text string | source string |
|---|---|
for spher- ical designs , Annals of mathematics (2013), 443–452. 19 Fixed-strength spherical designs Travis Dillon [10] Andriy V. Bondarenko and Maryna S. Viazovska, Spehrical designs via Brouwer fixed point theorem , SIAM Journal on Discrete Mathematics 24(2010), 207–217. [11] Fernando G. S. L. Brandão, Aram W. Harrow,... | https://arxiv.org/abs/2502.06002v1 |
S. Womersley, Efficient spherical designs with good geometric properties , Contemporary Com- putational Mathematics — A Celebration of the 80th Birthday of Ian Sloan, 2018, pp. 1243–1285. [33] Hung-Hsun Hans Yu, personal communication (2024). 20 Fixed-strength spherical designs Travis Dillon A. appendix A.1. spherical mo... | https://arxiv.org/abs/2502.06002v1 |
in [ 17]), but that approach doesn’t provide any geometric intuition. Proposition A.3.Qd k(1) = dim( Wd k) =/parenleftbigd+k−1 d−1/parenrightbig −/parenleftbigd+k−3 d−1/parenrightbig . Proof . The linear transformation evk,xev⊤ k,x:f/ma√s⊔o→ /an}⌊ra⌋ke⊔le{⊔evx,f/an}⌊ra⌋ke⊔ri}h⊔evxhas trace Tr(evxev⊤ x) = Tr(ev⊤ xevx) =... | https://arxiv.org/abs/2502.06002v1 |
arXiv:2502.06051v1 [cs.LG] 9 Feb 2025Nearly Optimal Sample Complexity of Offline KL-Regularized Contextual Bandits under Single-Policy Concentrability Qingyue Zhao∗†Kaixuan Ji∗‡Heyang Zhao∗§Tong Zhang¶Quanquan Gu‖ Abstract KL-regularized policy optimization has become a workhorse in learning -based decision mak- ing, whi... | https://arxiv.org/abs/2502.06051v1 |
2008;Ziebart,2010;Levine and Koltun ,2013;Fox et al. ,2015;Levine et al. ,2016;Haarnoja et al. , 2018;Richemond et al. ,2024;Liu et al. ,2024). The KL-regularized objective has also been widely used in the RL fine-tuning of large language models (LLMs) ( Ouyang et al. ,2022;Rafailov et al. , 2023,2024), where πrefis a p... | https://arxiv.org/abs/2502.06051v1 |
due to our proof technique, which leverages the strong c onvexity of KL regularization and the conditional non-negativity of thegap between thetruerewa rd andits pessimisticestimator to refine a mean-value-type risk upper bound to its extreme. This in tu rn leads to a novel covariance-based analysis, effectively bypassin... | https://arxiv.org/abs/2502.06051v1 |
bandit setting. The algorithmic idea of using pessimistic least-square estima tors under general function approximation inJin et al. (2021);Di et al. (2023) is similar to ours, but their suboptimality gap is bounded b y the sum of bonuses, the direct adaptation of which to our obje ctive cannot lead to the desired samp... | https://arxiv.org/abs/2502.06051v1 |
obtains her estimator /hatwidef. Assumption 2.1 (General Function Approximation) .There exists a known function class F: S×A→ [0,1] such that∃f∗∈Fsuch that f∗=r. We also need a standard condition to control the complexity o fFthrough the notion of covering number ( Wainwright ,2019, Definition 5.1). Definition 2.2 (ǫ-net... | https://arxiv.org/abs/2502.06051v1 |
performance gap upper bound under linear functi on approximation J(π∗)−J(π)≤/vextenddouble/vextenddoubleEρ×π∗[φ(s,a)]−ν/vextenddouble/vextenddouble Σ−1 off=:RHS, whereνis the reference vector, φ(s,a)∈Rdis the feature map, and Σ off=/summationtextn i=1φ(si,ai)φ(si,ai)⊤ is the sample covariance matrix. However, we can show... | https://arxiv.org/abs/2502.06051v1 |
with O(1) all-policy concentrability. Remark 2.12. Theorem 2.10shows that when ǫis sufficiently small, any algorithm for offline KL-regularized contextual bandits requires at least /tildewideΩ(ǫ−1) samples to output an ǫ-optimal policy. This sample complexity lower bound matches the sample compl exity upper bound in Theore... | https://arxiv.org/abs/2502.06051v1 |
ready to prove Theorem 2.8. Proof of Theorem 2.8.Following the proof of Zhao et al. (2024, Theorem 3.3), we know that there exists ¯γ∈[0,1] such that J(π∗)−J(/hatwideπ)≤ηG(¯γ)≤ηG(0), (2.6) where the first inequality holds due to Lemma 2.17and the second inequality holds due to the eventEand Lemma 2.16. The term G(0) can... | https://arxiv.org/abs/2502.06051v1 |
3.2 (All-Policy Concentrability) .Given a reference policy πref, there exists D >0 such that D2= sup(s,a)∈S×AD2 F((s,a);πref). Assumption 3.3 (Single-Policy Concentrability) .Given areference policy πref, there exists Dπ∗> 0 such that D2 π∗=E(s,a)∼ρ×π∗[D2 F((s,a);πref)]. Remark 3.4. Similar single-policy concentrabilit... | https://arxiv.org/abs/2502.06051v1 |
be concluded by plugging D2≥D2 π∗intoZhao et al. (2024, The- orem 4.3). This implies that the sample complexity lower bou nd in Theorem 3.7also holds for KL-regularized dueling contextual bandit with all-policy concentrability. Remark 3.9. Theorem 3.7shows that when ǫis sufficiently small, any algorithm for offline KL- reg... | https://arxiv.org/abs/2502.06051v1 |
pessimism with a newly identified covariance-ba sed observation, which in turn enables a neat refinement of a mean-value-type argument the its extre me point, showing that KL-PCB achieves a nearly optimal sample complexity under single-p olicy concentrability. The pair of novel techniques are decoupled from tricky algori... | https://arxiv.org/abs/2502.06051v1 |
Since the constructed πref(·|s) inKis a uniform distribution for any s∈S, the all-policy concentrability coefficient D2ofKdefined in Assumption 2.6satisfiesD2=O(1), whichindicatesthatthesingle-policy concentrabili ty coefficient D2 π∗defined in Assumption 2.7satisfiesD2 π∗≤D2=O(1). Thus, the proof of Zhao et al. (2024, Theorem... | https://arxiv.org/abs/2502.06051v1 |
Sci. Math. Hungar. 2299–318. Di, Q.,Zhao, H. ,He, J.andGu, Q.(2023). Pessimistic nonlinear least-squares value iterat ion for offline reinforcement learning. arXiv preprint arXiv:2310.01380 . Dud´ık, M.,Hofmann, K. ,Schapire, R. E. ,Slivkins, A. andZoghi, M. (2015). Contextual dueling bandits. In Conference on Learning T... | https://arxiv.org/abs/2502.06051v1 |
,Gu, Q.andYan, L. (2024). Enhancing multi- step reasoningabilities of language modelsthroughdirect q-function optimization. arXiv preprint arXiv:2410.09302 . Liu, Z.,Li, X.,Kang, B. andDarrell, T. (2019). Regularization mattersinpolicyoptimization. arXiv preprint arXiv:1910.09191 . M¨uller, A. (1997). Integral probabil... | https://arxiv.org/abs/2502.06051v1 |
Y. andGu, Q.(2024). Self-play preference optimiza- tion for language model alignment. arXiv preprint arXiv:2405.00675 . Xie, T.,Cheng, C.-A. ,Jiang, N. ,Mineiro, P. andAgarwal, A. (2021a). Bellman-consistent pessimism for offline reinforcement learning. Advances in neural information processing systems 346683–6694. Xie, ... | https://arxiv.org/abs/2502.06051v1 |
arXiv:2502.06188v1 [math.PR] 10 Feb 2025Nonasymptotic and distribution-uniform Koml´ os-Major-Tusn´ ady approximation Ian Waudby-Smith:, Martin Larsson;, and Aaditya Ramdas; :University of California, Berkeley ;Carnegie Mellon University Abstract We present nonasymptotic concentration inequalities for s ums of independ... | https://arxiv.org/abs/2502.06188v1 |
to the Gaussian law is insensitive t o distributional perturbations. These discussions surrounding distribution-uniformity of the cent ral limit theorem raise the question: is there a sense in which the Koml´ os-Major-Tusn´ ady approximatio ns are strongly distribution-uniform, and if so, for what class of distribution... | https://arxiv.org/abs/2502.06188v1 |
asopn1{qqin (2). This procedure of constructing a new probability space containing the relevant sequences is often given t he name of “the construction” and authors will often start with a sequence pXnq8 n“1onpΩ,F,Pqand say that “a construction exists” and then state the result in terms of prXnq8 n“1and prYnq8 n“1witho... | https://arxiv.org/abs/2502.06188v1 |
mean zero and Var rPαprYq “VarPαpXqfor each αPAas alluded to in Section1.1. 3 The central results of Koml´ os, Major, and Tusn´ ady [ 4,5] aredistribution-pointwise statements, which in the context of our setting means that A“ t9αuis taken to be a singleton. Focusing on the case of finite exponential moments, one of the... | https://arxiv.org/abs/2502.06188v1 |
is a sufficient condition for Sakhanenko regularity, we will show in Proposition 2.1that it is not necessary and that Xis uniformly Sakhanenko regular if and only if it is uniformly sub- exponential in a certain sense. To make this formal, recall the Bernstein parameter of XunderP: bpPq:“inf" bą0 :EP|X|qďq! 2bq´2VarPpXqf... | https://arxiv.org/abs/2502.06188v1 |
common probability space for infinite collections and applying it on doubly expon entially-spaced epochs. The application of Theorem 2.3to showing sufficiency of Sakhanenko regularity in Theorem 2.2can be found shortly after, while the proof of its necessity relies on a distribution -uniform analogue of the second Borel-C... | https://arxiv.org/abs/2502.06188v1 |
ppWpαq nq8 n“1qαPAisA-uniformly strongly approximated by pprVpαq nq8 n“1qαPAifĂWpαq n´rVpαq nvanishes A- uniformly almost surely in the sense of (13). As a shorthand, we write ĂWn´rVn“soAp1q, and we say that ĂWn´rVn“soAprnqfor some monotone real sequence prnq8 n“1ifr´1 npĂWn´rVnq “soAp1q. We now have the requisite defin... | https://arxiv.org/abs/2502.06188v1 |
may be clear how Corollary 3.3follows from the con- centration inequality in Theorem 3.2when instantiated with the same value of qą2, the same cannot be said for Theorem 3.1due to the additional factor of fpnq1{qappearing in the rate of convergence above. Indeed, the proof of Theorem 3.1crucially relies on Corollary 3.... | https://arxiv.org/abs/2502.06188v1 |
deviation inequality for |Y|, then use it within the integrated tail probability representation of the qthmoment for any integer qě3 to obtain an upper-bound on that moment, and ultimately re-writing the final expression in terms of th e Bernstein parameter as in ( 8). To this end, notice that the left-hand side of ( 15... | https://arxiv.org/abs/2502.06188v1 |
we write the expectation EP|Y|qas an integral of tail probabilities so that EP|Y|q“ż8 0Pp|Y|qězqdz ďż8 0Cexpt´tz1{qudz “Cqż8 0expt´tuuuq´1, where we used the change of variables z“uq. Using another change of variables given by u“w{t, we continue from the above and notice that EP|Y|qďCqż8 0expt´tuuuq´1du “Ct´qqż8 0e´wwq... | https://arxiv.org/abs/2502.06188v1 |
into doubly exponentially spac ed epochs. Recalling the definition of Λnfrom (16) and denoting rΛn:“ΛnprX,rYq, note that for any nP t1,2,...u, we can decompose its maximum over any interval ra,bs;aăbwherea,bare both positive integers as max aăkďbrΛk“max aăkďbtrΛk´rΛau `rΛa. (20) We will now break “time” (i.e. the natura... | https://arxiv.org/abs/2502.06188v1 |
ą0. Indeed, let Cą0 be arbitrary and notice that for any αPA, Pαˆ sup kěm|Xk| logkě4C˙ looooooooooooomooooooooooooon p‹q“rPα˜ sup kěm|rXk| logkě4C¸ ďrPαˆ sup kěm1 logk/vextendsingle/vextendsingle/vextendsinglerXk´rYk/vextendsingle/vextendsingle/vextendsingleě2C˙ looooooooooooooooooooomooooooooooooooooooooon p:q`rPαˆ su... | https://arxiv.org/abs/2502.06188v1 |
Cpnq qą0 in place of Cqbut this probability space, the constructed random variables, and the constant indeed depend on nPN. The refinement of Lifshits [ 6, Theorem 3.3] admits the same inequality for any nPNwhere the probability space, the random variables prX1,rX2,...q, and the new constant Cqno longer depend on n. We ... | https://arxiv.org/abs/2502.06188v1 |
k“1ρq k kfpkqfi fl ďε´qCq,f» –8ÿ k“2m´1ρq k kfpkq`1 2m2m´1´1ÿ k“1ρq k kfpkqfi fl, which completes the proof. 18 5.2 Proof of Theorem 3.1 In several instances throughout the proof of Theorem 3.1, we will apply a stochastic and uniform gener- alization of Kronecker’s lemma [ 13, Lemma 1] to show that certain sequences vanish... | https://arxiv.org/abs/2502.06188v1 |
that for every αPAandmě1, 8ÿ k“mVarrPαrprYk´rYpďq kq{k1{qs ď4CqEPαp|X|q1t|X|qąmuq, whereCqą0 depends only on qand not on αPA. Taking a supremum over αPAPPand the limit asmÑ 8, we apply the uniform Khintchine-Kolmogorov convergence theore m [13, Theorem 3] and conclude that Sn”řn k“1prYk´rYpďq kq{k1{qisA-uniformly Cauch... | https://arxiv.org/abs/2502.06188v1 |
random variables in piiq.We are tasked with showing that piiqisbothuniformlyCauchyanduniformlystochasticallynonincreasing. Beginningwiththeformer, observe that for any αPAand any εą0, Pα˜ sup k,něm/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglenÿ i“1Xi1t|Xi| ąi1{qu i1{q´kÿ i“1Xi1t|Xi| ąi1{qu i1{q... | https://arxiv.org/abs/2502.06188v1 |
8forqą2. Let prYnq8 n“1be independent Gaussian random variables on prΩ,rF,rPq with mean zero and rσ2 k:“VarPprYkq “VarPpX1t|X| ďk1{quq. LettingpYk:“ pσ{rσkq ¨rYkfor anykPN, we have that for any mě1, 8ÿ k“mVarrPprYk´pYkq k2{qď4CqEPp|X|q1t|X|qąmuq, whereCqis a constant depending only on q. Proof.First, note that since pY... | https://arxiv.org/abs/2502.06188v1 |
in a proba- bility space and sublinear expectation space. Chinese Journal of Applied Probability and Statistics , 34(6):577–586, 2017. URL https://arxiv.org/abs/1705.08333 .15 [4] J´ anos Koml´ os, P´ eter Major, and G´ abor Tusn´ ady. An appr oximation of partial sums of independent rv’-s, and the sample df. i. Zeitsc... | https://arxiv.org/abs/2502.06188v1 |
Robust Estimation With Latin Hypercube Sampling: A Central Limit Theorem for Z-estimators Faouzi Hakimi1,2 1Aix-Marseille University, 13284 Marseille, France 2Institut Mathématique de Toulouse, 31062 Toulouse, France Correspondence Faouzi Hakimi, Marseille, France Email: faouzi.hakimi@univ-amu.fr Funding informationLat... | https://arxiv.org/abs/2502.06321v1 |
vary uniformly in [0,1]so we have that, for jinJ1,dK={1,2, . . . ,d} and Xj∼U[0,1]. Indeed, one can always work under uniformity and then use the inverse transformation method [9] to place the support back on the original scale and retrieve the original distribution, as long as the sampling distribution of interest is ... | https://arxiv.org/abs/2502.06321v1 |
,1 n(πj(n)−u(n) j)T . (1) The corresponding LHS design of dimension dand sizenis then XLHS=(xLHS 1, . . . ,xLHS d). The LHS method leads 4 F.Hakimi to a good point repartition in the sub-projections of dimension 1. Indeed, a LHS verifies these two properties by definition: •[i∈J1,nK,min 1≤i′≤n(|x(i) j−x(i′) j|)≤2 n. •ma... | https://arxiv.org/abs/2502.06321v1 |
nrealizations of a random vector Xevolving in X⊂Òd, withn,d∈Î∗. Its law is parameterized by a vector θ∈Θ⊂Òq,q∈Î∗. Forx∈X ,θ∈Θ, let(x,θ)→ψθ(x)∈Òqbe a known measurable function such that ψθ(x)=(ψθ1(x), . . . ,ψθq(x))T. We also define the empirical mean of this function (X,θ)→Ψn(θ)∈Òqsuch thatΨn(θ)=1 nÍn i=1ψθ(x(i)). TheZ-... | https://arxiv.org/abs/2502.06321v1 |
of a random variable X=(X1, . . .Xd)evolving inX. Suppose also that the following assumptions are fulfilled: 1.ΨIIDn(ˆθIIDn)=1 nÍn i=1ψˆθIIDn(x(i))=0,[n∈Î∗; 2.there exists a unique θ0inΘsuch thatÅ(ψθ0(X))=Ψ(θ0)=0withθ0inΘ; 3.Å(||ψθ0(X)||2)<+∞; 4.Å(¤ψθ0(X))exists and is non-singular; 5.For any x∈ X and for any θin the ne... | https://arxiv.org/abs/2502.06321v1 |
the interior of a compact subset of Òq,q∈Î∗, andX=[0,1]d,d∈Î∗. For all θ∈Θandx∈X, assume(x,θ)→ψθ(x), whereψθ=(ψθ1(x), . . . ,ψθq(x))T∈Òq, is twice continuously differentiable in θ. Let(x,θ)→¤ψθ(x) ∈Mq,q(Ò)and(x,θ)→¥ψθ(x) ∈Tq,q,q(Ò)denote the first and second-order derivatives of ψθ, respectively. For anyn∈Î∗, let XLHS=(x... | https://arxiv.org/abs/2502.06321v1 |
way of unifying various statistical models, including linear regression, logistic regression and Poisson regression. To estimate the parameters of a GLM, one generally uses a Maximum Likelihood Estimator (MLE). It is therefore a special case ofZ-estimation supposing that the likelihood can be differentiated. Thus, the r... | https://arxiv.org/abs/2502.06321v1 |
in the case of an IID or a LHS design. We can thus conclude that ˆθnconverges in probability in θ0. Let us now verify that the conditions of application of Theorem 3 are fulfilled. First, we see that (XLHS,θ)→ΨLHSn(θ) is continuous and twice continuously differentiable in θ. Plus,Å(ψθ0(X))=Ψ(θ0)=0by construction. We also... | https://arxiv.org/abs/2502.06321v1 |
The MSE is also significantly lower. As shown previously, classic LHS designs allow better estimation performances than IID ones, regardless of the theoretical value of the estimated parameters. 14 F.Hakimi Figure 2 Average estimation variances of (θ0,1, . . .θ0,9)Taccording to the sampling size nfor IID and LHS designs... | https://arxiv.org/abs/2502.06321v1 |
sampling. Reliability Engineering & System Safety 89 2005;p. 305–330. [4] Viana FA. A tutorial on Latin hypercube design of experiments. Quality and reliability engineering international 2016;32(5):1975–1985. [5] Stein M. Large Sample Properties of Simulations Using Latin Hypercube Sampling. Technometrics 2 1987;p. 143... | https://arxiv.org/abs/2502.06321v1 |
AN ITERATIVE BLOCK MATRIX INVERSION (IBMI) ALGORITHM FOR SYMMETRIC POSITIVE DEFINITE MATRICES WITH APPLICATIONS TO COVARIANCE MATRICES∗ ANN PATERSON†, JENNIFER PESTANA‡,AND VICTORITA DOLEAN§ Abstract. Obtaining the inverse of a large symmetric positive definite matrix A ∈Rp×pis a continual challenge across many mathema... | https://arxiv.org/abs/2502.06377v1 |
(1.1) The key observation is that I−L⊤involves only the upper triangular part of A. Thus, if we wish to compute elements Hij,i≤jin the upper triangular part of H(which, sinceAis symmetric, also computes elements Hjiin the lower triangular part), we can work with triangular matrices only. This leads to the recursive for... | https://arxiv.org/abs/2502.06377v1 |
I+1 NsA−1 IAI,IcZIc(ZIc)⊤(AI,Ic)⊤A−1 I. (1.5) As for the simple Monte Carlo estimator (1.4), here Nsis the number of Gaussian samples zk∼ N (0,A−1), while ZIcrepresents the sub-matrix of Zin (1.4) formed from the rows indexed by Ic. When |I|= 1, the Block RBMC estimator becomes the simple RBMC estimator described in [1... | https://arxiv.org/abs/2502.06377v1 |
known as the precision matrix, in multivariate statistics and data science e.g., Gaussian process regression [3,§2]. A lot of the literature reviewed in section 1 focussed on the (partial) inversion of sparse symmetric positive definite matrices. The IBMI algorithm is a novel method which can obtain the inverse of both... | https://arxiv.org/abs/2502.06377v1 |
is certainly not the only choice. Other possible choices for the initial guess will be discussed at the end of subsection 2.2. The Two-Block Non-Overlapping Case Numerical evidence suggests the approximation in (2.3) may not be very accurate, as |˜Hij− H ij|i, j= 1, . . . p , may be large when |i−j|is large, i.e., elem... | https://arxiv.org/abs/2502.06377v1 |
I2AI2,Ic 2˜H(1,1) I1 −˜H(1,1) I1AIc 2,I2A−1 I2˜H(1,1) I1 ="˜H(1,2) I2˜H(1,2) I2,I1 ˜H(1,2) I1,I2˜H(1,1) I1# . This completes one full iteration, as both sets have been used to update the approxi- mate inverse ˜H. This iterative process then continues by incrementing rand iterating through the index sets k= 1,2 as d... | https://arxiv.org/abs/2502.06377v1 |
the remaining sets, namely I3andI4. As for the non-overlapping case, a visual representation of the resulting partitioning into the 2 ×2 structure for (2.3) is given in (2.10). A similar process is then repeated for the other three sets, I2,I3andI4, to complete one iteration. 8 A. PATERSON, J. PESTANA, V. DOLEAN A=· · ... | https://arxiv.org/abs/2502.06377v1 |
related to the error in the initial guess. AN ITERATIVE BLOCK MATRIX INVERSION ALGORITHM FOR SPD MATRICES 9 Algorithm 2.1 Iterative Block Matrix Inversion (IBMI) Algorithm Inputs: A,tol,Ikfork= 1, . . . , K , initial approximation ˜H(0,1) Ic 1ofHIc 1in (2.3). While error <tol fork= 1 : Kdo Determine Ic k. ifk= 1then Ge... | https://arxiv.org/abs/2502.06377v1 |
ρ(AI1,I2A−1 I2AI2,I1A−1 I1)<1 where ρ(·) is the spectral radius. It then follows, by similarity, that that ρ(A−1 I2AI2,I1A−1 I1AI1,I2)<1. Finally, since Ak→0 ask→ ∞ if and only if ρ(A)<1 for any square matrix A, we see that A−1 I2AI2,I1A−1 I1AI1,I2 2r 2→0 asr→ ∞ . Therefore, the two block non- overlapping case in Al... | https://arxiv.org/abs/2502.06377v1 |
2K(K−1) m3 +O 2(K−1)m3 =O1 3+ 2K2 m3 . Hence, the cost per iteration of Algorithm 2.1 is O (1 3+ 2K2)Km3 .The initial guess of the inverse of the Schur complement can also be considered here, but since we use the identity matrix there is no additional cost. When Ais partitioned according to the multi-block no... | https://arxiv.org/abs/2502.06377v1 |
was compared with the time taken for MATLAB’s inverse function inv() to invert the same ma- trices. It can be seen in Figure 1 that Algorithm 2.1 converges faster for covariance Kernel Type Covariance Matrix Exponential Kernel AEXP(x,x′) = exp−|x−x′| 5 RBF Kernel ARBF(x,x′) = exp−|x−x′|2 2(0.6)2 Inverse Quadratic Kerne... | https://arxiv.org/abs/2502.06377v1 |
to direct methods. The best approximated matrices came from the inverse quadratic kernel. The co- variance matrices produced by the exponential covariance kernel also had low errors for smaller covariance matrices, but the error did increase slightly as the dimension increased. This was not seen with the other covarian... | https://arxiv.org/abs/2502.06377v1 |
were used, Algorithm 2.1 took 476 iterations to converge, taking between 323 seconds (for two blocks) and 488 seconds (for six blocks). However, by introducing only a 5% overlap, the algorithm converged in 1 iteration and between 1.14 and 1.31 seconds. The number of blocks used when partitioning the covariance matrix w... | https://arxiv.org/abs/2502.06377v1 |
only a subset of the full inverse is required, such as in the literature discussed in section 2 and referenced in [2, 13, 18, 20]. Appendix A. Error Estimate. The following error was used as a stopping condition for Algorithm 2.1: (A.1) Error = ˜HIAI,Ic+˜HI,IcAIc 2, where ∥ · ∥ 2is the usual matrix norm induced by the ... | https://arxiv.org/abs/2502.06377v1 |
its application to short circuit study , Proc. PICA Conference, June, 1973, (1973). [13]L. Lin, C. Yang, J. C. Meza, J. Lu, L. Ying, and W. E ,SelInv—an algorithm for selected inversion of a sparse symmetric matrix , ACM Transactions on Mathematical Software (TOMS), 37 (2011), https://doi.org/10.1145/1916461.1916464. [... | https://arxiv.org/abs/2502.06377v1 |
REVISITING OPTIMAL PROPORTIONS FOR BINARY RESPONSES : INSIGHTS FROM INCORPORATING THE ABSENT PERSPECTIVE OF TYPE-I E RROR RATECONTROL A P REPRINT Lukas Pin MRC Biostatistics Unit University of Cambridge Cambridge, UK lukas.pin@mrc-bsu.cam.ac.uk Sofía S. Villar MRC Biostatistics Unit University of Cambridge Cambridge, U... | https://arxiv.org/abs/2502.06381v2 |
[stat.ME] 28 Feb 2025 Revisiting Optimal Proportions for Binary Responses A P REPRINT In Section 2, we present two optimal allocation proportions that were presented in Rosenberger et al. [2001] and serve as the foundation of our analysis. The first, Neyman allocation, maximizes the statistical power of a widely used W... | https://arxiv.org/abs/2502.06381v2 |
is often assumed to maximize statistical power [Berger et al., 2021, Friedman et al., 2015] and control the type-I error rate. In contrast, RAR dynamically adjusts allocation probabilities based on accumulating trial data. We assume patients enter the trial sequentially, with outcomes observed immediately, to explore t... | https://arxiv.org/abs/2502.06381v2 |
the optimization problem leads to the RSHIR allocation for the Wald test Z1 ρR1=√p1√p0+√p1. (6) Notice that to derive the optimal proportions, we utilized the theoretical variance of the test statistic Z1and the true success probabilities. However, these parameters are typically unknown in a real trial setting. 2.2 Tar... | https://arxiv.org/abs/2502.06381v2 |
10 0.3 0.3 6.3% 72.0% 66.8% 0.5 (0) 0.47 (0.1432) 0.48 (0.1336) 15 15 15 0.4 0.4 6.2% 64.7% 53.0% 0.5 (0) 0.47 (0.132) 0.48 (0.1063) 20 20 20 0.5 0.5 6.4% 61.9% 38.6% 0.5 (0) 0.47 (0.1261) 0.49 (0.0757) 25 25 25 0.6 0.6 6.0% 65.0% 26.6% 0.5 (0) 0.47 (0.1326) 0.49 (0.0490) 30 30 30 0.7 0.7 6.1% 71.9% 17.8% 0.5 (0) 0.47 ... | https://arxiv.org/abs/2502.06381v2 |
0.000.020.040.06 0.00 0.25 0.50 0.75 1.00 p1Type−I Error RateMethod ρCR 1 ρCR 0 ρN0n ρR0nn=50 Figure 1: Comparison of Type-I error rates of: the Neyman proportion testing with the Wald test ρN1, the RSHIR proportion testing with the Wald test ρN1, complete randomization testing with the Wald test ρCR1, complete random-... | https://arxiv.org/abs/2502.06381v2 |
at the cost of diminishing patient benefit gains (RSHIR) or power gains (Neyman). For Table 2 we used a burn-in period of six patients per arm ( B= 12 ), as recommend by Rosenberger et al. [2001], and 60% of the total trial size to demonstrate the the asymptotic behavior. In smaller trials, this trade-off often 5 Revis... | https://arxiv.org/abs/2502.06381v2 |
Agresti & Caffo correction and differnt burn-in periods ( B= 12 ,B= 30 ,B= 120 ) for two sample sizes, n= 50 (upper table) and n= 200 (lower table). 6 Revisiting Optimal Proportions for Binary Responses A P REPRINT Testing with Z1 p0p1 Type-I Error Rate or Power n1/n ENS ρCR ρN1 ρR1 ρCR ρN1 ρR1 ρCR ρN1ρR1 0.1 0.1 5.0% ... | https://arxiv.org/abs/2502.06381v2 |
the expense of inflating the type-I error rate, even in non-adaptive settings [Fleiss et al., 1981, page 30]. Rosenberger et al. [2001] suggest using the allocation proportion ρN1with the Z0test. However, combining ρN1with the score test does not effectively control the type-I error rate or maximize power, as shown in ... | https://arxiv.org/abs/2502.06381v2 |
4.1 Neyman-like Allocation In Section 3.4 we demonstrated that we can not simply combine use ρN1as an allocation rule to maximize power and test with the score test Z0to control the type-I error rate. To address this limitation, we propose deriving a Neyman-like allocation tailored for the Z0test, minimizing its varian... | https://arxiv.org/abs/2502.06381v2 |
to that of CR. In certain regions, ρn N0performs slightly better than CR, while in other regions, CR holds a marginal advantage. Similarly, for p0= 0.5, the power of ρn N0and CR remains almost identical, with only minor fluctuations in performance. For a higher value of p0= 0.7, the power of both ρn N0and CR continues ... | https://arxiv.org/abs/2502.06381v2 |
successfully controlled the type-I error rate. Even complete randomization exhibited inflation of type-I error. The 10 Revisiting Optimal Proportions for Binary Responses A P REPRINT 0.000.250.500.751.00 0.00 0.25 0.50 0.75 1.00 p1PowerMethod ρCR0ρN0n ρR0nPower for p 0=0.2 and n=50 152025303540 0.00 0.25 0.50 0.75 1.00... | https://arxiv.org/abs/2502.06381v2 |
at minimizing failures ρR1andρn R0, as well as maximizing power, ρN1andρn N0to CR. The analysis involved comparing test statistics Z0andZ1for each method. For each method, a burn-in period of 2patients per arm was implemented. Table 5 demonstrates the proportions ρCR,ρn R0andρn N0effectively control the type-I error ra... | https://arxiv.org/abs/2502.06381v2 |
not perform well. Additionally, one could investigate deriving optimal proportions for finite sample corrections, such as the Agresti & Caffo correction, i.e. using the adjusted estimators in the optimization problem. This work represents a significant step towards broader implementation of these methods. It addresses ... | https://arxiv.org/abs/2502.06381v2 |
Trials , 16:430, 2015. doi:10.1186/s13063-015- 0958-9. URL https://doi.org/10.1186/s13063-015-0958-9 . K. R. Eberhardt and M. A. Fligner. A comparison of two tests for equality of two proportions. The American Statistician , 31(4):151–155, 1977. doi:10.1080/00031305.1977.10479225. J. R. Eisele. The doubly adaptive bias... | https://arxiv.org/abs/2502.06381v2 |
Inference on the cointegration and the attractor spaces via functional approximation Massimo Franchi, Paolo Paruolo February 11, 2025 Abstract. This paper discusses semiparametric inference on hypotheses on the cointegration and the attractor spaces for I(1) linear processes, using canonical correlation analysis and fu... | https://arxiv.org/abs/2502.06462v1 |
fication of the CI parameter matrix β, and tests are informative of economic theory implications that go beyond the ones related to the dimension of col β, the cointegration rank. The present paper considers the hypotheses col b⊆colβ⊆colBin the semiparametric context of Franchi et al. (2024), henceforth FGP. They consi... | https://arxiv.org/abs/2502.06462v1 |
true and 0 otherwise. The space of right continuous functions f: [0,1]7→Rnhaving finite left limits, endowed with the Skorokhod topology, is denoted byDn[0,1], see Jacod and Shiryaev (2003, Ch. VI) for details, also abbreviated to D[0,1] ifn= 1. A similar notation is employed for the space of square integrable function... | https://arxiv.org/abs/2502.06462v1 |
Xt; see Johansen and Juselius (2014) for the discussion of this aspect in a VAR context. Theorem 2.2 (Dimensional coherence, FGP) .LetHbe a p×mfull column rank matrix; if Xt satisfies (2.1) with C(z)fulfilling Assumption 2.1, then the same holds for H′Xt, i.e. ∆H′Xt= G(L)εt, where G(z) :=H′C(z)satisfies Assumption 2.1w... | https://arxiv.org/abs/2502.06462v1 |
attractor space col ψof the following form: H1 0: col ψ⊆colA ⇔ colb⊆colβ (4.1) H2 0: col a⊆colψ ⇔ colβ⊆colB (4.2) where Aisp×m,aisp×q, both of full column rank, q≤s≤m, and b:=A⊥,B:=a⊥. The idea is to test the implication of hypotheses (4.1) and (4.2); these implications are discussed in the following in terms of ψonly.... | https://arxiv.org/abs/2502.06462v1 |
H2 01and rank( a′ ⊥ψ) = rank 0 1 0 1! = 1 = s−q, as in H2 02; hence H2 0holds. Consider next a=e2; one finds rank( a′ψ) = rank(0 ,1) = 1 = qwhich still falls under the null H2 01, and rank( a′ ⊥ψ) = rank( I2) = 2 > s−q, which falls under the alternative H2 12. This illustrates thatH2 1holds, with only one component of ... | https://arxiv.org/abs/2502.06462v1 |
criteria produce an estimate of the number of common trends; hence they can also be called estimators of the number of CT. 11 The estimators (both the ones based on criteria and the ones based on sequences of tests) are denoted as bs(yt),es(yt), and qs(yt), generically indicated as ˘ s(yt), to specify the observable va... | https://arxiv.org/abs/2502.06462v1 |
k=1(1−λk), while n=∞yields ∥τ(j)∥∞= 1−λj; these expressions are similar to the trace and λmaxstatistics in Johansen (1996). The procedure can be described as follows: consider hypothesis Hj:s=jversus HA j:s < j with test statistics Jn(τ(j)) and rejection region Rj,n,ηin the right tail, with significance level η. Start ... | https://arxiv.org/abs/2502.06462v1 |
adl` ag basis of L2[0,1]and let 1≥λ1≥λ2≥ ··· ≥ λp≥0be the eigenvalues of cca(xt, dt), see (5.2) and(5.3); then for (T, K)seq→ ∞ , λip≍( 1 i= 1, . . . , s K/T i =s+ 1, . . . , p, (6.1) 14 for(T, K)seq→ ∞ , and because K/T→0, λip→( 1 i= 1, . . . , s 0 i=s+ 1, . . . , p. (6.2) Moreover P(˘s=s)→1as(T, K)seq→ ∞ for any 0≤s≤... | https://arxiv.org/abs/2502.06462v1 |
the c.d.f and the quantile functions in Theorem 6.3 are available on the website of one of the authors. 16 6.3.Asymptotic properties of criteria on H1 0andH2 0.This section presents novel asymptotic properties of the criteria in (5.9), (5.10). Theorem 6.4 (Limit behavior of decision rules) .Let the assumptions of Theor... | https://arxiv.org/abs/2502.06462v1 |
0 11 0 0 0 0 0 0 0.0438 0.0438 0 0.0438 0.0438 0 12 0 0 0 0 0 0 0.0437 0.0437 0 0.0437 0.0437 0 13 0 0 0 0 0 0 0.0435 0.0435 0 0.0435 0.0435 0 14 0 0 0 0 0 0 0.0433 0.0433 0 0.0433 0.0433 0 15 0 0 0 0 0 0 0.0428 0.0428 0 0.0428 0.0428 0 16 0 0 0 0 0 0 0.0425 0.0425 0 0.0425 0.0425 0 17 0 0 0 0 0 0 0.0425 0.0425 0 0.042... | https://arxiv.org/abs/2502.06462v1 |
0 0.0171 0.0171 0 17 0 0 0 0 0 0 0.0335 0.0335 0 0.016 0.016 0 18 0 0 0 0 0 0 0.0323 0.0323 0 0.0154 0.0154 0 19 0 0 0 0 0 0 0.0312 0.0312 0 0.0144 0.0144 0 H2 0(q) in (7.4) z w v z w v z w v z w v q= 1 0 0 0 0 0 0 0.07 0.0399 0.0315 0.0579 0.0399 0.019 2 0 0 0 0 0 0 0.0737 0.0439 0.0305 0.061 0.0426 0.0188 3 0 0 0 0 0... | https://arxiv.org/abs/2502.06462v1 |
choice with probability 1, both under the null and under the alternative. •The tests based on qsofHj 0,j= 1,2, have empirical size bounded by the nominal level η1+η2= 10%, see (6.6), and power equal to one under Hj 11and in most cases bounded below by 1 −η2= 0.95 under Hj 01, see (6.7). •The results are similar for the... | https://arxiv.org/abs/2502.06462v1 |
0.5363 0.9554 3 1 0 1 1 0 1 0.9993 0.6677 0.9942 0.9995 0.8076 0.9921 4 1 0 1 1 0 1 1 0.8077 0.9997 1 0.9441 0.999 5 1 0 1 1 0 1 1 0.8885 1 1 0.9872 1 6 1 0 1 1 0 1 1 0.9419 1 1 0.9975 1 7 1 0 1 1 0 1 1 0.9732 1 1 0.9997 1 8 1 0 1 1 0 1 1 0.9861 1 1 0.9999 1 Table 3. Empirical power, s= 10: rejection frequency of (7.3)... | https://arxiv.org/abs/2502.06462v1 |
significance level and the number of Monte Carlo replications is set to N= 104. 22 16-Dec-2021 18-Nov-2022 23-Oct-2023 24-Sep-2024-0.3-0.2-0.100.10.20.30.4( p ; T ; b T = p c ) = ( 2 0 , 7 9 3 , 3 9 ) D K E U N O S WS Z U K A U C AH K J P S G B ZC H I N M A M XS A S K T A T H Figure 2. 20 World Markets currencies again... | https://arxiv.org/abs/2502.06462v1 |
CA, CH, DK, HK, IN, JP, MA, MX, NO, SA, SG, SK, SW, SZ, TA, TH, UK }.H1 0amounts to testing if there is a currency differential with the Euro that is stationary while H2 0checks if a cointegrating relation can be expressed as a linear combination of currency differentials. Results are reported in Table 6. The top part ... | https://arxiv.org/abs/2502.06462v1 |
of the nesting of subspaces and the property of dimensional coherence of the semiparametric approach in FGP for subsets of variables. The paper reports some novel results on a limit distribution of interest in the special one- dimensional case. The properties of the estimators and tests are compared with alternatives, ... | https://arxiv.org/abs/2502.06462v1 |
Let S=R1 0B(u)2du. By (10) in Abadir and Paruolo (1997), one has fS(s) =s−3 2√π∞X j=0ηjajexp −a2 j 2s! where ηj:= −1 2 j ,aj:= 2j+1 2. Next define ζ:= 1/S=g(S); by the transformation theorem S=g−1(ζ) = 1 /ζ,|dg−1(ζ)/dζ|=ζ−2, and hence, for z >0 fζ(z) =fS(h−1(z))z−2=z−1 21√π∞X j=0ηjajexp −a2 j 2z! . Note also that Fζ(t... | https://arxiv.org/abs/2502.06462v1 |
arXiv:2502.06514v2 [math.ST] 20 May 2025Fractional interacting particle system: drift parameter e stimation via Malliavin calculus Chiara Amorino∗, Ivan Nourdin†, Radomyra Shevchenko‡ May 21, 2025 Abstract We address the problem of estimating the drift parameter in a syste m ofNinteracting particlesdrivenbyadditivefrac... | https://arxiv.org/abs/2502.06514v2 |
main results 29 7.1 About the propagation of chaos: proofs . . . . . . . . . . . . . . . . . . . . . . . 29 7.1.1 Proof of Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7.1.2 Proof of Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 7.1.3 Proof of Corollary 2.6 . . . .... | https://arxiv.org/abs/2502.06514v2 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 8.2 Proof of Lemma 6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 8.3 Proof of Lemma 6.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 8.4 Proof of Lemma 6.9 . . . . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2502.06514v2 |
such as those between financial assets or neurons in neuroscience. Recently, IPS have also been applied to opinion dynamics (se e [26,52]), modeling how interactions among individuals influence opinion shifts. T his has been particularly relevant in understanding public opinion on COVID-19 vaccinations, as evidenced by s... | https://arxiv.org/abs/2502.06514v2 |
th e measureµθ,N tto the measure ¯µθ t=L(¯Xθ t) in the McKean-Vlasov SDE d¯Xθ t=p/summationdisplay m=1θmbm/parenleftbig¯Xθ t,¯µθ t/parenrightbig dt+σdBH t, t∈[0,T], (4) driven by an fBm BH, see also Section 2.1. This result allows for a type of averaging, similarly to the ergodic theorem in the asymptotic regime where ... | https://arxiv.org/abs/2502.06514v2 |
proposed in [ 60] for estimating the drift parameter in a fractional Ornstein-Uhlenbeck process. Consistency and a symptotic normality were proven in 4 this setting for H∈[1/2,3/4). These results were later extended in [ 61], using a novel method that established validity for any H∈(0,1). Their approach relies on Malli... | https://arxiv.org/abs/2502.06514v2 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.