas proxis for τindiv. Proposition 1. We have: |τ‹´τindiv|ďdTVpPYpXiqp1q,Yip0q,PYip1q,Yip0qq, and |τpop´τindiv|ďdTVpPYj... | https://arxiv.org/abs/2501.16933v3 |
Weighting approach with Nearest Neighbor pairings. Then, the second weakness of this Nearest Neighbor approach is that it 25 suffers from the curse of high dimensions: if the convergence speed in terms of samples n required will drop exponentially as the dimension of the features Xiincreases. This is a known weakness o... | https://arxiv.org/abs/2501.16933v3 |
being treated, conditionally onXi“x(as defined in Assumption 3): @xPX, πpxqdef“PpTi“1|Xi“xq, 27 and 1´πpxqdef“PpTi“0|Xi“xq. πpXiqis then called the propensity score of patient i. In this section, we assume that we have access to approximated propensity scores, via ˆπan approximation of π. We assume that ˆπis independen... | https://arxiv.org/abs/2501.16933v3 |
ErwpYip1q|Yip0qq|Xis, to obtain the estimator1 nřn i“1ˆppXiq. However, as opposed to (C)ATE estimation, we don’t have linearity of where, so that even if the conditional expectations are perfectly estimated ( ˆµt“µt), we won’t even have consistency. Regressing the conditional expectations makes us loose information on ... | https://arxiv.org/abs/2501.16933v3 |
mild assumptions. The as- sumption of Theorem 3.2 is much stronger, and requires a fast parametric rate of convergence. It will hold if for instance we perform logistic regression (Section 4.2.3) for a well-specified distributional regression problem. Under such assumptions, the asymptotic normality yields asymptotical... | https://arxiv.org/abs/2501.16933v3 |
Now,weremarkthatthequantitythatwewishtoestimatewritesas q1px,yq“PphypYiq|Xi“x,Ti“1q, wherehypYiq“wpYi|yq. Thus, our estimate ˆq1px,yqofq1px,yq“PpwpYi|yq|Xi“x,Ti“1qis: ˆq1px,yqdef“nÿ i“1ωipx,1qwpYi|yq, while our estimate ˆq0px,yqofq0px,yq“Ppwpy|Yiq|Xi“x,Ti“0qis: ˆq0px,yqdef“nÿ i“1ωipx,0qwpy|Yiq. In practice, these steps... | https://arxiv.org/abs/2501.16933v3 |
well estimated. These illustratetheshortcomingsofthenearestneighborapproachwhenthedimensionbecomeslarger, and the strangth of our distributional regression approach. 38 0.50.60.7 500 1000 3000 8000 Number of Patients (n)EstimationMethod DRF_AIPW_WR DRF_WR NearestNeigh_WRComparison of Estimations by Method, dim = 8(a)d“... | https://arxiv.org/abs/2501.16933v3 |
Gaussian, in our case). Treatment responses are then generated as multi dimensional Bernoulli random variables, of means σppXi,1´ Xi,2q2qandσppXi,1`Xi,2q2qfor respectively treated and non-treated individuals. 2. The second experiment tests for double robustness by mispecifying in the distributional regression. We perfo... | https://arxiv.org/abs/2501.16933v3 |
28 days). Computing the average treatment effect for each of these 3 outcomes lead respectively to the confidence intervals, where Y1,Y2,Y3are respectively our death, secondary effects and hospital- ization duration outcomes: ErY1p1q´Y1p0qsPr´ 0.0032,0.0033s, ErY2p1q´Y2p0qsPr´ 0.0021,0.0079s, and ErY3p1q´Y3p0qsPr´ 0.23... | https://arxiv.org/abs/2501.16933v3 |
latent space. 5. Distributional Random Forests, as described in Section 4.2. We estimate the Win Propor- tion(obtained with wpy,y1q“1tyąy1u) and the Loss Proportion (obtained with wpy,y1q“ 1tyĺy1u), and estimate the Win Ratio as the ratio between win et loss proportions. 6. Our doubly robust approach, as described in S... | https://arxiv.org/abs/2501.16933v3 |
from a Win Ratio analysis of the CRASH-3 dataset. 50 7 Conclusion and open directions In this paper, we have introduced a causal inference framework for hierarchical outcome comparison methods like Win Ratio or Generalized Pairwise Comparisons. Our goal is to make such methods more grounded, by offering new perspective... | https://arxiv.org/abs/2501.16933v3 |
In Sanjoy Dasgupta, Stephan Mandt, and Yingzhen Li, editors, Proceedings of The 27th International Conference on Artificial Intelligence and Statistics , volume 238 of Proceedings of Machine Learning Research , pages 1324–1332. PMLR, 02–04 May 2024. 35, 39 Gérard Biau and Luc Devroye. Lectures on the nearest neighbor m... | https://arxiv.org/abs/2501.16933v3 |
Moons, Linda M. Peelen, Mirjam J. Knol, and Arno W. Hoes. Reporting of treatment effects from randomized trials: A plea for multivariable risk ratios. Contemporary Clinical Trials , 32(3):399–402, may 2011. ISSN 1551-7144. 11 Xiaohan Guo and Ai Ni. Contrast weighted learning for robust optimal treatment rule estima- ti... | https://arxiv.org/abs/2501.16933v3 |
45 Clifford H Wagner. Simpson’s paradox in real life. The American Statistician , 36(1):46–48, 1982. 22 Frank Wilcoxon. Individual comparisons by ranking methods. Biometrics Bulletin , 1(6):80, dec 1945. ISSN 0099-4987. 8, 10 Anqi Yin, Ao Yuan, and Ming T. Tan. Highly robust causal semiparametric u-statistic with appli... | https://arxiv.org/abs/2501.16933v3 |
Xσ‹piqRBpukx,η{2q|kx˘‰ “ErPp@jPN1, XjRBpukx,η{2q|kxqs “Erp1´pkxqn1s ďp1´pminqn1 ÝÑn1Ñ80. This leads to: P` δpXi,Xσpiq,yqąε˘ ďp1´pminqn1, and thus P` δpXi,Xσpiq,yqÑ0˘ “1asn1Ñ8, leading to E“ δpXi,Xσpiq,Yip0qq‰ Ñ0. We thus have that Er|A1|s Ñ 0, and thus A1Ñ0in probability, since |A1| ď1almost surely. This concludes the ... | https://arxiv.org/abs/2501.16933v3 |
as nÑ 8. LetkPN1andxPSupppXq: @εą0,PpXPBpx,εqqą 0. First case: PpX“x|T“1q“pxą0. In that case, let Nx“|tℓPN1,Xℓ“xu. We have that Ppσpiq“k|Nx,Xk“x,σpjq“kq“1 Nx, andNxis a binomial random variable of parameters pn1,pxq. This leads to: Ppσpiq“k|Xk“x,σpjq“kq“n1ÿ N“12´n1pN xp1´pxqn1´N Nˆn1 N˙ “n1ÿ N“1pN xp1´pxqn1´N n1`1ˆn1`1... | https://arxiv.org/abs/2501.16933v3 |
thus are left with proving that τ‹ AIPWconverges in probability towards τ‹.‘ We have: τ‹ AIPW´τ‹“λ nnÿ i“1" wpYσ0piq|Yiq1´Ti p1´πpXiqq´τ‹* `1´λ nnÿ i“1" wpYi|Yσ1piqqTi πpXiq´τ‹* l jh n pIq `1 nnÿ i“1" p1´Tiqˆq1pXi,Yiq´λˆq1pXi,Yiq1´Ti p1´πpXiqq* l jh n pIIq `1 nnÿ i“1" Tiˆq0pXi,Yiq´p1´λqˆq0pXi,YiqTi πpXiq* l jh n pIIIq.... | https://arxiv.org/abs/2501.16933v3 |
arXiv:2501.16956v1 [math.ST] 28 Jan 2025The empirical median for estimating the common mean of heteroscedastic random variables Sirine Louati∗ January 29, 2025 Abstract We study the problem of mean estimation in the heteroscedast ic setting. In particular, we consider symmetric random variables having the same loca tio... | https://arxiv.org/abs/2501.16956v1 |
limit the damaging effect of a small number of outliers. In heteroscedastic esti mation, one would like to exploit the possible benefit of a small number of highly informative obse rvations. If the densities of the Xi’s and thus the scale parameters σi’s are known, the problem becomes easily tractable. The optimal estima... | https://arxiv.org/abs/2501.16956v1 |
good as that of the max imum likelihood estimator provided by the oracle, bearing in mind that the latter can ne ver be attained since the variances are unknown. Structure of the paper The paper is organized as follows. We will first analyze the em pirical median estimator in Section 2by establishing and upper bound and... | https://arxiv.org/abs/2501.16956v1 |
proof of which is given in Section 4.3. Lemma 1. Letδ∈(exp(−n),1)andS=/summationtextn i=1Vi, whereV1,...,V nare independent random variables with Vi∼ B(pi)for alli∈[[1,n]]. Ifpi∈[1 4,3 4]for everyi∈[[1,n]], then P/parenleftBigg S/greaterorequalslantE[S]+0.3/radicalBigg nlog/parenleftbigg2 δ/parenrightbigg/parenrightBig... | https://arxiv.org/abs/2501.16956v1 |
introduced an estimator tailored for heavy-tailed distri butions, leveraging advanced tail-bound techniques. In addition, the article by [ 23] delves into truncation-based methods for mean estimation under heteroscedasticity. The authors introduce an iterat ive estimator designed for datasets with a subset of variances... | https://arxiv.org/abs/2501.16956v1 |
estimation error of our estimator with th at of the empirical median estimator studied in [ 7] which does not make any assumption on σ1. This is equivalent to comparing our bound obtained in Theorem 1to 8e√ 2max/parenleftbigg log/parenleftbigg3 δ/parenrightbigg ,log(n+1)/parenrightbigg β−1max1/lessorequalslantj/lessore... | https://arxiv.org/abs/2501.16956v1 |
the unknown mean. Indeed, our lower bound on the estimation error does not depend on the√nsmallest variances. An interesting open problem is to find an estimator that would be sensitive to the smallest variances and thus achiev es optimality among all estimators. 4 Proofs Assumption Throughout the proofs, we assume that... | https://arxiv.org/abs/2501.16956v1 |
obtain P n/summationdisplay i=kt+1/parenleftBig 1Xi/greaterorequalslantt−P(Xi>t)/parenrightBig /greaterorequalslant(c−1√ 2)/radicalBigg (n−kt)log/parenleftbigg1 2δ/parenrightbigg /greaterorequalslant2δ, withc−1√ 2/lessorequalslant0.3andδ/lessorequalslant1 4such that log(1 2δ)/greaterorequalslant1 2log(1 δ). So with... | https://arxiv.org/abs/2501.16956v1 |
/lessorequalslantδ 2. We have that P/parenleftBigg ˜S−E[˜S]/lessorequalslant−c′′/radicalBigg nlog/parenleftbigg1 δ/parenrightbigg/parenrightBigg =P/parenleftBigg −˜S−E[−˜S]/greaterorequalslantc′′/radicalBigg nlog/parenleftbigg1 δ/parenrightbigg/parenrightBigg . Moreover, −˜S=/summationtextn i=1(−˜Vi)where(−˜Vi)are1 8-S... | https://arxiv.org/abs/2501.16956v1 |
and offering inva luable advice. References [1] S. Boucheron, G. Lugosi, and P. Massart. Concentration in equalities: A nonasymptotic theory of independence, 2013. [2] L. D. Brown. Admissible estimators, recurrent diffusions , and insoluble boundary value problems. The Annals of Mathematical Statistics , 42(3):855–903, 1... | https://arxiv.org/abs/2501.16956v1 |
Asymptotic properties and drift parameter estimations of the ergodic double Heston model based on continuous-time observations Mohamed Ben Alaya1, Houssem Dahbi∗1,2, and Hamdi Fathallah2 1Université de Rouen Normandie, Laboratoire de Mathématiques Raphaël Salem, Avenue de l’université, BP.12 F76801 Saint-Étienne-du-Rou... | https://arxiv.org/abs/2501.17100v1 |
the model to be more flexible. In the literature, among the many extensions proposed, Christoffersen et al. in [13] introduced a new extension named the double Heston model that specifies a two-factor structure for volatility in their model to explore the correlation between the volatility and the smile figure. In the ... | https://arxiv.org/abs/2501.17100v1 |
+ 1))e−δt, for allt∈[0,∞)and Borel measurable functions g:D→(−∞,∞), where (y0,x0)∈DandVis a chosen norm-like function on D, for more details see Theorem 5.1. We note that the stationarity result is similar to that of Jin et al. [26], and our aim in this work is to elaborate on all the mathematical details related to ou... | https://arxiv.org/abs/2501.17100v1 |
into a column vector, for more details, see, e.g., [24] and [35]. We use the notations Hx(f)and∇xffor the Hessian matrix and the gradient column vector of the functionfwith respect to x. we denote by C2(D,R)the set of twice continuously differentiable real- valued functions on Dand by C2 c(D,R)its subset of functions w... | https://arxiv.org/abs/2501.17100v1 |
eb22tE/parenleftig Y(2) t/parenrightig =E/parenleftig Y(2) 0/parenrightig −a2 b22/parenleftbig 1−eb22t/parenrightbig −b21/integraldisplayt 0eb22sE/parenleftig Y(1) s/parenrightig ds. (9) Hence, ifb11∈R+, theneb22sE(Y(1) s)is integrable and eb22tE(Y(2) t)converges and if b11∈R−−, then three cases appear: if b11=b2... | https://arxiv.org/abs/2501.17100v1 |
distri- bution asZ∞given by (13), then (Zt)t∈R+is strictly stationary. Proof. 1) The proof of the first assertion follows four main steps: Step 1: The first step is devoted to convert the asymptotic study for a triplet process ˜Z= (Y,U,V ) instead ofZ= (Y,X), where, for all t∈R+, Ut:=κ⊤/integraldisplayt 0e−θ(t−s)Ysdsan... | https://arxiv.org/abs/2501.17100v1 |
dz, for some constant C∈R. By variation of constants, we find the following particular solution of the inhomogeneous differential equation (25) w(2),p t(u) =−u3κ2e−/integraltextt 0f(2) z(u) dz/integraldisplayt 0e/integraltexts 0f(2) z(u) dze−θsds. Hence, a general solution of (25) takes the form w(2) t(u) =w(2),h t(u) ... | https://arxiv.org/abs/2501.17100v1 |
other hand, since the sequence (u(n))n∈Nis bounded (since it is convergent), then, provided the continuity of U′∋u∝⇕⊣√∫⊔≀→˜Kt(u), fort∈R+, Lebesgue’s dominated convergence theorem implies lim n→∞/integraldisplay∞ 0˜Ks(u(n)) ds=/integraldisplay∞ 0˜Ks(u) ds. which shows the continuity of U′∋u∝⇕⊣√∫⊔≀→˜g∞(u). It is worth t... | https://arxiv.org/abs/2501.17100v1 |
(iii) There exists c∈R++andd∈Rsuch that the inequality (AnV)(z)≤−cV(z) +d, z∈On, holds for all n∈N, where On:={z∈D:∥z∥1<n}, for eachn∈N,Andenotes the extended generator of the process Zt,n= (Yt,n,Xt,n)t∈R+given by Zt,n:=/braceleftigg Zt, t<T n, (0,0,n), t≥Tn, where the stopping time Tnis defined by Tn:= inf{t∈R+:Zt∈D\... | https://arxiv.org/abs/2501.17100v1 |
s/parenrightig e−b22(1−s) 1−iωσ2 12 2b22/parenleftbig 1−e−b22(1−s)/parenrightbigds dω =/integraldisplay Re−iωuexp iωy(2) 0e−b22 1 +ω2σ4 12 4b2 22(1−e−b22)2 exp −ω2σ2 12y(2) 0 2b22e−b22/parenleftbig 1−e−b22/parenrightbig 1 +ω2σ4 12 4b2 22(1−e−b22)2 ×E exp iω/integraldisplay1 0/parenleftig a2−b21Y(1) ... | https://arxiv.org/abs/2501.17100v1 |
˜ρ(˜Zt) = σ11/radicalig ˜Y(1) t 0 0 0 0σ12/radicalig ˜Y(2) t 0 0 ρ11σ21/radicalig ˜Y(1) tρ22σ22/radicalig ˜Y(2) t ¯ρ11σ21/radicalig ˜Y(1) t ¯ρ22σ22/radicalig ˜Y(2) t . (37) It is easy to check that ˜ρ(Zt)˜ρ⊤(Zt) = R(Zt). Using the relation (7.138) in Lipster and Shiryaev [29] applied on the process (Zt)... | https://arxiv.org/abs/2501.17100v1 |
Y(1) s/parenrightig5 2 ξ2(Ys)ds,˜V(4) T=/integraldisplayT 01 ξ(Ys)ds and ˜V(5) T=/integraldisplayT 0(Z(k) s)2 ξ(Ys)ds, for alli,j∈{1,2}such thati̸=jandk∈{1,2,3}. Hence, using the ergodicity theorem 5.1, we deduce that 1 T˜V(1) T,(α,β)a.s.−→E /parenleftig Y(i) ∞/parenrightigα/parenleftig Y(j) ∞/parenrightigβ ξ2(... | https://arxiv.org/abs/2501.17100v1 |
discussion is about the construction of a CLSE for the drift parameter τ= (a1,b11,a2,b21,b22,m,κ 1,κ2,θ)⊤based on continuous-time observation of (Zt)t∈[0,T], for someT∈R++. At first, we consider the CLSE ˇτT,Nbased on the following discretizated process (Yi N,Xi N)i∈{0,1,...,⌊NT⌋}, forN∈N∗and it is obtained by solving ... | https://arxiv.org/abs/2501.17100v1 |
T,N,with Γ(2) T,N= ⌊NT⌋ −⌊NT⌋/summationdisplay i=1Y(1) i−1 N−⌊NT⌋/summationdisplay i=1Y(2) i−1 N −⌊NT⌋/summationdisplay i=1Y(1) i−1 N⌊NT⌋/summationdisplay i=1/parenleftig Y(1) i−1 N/parenrightig2⌊NT⌋/summationdisplay i=1Y(1) i−1 NY(2) i−1 N −⌊NT⌋/summationdisplay i=1Y(2) i−1 N⌊NT⌋/summationdisplay i=1Y(1)... | https://arxiv.org/abs/2501.17100v1 |
T,G(2) T,G(3) T/parenrightig andfT= f(1) T f(2) T f(3) T . Hence, by the same arguments used in the proof of the invertibility of the matrices Γ(1) T,N,Γ(2) T,NandΓ(3) T,Nwith replacing the sum from 0to⌊NT⌋by the integral on [0,T], it is easy to check that GTis invertible. Note that this approximate CLSE is cons... | https://arxiv.org/abs/2501.17100v1 |
to check that T−1hTa.s.−→0,asT→ ∞. First, we have ⟨h(1) T,h(1) T⊤⟩,⟨h(1) T,h(2) T⊤⟩⟨h(1) T,h(3) T⊤⟩,⟨h(2) T,h(2) T⊤⟩,⟨h(2) T,h(3) T⊤⟩and⟨h(3) T,h(3) T⊤⟩are given, respec- tively, by H(1,1) T=σ2 11/integraldisplayT 0 Y(1) s−/parenleftig Y(1) s/parenrightig2 −/parenleftig Y(1) s/parenrightig2/parenleftig Y(1) s/p... | https://arxiv.org/abs/2501.17100v1 |
negative values with some positive probability. This solution was proposed by Higham and Mao [23] for the CIR process and they have shown the strong convergence of the scheme. In relation to this problem, many discretization schemes dedicated to the CIR process have been studied in recent years by Deelstra and Delbaen ... | https://arxiv.org/abs/2501.17100v1 |
0.0914 0.0661 0.0498 0.0998 0.0692 0.0699 0.0369 T= 1040.0061 0.0060 0.0103 0.0076 0.0053 0.0109 0.0075 0.0077 0.0041 As a result, according to our simulations, we observe that both the MLE and CLSE methods are consistent and have a Gaussian asymptotic distribution with a mean equal to 0. In addition, despite the advan... | https://arxiv.org/abs/2501.17100v1 |
A∈Mk,l,k,l∈N∗, defined on (Ω,F,(F)t∈R+,P), we have (Q(t)Mt,A)D−→(ηZ,A ),ast−→∞, whereZis ap-dimensional standard normally distributed random vector independent of (η,A). References [1] Aurélien Alfonsi. On the discretization schemes for the cir (and bessel squared) processes. Monte CarloMethods Appl., 11(4):355–384, 20... | https://arxiv.org/abs/2501.17100v1 |
A Horn and Charles R Johnson. Topicsinmatrixanalysis. Cambridge university press, 1994. [25] Jean Jacod and Albert Shiryaev. Limittheorems forstochastic processes, volume 288. Springer Science & Business Media, 2013. [26] Peng Jin, Jonas Kremer, and Barbara Rüdiger. Existence of limiting distribution for affine process... | https://arxiv.org/abs/2501.17100v1 |
Fundamental Computational Limits in Pursuing Invariant Causal Prediction and Invariance-Guided Regularization Yihong Gu1, Cong Fang2, Yang Xu2, Zijian Guo3, Jianqing Fan1∗ 1Princeton University,2Peking University, and3Rutgers University Abstract Pursuing invariant prediction from heterogeneous environments opens the do... | https://arxiv.org/abs/2501.17354v1 |
the set of environments. For each environment e∈ E, we observe ndata{(X(e) i, Y(e) i)}n i=1that are i.i.d. drawn from some distribution ( X(e), Y(e))∼µ(e)satisfying Y(e)= (β⋆ S⋆)⊤X(e) S⋆+ε(e)with E[X(e) S⋆ε(e)]≡0 (1.1) where β⋆is the true parameter that is invariant across different environment and S⋆= supp( β⋆) denote... | https://arxiv.org/abs/2501.17354v1 |
From a statistical viewpoint, the core difficulty is to distinguish whether a variable is truly important, or endogenously spurious among those statistically significant variables that contribute to predicting Y. This is where multi-environment comes into play. There is a considerable literature on estimating the param... | https://arxiv.org/abs/2501.17354v1 |
efficient algorithms in general? If not, can it be attainable under some additional conditions? Q2. Can we have benefits by designing methods that smoothly “interpolate” the estimators for the invariant causal model β⋆and the most predictive solution ¯β:= argminβP e∈EE(X,Y)∼µ(e) E[|Y−β⊤X|2]? 1.2 Computational Barrier T... | https://arxiv.org/abs/2501.17354v1 |
in polynomial time, are P(olynoimal- time) problems that are solvable in polynomial time. It is suspected, but is still a conjecture (Bovet et al., 1994; Fortnow, 2021), that P ̸=NP. This implies it is unlikely that there exists any polynomial-time algorithms for NP-hard problems. This paper proves the NP-hardness of E... | https://arxiv.org/abs/2501.17354v1 |
a better balance between prediction power and invariance, while partially circumventing the computational barriers as the second part of Q1. Given data from environments E, the population-level estimator with n=∞is the minimizer of the following objective function βk,γ= argmin β∈Rd1 |E|X e∈EE[|Y(e)−β⊤X(e)|2] +γdX j=1wE... | https://arxiv.org/abs/2501.17354v1 |
causal the solution is, it may lack some predictive power under the circumstances discussed before Q2. The estimator proposed in this paper leverages the invariance principle as an inductive bias for “soft” regularization instead of that for “hard” structural equation estimation and can alleviate the lack of predictive... | https://arxiv.org/abs/2501.17354v1 |
There is also a considerable literature on deriving statistical sub-optimality of computationally efficient algorithms using the reduction from the planted clique problem (Brennan & Bresler, 2019), such as sparse principle component (Berthet & Rigollet, 2013a,b; Wang et al., 2016), sparse submatrix recovery (Ma & Wu, 2... | https://arxiv.org/abs/2501.17354v1 |
instance of the problem, we use |x|to denote the size of its input and use XPto denote the set of all the problem instances. We use Sxto denote the set of solutions for the problem instance x. We use the notation x∈ XP,1if the answer to the instance xis 1(Yes). Clearly, we have x∈ XP,1⇐⇒ |S x| ≥1. The particular decisi... | https://arxiv.org/abs/2501.17354v1 |
of any polynomial-time algorithm for such a problem implies any NP problem can be solved within polynomial-time with high probability, that is, for any NP decision problem P, we can design a polynomial- time randomized algorithm eAsuch that ∀x∈ XP\ XP,1,eA(x) = 0 and ∀x∈ XP,1,P[eA(x) = 1] ≥1−0.01|x|−100. If the conject... | https://arxiv.org/abs/2501.17354v1 |
∀S⊆[d],either β(S∪¯S)=β(¯S) or sup e,e′∈[E]∥β(e,S)−β(e′,S)∥2>0! . (2.1) Problem 2.4 (Existence of Linear Invariant Set under Identification) .Problem ExistLIS-Ident is defined as the same problem as ExistLIS with the additional constraint that there exists a maximum invariant set S†. Note that S†can be an empty set, ... | https://arxiv.org/abs/2501.17354v1 |
maximum invariant set. Given XExistLIS-Ident ⊊XExistLIS ,ExistLIS may be potentially harder than ExistLIS-Ident . We will establish NP-hardness to both ExistLIS andExistLIS-Ident to rule out the possibility that the computational hardness is because of nonidentifiability, or in other words, computational difficulty can... | https://arxiv.org/abs/2501.17354v1 |
we use action ID in {0, . . . , 7}to represent the assignment for it. For example, for the clause v1∨ ¬v2∨ ¬v5and the action ID 6 with binary representation 110 means we let v1= True, ¬v2= True and¬v5= False. One will not adopt action ID 0 in a valid solution because a 3Sat valid solution should let each clause evaluat... | https://arxiv.org/abs/2501.17354v1 |
follows from the assumption β(2,S) j =β(1,S) j =1{j∈S}andd /∈S, and the inequality follows from the fact that A∈ {0,1}7k×7khence the L.H.S. is an integer. This indicates that β(1,S)̸=β(2,S) if|S| ≥1 and d /∈S. Given d∈S, we then obtain 5d+1 2k=u(2) d=h Σ(2) Sβ(2,S) Si |S|= 5d+1 27kX j′=11{j′∈S}= 5d+1 2(|S| −1), which i... | https://arxiv.org/abs/2501.17354v1 |
Guarantees The claim in Theorem 2.1 indicates a computational barrier exists in finding an exact invariant set. At first glance, it does not rule out the possibility that there exists some polynomial-time algorithm that can find an approximate solution whose prediction is relatively close to one of the non-trivial inva... | https://arxiv.org/abs/2501.17354v1 |
first transforms xintoy=T(x), then use algorithm Ato solve yand gets the returned ¯β, and finally output 1{eS∈ Sy}. It remains to verify ( a): the ⇒direction is obvious. For the ⇐direction, suppose |Sy| ≥1, the estimation error guarantee in Problem 2.5 indicates that ∥bβ−β(S†)∥∞≤ ∥¯β−β(S†)∥2≤s ∥¯β−β(S†)∥2 Σ λmin(Σ)(i) ... | https://arxiv.org/abs/2501.17354v1 |
first impose an additional restrictive assumption Condition 3.2 in the model (1.1) and see how the computational barrier can be circumvented under this condition. In the following Section 3.2, we shall consider a more general relaxation regime and establish a tradeoff between the additional assumption and computational... | https://arxiv.org/abs/2501.17354v1 |
the population-level minimizer βγofQ1,γ(β) for varying γfrom two perspectives. On the one hand, βγcan be interpreted as the distributionally robust prediction model over the uncertainty set Pγ(Σ, u): it minimizes the worst-case negative explained variance, or it is the maximin effects (Meinshausen & B¨ uhlmann, 2015; G... | https://arxiv.org/abs/2501.17354v1 |
two-dimensional uncertainty plane in covariance space further yields the two-dimensional uncertainty plane centered on the pooled least squares ¯βin the solution space after the affine transformation x→Σ−1x as shown in Fig. 1 (b). The uncertainty sets Θ γall lie in the same hyper-plane and their diameter scales linearl... | https://arxiv.org/abs/2501.17354v1 |
more computational budget is paid, the space of instances that can be solved enlarges and will finally coincide with that of EILLS or FAIR when k≥ |S⋆|. On the other hand, if the computational budget we can pay is relatively limited, one can still probably solve some problem instances with low-dimensional structures as... | https://arxiv.org/abs/2501.17354v1 |
following empirical-level penalized least squares bβk,γ= argmin βbQk,γ(β)z }| { 1 2n|E|X e∈E,i∈[n] Y(e) i−β⊤X(e) i2 +γ·dX j=1|βj|p bwk(j), with bwk(j) = inf S⊆[d],|S|≤k,j∈S1 |E|X e∈E bβ(e,S) S−bβ(S) S 2 bΣ(e) S.(3.7) The weighted L1-penalty aims at attenuating the endogenously spurious variables. This will be applied... | https://arxiv.org/abs/2501.17354v1 |
estimator lies in between the two. Turning to the high-dimensional regime, we have the following result. The main message is that the proposed estimator in (3.8) can handle the high-dimensional covariates in a similar spirit to Lasso (Tibshirani, 1997; Bickel et al., 2009) for the sparse linear model with the help of a... | https://arxiv.org/abs/2501.17354v1 |
environments defined as min e∈{4,5,6,7}R2 oos,ewith R2 oos,e= 1−P (X,Y)∈De(Y−bY(X))2 P (X,Y)∈DeY2(4.2) where bY(X) is the model’s prediction. Here we use the R2rather than the mean squared error in (4.1) to present the result to illustrate the challenge of this task, given most of the previous methods have negative out... | https://arxiv.org/abs/2501.17354v1 |
environments, the year 2010 as the validation environment ( D3), and the year 2020 as the test environment (D4), all on a daily timescale. The target is to predict a set of variables Y={j1, . . . , j w} ⊆[60], namely Yt= (Zt,j1, . . . , Z t,jw)⊤, using all the variables from the past seven days as covariates, namely, X... | https://arxiv.org/abs/2501.17354v1 |
principal components in high dimension. The Annals of Statistics , 41(4), 1780–1815. Bickel, P. J., Ritov, Y., & Tsybakov, A. B. (2009). Simultaneous analysis of lasso and dantzig selector. The Annals of statistics , 37(4), 1705–1732. Blanchet, J., Kang, Y., Murthy, K., & Zhang, F. (2019). Data-driven optimal transport... | https://arxiv.org/abs/2501.17354v1 |
A primer . John Wiley & Sons. Granger, C. W. (1969). Investigating causal relations by econometric models and cross-spectral methods. Econometrica: journal of the Econometric Society , (pp. 424–438). Gu, Y., Fang, C., B¨ uhlmann, P., & Fan, J. (2024). Causality pursuit from heterogeneous environments via neural adversa... | https://arxiv.org/abs/2501.17354v1 |
J., B¨ uhlmann, P., & Meinshausen, N. (2016). Causal inference by using invariant prediction: identification and confidence intervals. Journal of the Royal Statistical Society. Series B (Statistical Methodology) , (pp. 947–1012). Pfister, N., B¨ uhlmann, P., & Peters, J. (2019). Invariant causal prediction for sequenti... | https://arxiv.org/abs/2501.17354v1 |
Zhang, C.-H. & Zhang, T. (2012). A general theory of concave regularization for high-dimensional sparse estimation problems. Statistical Science , 27(4), 576–593. Zhang, Y., Wainwright, M. J., & Jordan, M. I. (2014). Lower bounds on the performance of polynomial-time algorithms for sparse linear regression. In Conferen... | https://arxiv.org/abs/2501.17354v1 |
problem ExistDIS-Unique as the same problem with the promise that the non-trivial distribution- invariant set is unique if exists. 28 The following lemma shows that (A.1) is equivalent to the full distribution invariance condition (Assumption 1 in Peters et al. (2016)) under the setting in Problem A.1. Lemma A.1. Under... | https://arxiv.org/abs/2501.17354v1 |
Theorem A.4. Consider the problem ExistLIS with the additional constraint that for any e∈[E], each row of matrix Σ(e)has no more than Cnon-zero elements for some universal constant C > 0. The above problem is NP-hard under deterministic polynomial-time reduction when E= 2. The proof idea is as follows. We first reduce ... | https://arxiv.org/abs/2501.17354v1 |
pa(j)⊆ {1, . . . , p }is the set of parents, or the direct causes, of the variable Zj, and the joint distribution ν(du) =Qp j=1νj(duj)over pindependent exogenous variables (U1, . . . , U p). For a given model M, there is an associated directed graph G(M) = ( V, E)that describes the causal relationships among variables,... | https://arxiv.org/abs/2501.17354v1 |
for any j∈S⋆=pa(d+ 1), and e, e′∈ E β(e) j=E[X(e) jY(e)] E[|X(e) j|2]=E[X(e′) jY(e′)] E[|X(e′) j|2]=β(e′) j. C Proofs for Computation Fundamental Limits C.1 Proof of Lemma 2.3 Proof of (2.5).We first establish the upper bound in (2.5). It follows from the definition of β(S)that ∥β(S)−β(S†)∥2 Σ=uSΣ−1 SuS+uS†Σ−1 S†uS†−2u... | https://arxiv.org/abs/2501.17354v1 |
the fact that ( X, Y) are multivariate Gaussian under which independence is equivalent to uncorrelatedness and the fact that bε(e)is also Gaussian, (b) follows from the fact that var(bε(e)) =E[|Y(e)|2]−2(β(e,S) S)⊤E[X(e) SY(e)] + (β(e,S) S)⊤Σ(e) Sβ(e,S) S =v(e)−(β(e,S) S)⊤Σ(e) Sβ(e,S) S. Proof of Theorem A.1. The proof... | https://arxiv.org/abs/2501.17354v1 |
5 dto 32 kto makeE[|Y(1)|2]≍E[|Y(2)|2]. We also change the coordinates of u(1) [7k+1]andu(2) [7k+1]accordingly. (c) We add a k−1multiplicative factor in u(1)andu(2)to letE[|Y(1)|2],E[|Y(2)|2]≍1. This will also result in all the β(e,S)andβ(S)being multiplied by the same k−1factor. Step 2. Verification of Parsimonious Re... | https://arxiv.org/abs/2501.17354v1 |
(k−1)21 32k(32k)2(7k+ 1) + 1⊤ d−7k−1H−1 d−7k−11d−7k−1 . 36 Here ( a) follows from the fact that Σ(1)is a block diagonal matrix. It follows from the identity 1⊤ ℓH−1 ℓ1ℓ= ℓ/(1 +ℓ) that 1<(k−1)21 32k(32k)2(7k+ 1) ≤E[|Y(1)|2] ≤(k−1)2 32k·(7k+ 1) +d−7k−1 d−7k−1 + 1 <256. Similarly, for E[|Y(2)|2], following from the fa... | https://arxiv.org/abs/2501.17354v1 |
2 ∥β(1,S)−β(S)∥2 Σ(1) S+∥β(2,S)−β(S)∥2 Σ(2) S ≥k−3/1280≥d−ϵ/1280. Step 3. Calculating the Gap between β(S)and β(S†).LetS†be arbitrary invariant set according to Definition 4 and Sbe any set that does not equal to S†. We keep adopting the notation S1=S∩[7k+1], S2= S\[7k+ 1], and divide it into two cases. Case 1. S2̸=∅... | https://arxiv.org/abs/2501.17354v1 |
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