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of Tse and Davison (2022). Interestingly, it also implies that universal inference remains asymptotically valid under model misspecification in high-dimensional settings. This does not contradict the findings of Park et al. (2023) since they focus on the finite-sample guarantees. Indeed, the empirical illustration in S... | https://arxiv.org/abs/2503.14717v1 |
this approach does not account for the fact that the summands are uncentered, the limiting distribution is always downward biased due to the definition of the optimization problem in (6). This property allows for the construction of a conservative confidence set. The resulting confidence set, obtained through studentiz... | https://arxiv.org/abs/2503.14717v1 |
significant improvement over the original universal inference in terms of asymp- totic conservativeness. The proposed method, based on studentization and bias correction, achieves exact coverage once the second-order bias condition is satisfied. Heuristically, for a d-dimensional inferential problem, this requirement c... | https://arxiv.org/abs/2503.14717v1 |
model misspecification. This result should be compared to Theorem 2, which requires (A7) for the validity. This robustness is a significant improvement over universal inference without studentization. Remark 3 (General power analysis of universal confidence set) .To the best of our knowledge, a comprehensive width anal... | https://arxiv.org/abs/2503.14717v1 |
D1 =−1 2σ2n EG[(Yi−θ⊤ GXi)2]−EG[(Yi−bθ⊤ 1Xi)2|D1]o =1 2σ2(bθ1−θG)⊤EG[XX⊤](bθ1−θG). To estimate the Gram matrix, we use the sample Gram matrix: (bθ1−θG)⊤bVG(bθ1−θG) =1 2σ2(bθ1−θG)⊤ n−1X i∈D2XiX⊤ i! (bθ1−θG). After scaling both sides by 2 σ2, the studentized and bias-corrected universal confidence set is defined as CIBC... | https://arxiv.org/abs/2503.14717v1 |
ultimately reaching extremely conservative as dapproaches n. This agrees with Theorem 2 in the regime where d=o(n); however, the theorem does not provide the validity guarantee beyond this regime. Interestingly, the confidence set remains valid even when dis comparable to n. Section S.6 of the supplement provides an in... | https://arxiv.org/abs/2503.14717v1 |
19 bound. The author also sincerely appreciates his numerous suggestions and corrections, as well as his encouragement to maintain this paper as a single-author work. The author wishes to thank Sivaraman Balakrishnan and Larry Wasserman for commenting on the manuscript. The author also acknowledges Woonyoung Chang for ... | https://arxiv.org/abs/2503.14717v1 |
Newey, W. K. and Robins, J. R. (2018). Cross-fitting and fast remainder rates for semipara- metric estimation. arXiv preprint arXiv:1801.09138 . Nguyen, H. D. (2020). Universal inference with composite likelihoods. arXiv preprint arXiv:2009.00848 . Park, B., Balakrishnan, S., and Wasserman, L. (2023). Robust universal ... | https://arxiv.org/abs/2503.14717v1 |
Ramdas, A. (2024). Time- uniform central limit theory and asymptotic confidence sequences. The Annals of Statis- tics, 52(6):2613–2640. Waudby-Smith, I. and Ramdas, A. (2024). Estimating means of bounded random variables by betting. Journal of the Royal Statistical Society Series B: Statistical Methodology , 86(1):1–27... | https://arxiv.org/abs/2503.14717v1 |
It then follows by Lemma 7, Φlog(1 /α) n1/2σθG,θG+h−n1/2 σθG,θG+hEG logpθG+h pθG D1 −Φlog(1 /α) n1/2∥h∥IG−n1/2 ∥h∥IGEG logpθG+h pθG D1 ≤min 1,2rn(h)√ 2πe . Next, (A2) implies that −1 2∥h∥2 VG−1 EG logpθG+h pθG D1 −1 ≤2ωG(∥h∥VG) ∥h∥2 VG. Assuming that 2 ωG(∥h∥VG)/∥h∥2 VG<1 (again, without loss of generalit... | https://arxiv.org/abs/2503.14717v1 |
This concludes the first result. In particular, universal inference is provably conservative when 2 log(1 /α)∥bθ1−θG∥2 VG ∥bθ1−θG∥2 IG!1/2 ≥q 2 log(1 /α)λmin(I−1/2 GVGI−1/2 G)> zα. When the model is correctly specified, that is G∈ P Θ, we have IG=VG, which implies λmin(I−1/2 GVGI−1/2 G) = 1. This leads to the bound inf... | https://arxiv.org/abs/2503.14717v1 |
sample variance of {ξi:i∈D2}. Now, we denote bθ1=θG+h. Then the conditional miscoverage rate can be bounded as PG(θG̸∈CIBC n,α|D1) ≤PG G× n[σ−1 θG,θG+hξ]>bσθG,θG+h σθG,bθ1zα−n1/2 σθG,θG+h1 2h⊤(bVG−VG)h −n1/2 σθG,θG+h EG logpθG+h pθG D1 +1 2h⊤VGh ∩Ωγ1δ(h)∩Ωε1δ(h) +PG(Ωc γ1δ(h)|D1) +PG(Ωc ε1δ(h)|D1) ≤1−Φ 1−ε−1√ 2... | https://arxiv.org/abs/2503.14717v1 |
Gsatisfies (A5) . Then lim sup n→∞sup G∈G|PG(θG̸∈CIStd n,α)− 1−EG[Φ(zα+rStd n,G)] |= 0. Proof of Theorem 3. The proof is analogous to that of Theorem 2, up to the definition of the remainder term rStd n,G. 32 S.5 Supporting Lemmas Lemma 4. Let{δn}∞ n=1be the sequence appearing in the definition of (A1) ,(A2) and (A4)... | https://arxiv.org/abs/2503.14717v1 |
depending onpsuch that, EG[|W|p|D1] =EG rpθG+h pθG−1 +1 2h⊤˙ℓG−1 2h⊤˙ℓG p |D1 ≤Cp EG rpθG+h pθG−1 +1 2h⊤˙ℓG p |D1 +EG 1 2h⊤˙ℓG p |D1 ≤Cp Lp n,G∥h∥p IG+ 2−p∥h∥p IG ≤2Cpmax Lp n,G,1 ∥h∥p IG(E.9) where we used (A4) . Hence we arrive EG g(W)2|D1 ≤22/pC2/p pmax L2 n,G,1 ∥h∥2 IGν2 G,p(δ) (E.10) under the assumpt... | https://arxiv.org/abs/2503.14717v1 |
n→ ∞ assuming (A1) —(A5) . This concludes the claim. Lemma 6. Let{δn}∞ n=1be the sequence appearing in the definition of (A1) ,(A2) ,(A4) and(A6) . For h∈Θ, assume max ∥h∥IG,∥h∥VG,∥h∥2 VG/∥h∥IG ≤δfor some δ∈(0, δn). Assume (A1) ,(A2) ,(A4) and(A6) . For any ε > 0, there exists a constant Cεonly depending on εsuch that... | https://arxiv.org/abs/2503.14717v1 |
i! (θG−bθ1). Thus, using D2to estimate the bias term VGleads to the exact cancellation of the second- order term. The miscoverage probability of the studentized and bias-corrected confidence set for this problem, as defined in Section 4, has the following equivalent form: PG(θG̸∈CIBC n,α) =PG −n−1X i∈D22(θG−bθ1)⊤εiXi>... | https://arxiv.org/abs/2503.14717v1 |
in Figure 2, despite the clear model misspecification. On the other hand, Universal Inference 1 now achieves coverage closer to the 1 −αlevel. However, it is incorrect to interpret it as Universal Inference 1 performing better. This result should be seen as a cautionary tale, as it is possible to construct a data-gener... | https://arxiv.org/abs/2503.14717v1 |
Testing Conditional Stochastic Dominance at Target Points∗ Federico A. Bugni Department of Economics Northwestern University federico.bugni@northwestern.eduIvan A. Canay Department of Economics Northwestern University iacanay@northwestern.edu Deborah Kim Department of Economics University of Warwick deborah.kim@warwick... | https://arxiv.org/abs/2503.14747v2 |
(2012), Shen and Zhang (2016), Goldman and Kaplan (2018), Qu and Yoon (2019)). Likewise, in wage discrimination studies, researchers may seek to compare wage dis- tributions across demographic groups while controlling for observed skill levels (Becker (1957), Canay et al. (2024), Bharadwaj et al. (2024)). The primary g... | https://arxiv.org/abs/2503.14747v2 |
Third, we show that the proposed critical value aligns with the one obtained from a permutation-based approach when the random variables YandXare both continuous, thus establishing a natural connection between our method and the broader literature on permutation-based inference. To the best of our knowledge, this resul... | https://arxiv.org/abs/2503.14747v2 |
finite-sample distribution. Although our null hypothesis differs, our critical value coincides with the corresponding quantile of this distribution. Notably, Hodges (1958) and McFadden (1989) proposed that the two-sample one-sided Kolmogorov–Smirnov test, originally designed for testing equality of two distributions, c... | https://arxiv.org/abs/2503.14747v2 |
1≤i≤nx}associated with the qxvalues of {Zi: 1≤i≤ nx}closest to z0. To define these samples formally, we introduce g-order statistics for the conditioning variable Z, where g(Z) :=|Z−z0|; see Reiss (1989, Section 2.1) and Kaufmann and Reiss (1992). For any two values z, z′∈ Z, we define the ordering ≤gas follows: z≤gz′i... | https://arxiv.org/abs/2503.14747v2 |
the case where YandXare continuously distributed, we show that this critical value is asymptotically sharp and cannot be improved, in the sense 6 formalized by Lemma 5.1 and the accompanying discussion. By contrast, when the random variables are discrete with finite support, the critical value can be refined. We detail... | https://arxiv.org/abs/2503.14747v2 |
the limit experiment. We start by deriving a result on the induced order statistics collected in the vector Snin (5). In order to do so, we make the following assumptions. Assumption 4.1. For any ε >0andz∈ Z,P{Z∈(z−ε, z+ε)}>0. Assumption 4.2. For any z∈ Zand sequence zk→z,supt∈R|FY(t|zk)−FY(t|z)| →0 andsupt∈R|FX(t|zk)−... | https://arxiv.org/abs/2503.14747v2 |
whenever P∈P0. Theorem 4.2 establishes the asymptotic validity of the test in (10). There are three main reasons why the inequality in (13) may be strict, resulting in the limiting rejection probability strictly below α. First, for distributions Pin the interior of P0, where the inequality in (1) holds strictly for som... | https://arxiv.org/abs/2503.14747v2 |
central role of the KS statistic in addressing such settings.1 1Although Goldman and Kaplan (2018, p. 146) state that their test remains valid—albeit conserva- tive—for discrete data, they do not provide a formal proof. In contrast, Theorem 4.2 establishes this property for the KS statistic. While this result may appea... | https://arxiv.org/abs/2503.14747v2 |
for the idea that qyshould be smaller when the density at z0is low. When fZ(z0) is small, the qyclosest observations to z0are likely to be “far” from z0. While one could replace the normality assumption with a non-parametric estimator of fZ(·), it is unfortunately impossible to adaptively choose CZandCYfor testing (1) ... | https://arxiv.org/abs/2503.14747v2 |
of support points, rather than for all discrete data. The computational cost of 13 cr α(qy, q) is also increasing in r, and so as rgets larger, the cost is higher and the benefits are lower. We propose to compute cr α(qy, q) numerically, by solving the following optimization problem: cr α(qy, q) = min x∈[clb, cub]∩T: ... | https://arxiv.org/abs/2503.14747v2 |
is the least favorable within the set of null distributions P0in (12) that satisfy stochastic dominance, a point first made by Lehmann (1951) and later reiterated by Hodges (1958), McFadden (1989), and Goldman and Kaplan (2018). We define the subset of continuous distributions in P0that satisfy FY(·|z0) =FX(·|z0) asP∗ ... | https://arxiv.org/abs/2503.14747v2 |
arguments for validity no longer apply. Alternative arguments that claim validity of permutation tests when invariance does not hold typically require q→ ∞ ; see Chung and Romano (2013), Canay et al. (2017), and Bugni et al. (2018), among others. In our setting, where qis fixed, such arguments do not directly apply. We... | https://arxiv.org/abs/2503.14747v2 |
establish an analog of Lemma 5.2 for discrete data, we do not pursue such an extension, as randomized tests are rarely used in practice. It is possible to improve upon cα(qy, q) when Sis discrete, though this comes at the cost of additional computational complexity. This insight led to the refined test for discrete dat... | https://arxiv.org/abs/2503.14747v2 |
Sℓ ndepends on zℓthrough the induced order statistics, mean- ing the test statistic T(Sℓ n) is influenced by the choice of the target point zℓ. Second, the critical value cα(qy, q) is a function of ( qy, q), which, through the data-dependent rules introduced in Section 5.1, implicitly depends on zℓ. The dependence on α... | https://arxiv.org/abs/2503.14747v2 |
each simulation design. Designs 1 to 4 are based on the location-scale model in (27), while Designs 5 and 6 are discrete designs as described in the main text. Cases “a” to “c” are under the null, while case “d” is under the alternative. •Case (b) : the null hypothesis holds with strict inequality. •Case (c) : the null... | https://arxiv.org/abs/2503.14747v2 |
Recall that this refined test coincides with the original test in (10) when both samples have infinitely many support points r. Design1 2 3 a b c a b c a b c q∗ y79.54 78.63 34.84 79.51 79.53 34.85 45.48 44.04 20.05 q∗ x79.52 79.52 34.86 79.72 89.57 34.92 45.49 45.48 20.05 Design4 5 6 a b c a b c a b c q∗ y82.97 82.99 ... | https://arxiv.org/abs/2503.14747v2 |
statistic with a deterministic criti- cal value, without the need for kernels, local polynomials, bias correction, or bandwidth selection. Furthermore, our test admits a clear interpretation in the limit experiment, which allows us to connect it with classical analytical critical values and permutation- based tests. In... | https://arxiv.org/abs/2503.14747v2 |
such that ˜P{E1(ε)∩ En,2(ε)∩En,3} ≥1−δfor all n≥N(δ). Let ε1= inf{∥˜d−d∥/2 :d <˜d∈ D} >0. By Lemma B.2,∃ε2>0 such that, for i̸=j= 1, . . . , q , ˜P{{|˜Si−˜Sj|< ε2} ∩ { ˜Si,˜Sj∈ Dc}}< δ/(9q(q−1)), ˜P{∪d∈D{|˜Si−d|< ε2} ∩ { ˜Si∈ Dc}}< δ/(9q(q−1)). Finally, set ε= min {ε1, ε2}>0 for the remainder of the proof. By elementar... | https://arxiv.org/abs/2503.14747v2 |
as in (22), cp α(s) = cp α. 26 To establish this, consider the following derivation, cp α(s) = inf x∈R( 1 |G|X π∈GI{T(sπ)≤x} ≥1−α) (1)= inf x∈R( 1 |G|X π∈GI{T∗(R(sπ))≤x} ≥1−α) (2)= inf x∈R( 1 |G|X π∈GI T∗((Q¯π)π)≤x ≥1−α) (3)= inf x∈R( 1 |G|X π∈GI{T∗(Qπ)≤x} ≥1−α) (4)=cp α. (A-8) Here, (1) follows by (A-6), (2) follows ... | https://arxiv.org/abs/2503.14747v2 |
xfirstg-order statistics ( Zℓ,(1), . . . , Z ℓ,(qℓx)). Let Endenote the following event: En=Eny,y∩Enx,xwhere Eny,y:=(L\ ℓ=1Mℓ y=∅) and Enx,x:=(L\ ℓ=1Mℓ x=∅) . In words, Enmeans that the subsets of the data used in each of the Ltests have no observations in common. We begin by showing that P{En} →1. (B-10) Since the two... | https://arxiv.org/abs/2503.14747v2 |
ε}]ny−u (2)= 1−P{|Z−zℓ| ≥ε}ny(3)→1, (B-16) as desired, where (1) holds by the fact that {Zi: 1≤i≤ny}are identically distributed, (2) by the Binomial Theorem, and (3) by Assumption 4.1. For the inductive step, we assume 29 Zℓ ny,(j)−zℓ=op(1) for j∈ {1, . . . , q y−1}, and prove that Zℓ ny,(j+1)−zℓ=op(1). For this, consi... | https://arxiv.org/abs/2503.14747v2 |
t. The first claim is immediate. For the second, it follows from the fact that the continuity of the CDFs implies U= (0,1). By elementary properties of cdfs, lim t→−∞ FX(t|z0) = 0 and lim t→∞FX(t|z0) = 1. By the intermediate value theorem, for any u∈(0,1), there exists t∈Rsuch that u=FX(t|z0), implying u∈ U, as desired... | https://arxiv.org/abs/2503.14747v2 |
a sequence of random variables that satisfy Vnp→V, where V is a random variable whose cdf is discontinuous at a finite set of points D. Furthermore, assume supt∈R|FVn(t)−FV(t)| →0. Then, P{{V∈ D} ∩ { Vn̸=V}} → 0. Proof. Fixδ >0 arbitrarily. It suffices to find N=N(δ) such that P{V∈ D, Vn̸=V}< δfor alln > N . Set ¯ε:={m... | https://arxiv.org/abs/2503.14747v2 |
z0}distributed according to Bernoulli (0.5). For α= 5%, we get E˜P[ϕCvM(˜S)]≈12.5% and E˜P[ϕAD(˜S)]≈12.5%, as desired. A notable feature of the examples above is that the data are discrete. This naturally raises the question of whether the validity of these tests can be recovered in settings with continuously distribut... | https://arxiv.org/abs/2503.14747v2 |
variable models. Journal of the American statistical Association ,97284–292. Anderson, G. (1996). Nonparametric tests of stochastic dominance in income distri- butions. Econometrica: Journal of the Econometric Society 1183–1193. 36 Andrews, D. W. andShi, X. (2017). Inference based on many conditional moment inequalitie... | https://arxiv.org/abs/2503.14747v2 |
Academic press. Hodges, J. (1958). The significance probability of the smirnov two-sample test. Arkiv f¨ or matematik ,3469–486. H´ajek, J. andˇSid´ak, Z. (1967). Theory of Rank Tests . Academic Press, New York. Kaufmann, E. andReiss, R.-D. (1992). On conditional distributions of nearest neigh- bors. Journal of Multiva... | https://arxiv.org/abs/2503.14747v2 |
Hazard Rate for Associated Data in Deconvolution Problems: Asymptotic Normality Mohammed Essalih Benjrada1,∗ Abstract In reliability theory and survival analysis, observed data are often weakly dependent and subject to additive measurement errors. Such contamination arises when the underlying data are neither independe... | https://arxiv.org/abs/2503.14759v1 |
Definition 1. A finite family of random variables {Xi}n i=1is said to be positively associated (or simply associated ) if for any two real-valued, coordinate-wise increasing functions Φ 1(·) and Φ 2(·) defined on Rn, the following holds: Whenever Eh Φ2 j({Xi}n i=1)i <+∞forj= 1,2, the covariance satisfies: Cov (Φ 1({Xi}... | https://arxiv.org/abs/2503.14759v1 |
time to failure or death). The hazard rate function λ(x) is defined as: λ(x) := lim dx→0P(x≤X < x +dx|X≥x) dx=f(x) 1−F(x), where: •f(x) is the probability density function (PDF) of X, •F(x) is the cumulative distribution function (CDF) of X. The quantity λ(x)dxrepresents the probability that an organism or component, w... | https://arxiv.org/abs/2503.14759v1 |
errors are present) as follows: ˆλn(x) =ˆfn(x) 1−ˆFn(x), (2) where ˆfn(x) denotes the kernel-type density estimator, and ˆFn(x) represents the empirical distri- bution function. In developmental process fields, convolution models are not only of theoretical interest but also have practical implications. Ignoring measur... | https://arxiv.org/abs/2503.14759v1 |
The error density function r(·) is assumed to be known. Additionally, its characteristic function ϕr(·) satisfies: 1.|ϕr(t)|>0 for all t∈R. 2. lim t→+∞tβϕr(t) =β1, where βis an even number and β1is a positive constant. (H2) Properties of the Kernel Function The kernel function k(·) is a bounded density with an even Fou... | https://arxiv.org/abs/2503.14759v1 |
this case, the process {Xj}is positively associated, and its covariance function is given by: Cov(X1, Xj) =ϕj−1 1. Thus, Condition ( H5)-2 is satisfied whenever 0 < ϕ 1<1. Indeed, in this case, the covariance decays exponentially as j→+∞: Cov(X1, Xj) =O(e−c(j−1)), 10 where c=−logϕ1. Hence, since the series X j≥2jµe−c(j... | https://arxiv.org/abs/2503.14759v1 |
is needed: ˜E(λn(x)) =E[fn(x)] 1−E[Fn(x)]. We give the following intermediate result. 12 Proposition 9. Under the assumptions of Theorem 6, we have E(λn(x)) = ˜E(λn(x)) +O((nh2β n)−1). Next Proposition 10. Under the assumptions of Theorem 6 and the assumption that fandFare in C2(R), we get ˜E(λn(x))−λ(x) =h2 n 2 λ′′(x... | https://arxiv.org/abs/2503.14759v1 |
assumption. Examples include triangular densities, their convolutions with arbitrary densities, and uniform densities. These are informally referred to as non-standard errors. In such cases, ridge-parameter regularization is employed to estimate ϕr(t). For further details on ridge-parameter regularization in deconvolut... | https://arxiv.org/abs/2503.14759v1 |
W(a, b), where ais the shape parameter andbis the scale parameter. Since the scale parameter does not affect the shape of the hazard rate, we fix b= 1. We have the following cases: - When a >1, the Weibull distribution exhibits an IHR. - When a <1, it exhibits a DHR. - When a= 1, it reduces to the exponential distribut... | https://arxiv.org/abs/2503.14759v1 |
the computational complexity of the estimation process, substantial resources are required. 17 0.00.51.01.52.0 0 1 2 3 xHazard EstimateNSR 0 0.1 0.2 0.5n=1000 0.00.51.01.52.0 0 1 2 3 xHazard EstimateNSR 0 0.1 0.2 0.5n=2000 0.00.51.01.52.0 0 1 2 3 xHazard EstimateNSR 0 0.1 0.2 0.5n=5000(a)W(1,1) 012345 0 1 2 3 xHazard E... | https://arxiv.org/abs/2503.14759v1 |
0.938 0.0871 0.931 2000 0.0561 0.945 0.0578 0.941 0.0612 0.937 5000 0.0521 0.958 0.0571 0.945 0.0688 0.946 2000, and 5000) under the contamination scenario described earlier. The coverage probabilities (CP) and average lengths ( AL) of these confidence intervals are summarized in Table 1. Standard Normal Quantiles-3 -2... | https://arxiv.org/abs/2503.14759v1 |
n,1→0 as n→+∞. The main task is to calculate the asymptotic value of I′′ n,1. Applying the dominated convergence theorem and using the result from (12), we obtain: lim n→+∞I′′ n,1=Zx −∞Z+∞ −∞L(u)du g(s)ds =Z+∞ −∞L(u)du G(x). The desired result is established for l= 1. Next, for l= 2, we proceed as follows: h2β nE ... | https://arxiv.org/abs/2503.14759v1 |
find Cov ˜Mhn,1(x),˜Mhn,i(x) ≤C h2β−2 n. This means that h2β nρnX i=2 Cov ˜Mhn,1(x),˜Mhn,i(x) =O ρnh2 n . Using the fact that h2 nρn→0 asn→+∞, gives h2β nρnX i=2 Cov ˜Mhn,1(x),˜Mhn,i(x) =o(1). Concerning the second contribution (for ρn+ 1≤i≤n), we make use of Lemma 16. In- deed, it is mentioned in Section 1 tha... | https://arxiv.org/abs/2503.14759v1 |
By applying the δ-method theorem due to Doob Doob (1935), we obtain Var(λn(x)) = Var( τ(θn(x)))→ ▽τ(θ(x))TΣθ(x)▽τ(θ(x)), where Σθ(x) = lim n→∞Σθn(x) and the gradient vector ▽τ(θ(x)) is given by ▽τ(θ(x)) := ∂λ(x) ∂f(x) ∂λ(x) ∂(1−F(x)) = 1 1−F(x) f(x) (1−F(x))2 . (23) Thus, we obtain the asymptotic variance: lim ... | https://arxiv.org/abs/2503.14759v1 |
, s (pn+qn)}. For 1≤s≤τn, define the random variables: ηs=(s−1)(pn+qn)+pnX i=(s−1)(pn+qn)+1Uiand ζs=s(pn+qn)X j=(s−1)(pn+qn)+pn+1Ui. The remaining block is: ϑτn=nX i=τn(pn+qn)+1Ui. 31 Thus, we can write: 1√nnX i=1Ui=1√n"τnX s=1ηs+τnX s=1ζs+ϑτn# =1√n[In,1+In,2+In,3]. The goal now is to show: 1 nE I2 n,2 →0,1 nE I2 n,... | https://arxiv.org/abs/2503.14759v1 |
the Lindeberg-Feller condition of the asymptotic normality under independence. First, we proceed to prove assertion (32). Lemma 19 hereinafter is in order: Lemma 19. Under Conditions ( H1), (H3)–(H4), (H5)-2, and ( H6), for sufficiently large n, we have: i)τn nVar(η1)→σ2(x), ii)1 nP 1≤i<j≤τnCov(ηi, ηj)→0, iii) Var 1√n... | https://arxiv.org/abs/2503.14759v1 |
nh2(β+1) n+∞X i=pn|Cov ( X1, Xi)|. Finally, from the fact thatτnpn n→1 asn→+∞and Condition ( H6), we conclude that Iτn(t)≤Ct2 h2(β+1) n+∞X i=pn|Cov ( X1, Xi)| →0. The proof of (33) is finished. Now, we establish (34). From Lemma 17, we can see that |Whn(x)| ≤C hβ nand|Mhn(x)| ≤C hβ n. This leads to |η1| ≤Cpn hn(α1+hnα2... | https://arxiv.org/abs/2503.14759v1 |
arXiv:2503.14763v1 [math.ST] 18 Mar 2025The Field Equations of Penalized non-Parametric Regressio n Sven Papperta,∗ aChair of Econometrics, Department of Statistics, TU Dortmu nd University, Germany Abstract We view penalized risks through the lens of the calculus of va riations. We consider risks comprised of a fitness... | https://arxiv.org/abs/2503.14763v1 |
that under suitable regulatory conditions, ˆQ(Y,X;g) is a consistent and unbiased estimator for the MSE risk: Q[g] =E[(Y1−g(X1))2] =/integraldisplay (y−g(x))2fY,X(y, x)dydx. (1) The aim of this work is to find the function that minimizes risk s as the one in Eq. 1and risks which are penalized by a functional of the grad... | https://arxiv.org/abs/2503.14763v1 |
fill these cells for the case where the pe nalty is gradient-based.2In the related literature different choices for the penalty are available. Taking the i ntegral of the ℓ1orℓ2norm of ∇g(or versions thereof) are popular choices, see e.g. Drucker and Le Cun (1991,1992),Rosasco et al. (2013),Follain and Bach (2024), Ding ... | https://arxiv.org/abs/2503.14763v1 |
examples for gradient-based penalization from the r elated literature, including deep learning, non- parametric regression and image denoising, and embed them i n our definition of a risk functional, Def. 1. In Sect. 2we derive the field equations for risk functionals, represen ted as the sum of a fitness-term and a gradi... | https://arxiv.org/abs/2503.14763v1 |
be a minimum, f romRindler (2018), Prop. 3.3) . LetΓ :W2,p(Ω)→Rbe given as Γ[φ] =/integraltext f(x, φ(x),∇φ(x))dx. If the following conditions hold, the solution to the Euler-Lagrange equation of Γis a minimum of Γ: Let x∈Ω,s∈Randu∈Rp. i)|∂ ∂sf(x, s, u )| ≤C(1 +|s|p−1+||u||p−1)and||∂ ∂uf(x, s, u )|| ≤C(1 +|s|p−1+||u||p... | https://arxiv.org/abs/2503.14763v1 |
p), is given as Q(β) =E Y1−(β0+p/summationdisplay j=1βiX1j)2 +λ/hatwidestPen(β). Since/hatwidestPen(β) only depends on deterministic β, the penalty takes the same form in the empirical and the non- empirical risk. This is a major difference to non-parametric penalization. Depending on the choice of the penalty, the ... | https://arxiv.org/abs/2503.14763v1 |
(log-)Likelihood, MAE or the Energy Distance. Exam ples for Pen are integrals of ℓ2orℓ1norms (or versions thereof) of the gradient. Definition 1 (Risk Functional) .Let/hatwidestPen : Wk,p(Ω)×Ω→RandˆL:Wk,p(Ω)×Ω×R→R. The empirical risk functional evaluated at Y∈R,X∈Ω and g∈Wk,p(Ω), is given as ˆQ(Y,X, g) =1 nn/summationdi... | https://arxiv.org/abs/2503.14763v1 |
denotes the m-th order derivative, cf. Wahba (1990), Chapter 1. Choosing /hatwidestPen(g, X) =g(m)andν=λ, we may embed the smoothing spline risk into the risk functional fra mework too. Risks with m-th order derivatives may be called m-th-order risk functionals . Additionally, we recognize that the empirical fitness ter... | https://arxiv.org/abs/2503.14763v1 |
on the explicit choice of the kernel K. In its general form, the risk can not be embedded into the risk functional. Another gradient-based penalization approach, which can n ot be represented as a risk functional is the one by 5In the original paper, the notation is different and the risk a nd penalty are given for mult... | https://arxiv.org/abs/2503.14763v1 |
=E(Y|X= 0) as auxiliary boundary condition when necessary. The nec essity occurs e.g. in the limiting case λ→ ∞ . This particular choice will be motivated in the following e xample of the squared- ℓ2penalized MSE-risk. It would also be possible to impose boun dary conditions such as g=E(Y|X=·) on∂Ω a priori. Imposing n... | https://arxiv.org/abs/2503.14763v1 |
one-dimensional version of the equation is given as E(Y|X=x) =−λg′′(x)−λ(log(fX(x)))′g′(x) +g(x)x∈Ω g′(0) = g′(1) = 0(15) This equation is closely related to the Hermite differential equation and may be solved by a power-series ansatz. For the special case where additionally fX(x) = 1, the resulting ODE can be so... | https://arxiv.org/abs/2503.14763v1 |
estimator. 4.2. Analysis of other penalization and regularization met hods While we tried to render the results in this paper as general a s possible, there are penalization/regularization methods where it is not clear how they may be analyzed using th is framework. First and foremost, regularization 14 methods which a... | https://arxiv.org/abs/2503.14763v1 |
feedforward MLP, cf. Hornik (1991). If the solution to the Euler-Lagrange equation lies in the function space which the learner can universally approximate, then we pres ume that the learner converges to the minimum of the Euler-Lagrange equation for the MSE with gradient-base d penalization. However, for now, this is ... | https://arxiv.org/abs/2503.14763v1 |
(1):69–82, 1970. K. Hornik. Approximation capabilities of multilayer feedf orward networks. Neural networks , 4(2):251–257, 1991. M. Kohler and S. Langer. On the rate of convergence of fully co nnected deep neural network regression estimates. The Annals of Statistics , 49(4):2231–2249, 2021. F. Rindler. Calculus of va... | https://arxiv.org/abs/2503.14763v1 |
arXiv:2503.14978v1 [math.ST] 19 Mar 2025Inferring diffusivity from killed diffusion Richard Nickl and Fanny Seizilles University of Cambridge March 20, 2025 Abstract We consider diffusion of independent molecules in an insulat ed Euclidean domain with unknown diffusivity parameter. At a random time and position, the mol ec... | https://arxiv.org/abs/2503.14978v1 |
killed diffusion 1 1 4.1 Construction of the prior and posterior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Numerical illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3 Posterior consistency theorems . . . . . . . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2503.14978v1 |
non-zero only when Xtis at∂Ω, andν(x),x∈∂Ω,is the inward pointing normal vector. To avoid technicalitieswe assumethat Ω isconvexwith smoothboundary– the existenceofaprocess( Xt) with almost surely continuous sample paths is then shown in [40, 28]. One can mode l a further drift term ∇U(Xt) in the above SDE (see after (... | https://arxiv.org/abs/2503.14978v1 |
sion, is enough to consistently identify D.This is shown to be true as long as a) the initial state φof the system satisfies the mild identifiability Condition 5.5 (of being either close to equilibrium, or appropriately prepared by th e experimenter), and b) if the binding potentialqis not too ‘aggressive’ (i.e., small e... | https://arxiv.org/abs/2503.14978v1 |
more discussion. 1.2 Basic notation TheLp, H¨ older,L2- andLp-Sobolev spaces over a bounded domain Ω in Rdwith smooth boundary are denoted by Lp(Ω),Ca(Ω),Ha(Ω),Wa,p(Ω), respectively. We denote by ∝ba∇dbl · ∝ba∇dblBthe norm on the resulting Banach space B, and∝a\}b∇acketle{t·,·∝a\}b∇acket∇i}htL2denotes the inner product... | https://arxiv.org/abs/2503.14978v1 |
Λ (such as its density w.r.t. some dominating measure) by standard approximation theoretic methods, see Subsection 2.3. 4 To proceed, consider Krealisations Y1=y1,...,YK=ykof (7). The probability density of each observation Yiis given by pi,Λ,n(yi) =e−nΛ(Bi)(nΛ(Bi))yi yi!, yi∈N∪{0} ≡N0, and, by independence the resulti... | https://arxiv.org/abs/2503.14978v1 |
bins used, hence the previous theorem is stated in this more flexib le form. The uniform bound on Λ( O) can be replaced by a condition that the posterior concentrates o n uniformly bounded intensities with frequentist probability approaching one. 2.3 Contraction rates for intensity densities Theℓ1-metric on the bin-coun... | https://arxiv.org/abs/2503.14978v1 |
of disjoint support of the bins is crucial to retain independence of the Ksamples in (7), and obtaining faster rates will require rather differe nt techniques based on general functionals of Poisson point processes that we d o not wish to develop here. 3 Derivation of the observation model and proof of Theorem 1. 1 Inav... | https://arxiv.org/abs/2503.14978v1 |
whereas we observe the actual binding locationXSof the molecule. This is a well-defined random variable since Sis a stopping time for the process (Xt) (e.g., Sections 3.3 and 22 in [3]). Since all particles will bind at some rand om point in time, intuitively, the distribution of their binding locations should equal the... | https://arxiv.org/abs/2503.14978v1 |
dominated convergence theorem we have /integraldisplay /integraldisplay R1Ah(w,·)1Bh(w,y)dPY(y)dPW(w) =/integraldisplay 1Ah(w,·)PY/parenleftbig/integraldisplayt+h tq(w(s))ds>Y/parenrightbig dPW(w) =/integraldisplay 1Ah(w,·)/parenleftbig 1−e−/integraltextt+h tq(w(s))ds/parenrightbig dPW(w) =h/integraldisplay 1Ah(w,·)1 h... | https://arxiv.org/abs/2503.14978v1 |
of molecules having bound in a measurable subsetAof Ω. Concretely, we take independent and identically distributed ra ndom variables T1,T2,..., Ti(A) =/braceleftBigg 1,if molecule ihas bound in A 0,otherwise such thatP(Ti(A) = 1) = Λ D(A), and obtain a mixed binomial process N(A) =nmol/summationdisplay i=1Ti(A), Ameasu... | https://arxiv.org/abs/2503.14978v1 |
denote the resulting law PΛDof the observation vector by PDin this section. 4.1 Construction of the prior and posterior Our goal is to make inference on Dand hence we construct a prior directly for this parameter which induces a pushforward prior for the intensity measure Λ Dvia (27). We start with a Gaussian random fie... | https://arxiv.org/abs/2503.14978v1 |
our results – a more c omprehensive numerical investigation of the algorithm proposed here will be undertaken in future work. Figure 2: Problem setting 13 Synthetic data We take the initial condition φand ground truth D0such thatφ(x,y)∝exp(cos(3π(x2+ y2))),W0(x,y) = 5e−10x2−10(y−0.4)2for (x,y) in the interior of the me... | https://arxiv.org/abs/2503.14978v1 |
b>d/2, Proposition 5.2, the Sobolev imbedding, and our hypotheses which ensure φ∈˜Hb Di,q. Then we have LD1,q(¯u) =−φ−LD2,quD2+∇·(D2−D1)∇uD2=∇·(D2−D1)∇uD2 so that ¯uequalsL−1 D1,q[ϕ] withϕ=∇ ·(D2−D1)∇uD2which lies in ˜Hb D1,q,b > d/2,by what precedes, Proposition 5.2, and since D1−D2vanishes near ∂Ω. We thus obtain fro... | https://arxiv.org/abs/2503.14978v1 |
|ΛD(Bi)−ΛD0(Bi)| ΛD0(Bi)≤Λ−1 min1 |Bi|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay Bi(λD−λD0)(x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Λ−1 min||λD−λD0||∞ ≤c||D−D0||Ha′≤c′∝ba∇dblW−W0∝ba∇dblHa′ for somec′>0. The condition (13) then follows from Lemma 4.4 with εna constant m... | https://arxiv.org/abs/2503.14978v1 |
last expression equals zero. By the usual Poincar´ e inequality (p.292 in [10]) this implies first that u=constand then that |Ω00|/|Ω|= 0, a contradiction since Ω 00has positive Lebesgue measure. The result follows. We deduce from Parseval’s identity that for any ϕ∈H2we have LD,qϕ=−∞/summationdisplay j=1λj,D,qej,D,q∝a\}... | https://arxiv.org/abs/2503.14978v1 |
5.2) in [12] then allows us to represent the solution via the Feynman-Kac formula uD,q(x) =/integraldisplay∞ 0v(t,x)dt=Ex/integraldisplay∞ 0ϕ(Xt)exp/braceleftBig −/integraldisplayt 0q(Xs)ds/bracerightBig dt, x∈Ω, (46) whereXsis the Markov process from (1) started at X0=x. Lemma 5.3. Assume Conditions 4.1, 4.2 and that ... | https://arxiv.org/abs/2503.14978v1 |
many b inding events in the region Ω 00. The hypothesis a) is somewhat more subtle to interpret: It essent ially requires the domain Ω and operator LD0,0=∇·(D0∇) to satisfy the hotspots conjecture (see Sections 2.2.2 and 3.7 in [ 33] for concrete examples and relevant references). We also notice that e1∈H2 ν⊂L∞(cf. Pro... | https://arxiv.org/abs/2503.14978v1 |
λ′ j=λj+qmin, j∈N∪{0}, shifted byqmin, including λ0= 0 corresponding to e0= 1. The operator LD1,qminthus has inverse L−1 D1,qmin onL2(Ω) with spectral representation u=L−1 D1,qmin[ϕ] =−∞/summationdisplay j=01 λ′ jej∝a\}b∇acketle{tej,ϕ∝a\}b∇acket∇i}ht, ϕ∈L2, solving∇·(D∇u)−qminu=ϕsubjecttoNeumannboundaryconditions. Nows... | https://arxiv.org/abs/2503.14978v1 |
pΛ0(Y1,...,YK)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketrightbigg/bracketrightbigg using also Jensen’s inequality and Fubini’s theorem in the last step. We d efine random variables Zi:= logpΛ pΛ0(Yi)−EΛ0logpΛ pΛ0(Yi) =YilogΛ(Bi) Λ0(Bi)−nΛ0(Bi)logΛ(Bi) Λ0(Bi), i= 1,...,K, and estimate the expectatio... | https://arxiv.org/abs/2503.14978v1 |
constraint’ (24), hence cannot be use d itself in the context of our inverse problem since the stability estimat e Theorem 5.6 does not apply to it. A natural statistic is the normalised bin count, which gives the rando m vector in ( R+)Kwith entries ˆΛi=Yi n, i= 1,...,K, (53) for data from (7). Now consider the averag... | https://arxiv.org/abs/2503.14978v1 |
Richard Bass. Diffusions and Elliptic Operators . en. Probability and its Applications. New York: Springer-Verlag, 1998. isbn: 978-0-387-98315-8. doi:10.1007/b97611 . [3] Richard F. Bass. Stochastic Processes . Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge:CambridgeUniversityPress,2011. isbn:9... | https://arxiv.org/abs/2503.14978v1 |
arXiv:1804.03616 [stat]. Mar. 2020. [22] Alec Heckert, Liza Dahal, Robert Tjian, and Xavier Darzacq. “R ecovering mixtures of fast-diffusing states from short single-particle trajectories”. In: eLife11 (Sept. 2022). Ed. by Ihor Smal, Anna Akhmanova, and Maarten W Paul. Publisher: eLife Sciences Publicatio ns, Ltd, e7016... | https://arxiv.org/abs/2503.14978v1 |
A Note on Local Linear Regression for Time Series in Banach Spaces Florian Heinrichs f.heinrichs@fh-aachen.de FH Aachen Heinrich-Mußmann-Straße 1 52428 Jülich, Germany Abstract This note extends local linear regression to Banach space-valued time series for estimating smoothlyvaryingmeansandtheirderivativesinnon-statio... | https://arxiv.org/abs/2503.15039v1 |
are referred to as “scalar-on-function” regression. Local linear regression was used for the estimation of νunder the assumption of i.i.d. observations (Benhenni et al., 2007; Baíllo and Grané, 2009; Boj et al., 2010; Barrientos-Marin et al., 2010; Berlinet et al., 2011; Ferraty and Nagy, 2022; Bhadra and Srivastava, 2... | https://arxiv.org/abs/2503.15039v1 |
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