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+n 2dlog 2πσ2, V(qt, αt) =1 dZ1 2σ2∥y−Xθ∥2qt(θ)dθ+1 dDKL(qt∥g(·, αt)⊗d) +n 2dlog 2πσ2. Let Π X∈Rn×nbe the orthogonal projection onto the column span of X. Then, applying the above forms with D KL(qt∥g(·, αt)⊗d)≥0 and noting that ∥(I−ΠX)(y−Xθ)∥2=∥(I−ΠX)y∥2which does not depend onθ, we have V(q0, α0)−V(qt, αt) ≤1 dZ1 2σ2... | https://arxiv.org/abs/2504.15558v1 |
equations ω(α) =δ/(σ2+ mse( α)) and ω∗(α) =δ/(σ2+ mse ∗(α)), we have ω(α)−1=s2+oδ(1), ω ∗(α)−1=s2+oδ(1). (216) We recall from Lemma 4.12 that ω(α), ω∗(α) must be continuous functions of α∈O. We now argue via the implicit function theorem that for all δ > δ 0sufficiently large, these are in fact continuously-differentia... | https://arxiv.org/abs/2504.15558v1 |
(56) in the scalar channel model with fixed noise variance s2. [Note that fixing an arbitrary point α0∈Oand integrating this gradient approximation over α∈O, this also implies F(α) =G(α) + (F(α0)−G(α0)) +oδ(1), i.e.Fapproximately coincides with Gup to an additive shift.] Furthermore, the above continuous-differentiabil... | https://arxiv.org/abs/2504.15558v1 |
subset S⊂RK, there exist constants B(S), c0(S)>0 depending on the fixed values {p1, . . . , p K},{ω1, . . . , ω K}andSsuch that for all α∈S, we have ωk 2(θ−αk)2≥ωkmax 2(θ−αkmax)2+c0(S)θ2for any θ > B (S) and all k̸=kmax. This implies there exists a constant C(S)>0 for which ⟨1ι̸=kmax⟩ ≤C(S)e−c0(S)θ2for all θ > B (S) an... | https://arxiv.org/abs/2504.15558v1 |
the preceding example show ⟨|ωι− ⟨ωι⟩|2⟩ ≤C′(S)e−c0θ,⟨|ωι(µι−θ)− ⟨ωι(µι−θ)⟩|2⟩ ≤C′(S)(1 + θ)2e−c0θ, implying via Cauchy-Schwarz that each order-2 derivative in (225) is bounded over α∈Sandθ > B . Similarly it is bounded over α∈Sandθ <−B, hence also for all α∈Sandθ∈R. The same argument applies to bound the mixed cumulan... | https://arxiv.org/abs/2504.15558v1 |
of g(·) andR(·). Since {(θt,ηt,bαt)}t∈[0,T]={(θt M,ηt M,bαt M)}t∈[0,T]a.s. for all large n, d, this implies that (226) holds also with {(θt,ηt,bαt)}t∈[0,T]in place of {(θt M,ηt M,bαt M)}t∈[0,T]. Furthermore, the deterministic limit process {αt M}t∈[0,T]must satisfy ∥αt M∥ ≤Mfor all t∈[0, T], so the solution up to time ... | https://arxiv.org/abs/2504.15558v1 |
λ+xµ(dx) cinit η(s) =Z(E(θ∗)2x+ 1)λx (λ+x)2e−(λ+x)sµ(dx), ctti η(∞) =Z(E(θ∗)2x+ 1)λ2 (λ+x)2µ(dx) +δ−1 ctti η(τ) =ctti η(∞) +Zx λ+xe−(λ+x)τµ(dx), rtti η(τ) =Z xe−(λ+x)τµ(dx). These functions ctti θ, ctti ηhave the forms (33) for the positive, finite measures µθ(da) =a−1µ(d(a−λ)) and µη(da) = [( a−λ)/a]µ(d(a−λ)) supporte... | https://arxiv.org/abs/2504.15558v1 |
ε)−1−ε), so the coefficient of θ2 jis negative for sufficiently small ε >0. Then by Lemma C.1,qφj(θj) satisfies the univariate LSI (17) with constant CLSI:= (4 /c0) exp(8 r2 0(c0+C)2/(πc0)). So the product law qφsatisfies the LSI with the same constant by tensorization, and Eφ∼µEntθ∼qφf(θ)2≤CLSIEφ∼µEθ∼qφ∥∇f(θ)∥2 2=CLSI... | https://arxiv.org/abs/2504.15558v1 |
and the last inequality applies (234). Then we have sup t∈[0,T]E(ut−˜ut)2≤3h sup t∈[0,T]E(ut−vt)2+ sup t∈[0,T]E(vt−˜vt)2+ sup t∈[0,T]E(˜ut−˜vt)2i ≤6C0γ+ 12ε+ 3Nmax i=1E(Xi−˜Xi)2≤6C0γ+ 12ε+ 3(T/γ+ 1)ε. The conclusion follows by choosing γ=√ Tε. References [1] Greg CG Wei and Martin A Tanner. A Monte Carlo implementation... | https://arxiv.org/abs/2504.15558v1 |
88(1):76–82, 2011. [19] Po-Ru Loh, George Tucker, Brendan K Bulik-Sullivan, et al. Efficient Bayesian mixed-model analysis increases association power in large cohorts. Nature genetics , 47(3):284–290, 2015. 76 [20] Gerhard Moser, Sang Hong Lee, Ben J Hayes, et al. Simultaneous discovery, estimation and prediction anal... | https://arxiv.org/abs/2504.15558v1 |
model. Journal of Physics A: Mathematical and General , 27(17):5749, 1994. [38] G Ben Arous and Alice Guionnet. Large deviations for Langevin spin glass dynamics. Probability Theory and Related Fields , 102:455–509, 1995. 77 [39] G Ben Arous and Alice Guionnet. Symmetric Langevin spin glass dynamics. The Annals of Prob... | https://arxiv.org/abs/2504.15558v1 |
Theory , pages 635–678. PMLR, 2021. 78 [57] Toshiyuki Tanaka. A statistical-mechanics approach to large-system analysis of CDMA multiuser de- tectors. IEEE Transactions on Information theory , 48(11):2888–2910, 2002. [58] Andrea Montanari and David Tse. Analysis of belief propagation for non-linear problems: The exampl... | https://arxiv.org/abs/2504.15558v1 |
Protter. Stochastic differential equations . Springer, 2005. [77] Amir Dembo and Jean-Dominique Deuschel. Markovian perturbation, response and fluctuation dissi- pation theorem. Annales de l’IHP Probabilit´ es et statistiques , 46(3):822–852, 2010. [78] Xian Chen and Chen Jia. Mathematical foundation of nonequilibrium ... | https://arxiv.org/abs/2504.15558v1 |
arXiv:2504.15753v1 [math.OC] 22 Apr 2025 1 Markov Kernels, Distances and Optimal Control: A Parable of Linear Quadratic Non-Gaussian Distribution Steering Alexis M.H. Teter, Wenqing Wang, Sachin Shivakumar, Abhish ek Halder, Senior Member, IEEE Abstract —For a controllable linear time-varying (LTV) pair(At,Bt)andQtposi... | https://arxiv.org/abs/2504.15753v1 |
bility Gramian Γtt0, i.e., Γtt0:=/integraldisplayt t0ΦtτBτB⊤ τΦ⊤ tτdτ≻0,0≤t0≤t <∞. As expected, (3) reduces to (1) for (At,Bt) = (0,I). Both (1) and (3) are instances of κthat are transition probabilities, and satisfy κ≥0,/integraltext Rnκdy= 1. They solve 1endowed with the topology of weak convergence 2 Kolmogorov’s f... | https://arxiv.org/abs/2504.15753v1 |
a.k .a. the action of the Green’s function/integraldisplay Rnκ(t0,x,t,y)/hatwideϕ0(y)dy, whereκsolves (6), and /hatwideϕ0is a suitable measurable function. In other words, the state cost-to-go manifests as a reaction rate in the PDE for the Markov kernel . Then, knowing a closed- form formula for κfacilitates the solut... | https://arxiv.org/abs/2504.15753v1 |
solving an associated deterministic optimal control problem. Motivated by the structural observations made in Sections II and III, we next follow the computational template: Marko v kernel←−distance function←−deterministic optimal control problem, to derive the Markov kernel for the Itˆ o diffusion ( 4) with rate of ki... | https://arxiv.org/abs/2504.15753v1 |
=x, zτ(τ=t) =y. Notice that in objective (12), the state cost q is the rate of creation or killing of probability mass. Let us verify this postulate for known cases discussed before. Heat kernel (1).Hereq≡0, anddwτ/ma√sto→udτin (2) gives the controlled ODE ˙zτ=√ 2uτwhere dot denotes derivative w.r.t.τ∈[t0,t]. This lead... | https://arxiv.org/abs/2504.15753v1 |
not a transitio n probability, the normalization condition no longer holds, and the pre-factor c(t,t0) =n/productdisplay i=1/parenleftbiggωi 4πsinh(ωi(t−t0))/parenrightbigg1/2 (20) does not follow from there. However, having determined (19) , the pre-factor (20) can be obtained by substituting (9) with (19) in (6). See... | https://arxiv.org/abs/2504.15753v1 |
notations. Proposition 1. [29, p. 140-141] Suppose there exists sym- metric matrix K1such that the solution map3Π(τ,K1,t)of 3The mapping Π(τ,K1,t)is understood as the solution of (25a) at any τ∈[t0,t]solved backward in time with initial condition (25b).the Riccati matrix ODE initial value problem ˙Kτ=−A⊤ τKτ−KτAτ+KτˆBτ... | https://arxiv.org/abs/2504.15753v1 |
g κin the postulated form (9), i.e., in the form κ(t0,x,t,y) =c(t,t0)exp/parenleftBigg −1 2/parenleftbiggx y/parenrightbigg⊤ Mtt0/parenleftbiggx y/parenrightbigg/parenrightBigg ,(32) is to compute the pre-factor c(t,t0). Unique identification of c(t,t0)serves the dual purpose of uniquely determining the kernel as well a... | https://arxiv.org/abs/2504.15753v1 |
(38c) in unknown primal-dual pair/parenleftbig ρu opt(t,x),S(t,x)/parenrightbig , i.e., the optimally controlled joint state PDF and the value function . The Hopf-Cole transform [34], [35] given by /parenleftbig ρu opt,S/parenrightbig /ma√sto→(/hatwideϕ,ϕ) :=/parenleftbig ρu optexp(−S),expS/parenrightbig , (39) recasts... | https://arxiv.org/abs/2504.15753v1 |
presence of killing of probability mass at a rate that is convex quadratic with time-varying weight matrix. The resulting kernel is parameterized by the LTV matrix pair (At,Bt)and the killing weight Qt/{ollowsequal0, in terms of the solution of a Riccati matrix ODE initial value problem. The derived kernel has relevanc... | https://arxiv.org/abs/2504.15753v1 |
coth( ωiτ)/parenrightbigg/parenleftbigg√ωixi√ωiyi/parenrightbigg is a positive definite quadratic form, we have 0≤exp/parenleftBigg −1 2ωi/parenleftbig x2 i+y2 i/parenrightbig cosh(ωiτ)−2ωixiyi 2sinh(ωiτ)/parenrightBigg ≤1. Hence, using the dominated convergence theorem [43, Thm. 1.13], we exchange the limit and integra... | https://arxiv.org/abs/2504.15753v1 |
c(t,t0)up to a constant a. Combining (9) with (29), we get logκ= logc(t,t0)−1 2/parenleftbiggx y/parenrightbigg⊤ Mtt0/parenleftbiggx y/parenrightbigg . (63) Applying ∂tto both sides of (63) and rearranging, gives ∂tκ=κ/parenleftBigg ˙c c−1 2/parenleftbiggx y/parenrightbigg⊤ ˙Mtt0/parenleftbiggx y/parenrightbigg/parenri... | https://arxiv.org/abs/2504.15753v1 |
turn made possible by the fact that M11 being a sum of two positive definite matrices (see (30)), is positive definite. Now the idea is to evaluate the limits in (77). For the exponential of negative quadratic term, using (30), we find M22(t,t0)−M⊤ 12(t,t0)M−1 11(t,t0)M12(t,t0) =ˆΓ−1 tt0−ˆΓ−1 tt0ˆΦtt0/parenleftig ˆΦ⊤ tt0... | https://arxiv.org/abs/2504.15753v1 |
1163–1178, 2021. [3] Y . Chen, T. T. Georgiou, and M. Pavon, “Stochastic contro l liaisons: Richard Sinkhorn meets Gaspard Monge on a Schrodinger bridg e,”Siam Review , vol. 63, no. 2, pp. 249–313, 2021. [4] A. M. Teter, Y . Chen, and A. Halder, “On the contraction co efficient of the Schr¨ odinger bridge for stochastic... | https://arxiv.org/abs/2504.15753v1 |
Occasion of the Case Centennial Celebration . Springer, 1982, pp. 11–27. [22] R. S. Strichartz, “Sub-Riemannian geometry,” Journal of Differential Geometry , vol. 24, no. 2, pp. 221–263, 1986. [23] M. Gromov, “Carnot-Carath´ eodory spaces seen from wit hin,” in Sub- Riemannian geometry . Springer, 1996, pp. 79–323. [24... | https://arxiv.org/abs/2504.15753v1 |
. Zhou, Stochastic controls: Hamiltonian systems and HJB equations . Springer Science & Business Media, 2012, vol. 43. [43] E. M. Stein and R. Shakarchi, Real analysis: measure theory, integration, and Hilbert spaces . Princeton University Press, 2009. [44] J. Zinn-Justin, Quantum field theory and critical phenomena . O... | https://arxiv.org/abs/2504.15753v1 |
Multiscale detection of practically significant changes in a gradually varying time series Patrick Bastian, Holger Dette Ruhr-Universit¨ at Bochum Fakult¨ at f¨ ur Mathematik 44780 Bochum, Germany Abstract In many change point problems it is reasonable to assume that compared to a benchmark at a given time point t0the ... | https://arxiv.org/abs/2504.15872v1 |
approximately (see for instance Aston and Kirch, 2012; Hotz et al., 2013, for some examples), there exist also many situations where it is not reasonable to assume that the mean function µin model (1.1) is piecewise constant over the full period because it is continuously smoothly between potential jumps points. Exampl... | https://arxiv.org/abs/2504.15872v1 |
are either obtained by asymptotic theory (which shows that a properly scaled version of the estimate converges weakly to a Gumbel type distribution) or by resampling based on a Gaussian approximation. As a consequence, the resulting procedure suffers from several deficits First, it is conservative in finite samples, pa... | https://arxiv.org/abs/2504.15872v1 |
global mean 4 temperature anomalies, a reasonable choice is ∆ = 1 .5 degrees Celsius corresponding to the Paris Agreement adopted at the UN Climate Change Conference(COP21) in Paris, 2015. In other circumstances the choice of ∆ might not be so obvious and has to be carefully discussed for each application. We defer fur... | https://arxiv.org/abs/2504.15872v1 |
|k−j|=c k>j≥⌊nt0⌋√c ˆµ⌊nt0⌋ 0−ˆµk j −Γn(c)−d∞ , which reduces to the statistic ˆTn,∆in (2.5), if the centering term d∞defined in (2.3) is replaced by the threshold ∆. Theorem 2.2. Grant assumptions (A1) and (A2). We then have ˆTnd−→Td∞ (2.6) where Td∞:=σlim ϵ↓0sup (s,t)∈Aϵ,d∞n s(s, t)√ t−sB(t0) t0−B(t)−B(s)√t−s −Γ(... | https://arxiv.org/abs/2504.15872v1 |
assumptions (A1) and (A2) and be satisfied and consider local alterna- tives of the form (2.14) withβn=n−1/2, then ˆTn,∆d→σsup t0≤s<t≤1√ t−sB(t0) t0−B(t)−B(s)√t−s−Γ(t−s) +1√t−sZt sh(x)dx We collect some observations that follow from this result in the following remark. 8 Remark 2.5. (1) By Theorem 2.4 the test (2.13)... | https://arxiv.org/abs/2504.15872v1 |
question when a deviation from the reference value µt0 0is practically significant, which is related to the specification of the effect size (see Cohen, 1988). While in many situations, such as in the climate data example mentioned before, this specification is quite obvious, there are other applications where this cho... | https://arxiv.org/abs/2504.15872v1 |
> t∗(3.4) where ˜t > t∗is the smallest point with a jump of the function µat˜t(if there are no jumps fort > t∗we set ˜t= 1). Theorem 3.2. Let Assumptions (A1) - (A3) be satisfied and let cnsatisfy c3/2 n≲np log(n). Then (a) If t∗∈(t0,1]is a jump discontinuity satisfying (3.4), then ˆt=t∗+OPcn n . (b) If t∗=∞, we have... | https://arxiv.org/abs/2504.15872v1 |
parameter asatisfies a≤128/81≃1.58 and that we expect an increasing number of rejections for larger values of a, which yield increasing values for the difference d∞−∆. Note that all three tests are conservative in the sense that the empirical size is smaller than 5% at the boundary of the hypotheses (2.2) defined by d∞... | https://arxiv.org/abs/2504.15872v1 |
runs. We observe that the estimator introduced in (B¨ ucher et al., 2021) does not detect a relevant deviation in more than 90% of all cases for many of the considered settings while our estimator detects 15 such a deviation almost always. However, due to the noise of the error process there exist also cases where ˆtun... | https://arxiv.org/abs/2504.15872v1 |
fChange (see Sonmez et al., 2025) on Github and the sample size varies between 100 and 150, depending on the weather station. For each weather station we test the hypotheses (2.2) for different thresholds ∆ ∈ {0.5,1,1.5}, where t0is chosen such that the years 1 , ..., nt 0 correspond to the time frame until the year 19... | https://arxiv.org/abs/2504.15872v1 |
detects larger differences than the test proposed in (B¨ ucher et al., 2021). In particular we are able to detect changes in Boulia and Hobart where the method (B¨ ucher et al., 2021) does 18 not detect any relevant deviation. While at Hobart the difference in the value ˆδ0.05is small, it is larger than 1 degree Celsiu... | https://arxiv.org/abs/2504.15872v1 |
>0 we have |µ(i/n)−µt0 0| ≥∆ +ρfor all jn≤i≤kn. Consequently, using Lemma 5.1, we obtain 1 k0k0X i=1µ(i/n)−1 cknX i=jn+1µ(i/n) ≥∆ +ρ−O(n−1) which yields ˆTn,∆→ ∞ by an application of the triangle inequality. 21 5.1.1 Proof of Lemma 5.1 - 5.4 Proof of Lemma 5.1. Let us first assume that µhas no discontinuities, then ¯µk... | https://arxiv.org/abs/2504.15872v1 |
anyδ1>0, a set A1with probability 1 −δ1on which there exists δ2>0 such that sup (s,t)∈Abn∩{k0/n,..., 1}2 |s−t|≥cn/n s(j/n, k/n )√cˇB(k0) k0−ˇB(k)−ˇB(j)√c −Γn(c) = sup (s,t)∈Abn∩{k0/n,..., 1}2 |s−t|≥δ2 s(j/n, k/n )√cˇB(k0) k0−ˇB(k)−ˇB(j)√c −Γn(c) holds. Exactly the same arguments also yield a set A2with probabil... | https://arxiv.org/abs/2504.15872v1 |
the inequality follows by the Lipschitz continuity of µt sinsandt. Standard arguments then yield ˆσ2=1 ⌊n/m⌋ −1⌊n/m⌋−1X i=1 2B(im)−B((i−1)m)−B((i+ 1)m)2 2m+OP(n−1/3) which in turn yields the desired statement by noting that Zim=B(im)−B((i−1)m) is a triangular array of independent N(0, σ2) variables. 5.5 Proof of Theo... | https://arxiv.org/abs/2504.15872v1 |
Series B: Statistical Methodology , 81(3):649–672. Berkes, I., H¨ ormann, S., and Schauer, J. (2011). Split invariance principles for stationary processes. The Annals of Probability , 39(6):2441 – 2473. 30 Bogachev, V. (2015). Gaussian Measures . Mathematical Surveys and Monographs. American Mathematical Society. B¨ uc... | https://arxiv.org/abs/2504.15872v1 |
uller, H.-G. (1992). Change-Points in Nonparametric Regression Analysis. The Annals of Statistics , 20(2):737 – 761. Page, E. S. (1955). A test for a change in a parameter occurring at an unknown point. Biometrika , 42(3-4):523–527. Schmidt-Hieber, J., Munk, A., and D¨ umbgen, L. (2013). Multiscale methods for shape co... | https://arxiv.org/abs/2504.15872v1 |
Deep learning of point processes for modeling high-frequency data∗ Yoshihiro Gyotoku†1,3, Ioane Muni Toke‡2,3, and Nakahiro Yoshida§1,3 1University of Tokyo, Graduate School of Mathematical Sciences¶ 2Universit´ e Paris-Saclay, CentraleSup´ elec, Math´ ematiques et Informatique pour la Complexit´ e et les Syst` emes‖ 3... | https://arxiv.org/abs/2504.15944v1 |
belongs to the family FT. Three examples of setting ( a, b) will be provided later. A family FTwe are interested in in this article is a deep neural network the size of which is increasing to infinity as T→ ∞ . However, the result we will obtain is more general and not confined to the case of deep learning (DL). The ai... | https://arxiv.org/abs/2504.15944v1 |
prediction error has been a hot topic in the nonparametric statistical approaches to it. Among these efforts, several survey papers (e.g., [26, 6, 4, 5]) provide a comprehensive overview of the state of the art, offering valuable insights into the key open questions and major developments in the field. More specificall... | https://arxiv.org/abs/2504.15944v1 |
argu- ment “ x”. Denote by Aa family of pairs of bounded measurable mappings ( a, b) onXsuch that sup (a,b)∈A |a| ∞∨ ∥b∥∞ ≤F for some positive constant F. The true function ( a∗, b∗) correspond the true structure is assumed to satisfy ( a∗, b∗)∈A, as well as FT⊂A. We consider an estimator ( baT,bbT) for ( a∗, b∗) fro... | https://arxiv.org/abs/2504.15944v1 |
combined with an error bound of the approximation by the machine FTand an estimate of its covering number NT. Schmidt- Hieber [22] considered a deep neural network with ReLU activation function and presented a covering number when the network is fitted under a sparse condition. The shifted ReLU activation function σv:R... | https://arxiv.org/abs/2504.15944v1 |
we obtain the following estimate of the risk in the prediction withFTspecified above, if combined with the properties (3.3)-(3.4) ( pℓare bounded by s). Theorem 3.1. Letξ >0. If∆T≤C0ϕTL(logT)4(T≥T0)for some positive constants C0 andT0>1, then there exists a constant Csuch that RT≤CϕTL(logT)4(3.5) forT≥T0whenever T≥ξ(lo... | https://arxiv.org/abs/2504.15944v1 |
of [17]. We start by considering the 8- dimensional point process ( Ni,ki)i∈I,ki∈Ki. We define the ratio functions ri,ki 1(x, y) =λi,ki(x, y)P j∈I,kj∈Kjλj,kj(x, y)and ˜ ri,ki 1(x, y) =ri,ki 1(x, y) r0,0 1(x, y). (4.5) Obviously,P i∈I,ki∈Kiri,ki 1= 1, ˜ r0,0 1= 1 andP i∈I,ki∈Ki˜ri,ki 1=1 r0,0 1. In this first estimation... | https://arxiv.org/abs/2504.15944v1 |
k∈Kexp(li,k 2(Yt))dNi,ki t. (4.12) 4.1.4 Fitting results As a first illustration, we simulate the model (4.1)-(4.4) for an horizon T= 128 ,000 (note that given the above definitions, a sample has roughly 2 Tpoints in each of the 4 dimensions of the process in this model). We then fit the model with our two estimation m... | https://arxiv.org/abs/2504.15944v1 |
by the one-step estimation method with ˆpi,ki 1(x, y) =exp(ˆli,ki 1(x, y)) P j∈I,kj∈Kjexp(ˆlj,kj 1(x, y)), (4.14) and by the two-step estimation method with ˆpi,ki 2(x, y) =exp(ˆli 2(x))P j∈Iexp(ˆlj 2(x))exp(ˆli,ki 2(y))P k∈Kiexp(ˆli,k 2(y)). (4.15) Figure 4 plots the true functions pi,ki(x, y) and the estimated functi... | https://arxiv.org/abs/2504.15944v1 |
(m= 1 for the one-step ratio method, m= 2 for the two-step ratio method). For each covariate Z∈ {X0, X1, Y}, we compute the 1% and 99% empirical quantiles qZ 0.01andqZ 0.99on the (full) data and define a 1-dimensional regular grid of sizeG+ 1: GZ= qZ 0.01+gqZ 0.99−qZ 0.01 M:g= 0, . . . , G (4.18) The uniform L2-type ... | https://arxiv.org/abs/2504.15944v1 |
nL 1=nL 2,1=nL 2,2andnN 1=nN 2,1=nLN2,2as we did above, then the two-step estimation has a much larger number of parameters since we use 1 + # I= 5 networks very close in shape to the single one used for the one-step estimation method. In the case nL 1=nL 2,1=nL 2,2= 8 and nN 1=nN 2,1=nLN2,2= 64, this represents 33,991... | https://arxiv.org/abs/2504.15944v1 |
pressure on the price. Let X1 t−be the sign of the last trade, i.e. X1 t−=−1 if the last transaction occured on the bid side of the limit order book, X1 t−= 1 if the last transaction occured on the ask side of the limit order book. It is well-known in high-frequency finance that the series of trade signs have long-memo... | https://arxiv.org/abs/2504.15944v1 |
to observe a market order on the side i with price-changing character kwhen the LOB has imbalance x0, spread x2and the last traded order was if sign x1.) Figure 7 plots the fitting results. The first row plot the fitted probabilities ˆ pi,ki 1(x0, x1, x2) (one-step estimation). From left to right, the four columns plot... | https://arxiv.org/abs/2504.15944v1 |
ak(x)−a∗(x) + bk(x)−b∗(x) . 5.2 Estimate of Φ(5.4) T Define rk Tby rk T= ( T−1(logT)2logNT)1/2 ∨ h−1Zh 0E −λ∗(Xt)· ak(Xt)−a∗(Xt) + bk(Xt)−b∗(Xt) dt1/2 (5.5) 20 fork∈ {1, ...,NT}. The random number rk Tisrk Twith kplugged into k. We have Φ(5.4) T = T−1EZT 0bUT(Xt)−bUT(Xt) dt ≤ T−1EZT 0 Uk T(Xt)−Uk T(Xt) dt... | https://arxiv.org/abs/2504.15944v1 |
ϵ)≤4 +16γ−1 1+ϵ−1 1−exp −γ 1+ϵ−1. (5.24) 23 We know r−1 T≤(5.13)T1/2(logT)−1(logNT)−1/2=T1/2(logNT)1/2(logT)−1(logNT)−1 ≤(5.17)z−1(logT)−1(logNT)−1x (5.25) and hence (rT)−2ϵ 1+ϵ≤ z−2ϵ 1+ϵx2ϵ 1+ϵ(logT)−2ϵ 1+ϵ(logNT)−2ϵ 1+ϵ ≤(5.15)z−2ϵ 1+ϵx2ϵ 1+ϵ(logNT)−2ϵ 1+ϵ. (5.26) We have (rT)2T≥(5.13)(logT)2logNT≥(5.15) (5.16)4∥U∥... | https://arxiv.org/abs/2504.15944v1 |
≤2dN∥λ∗∥∞ baT−ak ∞ ≤d (baT,bbT),(ak, bk) ≤δ, we obtain Φ(5.3) T ≤ T−1EZT 0 ak(Xt)−a∗(Xt) ·deNt +δ = E rk TFT−1 × rk TFh−1ZT 0 ak(Xt)−a∗(Xt) ·deNt +δ ≤(5.8)E FT−1 T−1(logT)2logNT1/2+C∗2bE1/2 T+δ |MT| +δ ≤ C∗2FT−1R1/2 T E[|MT|2]1/2+FT−1 T−1(logT)2logNT1/2+δ E[|MT|] +δ, (5.43) where MT= (rk T)−1F−1h... | https://arxiv.org/abs/2504.15944v1 |
We haveZ∞ C9T1/2(logT)(logNT)1/22xP Mk T ≥x,Ω(k, T) dx ≤(5.47)Z∞ C9T1/2(logT)(logNT)1/24xexp −x2 2C12F−2h−1TlogTψ2F2x C12rk TTlogT dx ≤(5.51)Z∞ C9T1/2(logT)(logNT)1/24xe−C16T−1/2(logNT)1/2xdx. (5.54) Applying Lemma 5.5 in the case where q= 1, p= 1 and C=C16T−1/2(logNT)1/2, we obtain the estimateZ∞ C9T1/2(logT)(lo... | https://arxiv.org/abs/2504.15944v1 |
Muzy, J.F.: Modelling microstructure noise with mutually exciting point processes. Quantitative Finance 13(1), 65–77 (2013) [3] Bowsher, C.G.: Modelling security market events in continuous time: Intensity based, multivariate point process models. Journal of Econometrics 141(2), 876–912 (2007) [4] DeVore, R., Hanin, B.... | https://arxiv.org/abs/2504.15944v1 |
books. Quantitative Finance 19(4), 549–570 (2019) [26] Suh, N., Cheng, G.: A survey on statistical theory of deep learning: Approximation, training dynamics, and generative models (2024). DOI 10.48550/ARXIV.2401.07187. URL https://arxiv.org/abs/2401.07187 [27] Suzuki, T., Nitanda, A.: Deep learning is adaptive to intri... | https://arxiv.org/abs/2504.15944v1 |
7.95e-02 7.33e-02 7.19e-02 7.65e-02 7.87e-02 9.89e-02 1.43e-01 1.69e-01 7.34e-02 7.05e-02 7.19e-02 6.50e-02 6.39e-02 6.19e-02 6.33e-02 8.63e-02 1.34e-01 6.87e-02 6.86e-02 6.05e-02 5.95e-02 5.48e-02 5.30e-02 5.79e-02 5.63e-02 8.33e-02 6.52e-02 6.89e-02 5.87e-02 5.67e-02 5.47e-02 5.37e-02 5.04e-02 5.01e-02 6.40e-02 5.91e... | https://arxiv.org/abs/2504.15944v1 |
arXiv:2504.15946v1 [math.ST] 22 Apr 2025Thee-Partitioning Principle of False Discovery Rate Control Jelle Goeman* Rianne de Heide†Aldo Solari‡ April 23, 2025 Abstract We present a novel necessary and sufficient principle for Fal se Discovery Rate (FDR) control. This e-Partitioning Principle says that a procedure control... | https://arxiv.org/abs/2504.15946v1 |
methods, Blanchard and Roquain (2008) formulated two quite general sufficient conditions, self-consistency and dependence control, under which, if both hold, FDR contr ol is guaranteed. This Self-Consistency Prin- ciple simplifies the proof of well-known FDR-controlling pr ocedures such as BH (Benjamini and Hochberg, 199... | https://arxiv.org/abs/2504.15946v1 |
the rejections. Next, w e formulate the general e-Partitioning procedure and establish that its necessity and sufficiency: the actual e-Partitioning Principle. Sections 5 and 6 explore existing methods for FDR control and investigate whether th ey can be improved. Section 7 shows how the e- Partitioning Principle allows... | https://arxiv.org/abs/2504.15946v1 |
of M. Suppose that we have m hypotheses H1,...,H m⊆Mof interest. For every P∈M, some hypotheses are true and others false; let NP={i: P∈Hi}be the index set of the true hypotheses for P. If we choose to reject the hypotheses with indices in R⊆[m], we say that we make |R|discoveries, of which|R∩NP|are false. The false di... | https://arxiv.org/abs/2504.15946v1 |
others are false. We will generally ignore H∅, since it is impossible to make false discoveries if H∅is true; define M= 2[m]\{∅} as the collection indexing the partitioning hypotheses of int erest. To adapt the Partitioning Principle for FDR control, we comb ine partitioning with the concept of the e- value, building on... | https://arxiv.org/abs/2504.15946v1 |
R∈Rand every S∈ M , we have, αeS= max U∈R|U∩S| |U|∨1≥|R∩S| |R|∨1, soR∈ Rα(E). This proves the “only if” part. It is worth making explicit that Theorem 1 is not tied to our no vel Definition 2 of simultaneous FDR, as Corollary 1 below, which combines Theorem 1 and Lemma 1, ma kes clear. In fact, rewriting an FDR- control... | https://arxiv.org/abs/2504.15946v1 |
1, there may be room for uniform improvement of the method as described above. In Wang and Ram das (2022) this is done by boosting the e-values using a truncation function. However, in Section 9 we will see that in some cases there are good reasons to forgo such a truncation. Finally, it is worth noting that, in the pr... | https://arxiv.org/abs/2504.15946v1 |
motivating its name of eBH+. First, we must formally define what we mean by a uniform improv ement of a classical FDR-control procedure by a simultaneous FDR-control procedure. Definition 3 makes this explicit. A simultaneous procedure R that uniformly improves a classical procedure Smust always allow rejection of the sa... | https://arxiv.org/abs/2504.15946v1 |
e2<1/α. Translated top-values, these properties of the procedure of (4) are shared by adaptive FDR control procedures that plug in an estimate ˆπ0ofπ0,P=|NP|/minto a procedure controlling FDR at π0,Pα(Storey et al., 2004; Benjamini et al., 2006; Blanchard and Roquain, 2009). For su ch procedures, rejection of a hypothe... | https://arxiv.org/abs/2504.15946v1 |
jamini and Yekutieli, 2001), which is valid for any distribution of the p-values. The second is the procedure of Su (2018) based on his based on the FDR Linking Theorem. The Su method is valid under the PRDN ass umption, a weaker variant of the PRDS assumption onderlying BH. We will place these two methods in the conte... | https://arxiv.org/abs/2504.15946v1 |
ForS/\⌉}atio\slash= [m], it follows that /ceilingleftbigg|S|h|S|pi α/ceilingrightbigg ≤/ceilingleftbigg|S|h|S|(i+1) mhm/ceilingrightbigg ≤m−1for alli≤m−1. Hence, for all S/\⌉}atio\slash= [m], we obtain αeS≥/summationdisplay i∈S\{m}1/ceilingleftBig|S|h|S|pi α/ceilingrightBig≥|S\{m}| m−1=|S∩R| |R|. In caseS= [m], we have... | https://arxiv.org/abs/2504.15946v1 |
= 1. 1Though we use eboth forp-to-ecalibrators (as a function) and for exp(1) (as a constant), this should lead to no confusion. 11 Corollary 2. For allP∈HS, eS= min(lα/pS,1/α) (8) is ane-value under PRDN. We can combine the e-values from (8) into a suite E= (eS)S∈M and use the general e-Partitioning procedure (1). We ... | https://arxiv.org/abs/2504.15946v1 |
other 50 can essentially be ignor ed by the-Partitioning procedure. Suppose that the hypotheses H1:θ1=θ2, has ane-value of 4/α,H2:θ1=θ3andH3:θ1=θ4both have e-values of1/α, while the other three hypotheses we have e-values of 0, then eBH+ would only reject R={1}when restricted combinations would not be taken into accoun... | https://arxiv.org/abs/2504.15946v1 |
FDR is controlled over the maximum over all sets in Rα(E), the researcher is allowed to choose the final rejected set from am ong the collection Rα(E)in any desired way, using all information available. Control of FDR according t o Definition 2 is simultaneous over the sets inRα(E), and can be used much like simultaneous... | https://arxiv.org/abs/2504.15946v1 |
the error rate to the data. To achieve this, we will e xtend the Definition 2 of FDR control further, building on the work of Gr¨ unwald (2024) and Koning (2023). Definition 4 (FDR control with post hoc α).For every α∈(0,1], letRα⊆2[m]. Then(Rα)α∈(0,1], controls FDR with post hoc αif, for every P∈M, EP/parenleftbigg sup ... | https://arxiv.org/abs/2504.15946v1 |
[r]can also be done in O(mlogm)time following essentially the same reasoning as for sets of the form [r], adapting the definition of gas appropriate. 15 0.125 0.25 0.375 0.500.20.40.60.81 Signal strength APowerPositive dependence eBH eBH+0.125 0.25 0.375 0.500.20.40.60.81 Signal strength APowerNegative dependence eBH eB... | https://arxiv.org/abs/2504.15946v1 |
the e-Partitioning Principle brings post hoc flexibility to FDR control on a scale that was previously only known in FDP contr ol. Rather than only a single rejection set, researchers have a choice of many rejection sets to choose fr om, and they may use all the data to decide which one to report, while still retaining ... | https://arxiv.org/abs/2504.15946v1 |
(2021). Inflated false disco very rate due to volcano plots: problem and solutions. Briefings in bioinformatics 22 (5), bbab053. Ebrahimpoor, M., P. Spitali, K. Hettne, R. Tsonaka, and J. Go eman (2020). Simultaneous enrichment anal- ysis of all possible gene-sets: unifying self-contained an d competitive methods. Briefin... | https://arxiv.org/abs/2504.15946v1 |
Royal Statistical Society Series B: Statisti cal Methodology 86 (1), 122–154. Rosenblatt, J. D., L. Finos, W. D. Weeda, A. Solari, and J. J. G oeman (2018). All-resolutions inference for brain imaging. Neuroimage 181 , 786–796. Shafer, G. (2021). Testing by betting: A strategy for statis tical and scientific communicati... | https://arxiv.org/abs/2504.15946v1 |
Bayesian Parameter Identification in the Landau-de Gennes Theory for Nematic Liquid Crystals Heiko Gimperlein§Ruma R. Maity¶Apala Majumdar‖Michael Oberguggenberger∗∗ Abstract This manuscript establishes a pathway to reconstruct material parameters from measurements within the Landau-de Gennes model for nematic liquid c... | https://arxiv.org/abs/2504.16029v1 |
can we estimate the unknown/uncertain LdG model inputs for benchmark problems? We focus on inverse UQ problems in the LdG theory in this manuscript. Our goal is to establish an algorithmic pathway from optical measurements and experimentally measured dielectric data to experimental measurements of the Q-tensor order pa... | https://arxiv.org/abs/2504.16029v1 |
in their work. In [7], the authors study the solution landscape of the LdG Euler-Lagrange equations, for this benchmark problem, with additive and multiplicative noise wherebythisnoisecapturesuncertaintiesinaholisticsense. Theauthorsconcludethatthedeterministic LdG predictions are fairly robust, notwithstanding the add... | https://arxiv.org/abs/2504.16029v1 |
1. inferring the dielectric parameters embedded in 𝜀from optical measurements; 2. determining the LdG Q-tensor from 𝜀; 3. andusingtheLdGframeworktodeterminetheNLCmaterialparametersfrommeasurements/computations of the equilibrium values of the LdG Q-tensor order parameter. Thispaperisprimarilydevotedtothelaststepinthe... | https://arxiv.org/abs/2504.16029v1 |
uniaxial, i.e., the NLC sample has a single distinguished director so that all directions perpendicular to the NLC director are physically equivalent. Next,wedescribeathoughtexperimenttomeasurethedielectricpermittivitiesinthematrixequation above. Onecouldtakeathinslab-based3Dgeometry,whereinthetopandbottomsurfacesareun... | https://arxiv.org/abs/2504.16029v1 |
solutions of the corresponding Euler-Lagrange equations, which are a system of five nonlinear and coupled elliptic partial differential equations [28]: 𝐿ΔQ=𝐴Q−𝐵 QQ−1 3 trQ2 I +𝐶 trQ2 Q. (2.5) 6 For a given non-homogeneous Dirichlet boundary condition Q𝑏∈𝐻1 2(𝜕Ω;𝑆3 0),and the admissible set,A(Q𝑏):={Q∈𝐻1(... | https://arxiv.org/abs/2504.16029v1 |
obtain the equivalent system of two coupled and nonlinear partial differential equations: −𝛼ΔQ+(|Q|2−𝛽)Q=0inΩ, Q=Q𝑏on𝜕Ω,(2.10) where𝛼:=𝐿 2𝐶𝜆2and𝛽:=𝐵2 4𝐶2.If the temperature, 𝐴=−𝐵2 3𝐶, and the domain length-scale 𝜆are given, the parameters 𝛼,𝛽determine𝐶=−3𝐴 4𝛽, which allows us to compute 𝐵=√︁ 4𝐶2𝛽... | https://arxiv.org/abs/2504.16029v1 |
covariance matrix Σ𝐸is usually based on the empirical covariance matrix of the observed data [39] or, simpler, on the identity matrix multiplied by the empirical variance of 𝑌obs. For multiple observation variables, say a bivariate 𝐸=(𝐸1,𝐸2), the errors are combined to give 𝜋err(𝐸)∝exp −1 2(𝐸𝑇 1Σ−1 𝐸1𝐸1+𝐸�... | https://arxiv.org/abs/2504.16029v1 |
𝑁𝑁∑︁ 𝑛=1𝑓(𝜉𝑛)=∫ 𝑆𝑓(𝑥)𝜋(𝑥)d𝑥. Thisisoftenreferredtoastheergodictheorem[35,Theorem6.63]. Therefore,theendpieces 𝜉𝑀,···,𝜉𝑁 forsufficientlylarge 𝑀(aftertheso-calledburn-inphase)aretreatedasasampleofthelimitingdistribution 𝜋. By the ergodic theorem, the sample mean, moments and quantiles of 𝜉𝑀,···,𝜉𝑁co... | https://arxiv.org/abs/2504.16029v1 |
be checked visually and is evident for all chains in the figures in Sections 5 and 6. Thefinalquestionconcernstheconvergenceofthechaintoitsstationarydistribution. Thismeansthat subsamples𝜉𝐾,...,𝜉𝐾+𝑀and𝜉𝐿,...,𝜉𝐿+𝑀, with sufficiently large 𝐾and𝐿 > 𝐾+𝑀, should have the samedistribution. Thiscanbetestedbynonp... | https://arxiv.org/abs/2504.16029v1 |
subject to the Dirichlet boundary condition Q𝑏defined above, admits two classes of solutions for small enough 𝛼(for𝜆≥𝜆𝑐 where𝜆𝑐is estimated to be in the nanometre range from the results in [21]): 1. diagonal solutions: the planar director, modelled by nin (2.6), roughly aligns along one of the squarediagonalsand... | https://arxiv.org/abs/2504.16029v1 |
phase for all cases. Next, we use the diagonal and rotated solutions of (4.1) (for the benchmark example in Section 4) with𝛼∗=0.004and𝛽∗=1as the reference solution and use the inverse problem approach in Section 3 to reconstruct the value of 𝛼. For each choice of the prior distribution of 𝛼(UP or GP), we plot the M... | https://arxiv.org/abs/2504.16029v1 |
(𝛼,𝛽)in (2.10) can be reconstructed simultaneously. We follow the same strategy as described at the beginning of Section 5, using solutions to (2.10) 16 (a) Reference solution (b) Markov chain for 𝛼(UP) (c) Histogram of 𝛼(UP) (d) Markov chain for 𝛼(GP) (e) Histogram of 𝛼(GP) Figure 6: Rotated solution, reference ... | https://arxiv.org/abs/2504.16029v1 |
R4, UP 0.0041191 0.0041147 0.6085217 0.6077666 0.9258516 22% R4, GP 0.0040311 0.0040217 0.6023924 0.6016717 0.9064120 19% Table 4: Sample statistics for posterior distributions, reference values 𝛼∗=0.004,𝛽∗=0.6.Type of chain mean𝜇𝛼median𝑚𝛼mean𝜇𝛽median𝜇𝛽correlation𝜌𝛼𝛽AR D1, UP 0.0008587 0.0008478 1.4044713 ... | https://arxiv.org/abs/2504.16029v1 |
reference value 𝛼∗=0.01;(b) Markov chain and (c) histogram of the posterior distribution of parameter 𝛼. The figures show that the Markov chain does not converge for data Qobsconstructed from 𝛼∗=1. It somewhat converges around 𝛼∗=0.1, though with a badly mixing Markov chain, and finally converges when𝛼∗=0.01(and s... | https://arxiv.org/abs/2504.16029v1 |
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