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, 7:421–444, 1997. URL https://api.semanticscholar. org/CorpusID:7104716 . Jiahua Chen and Zehua Chen. Extended bayesian information criteria for model selection with large model spaces. Biometrika , 95(3):759–771, 2008. ISSN 00063444, 14643510. URL http://www.jstor.org/stable/20441500 . Emilie Chouzenoux, Jean-Christo... | https://arxiv.org/abs/2503.20401v1 |
Statistics and Computing , 34(1):53, 2024. Shu Kay Ng, Thriyambakam Krishnan, and Geoffrey J McLachlan. The em algorithm. Handbook of computational statistics: concepts and methods , pages 139–172, 2012. Edouard Ollier. Fast selection of nonlinear mixed effect models using penalized likelihood. Computational Statistics... | https://arxiv.org/abs/2503.20401v1 |
Concentration inequalities for the sum in sampling without replacement: an approach via majorization Jianhang Ai * Ond ˇrej Ku ˇzelka†Christos Pelekis‡ March 27, 2025 Abstract LetP= (x1, . . . , x n)be a population consisting of n≥2real numbers whose sum is zero, and let k < n be a positive integer. We sample kelements... | https://arxiv.org/abs/2503.20473v1 |
the elements in our sample. More concretely, choose a set I∈ [n] k uniformly at random and let XP=X i∈Ixi. How much does XPfluctuate from its expected value? A bit more precisely, let t≥0be fixed and observe that E(XP) = 0 . What is an upper and a lower bound on the tail probability P(XP≥t)? The problem is of fundamen... | https://arxiv.org/abs/2503.20473v1 |
an upper bound on P(XP≥t)which requires information on the absolute deviation of the population. Our bounds regarding the first case read as follows. Theorem 3. LetP= (x1, . . . , x n)be a population of size n≥2such thatPn i=1xi= 0andPn i=1|xi|>0. LetXPdenote the sum of k∈[n−1]elements that are sampled without replacem... | https://arxiv.org/abs/2503.20473v1 |
= 2·E(X+). (1) The proof of Theorem 3 is based upon the following observation. Lemma 1 (Folklore) .LetXbe a real-valued random variable such that E(X) = 0 . Then P(X > 0) =E(|X|) 2·E(X|X > 0). Proof. Observe that it holds E(X+) =E(max{0, X}) =E(X|X≥0)·P(X≥0) and E(X|X≥0)·P(X≥0) =E(X|X > 0)·P(X > 0). The desired result ... | https://arxiv.org/abs/2503.20473v1 |
x2<α i; hence xj<α iforj≥3and thusPi j=1xi= (x1+x2) +Pi j=3xj< α, a contradiction. Continuing in this manner we see thatPℓ j=1xj≥ℓ·α i, for all ℓ∈ {1, . . . , i}. We now claim that it holds xi+1≥ −α n−i. Indeed, if this is not the case then xi+1<−α n−i and thusPn ℓ=i+1xℓ<−α, a contradiction. Continuing in this manner w... | https://arxiv.org/abs/2503.20473v1 |
{(1−1 s)s}s≥1is increasing implies that C≥1 4·1 4=1 16, as desired. Now we estimate Bfrom below. Since it clearly holds1 12i+1−1 12n≥0and1 12s≤1 12, for s∈ {m, i−m, k−m, n−i−k+m}, we conclude that B≥ −4 12=−1 3. The result follows. Assume now that the parameters n, i, k, m satisfy n−i−k+m= 0. In this case it holds n−i=... | https://arxiv.org/abs/2503.20473v1 |
E(Hi) =ik n= 1. We may assume that i >1; ifi= 1, then k=nand A1>1 2·k n. Now observe that, since i >1, we may write Ai:=ik 2i(ik−i)· P(Hi= 0) +X j≥2(j−1)·P(Hi=j) =k 2i(k−1)· P(Hi= 0) + E(Hi)−P(Hi= 1)−P(Hi≥2) =k i(k−1)·P(Hi= 0) =k i(k−1)· ik−i k ik k=ik−i−k+ 1 i2(k−1)·P(Hi= 1). Since the median of H1is equal t... | https://arxiv.org/abs/2503.20473v1 |
first thatt α−t≥1, so that t∈[α/2, α). Observe that in this case it holds f(0) = 1 , f(1) = 1 andf(−1) =−1; thus E(f(YP∗)) = f(−1)·k(n−k) n(n−1)+f(0)· 1−2k(n−k) n(n−1) +f(1)·k(n−k) n(n−1) = 1−2k(n−k) n(n−1). Now suppose thatt α−t<1, so that t∈(0, α/2). Then f(0) =t α−t, f(1) = 1 andf(−1) =−1; hence E(f(YP∗)) =t α−t·... | https://arxiv.org/abs/2503.20473v1 |
in instances where one has only access to the mean and the absolute deviation of the population. 12 (a)ε= 0.001 (b)ε= 0.005 (c)ε= 0.01 Figure 1: Comparison of the bounds for different values of ε 13 A A lower bound We present a lower bound on P(XP≥t), for t > 0, which applies to moderate values of the parameter t >0. P... | https://arxiv.org/abs/2503.20473v1 |
arXiv:2503.20495v1 [math.ST] 26 Mar 2025REVISITING GENERAL SOURCE CONDITION IN LEARNING OVER A HILBERT SPACE NAVEEN GUPTA AND S. SIVANANTHAN Abstract. In Learning Theory, the smoothness assumption on the target function (known as source condition) is a key factor in establishing t heoretical convergence rates for an es... | https://arxiv.org/abs/2503.20495v1 |
of examples, it does not encompass all index 1 2 N. GUPTA AND S. SIVANANTHAN functions (see [ 18]). For instance, non-smooth index functions can easily be c onstructed that are not expressible as such a product. Consequently, th is restrictive source condition might fail to encompass the true solution, potentially limi... | https://arxiv.org/abs/2503.20495v1 |
as: inf w∈HEz(w),Ez(w) =1 nn/summationdisplay i=1(/an}⌊ra⌋ketle{tw,wi/an}⌊ra⌋ketri}htH−yi)2. (2.2) REVISITING GENERAL SOURCE CONDITION IN LEARNING OVER A HILB ERT SPACE 3 Note that if we consider H=L2([0,1]), then the problem ( 2.1) provides a solution to the functional linear regression (FLR) model [ 21]. Therefore, t... | https://arxiv.org/abs/2503.20495v1 |
reproducing kernel K: H×H→Rgiven asK(w,w′) =/an}⌊ra⌋ketle{tw,w′/an}⌊ra⌋ketri}htH. This space is equipped with the inner product /an}⌊ra⌋ketle{tf,g/an}⌊ra⌋ketri}htHK=/an}⌊ra⌋ketle{t/an}⌊ra⌋ketle{tw′,·/an}⌊ra⌋ketri}htH,/an}⌊ra⌋ketle{tw′′,·/an}⌊ra⌋ketri}htH/an}⌊ra⌋ketri}htHK=/an}⌊ra⌋ketle{tw′,w′′/an}⌊ra⌋ketri}htH,∀f(·) =/... | https://arxiv.org/abs/2503.20495v1 |
function φsuch thatf∈ R(φ(T)). Therefore, the natural way to describe the smoothness of f† His in terms of the decay of its Fourier coefficients, given by ∞/summationdisplay i=1/an}⌊ra⌋ketle{tfρ,ψi/an}⌊ra⌋ketri}ht2 ρH φ2(µi)<∞, (2.11) whereφis an index function on the interval [0 ,s] (s≥κ2). So in terms of source conditi... | https://arxiv.org/abs/2503.20495v1 |
(∗) ≤1 λν′/⌊ard⌊l(T−Tx)(T+λI)−1/⌊ard⌊lL(HK)/⌊ard⌊lf† H/⌊ard⌊lHK+ν′−1/summationdisplay i=11 λi+ν′−i/⌊ard⌊l(T−Tx)(T+λI)−1/⌊ard⌊lL(HK)/⌊ard⌊lf† H/⌊ard⌊lHK =ν′ λν′/⌊ard⌊l(T−Tx)(T+λI)−1/⌊ard⌊lL(HK)/⌊ard⌊lf† H/⌊ard⌊lHK, 8 N. GUPTA AND S. SIVANANTHAN where (∗) follows from the fact that λ(T+λI)−1=I−T(T+λI)−1. Applying Lemma 3... | https://arxiv.org/abs/2503.20495v1 |
can be applied to any reproducing kernel Kthat satisfies the necessary assumptions. Specifically, ifKis a Mercer kernel with eigenvalues µi, i∈Nsatisfying appropriate polynomial decay conditions, then our results remain valid. This generaliza tion is crucial as it ensures that our theoretical findings are not merely confin... | https://arxiv.org/abs/2503.20495v1 |
Group, Dordrecht, 1996. [8]Evgeniou, T., Pontil, M., and Poggio, T. Regularization networks and support vector machines. Adv. Comput. Math. 13 , 1 (2000), 1–50. [9]Guo, Z.-C., Lin, S.-B., and Zhou, D.-X. Learning theory of distributed spectral algorithms. Inverse Problems 33 , 7 (2017), 074009, 29. [10]Gupta, N., Sivan... | https://arxiv.org/abs/2503.20495v1 |
STOCHASTIC TRANSPORT MAPS IN DIFFUSION MODELS AND SAMPLING XICHENG ZHANG Abstract. In this work, we present a theoretical and computational framework for construct- ing stochastic transport maps between probability distributions using diffusion processes. We begin by proving that the time-marginal distribution of the s... | https://arxiv.org/abs/2503.20573v1 |
a simple prior distribution (e.g., Gaussian) into the posterior distribution. Once the map is learned, posterior samples can be generated efficiently [19,30]. Challenges in Constructing Transport Maps. While transport maps offer a promising approach to sampling, constructing them is often non-trivial. Indeed, the exist... | https://arxiv.org/abs/2503.20573v1 |
1.2.Diffusion Models. Diffusion models are a class of generative models that have gained sig- nificant attention in recent years due to their ability to generate high-quality samples, particularly in tasks such as image synthesis, denoising, and data reconstruction (see [6,35] for comprehensive surveys). These models a... | https://arxiv.org/abs/2503.20573v1 |
a unified and rigorous theoretical foundation for various diffusion models. In our framework, the traditional procedures of adding noise and denoising are replaced by a direct learning approach. Specifically, we propose using a neural network to learn the transport map from the Gaussian distribution to the data/target ... | https://arxiv.org/abs/2503.20573v1 |
1[0,1](x). (1.7) Based on ODE (1.4) and Euler’s algorithm, we simulate 5000-trajectories. The following figure illustrates the transport map x7→Xt(x)as defined by the ODE in (1.4): •The left-hand plot displays 5000 simulated trajectories. •The right-hand plot shows the kernel density estimate of the 5000 terminal point... | https://arxiv.org/abs/2503.20573v1 |
challenging. In a recent work [1], Albergo, Boffi, and Vanden-Eijnden introduced a general framework for generative models: stochastic interpolants. Let x0∼ρ0(x)dx,x1∼ρ1(x)dx, and z∼N(0,Id) be independent random variables, and let I∈C2([0,1];C2(Rd×Rd)d) satisfy the conditions I(0, x0, x1) =x0, I(1, x0, x1) =x1,|∂tI(t, ... | https://arxiv.org/abs/2503.20573v1 |
we establish strong convergence in Lp-spaces. In Section 5, we apply our theoretical results to two concrete sampling problems: high-dimensional anisotropic funnel distributions and Gaussian mixture distributions. Numerical experiments demon- strate the effectiveness of our proposed algorithm. Finally, in the Appendix,... | https://arxiv.org/abs/2503.20573v1 |
where for i= 1,2, L(i) sf(x) = tr( a(i)(s, ω, x )· ∇2f(x)) +b(i)(s, ω, x )· ∇f(x). Since X(1)andX(2)are independent, by Fubini’s theorem, we have Eh L(1) sf(·+X(2) s)(X(1) s) +L(2) sf(·+X(1) s)(X(2) s)i =Eh tr (a(1)(s, X(1) s) +a(2)(s, X(2) s))· ∇2f(Xs)i +Eh (b(1)(s, X(1) s) +b(2)(s, X(2) s))· ∇f(Xs)i =E tr(a(s, Xs)... | https://arxiv.org/abs/2503.20573v1 |
11 =Z Rdf(x)Eh a(1)(t, x−X(2) t)ρ(t, x−X(2) t)i dx. Since Xthas density Eρ(t, x−X(2) t), from the above two equalities, it is easy to see that Eh a(2)(t, X(2) t)|Xt=xi =Eh a(2)(t, X(2) t)ρ(t, x−X(2) t)i Eρ(t, x−X(2) t), and Eh a(1)(t, X(1) t)|Xt=xi =Eh a(1)(t, x−X(2) t)ρ(t, x−X(2) t)i Eρ(t, x−X(2) t). This gives (2.10)... | https://arxiv.org/abs/2503.20573v1 |
from Absolutely Continuous Distributions. In this subsection, we use forward ODE starting from any absolutely continuous distribution to generate the sample. For this aim, we assume that µ(dx) =ρ0(x)dx, a t= 0, ηt=βtη, η∼ν, where ρ0is aC1-density function (for examples, Gaussian or stable distribution). Suppose that σt... | https://arxiv.org/abs/2503.20573v1 |
and [11, page 95, Theorem 3.1]. (iii) Fix 0 ⩽t0< t1<1. By (3.15), the definition of bs(x) and Jensen’s inequality, we have ∥Xt1(ξ0)−Xt0(ξ0)∥p⩽Zt1 t0∥bs(Xs(ξ0))∥pds=Zt1 t0Z Rd|bs(x)|pφs(x)dx1/p ds ⩽Zt1 t0|(logσs)′| Z RdE |x−η|pρs(x−βsη) Eρs(x−βsη)φs(x)dx!1/p ds =Zt1 t0|(logσs)′|Z RdZ Rd|x−y|pρs(x−βsy)ν(dy)dx1/p ds... | https://arxiv.org/abs/2503.20573v1 |
is easy to see that ∇xDt(x) =E[(η⊗η)Ct(x, η)] ϵσtECt(x, η)−E[ηCt(x, η)]⊗E[ηCt(x, η)] ϵσt(ECt(x, η))2. SinceEeλ|η|<∞for any λ >0, by the dominated convergence theorem, the function [0 ,1)×Rd∋ (t, x)7→ECt(x, η) is continuous and strictly positive. Hence, inf t∈[0,t0],|x|⩽RECt(x, η)>0. The desired estimate (3.24) follows.... | https://arxiv.org/abs/2503.20573v1 |
ds =Zt1 t0|(logσs)′| EZ RdZ Rd|x−y|pϕϵβsσs(x−βsy−σsξ0)ν(dy)dx1/p ds. By the change of variables x−βsy−σsξ0=σszandβs+σs= 1, we obtain Zt1 t0bs(Xs)ds p⩽Zt1 t0|σ′ s| EZ RdZ Rd|z+ξ0−y|pϕϵβs/σs(z)ν(dy)dz1/p ds =Zt1 t0|σ′ s| p ϵβs/σsξ+ξ0−η pds, where ξ∼N(0,Id) is independent of ξ0andη. Hence, ∥Xt1−Xt0∥p⩽ Zt1 t0bs(Xs)ds p... | https://arxiv.org/abs/2503.20573v1 |
(3.36), we obtain (3.37). Moreover, by (3.35), we also have φt(x) =Eh e−|x−βtη|2/(2ℓtσt)i (2πℓtσt)d/2=e−|x|2/(2ℓt)Eh e(βtσt|η|2−βt|x−η|2)/(2ℓtσt)i (2πℓtσt)d/2, which, by (3.39), gives (3.38). □ Remark 3.15. Ifλ=ϵ=γ= 1, then ℓt=ϵβt+γσt= 1, and (3.37) reduces to bt(x) =σ′ tEh ξe(βt|x−√σtξ|2+σt|ξ|2)/2ρ1(x−√σtξ)i √σtE e(β... | https://arxiv.org/abs/2503.20573v1 |
∗∥r PN i=1(C(ξi)−EC(ξ)) q NEC(ξ). 22 XICHENG ZHANG On the other hand, for independent random variables ( Xi)i=1,...,dwith zero mean, by Burkh¨ older’s inequality (see [31, Theorem 6.3.6]), E NX i=1Xi p ⩽CpN(p/2)∨1−1NX i=1E|Xi|p, p∈[1,∞). (4.3) Using this inequality, we derive that PN i=1(ξiC(ξi)) PN i=1C(ξi)−E(ξC(ξ)) E... | https://arxiv.org/abs/2503.20573v1 |
bN t(x) =σ′ tPN i=1(x−ηi)Ct(x, ηi) σtPN i=1Ct(x, ηi). We also consider the following SDE: dXN t=bN t(XN t)dt+p ϵβ′ tdWt, XN 0=ξ0∼N(0, γId). Clearly, we have XN t=Xy t|y=η. LetµN tbe the law of XN t. Under the assumptions of Theorem 4.4, we have ∥µN t−µt∥2 var⩽8K2 NE eβt(2|η|2+E|η|2)/(2ϵσt) 2|η|2+E|η|2! . In fact, we al... | https://arxiv.org/abs/2503.20573v1 |
ϵσtδd/qλ(3/2−2/q)d! lnN N, t∈(0,1), where ℓt=ϵβt+γσtand Vt:=Z Rdeq2|z|2/ℓteqE|η|2/(4ℓtσt)ρq 1(z)dz. 26 XICHENG ZHANG Proof. Following the proof of Theorem 4.4, we have E ∥µy t−µt∥2 var|y=ξ ⩽Zt 0β′ s ϵσ2sZ RdE|EN s(x,ξ)|2φs(x)dxds, (4.12) where φs(x) is given by (4.10) and EN s(x,ξ) :=E[ξCs(x, ξ)] ECs(x, ξ)−PN i=1[ξiC... | https://arxiv.org/abs/2503.20573v1 |
18: end for 19: return X{Final samples } 28 XICHENG ZHANG Using the above Algorithm, where the parameters are choosen as follows: βt= sin2πt 2 , N = 100 ,000, ϵ= 1, γ= 2,num steps = 100 , we generated 10,000 sample points and plot the histogram below when the dimensions are 2, 7, 10, 15, and 20. For the high-dimensio... | https://arxiv.org/abs/2503.20573v1 |
the covariance matrix Σ is potentially degenerate. We set d= 100, m= 0, and let Σ be a degenerate covariance matrix. 30 XICHENG ZHANG We generated 2,000 sample points and plotted their scatter plots along the (0 ,1)-dimension and (23,54)-dimension, as shown below. In the plots, the red circular line represents the true... | https://arxiv.org/abs/2503.20573v1 |
−e−c0t2=x, we have I=Z1 0 1−xN dln(1−x)−1 c0p/2 =NZ1 0ln(1−x)−1 c0p/2 xN−1dx=:I1+I2, where I1:=NZ1 1/Nlnx−1 c0p/2 (1−x)N−1dx and I2:=NZ1/N 0lnx−1 c0p/2 (1−x)N−1dx. ForI1, we have I1⩽lnN c0p/2 NZ1 1/N(1−x)N−1dx=lnN c0p/2 (1−1/N)N⩽(lnN)p/2 cp/2 0e. ForI2, we have I2⩽Nc−p/2 0Z1/N 0 lnx−1p/2dx=Nc−p/2 0Z∞ ln... | https://arxiv.org/abs/2503.20573v1 |
Ho, A. Jain, and P. Abbeel. Denoising Diffusion Probabilistic Models. In Proceedings of the 37th International Conference on Machine Learning (ICML) , 2020. [15] M. H. Kalos and P. A. Whitlock. Monte-Carlo Method . WILEY-VCH Verlag GmbH & Co., KGaA, Weinheim, 2008. [16] I. Kobyzev, S. Prince, and M. Brubaker. Normalizi... | https://arxiv.org/abs/2503.20573v1 |
PARAMETER ESTIMATION FOR FRACTIONAL AUTOREGRESSIVE PROCESS WITH PERIODIC STRUCTURE A P REPRINT Cai Chunhao School of mathematics(Zhuhai) Sun Yat-sen University Zhuhai,People’s Republic of China caichh9@mail.sysu.edu.cnShang Yiwu School of mathematical sciences Nankai University Tianjin,People’s Republic of China shangy... | https://arxiv.org/abs/2503.20736v1 |
ϕi(u). Additionally, we will prove that this estimator is asymptotically normal. Time series models with long memory show long range dependencies between distant observations, posing challenges to traditional statistical analysis and forecasting. In the PFAR model, long memory comes from fractional Gaussian noise, wher... | https://arxiv.org/abs/2503.20736v1 |
set with the following expression, Θl u∗={ϕ(u)∈R;the roots of 1−ϕ(v)z= 0have modulus ≥1 +l} We define the set Θl uas the Cartesian product Θl⋆ u×[d1, d2], where lis a positive constant and [d1, d2]∈(0,1). A1:ϕ(u)∈(−1,1)andH∈(0,1). Notation: ByL− →andP− →, respectively, we denote convergence in law and convergence in pr... | https://arxiv.org/abs/2503.20736v1 |
I(λ) =1 2πn|nX t=1Ytexp(itλ)|2, (11) λj=2πj n, j∈{1,2, ...m}, (12) aj= log(2 sinλj 2),am=1 mmX j=1aj, S m=mX j=1(aj−am)2. (13) We estimate d by regressing logI(λj)with respect to aj, such that ˆdn=−1 2SmmX j=1(aj−am) logI(λj), (14) The estimator ˆHnis defined by ˆHn=ˆdn+1 2, (15) Remark 2. There are several semi-parame... | https://arxiv.org/abs/2503.20736v1 |
2+∞X i=1,3,5,...ϵH 2n+2−i[ϕ(2)ϕ(1)]i−1 2ϕ(2), (27) X2n+1=∞X i=0,2,4,...ϵH 2n+1−i[ϕ(2)ϕ(1)]i 2+∞X i=1,3,5,...ϵH 2n+1−i[ϕ(2)ϕ(1)]i−1 2ϕ(1), (28) 5 A P REPRINT From equations (27) and (28), the vectors (X1, X2, ..., X 2n−1)and(X2, X4, ..., X 2n)have the following form X1 X3 ... X2n−1 = P∞ i=0,2,4,...ϵH 1−i[... | https://arxiv.org/abs/2503.20736v1 |
2(2)ϕk1 2(1) E ϵH 2n+1−s1ϵH 2n+1−k1 +ϕ2(2)∞X s2=1,3,5...∞X k2=1,3,5... ϕs2−1 2(2)ϕk2−1 2(1) E ϵH 2n+1−s1ϵH 2n+1−k1 + 2ϕ(2)∞X s3=0,2,4...∞X k3=1,3,5... ϕs3 2(2)ϕk3−1 2(1) E ϵH 2n+1−s3ϵH 2n+1−k3 (40) (41) and the covariance of ϵH (n−s)T+uandϵH (n−k)T+uis Cov(ϵH 2n+1−s, ϵH 2n+1−k) =1 2 |(s−k) + 1|2H−2|(s−k)T|2H+|(s−k)T−1... | https://arxiv.org/abs/2503.20736v1 |
gaussian noise. Proof. According to the definition of Yn, we can obtain Yn=X2n+X2n+1 =X i=0,2,4,...ϵH 2n+1−i[ϕ(1)ϕ(2)]i 2+X i=1,3,5,...ϵH 2n−i[ϕ(1)ϕ(2)]i−1 2ϕ(2) +X i=1,3,5,...ϵH 2n+1−i[ϕ(1)ϕ(2)]i−1 2ϕ(1) +X i=0,2,4,...ϵH 2n−i[ϕ(1)ϕ(2)]i 2 =X i=0,2,4,...ϵH 2n+1−i[[ϕ(1)ϕ(2)]i 2+ [ϕ(1)ϕ(2)]i−2 2ϕ(2)] +X i=1,3,5,...ϵH 2n+... | https://arxiv.org/abs/2503.20736v1 |
n,HT B2j2−1U(1) n,H =ϕ(1) nnX j1=1nX j2=1[ϕ(1)ϕ(2)]j1+j2−2nX r1=j1nX r2=j2Ω−1 r1,r2Ω∗ 2r1−1−2j1,2r2−2j2(66) where Ω∗ 2r1−1−2j1,2r2−2j2is the covariance matrix of vector (ϵH 1, ϵH 2, ..., ϵH 2n). Since Ω∗ i,jdecreases as |i−j|increases, we have Case 1: 2r1−2j1−2r1+ 2j2−1≥1: E(Bn n)≤ϕ(1) nnX j1=1nX j2=1[ϕ(1)ϕ(2)]j1+j2... | https://arxiv.org/abs/2503.20736v1 |
n,HΣn,HΩ−1 2 n,H, then there exists C1∈Rn×1such that Ω−1 2 n,HΣn,HΩ−1 2 n,HC1=ξ∥C1∥2. (91) 14 A P REPRINT Taking C2= Ω−1 2 n,HC1, we obtain from this last equation that CT 2Σn,HC2=ξ Ω−1 2 n,HC2 2 , (92) By the equation (90), we deduce that CT 2Ωn,HC2≤1 D(1) ϕ(1),ϕ(2)ξ Ω−1 2 n,HC2 2 , (93) and ξ≥CT 2Ωn,HC2 Ω−1 2 n,HC2 2... | https://arxiv.org/abs/2503.20736v1 |
By classifying the magnitudes of 2r3+ 1−2j2−2r4and2r1+ 1−2j1−2r2, We can divide the summation into the following four parts. (1)2r3+ 1−2j2−2r4>1⇔r3> r4+j2 (2)2r3+ 1−2j2−2r4<1⇔r3< r4+j2 (3)2r1+ 1−2j1−2r2>1⇔r1> r2+j1 (4)2r1+ 1−2j1−2r2<1⇔r1< r2+j1 By the symmetry of Ωi,jand the monotonicity of ρϵH(i−j), we only need to co... | https://arxiv.org/abs/2503.20736v1 |
0 when ris odd), with D2,H ϕ(1),ϕ(2)being another constant related to H, ϕ(1), ϕ(2). Secondly, we analyze Rr 3. Starting from the k-th term of the expansion in equation (2), we have lim n→∞E n−r 2r k AkBr−k =r k lim n→∞E n−r 2 nX j=1[ϕ(1)ϕ(2)]j−1B2j−2U(1) n,H k nX k=0[ϕ(1)ϕ(2)]k−1B2k−1U(1) n,H!r−k We c... | https://arxiv.org/abs/2503.20736v1 |
1 n˜Φn∗ 1(2)A(2) n(H)Φn−1 1(1)P− − − − → n→∞k(2) H,ϕ(2)(125) n−3 2˜Φn∗ 1(2)A(3) n(Hn)Φn−1 1(1) = OP(1) (126) where k(1) H,ϕ(2),k(2) H,ϕ(2)are constants and A(1) n(H),A(2 n(H),A(3) n(H)are A(1) n(H) =−Ω−1 n,H∂Ωn,H ∂HΩ−1 n,H (127) A(2) n(H) = Ω−1 n,H∂2Ωn,H ∂2HΩ−1 n,H (128) + 2Ω−1 n,H∂Ωn,H ∂HΩ−1 n,H∂Ωn,H ∂HΩ−1 n,H A(3) n(... | https://arxiv.org/abs/2503.20736v1 |
to a normal distribution 1√n˜Φn∗ 1(2) Ω−1 n−1(H) Φn−1 1(1)P− − − − → n→∞N(0, σH ϕ(1),ϕ(2)), (144) when n→ ∞ , the reminder R(1) nconverges to 0. Thus, we can rewrite equation (142) as follows √m(ˆϕn(2)−ϕ(2)) =√m1 n˜Φn∗ 1(2) Ω−1 n−1(H) Φn−1 1(1) 1 nΦn−1∗ 1(1) Ω−1 n−1(ˆHn) Φn−1 1(1)+R(1) n (145) =√m√n1 σH ϕ(1),ϕ(2)... | https://arxiv.org/abs/2503.20736v1 |
does not depend on 1 +e−2λi ϕ(1)[1−e−2λiϕ2(2)ϕ2(1)]+e−λi(1+ϕ(1)) 1−e−2λiϕ2(2)ϕ2(1) 2 , and the modulus is bounded. Proposition 5.2. We let lnbe the log-likelihood function of a stationary process (Yn)n∈N. We assume that gH,ϕ(λ) satisfies the regularity conditions and let B(θ, R)(open ball of center θand radius R) for s... | https://arxiv.org/abs/2503.20736v1 |
of θ. When gH,ϕ(λ)satisfy regular condition, we have a asymptotic normal distribution of ˜θnthat √n(˜θn−θ)P− − − − → n→∞N(0,I(θ)−1), Proof. We will discuss the consistency and asymptotic normality of one-step estimator. (1) Consistency of ˜θn Observing equation (160) , the first and second terms on the right-hand side ... | https://arxiv.org/abs/2503.20736v1 |
simulation of initial estimator and one-step estimator where θ= (0.7,0.6,0.6)form= [n3 5],n= 100 . 27 A P REPRINT Figure 2: The simulation of initial estimator and one-step estimator where θ= (0.7,0.6,0.6)form= [n3 5],n= 1000 . 28 A P REPRINT Figure 3: The simulation of initial estimator and one-step estimator where θ=... | https://arxiv.org/abs/2503.20736v1 |
H, at the same time, we found that as the sample size increases, the estimation becomes more efficient. According to [Hariz et al., 2024] and our simulations, the one-step estimation also has a faster running speed. 34 A P REPRINT 7 Conclusions This paper presents a study on parameter estimation for a periodic fraction... | https://arxiv.org/abs/2503.20736v1 |
pages 129–157. University of California Press, 1956. doi:10.1525/9780520313880-014. Yu A Kutoyants and Anastasia Motrunich. On multi-step mle-process for markov sequences. Metrika , 79:705–724, 2016. doi:10.1007/s00184-015-0574-4. Arnaud Gloter and Nakahiro Yoshida. Adaptive estimation for degenerate diffusion processe... | https://arxiv.org/abs/2503.20736v1 |
arXiv:2503.20852v1 [stat.OT] 26 Mar 2025TEACHABLE NORMAL APPROXIMATIONS TO BINOMIAL AND RELATED PROBABILITIES OR CONFIDENCE BOUNDS LUTZ MATTNER Abstract. The text below is an extended version of an abstract for a talk , with approximately the same title, to be held at the 7thJoint Statistical Meeting of the Deutsche Ar... | https://arxiv.org/abs/2503.20852v1 |
quite wide ly teachable, whereas others might only be of interest for anybody trying to improve the th eorems, or for the teacher of the theorems to make sure he knowns what he is talking about. All proofs are collected in section 5. They mainly consist of references to the literature, so clearly there is not much of o... | https://arxiv.org/abs/2503.20852v1 |
(5) hold more generally than stated above: We call any law of the form ∗n j=1Bpjwith p∈[0,1]naBernoulli convolution (also known as aPoisson binomial law ), write ˇPfor the reflection of any law P, defined by ˇP(B):=P(−B) forB⊆Rmeasurable, and we let here Pdenote the set of all limit laws (say with respect to weak converg... | https://arxiv.org/abs/2503.20852v1 |
to 0 .6379 and 0 .3440 changed to 0 .3190, apply with the exception of 1.2(e) . Thus the analogues of ( 4) and ( 5) are (8)0.1386 1∨σ≤1 2√ 1 + 12 σ2≤ /bardblF−G/bardbl∞≤0.5410 ∧0.3190 σ≤0.5410 1∨σ and (9)0.2773 1∨σ≤1√ 1 + 12 σ2≤max IL.H.S.( 6)≤1∧0.6379 σ≤1 1∨σ. (b)Better upper bounds for /bardblF−G/bardbl∞and max IL.H.... | https://arxiv.org/abs/2503.20852v1 |
O(1/n) for /hatwidepbounded away from 0 and 1, so the valid bound pis a good substitute. (e)In the example of x= 0, we have p(0) = 1 −(1−β)1 n<−log(1 −β) nalways ,≈and<3 nifβ= 0.95, (15) /tildewidep(0) = 0 , (16) p(0) =z/radicalBig 1 +z2 9+ 1 +z2 3 nalways ,=3.777. . . nifβ= 0.95. (17) 5.Proofs Proofs for Theorem 1.1an... | https://arxiv.org/abs/2503.20852v1 |
example Mattner and Schulz (2018, p. 733, (18) with h= 1), and (1 + 12 σ2)−1/2≥13−1/2(1∨σ)−1, 13−1/2= 0.277350 . . .. 4. If FandGare distribution functions with identical means and varian ces, and with G normal, then /bardblF−G/bardbl∞<0.5409365 . . .holds by Chebotarev et al.’s (2007) refinement of Bhattacharya and Ran... | https://arxiv.org/abs/2503.20852v1 |
mathematica l study of the Laplace– Gaussian law. Acta Mathematica 77, no. 1, 1–125, https://doi.org/10.1007/BF02392223 . (1956). A moment inequality with an application to the central limit theorem. Skandinavisk Aktuarietid- skrift 39, 160–170. Norbert Henze (2019). Zentraler Grenzwertsatz für die Binomialverteilung: ... | https://arxiv.org/abs/2503.20852v1 |
arXiv:2503.21137v1 [math.ST] 27 Mar 2025Variable selection via thresholding Ka Long Keith Ho1and Hien Duy Nguyen1,2 1Joint Graduate School of Mathematics for Innovation, Kyush u University, Fukuoka, Japan. 2Department of Mathematics and Physical Science, La Trobe Un iversity, Melbourne, Australia. March 28, 2025 Abstra... | https://arxiv.org/abs/2503.21137v1 |
[2005], among other early pioneers. Detailed accounts of the shrinkage estimat ion literature can be found in the texts of Bühlmann and van de Geer [2011], Hastie et al. [2015] , Rish and Grabarnik [2014], and van de Geer [2016]. Typically, a shrinkage estimator ˆβn=/parenleftBig ˆβn,1,...,ˆβn,p/parenrightBig⊤ will be ... | https://arxiv.org/abs/2503.21137v1 |
with con- nections between such asymptotics and information criteri a, reported in Nguyen [2024]. The proofs are further made possible using the random empirical proces ses theory of van der Vaart and Wellner [2007], as presented in van der Vaart and Wellner [2023]. Alo ng with our theory, we also provide some illustra... | https://arxiv.org/abs/2503.21137v1 |
anticipate that Rn,kwill converge to some limiting risk functional rk, defined for each β∈Rpby rk(β) =E/bracketleftBig/braceleftbig Y−X⊤Tk(β,β0)/bracerightbig2/bracketrightBig , whose minimum value we can write as ψk= min β∈Rprk(β). We note that this minimum and its empirical variant exist bec ause bothRn,kandrkare quad... | https://arxiv.org/abs/2503.21137v1 |
4 h3(x−δk)3ifδk<b≤δk+h 2, 4 h3(x−δk−h)3+1 ifδk+h 2<δk+h, 1 ifb≥δk+h. We can check that tk=τk◦ |·|satisfy Assumption 2, as required. For the remainder of the manuscript, we will use this spline construction as our choi ce for thresholding. However, as will be apparent in the sequel, the thresholding function serves as a... | https://arxiv.org/abs/2503.21137v1 |
that√n(Rn−r)converges weakly in ℓ∞(Θ), asn→ ∞. Denoting weak convergence by/squiggleright, we make the following assumptions. Assumption 4. The variables Xandǫhave finite fourth moments: i.e., E/bracketleftbig /ba∇dblX/ba∇dbl4/bracketrightbig <∞and E[ǫ4]<∞. Assumption 5. The estimator ˆβnconverges in distribution to β0:... | https://arxiv.org/abs/2503.21137v1 |
that the least squares loss ˜r(β) =E/bracketleftBig/braceleftbig Y−X⊤β/bracerightbig2/bracketrightBig has the same minimum as that of the relevant loss function ˜rR(βR) =E/bracketleftBig/braceleftbig Y−X⊤ RβR/bracerightbig2/bracketrightBig , where the relevant set is Sc 0=Sc K. However, we note that that this minimum i... | https://arxiv.org/abs/2503.21137v1 |
Assumptions 1–5,√n(¯βn,R−β0,R)/squigglerightZRandlimn→∞P(¯βn,I=0) = 1. The result of Theorem 4 therefore justifies that the threshol ding process is a one-step improve- ment to any non-sparse estimator that converges in distribu tion, producing a sparse estimator that has the same asymptotic distribution in the relevant... | https://arxiv.org/abs/2503.21137v1 |
estimated to be irrelevant. The two latter indices can be characterized as t he average false negative rate (FNR) and average true negative rate (TNR), respectively, as perc entages where FNR= 1−/vextendsingle/vextendsingle/vextendsingleˆSc n∩Sc 0/vextendsingle/vextendsingle/vextendsingle |Sc 0|, and TNR =/vextendsingl... | https://arxiv.org/abs/2503.21137v1 |
like the LASSO, but instead me rely shrink weak signals arbitrarily close to zero. As such, many practitioners have taken to simp ly conduct arbitrary thresholding of small signals as an ad hoc approach to variable selection, wh ich our thresholding approach now makes rigorous. In this section, we consider the same sim... | https://arxiv.org/abs/2503.21137v1 |
patients: the logar ithm of cancer volume (X1), logarithm of weight (X2), age(X3), logarithm of the amount of benign prostatic hyperplasia (X4), seminal vesicle invasion (X5), logarithm of capsular penetration (X6), Gleason score (X7), and percentage Gleason scores of four or five (X8). The response variable is the loga... | https://arxiv.org/abs/2503.21137v1 |
selection applies. Thirdly, our current theory makes few restrictions on the class of allowable penalty functions, and in effect, makes no recommendations regarding the design of an efficient penalty function. Similar questions of optima l penalization appears in the setting of model selection oracle bounds, manifesting a... | https://arxiv.org/abs/2503.21137v1 |
that E/bracketleftbig¯L(X,Y)2/bracketrightbig ≤C/braceleftBiggp/summationdisplay j=1p/summationdisplay j′=1E/bracketleftbig |XjXj′|2/bracketrightbig +p/summationdisplay j=1/bracketleftbig E|Xjǫ|2/bracketrightbig/bracerightBigg <∞, and thus (ii) holds, implying that LisP-Donsker. Next, to check that ¯∆nP−→0, we define ∆n... | https://arxiv.org/abs/2503.21137v1 |
class (cf. van der Vaart and Wellner, 2023, Sec. 2.1 ), we have our required result. Proposition 2. Under Assumptions 2 and 4, (1B) /squigglerightR, whereR: Ω→ℓ∞(Θ). It remains to show that (1C) /squigglerightZ⊤∇2g(·,β0). To this end, we follow the suggestion from van der Vaart and Wellner [2023, Sec. 3.13] and apply t... | https://arxiv.org/abs/2503.21137v1 |
survey. Journal de la Société Française de Statistique , 160:1–106, 2019. Jean-Patrick Baudry, Cathy Maugis, and Bertrand Michel. Sl ope heuristics: overview and imple- mentation. Statistics and Computing , 22:455–470, 2012. Lucien Birgé and Pascal Massart. Minimal penalties for Gaus sian model selection. Probability t... | https://arxiv.org/abs/2503.21137v1 |
thresholding est imators in high-dimensional gaussian regression models. Econometric Reviews , 35:1412–1455, 2016. 27 Gideon Schwarz. Estimating the dimension of a model. Annals of Statistics , 6:461–464, 1978. Alexander Shapiro, Darinka Dentcheva, and Andrzej Ruszczy nski. Lectures on Stochastic Pro- gramming: Modelin... | https://arxiv.org/abs/2503.21137v1 |
A Computational Theory for Efficient Mini Agent Evaluation with Causal Guarantees Hedong YAN∗ Institute of Computing Technology Chinese Academy of Sciences No.6 Kexueyuan South Road, Zhongguancun, Haidian District, Beijing, China herdonyan@outlook.com Abstract In order to reduce the cost of experimental evaluation for ... | https://arxiv.org/abs/2503.21138v5 |
conducted. This proliferation of experiments raises ethical concerns regarding participant exposure to potential harm during the randomization process. For instance, as highlighted by Zhou et al. [2024b], the automation and scalability of A/B testing frameworks, while beneficial for rapid experimentation, necessitate c... | https://arxiv.org/abs/2503.21138v5 |
strategy, our evaluation model reduced 89.4% evaluation errors. The evaluation acceleration ratio is ranged from 1000 times to 10000000 times comparing with experimental or simulation based evaluation in the 12 scenes. 2 Notation Variable Name Description S evaluation subject the operable and indivisible unit that we (... | https://arxiv.org/abs/2503.21138v5 |
and metric may introduce the confounding bias if the evaluation task is causality sensitive, such as individual medicine. Causal inference is often used to handle hidden common cause from data. However, applying existed methods to handle unmeasured hidden common cause problem in evaluation faced those challenges: •Hidd... | https://arxiv.org/abs/2503.21138v5 |
from 0 to 1. 4 In practice, naandnbare typically zero for infinite evaluation subject (non-positivity Imbens and Rubin [2015]); in such cases, upper bound of generalized effect error is given by Theorem 4. Theorem 4. Causal bound with non-positivity . Given unbiased evaluation model ˆf, then∀sa∈ S,∀sb∈S, P(Egen(∆ˆf)<2(... | https://arxiv.org/abs/2503.21138v5 |
then computed using predefined proxy functions, facilitating the development of tailored evaluation models. For the assessment of unseen subjects, the process involves computing proxy values, determining the subject type, selecting the appropriate vectorization function, and inputting these into the corresponding evalu... | https://arxiv.org/abs/2503.21138v5 |
are listed in the appendix D, E, F, G. 6 6.1 Evaluation model To demonstrate the generalizability of our evaluation model in agent space, we consider 11 distinct scenes. In these scenes, computational evaluation can be helpful to reduce experimental cost, shorten simulation time, or enhance the efficiency of advertisin... | https://arxiv.org/abs/2503.21138v5 |
The metric is Return of Invest of a subject in a time slot (10 days) in future. The condition is the stock’s time-series variables in the last time slot. In this scene, we not only measure the subject and its metric, but also consider the pre-trade variables (condition) of the subject in an attempt to reduce evaluation... | https://arxiv.org/abs/2503.21138v5 |
vector, subject vector, and proxy metrics for 8 (a) Shapley Value of Subject and Proxy when metric is ROC-AUC. (b) Shapley Value of Subject and Proxy when metric is ACC. (c) Shapley Value of Subject and Proxy when metric is RMSE. (d) Shapley Value of Subject and Proxy when metric isR2. Figure 4: Shapley value of subjec... | https://arxiv.org/abs/2503.21138v5 |
Md Sajidul Huq Tanjil, Dena Weitzman, Tinglong Dai, Brandie D Wagner, David H Cherwek, Nathan Congdon, and Khairul Islam. Autonomous artificial intelligence increases real-world specialist clinic productivity in a cluster-randomized trial. NPJ Digit. Med. , 6(1):184, October 2023. Josh Abramson, Jonas Adler, Jack Dunge... | https://arxiv.org/abs/2503.21138v5 |
YLearn: A Python Package for Causal Inference. https://github.com/DataCanvasIO/YLearn, 2022. Version 0.2.x. 11 Ben Bogin, Kejuan Yang, Shashank Gupta, Kyle Richardson, Erin Bransom, Peter Clark, Ashish Sabharwal, and Tushar Khot. SUPER: Evaluating agents on setting up and executing tasks from research repositories. In ... | https://arxiv.org/abs/2503.21138v5 |
Li, Yaohui Wang, Yi Yu, Yi Zheng, Yichao Zhang, Yifan Shi, Yiliang Xiong, Ying He, Ying Tang, Yishi Piao, Yisong Wang, Yixuan Tan, Yiyang Ma, Yiyuan Liu, Yongqiang Guo, Yu Wu, Yuan Ou, Yuchen Zhu, Yuduan Wang, Yue Gong, Yuheng Zou, Yujia He, Yukun Zha, Yunfan Xiong, Yunxian Ma, Yuting Yan, Yuxiang Luo, Yuxiang You, Yux... | https://arxiv.org/abs/2503.21138v5 |
Leskovec. Mlagentbench: evaluating language agents on machine learning experimentation. In Proceedings of the 41st International Conference on Machine Learning , ICML’24. JMLR.org, 2024. Nguyen Huu Tiep. Lattice-physics (PWR fuel assembly neutronics simulation results). UCI Machine Learning Repository, 2024. DOI: https... | https://arxiv.org/abs/2503.21138v5 |
. Ang Li and Judea Pearl. Bounds on causal effects and application to high dimensional data. Proceed- ings of the AAAI Conference on Artificial Intelligence , 36(5):5773–5780, Jun. 2022. doi: 10.1609/ aaai.v36i5.20520. URL https://ojs.aaai.org/index.php/AAAI/article/view/20520 . Fengzi Li. Causal effect engine. https:/... | https://arxiv.org/abs/2503.21138v5 |
Kevin Schulman. Use of voice-based conversational artificial intelligence for basal insulin prescription management among patients with type 2 diabetes: A randomized clinical trial. JAMA Network Open , 6(12): e2340232–e2340232, 12 2023. ISSN 2574-3805. doi: 10.1001/jamanetworkopen.2023.40232. URL https://doi.org/10.100... | https://arxiv.org/abs/2503.21138v5 |
David Schnurr, John Schulman, Daniel Selsam, Kyla Sheppard, Toki Sherbakov, Jessica Shieh, Sarah Shoker, Pranav Shyam, Szymon Sidor, Eric Sigler, Maddie Simens, Jordan Sitkin, Katarina Slama, Ian Sohl, Benjamin Sokolowsky, Yang Song, Natalie Staudacher, Felipe Petroski Such, Natalie Summers, Ilya Sutskever, Jie Tang, N... | https://arxiv.org/abs/2503.21138v5 |
networks by decentralized control. Eur. Phys. J. Spec. Top. , 225(3):569–582, May 2016. Patrick Schwab, Lorenz Linhardt, Stefan Bauer, Joachim M. Buhmann, and Walter Karlen. Learning counterfactual representations for estimating individual dose-response curves. Proceedings of the AAAI Conference on Artificial Intellige... | https://arxiv.org/abs/2503.21138v5 |
Karvanen. Causal effect identification from multiple in- complete data sources: A general search-based approach. Journal of Statistical Software , 99 (5):1–40, 2021. doi: 10.18637/jss.v099.i05. URL https://www.jstatsoft.org/index.php/ jss/article/view/v099i05 . L. G. Valiant. A theory of the learnable. In Proceedings o... | https://arxiv.org/abs/2503.21138v5 |
the identifiability of the post-nonlinear causal model. In Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence , UAI ’09, page 647–655, Arlington, Virginia, USA, 2009. AUAI Press. ISBN 9780974903958. Yang Zhao and Qing Liu. Causal ml: Python package for causal inference machine learning... | https://arxiv.org/abs/2503.21138v5 |
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