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1. As a byproduct of the above proof, we find that, for any δ >0and distinct points (x1, . . . , x n)∈[0,1]nin a set of n-dimensional full Lebesgue measure, there exist infinitely many θ > M(for any arbitrarily large M > 0) at which the product likelihood takes a value greater than (2−δ)n. Because max x∈[0,1]fθ(x) =2θ ... | https://arxiv.org/abs/2504.11360v2 |
x∈[0,1]andθ, θ′∈[0, M](recall that ε >0has been fixed beforehand). Now fix x1:∞∈Ω⋆. Therefore, there exists N∈Nsuch that max θ∈TnnY i=1(fθ(xi) +ε)< bn for all n≥N. Choosing n≥N∨Nδand denoting θn∈arg max θ∈[0,M]Qn i=1fθ(xi),17we have that |θn−θ|< δfor some θ∈ Tn, and so nY i=1fθn(xi)≤nY i=1(fθ(xi) +ε)< bn. Because this ... | https://arxiv.org/abs/2504.11360v2 |
:dh(fθ, f0)< ε}for some small ε >0. Then, for δ >0chosen small enough, for all θ∈Ac εwe have F∞ 0 nY i=1 fθ(Xi)1/2+δ ≥e−nb/2! ≤enb/2Z1 0p fθ(x)fθ⋆(x) dx+δn =enb/2 1−d2 h(fθ, fθ⋆)/2 +δn ≤enb/2(1−ε2/2 +δ)n ≤enb/2(1−ε2/4)n ≤enb/2e−nε2/4, which, choosing b >0small enough, is smaller than e−nCεfor some Cε>0. Let Mn=en... | https://arxiv.org/abs/2504.11360v2 |
G. (2013). Bayesian asymptotics with misspecified models. Statistica Sinica, pages 169–187. (Cited on page 2.) Diaconis, P. and Freedman, D. (1986a). On inconsistent Bayes estimates of location. The Annals of Statistics , pages 68–87. (Cited on page 2.) Diaconis, P. and Freedman, D. (1986b). On the consistency of Bayes... | https://arxiv.org/abs/2504.11360v2 |
of gravitational-wave parameters. Physical Review D , 109(8):083002. (Cited on page 2.) Miller, J. W. (2021). Asymptotic normality, concentration, and coverage of generalized posteriors. Journal of Machine Learning Research , 22(168):1–53. (Cited on page 2.) Rousseau, J. and Mengersen, K. (2011). Asymptotic behaviour o... | https://arxiv.org/abs/2504.11360v2 |
Bringing closure to FDR control: beating the e-Benjamini-Hochberg procedure Ziyu Xu* Lasse Fischer†Aaditya Ramdas‡ April 23, 2025 Abstract False discovery rate (FDR) has been a key metric for error control in multiple hypothesis testing, and many methods have developed for FDR control across a diverse cross-section of ... | https://arxiv.org/abs/2504.11759v2 |
results of Ignatiadis et al. (2025) who show that every FDR controlling procedure is an instance of the eBH procedure with compound e-values, a fact that we return to later in this work. Problem setup. The standard multiple testing problem assumes that there are Knull hypotheses H1, . . . ,HKa scientist wishes to test,... | https://arxiv.org/abs/2504.11759v2 |
all procedures controlling these metrics (Theorem 3). This char- acterizes multiple testing procedures by single e-values for the intersection hypotheses and often yields improvements of existing procedures. While closure principles for FWER con- trol (Marcus et al., 1976) and FDP probability bounds (Goeman and Solari,... | https://arxiv.org/abs/2504.11759v2 |
first define self-consistent candidate discovery sets CSC, and then note that ReBHis the largest such set. Formally, CSC:= R⊆[K] : min i∈REi≥K/(α|R|) , ReBH=argmax R⊆CSC|R|. 4 Proposition 2. CSC⊆C, and consequently ReBH⊆ReBH. Proof. Consider the following derivation for arbitrary AandR: EA=|A|−1X i∈AEi ≥ |A|−1|A∩R|mi... | https://arxiv.org/abs/2504.11759v2 |
K−1for every j∈ReBHm, which is again a much more stringent condition. Since K/(K−1)≤2, with equality only for K= 2, the latter threshold matches that of the closed eBH procedure for K= 2, but is more stringent otherwise. It can be easily shown that the closed eBH procedure indeed equals the minimally adaptive eBH proce... | https://arxiv.org/abs/2504.11759v2 |
. , K }, we can 7 compute dFDR(Rk)as a maximization over a O(K2)sized table which also only take O(K2)time to compute. Post-hoc validity A stronger form of control on the FDP has been considered in prior literature (Gr¨unwald, 2024; Xu et al., 2024; Koning, 2024) that the eBH (or any e-self-consistent) procedure satisf... | https://arxiv.org/abs/2504.11759v2 |
of (1) and be denoted as ˜ReBH:=ReBH(eE). Let ˜ReBH:=ReBH(eE)now be the discovery set that results from applying the eBH procedure. Proposition 6. We have that ˜ReBH⊆˜ReBH. Proof. The proof is nearly identical to that of the proof of Proposition 2, albeit using the e-values defined in (9). EA=K−1X i∈AeEi≥K−1|A∩˜R|min i... | https://arxiv.org/abs/2504.11759v2 |
procedure Rwhere R= R(X) =D(X)αfor a specific α∈[0,1]. Proof. The theorem follows from (¯EA)A⊆[K]being a valid e-collection due to Dhaving post-hoc FDP control. Further, we note that ¯EA≥EA(as defined in (12)) since EAis included in the supremum taken in ¯EA. As a result, the resulting e-closed procedure Rprovides FDR ... | https://arxiv.org/abs/2504.11759v2 |
the largest rejection set R∈C(F)contains all indices i∈[K]such that EA≥1/αfor allAwithi∈A. This is the e-Holm procedure (V ovk and Wang, 2021; Hartog and Lei, 2025). Consequently, we could decide based on the data if we want to control the FDR with the eBH procedure or to control the FWER with the e-Holm procedure with... | https://arxiv.org/abs/2504.11759v2 |
more powerful than the eBH procedure. Boosting. Wang and Ramdas (2022) observed that one can increase the power of an e-value when testing against a finite set of rejection thresholds. In the most general setting of assuming arbitrary dependence, if one knows the marginal distribution of an e-value under the null, one ... | https://arxiv.org/abs/2504.11759v2 |
average FDR and true positive rate (TPR) :=E[|(A∗)c∩R|/|(A∗)c|]of eBH, minimally adaptive eBH (Ignatiadis et al., 2024), and eBH over n= 1000 trials. We plot the results in Figure 1, and can see that the eBH procedure improves noticeably over the standard and minimally adaptive eBH procedures across all settings, while... | https://arxiv.org/abs/2504.11759v2 |
so that more discoveries are made. There are generally two areas in this line: (1) boosting with no assumptions on the e-value distributions and (2) boosting e-values by using explicit knowledge about the underlying (marginal or joint) distribution. For the first area, we have already discussed the connection between o... | https://arxiv.org/abs/2504.11759v2 |
sequentially rejective multiple test procedures. Statistics in Medicine , 28(4):586–604, 2009. X. Cui, T. Dickhaus, Y . Ding, and J. C. Hsu. Handbook of Multiple Comparisons . CRC Press, 2021. S. Dandapanthula and A. Ramdas. Multiple testing in multi-stream sequential change detection. arXiv:2501.04130, 2025. P. Gablen... | https://arxiv.org/abs/2504.11759v2 |
arXiv:2504.11834v2 [math.ST] 21 Apr 2025Estimation and inference in error-in-operator model Vladimir Spokoiny∗ Weierstrass Institute and HU Berlin, Mohrenstr. 39, 10117 Berlin, Germany spokoiny@wias-berlin.de April 22, 2025 Abstract Many statistical problems can be reduced to a linear inverse problem in which only a no... | https://arxiv.org/abs/2504.11834v2 |
. . . . . . . . 19 3.2.1 Spectral cut-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2.2 Approximation spaces and truncation penalties . . . . . . . . . . . 20 3.2.3 Estimation risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2.4 Regularity of A∗and smoothness of θ∗. . . . . . . . ... | https://arxiv.org/abs/2504.11834v2 |
Ais smooth and the dimensions p,qare large or infinite. Hoffmann and Reiss (2008) considered simultaneous wavelet estimation of the signal θand the operator A.Trabs (2018) extended these results to Bayesian inference using a param etric assumption about the unknown operator A=Aθand provided some examples from imaging and... | https://arxiv.org/abs/2504.11834v2 |
to be random, often i.i.d. This particularly concerns econometric, biologica l, chemical, sociological studies etc. Under linear parametric assumption f(x) =/summationtext jθjψj(x) =Ψ(x)⊤θand noise homogeneity, the log-likelihood reads exactly as in the cas e of a deterministic design: L(θ) =−1 2σ2/ba∇dblY−Ψ(X)⊤θ/ba∇db... | https://arxiv.org/abs/2504.11834v2 |
assumption yields /BXUm= 0 for each m. Introduce the linear operator /hatwideA: /CAp→ /CAqwith /hatwideAdef=/parenleftbiggn/summationdisplay i=1ψj(Xi)φm(Wi)/parenrightbigg j≤p,m≤q(1.6) and define A∗= /BX/hatwideA. The operator A∗expresses the joint distribution of the regressor Xand the instrument W. Further we consider... | https://arxiv.org/abs/2504.11834v2 |
spectral cut-off to achieve the optimal rate of estimation. A data-driven choice of the 8 Estimation and inference in error-in-operator model penalty parameters is not discussed in this paper. Note howe ver, that a standard cross- validation approach is well applicable in this setup, and th e established Wilks expansion... | https://arxiv.org/abs/2504.11834v2 |
and Reiss (2008);Trabs(2018) with kmj=kmfork2 1≤...≤k2 qresults in /ba∇dblA/ba∇dbl2 K=q/summationdisplay m=1k2 m/ba∇dblAm/ba∇dbl2, whereA⊤ mis themth row ofA,Am∈ /CAp. Define the penalized MLE /tildewideυG, its population counterpart υ∗ G, and the truth υ∗as /tildewideυG= (/tildewideθG,/tildewidezG,/tildewideAG) = argma... | https://arxiv.org/abs/2504.11834v2 |
. Our results state root- Nthe accuracy of estimation. With some δ0<1, define the radius R=δ0µ√ Nand the local set Υ◦def=/braceleftbig (θ,z,A):/ba∇dblDθ/ba∇dbl ≤R,/ba∇dblz/ba∇dbl ≤R,/ba∇dbl(A−A∗)D−1/ba∇dblFr≤δ0/bracerightbig . (2.5) Later we use δ0≤1/10 and assume that the full dimensional estimator /tildewideυGis limit... | https://arxiv.org/abs/2504.11834v2 |
αQensures αQdef=/ba∇dblQD−1/ba∇dbl(3/4)κ2τ3(C4s/u1D561D+b2 D)/radicalbig RQ<1, then (1−αQ)2RQ≤ /BX/braceleftbig /ba∇dblQ(/tildewideθG−θ∗)/ba∇dbl21IΩ(x)/bracerightbig ≤(1+αQ)2RQ.(2.15) 14 Estimation and inference in error-in-operator model A great benefit of the suggested approach is that the scope of t he results presen... | https://arxiv.org/abs/2504.11834v2 |
Var/parenleftbig F−1 G∇ζ/parenrightbig θ≤2κ2(Cω+4Cνδ2 0)(κ−2D2+G2)−1, (2.19) and for any Qlinear mapping Qon /CAp/BX/vextenddouble/vextenddoubleQ/parenleftbig F−1 G∇ζ/parenrightbig θ/vextenddouble/vextenddouble2≤2κ2(Cω+4Cνδ2 0)tr/braceleftbig Q(κ−2D2+G2)−1Q⊤/bracerightbig .(2.20) Moreover, with ˘ξGfrom(2.13)/BX/ba∇dbl˘... | https://arxiv.org/abs/2504.11834v2 |
should be applied. In the rest of this section, we assume a smooth operator A∗and focus on ridge penalization for the target parameter θonly, that is, K ≡0; cf.Noskov et al. (2025). Our study includes the case with p=q=∞. Introduce the ordered eigenvalues N1≥ N2≥...≥NpofD2=A∗⊤A∗. By (2.15) and (2.14) of Theorem 2.1, th... | https://arxiv.org/abs/2504.11834v2 |
a fixed cut-off parameter 20 Estimation and inference in error-in-operator model J, consider the truncation penalty G2 J= diag{G2 1,...,G2 p}withg2 j= 0 forj≤Jand g2 j=∞forj >J. Effectively, this penalty enforces /tildewideθG,j=θ∗ G,j= 0 forj >J. Proposition 3.3. LetN1≥N2≥...≥Npbe the ordered eigenvalues of D2. Let also G... | https://arxiv.org/abs/2504.11834v2 |
smallest eigenvalue. If the basis vectors ejin /CApare defined as the ordered eigenvectors of D2 thenNj= /D2jcoincide with the jth eigenvalue of D2. In general, Njand /D2jmight 22 Estimation and inference in error-in-operator model be different. For mildly/severely ill-posed problems, these values rapidly decrease with j... | https://arxiv.org/abs/2504.11834v2 |
(3.9). Further, as w2 j=Cwj2βincrease with j, it holds /ba∇dbl( /C1p−ΠJ)θ∗/ba∇dbl2=p/summationdisplay j=J+1/an}b∇acketle{tθ∗,ej/an}b∇acket∇i}ht2≤w−2 Jp/summationdisplay j=J+1w2 j/an}b∇acketle{tθ∗,ej/an}b∇acket∇i}ht2≤w−2 J=C−1 wJ−2β. This yields R/lessorsimilartr(D−2 J)+w−2 J≤N−1 1J2s+1+C−1 wJ−2β/lessorsimilarC−2s+1 1+2... | https://arxiv.org/abs/2504.11834v2 |
corresponds to B= /BY−1/2V2/BY−1/2andx≈lognleading to the bound/C8/parenleftbig /ba∇dbl /BY−1/2∇ζ/ba∇dbl>z(B,x)/parenrightbig ≤3/n. The value /u1D561G= tr( /BY−1V2) can be called the effective dimension ; seeSpokoiny (2017). We also assume that the log-likelihood L(υ) or, equivalently, its deterministic part/BXL(υ) is a... | https://arxiv.org/abs/2504.11834v2 |
blocks Fθθ,FηηofFbe positive definite. Define Φθθdef=Fθθ−FθηF−1 ηηFηθ, Φ ηηdef=Fηη−FηθF−1 θθFθη. IfΦθθorΦηηis also positive definite then Fis positive definite as well. It holds /parenleftigg FθθFθη FηθFηη/parenrightigg−1 =/parenleftigg/C1p0 −F−1 ηηFηθ /C1q/parenrightigg /parenleftigg Φ−1 θθ0 0F−1 ηη/parenrightigg /p... | https://arxiv.org/abs/2504.11834v2 |
( B.8) with ρxz=αxz βxxβzz, ρxτ=αxτ βxxβττ, ρτz=ατz βττβzz, Now we apply Lemma B.2and note that by ( B.9) (1−ρxz−ρxτ)Fxx−κ−2D2 xx≥(1−ρxz−ρxτ)D2 xxβ2 xx−κ−2D2 xx≥0, (1−ρxz−ρzτ)Fzz−κ−2D2 zz≥(1−ρxz−ρzτ)D2 zzβ2 zz−κ−2D2 zz≥0, (1−ρxτ−ρzτ)Fττ−κ−2D2 ττ≥(1−ρxτ−ρzτ)D2 ττβ2 ττ−κ−2D2 ττ≥0. This implies the assertion. C Error-in-O... | https://arxiv.org/abs/2504.11834v2 |
4= 3N−1µ−2. Proof.Fixu= (α,h,∆) withα∈ /CAp,h∈ /CAq, and∆∈ /CAq×ps.t. u⊤D2u=/ba∇dblDα/ba∇dbl2+/ba∇dblh/ba∇dbl2+µ2/ba∇dbl∆/ba∇dbl2 Fr= 1. (C.13) Considerf(υ+tu) =f(θ+tα,z+th,A+t∆). The fourth derivatived4 dt4f(υ+tu) does not depend on υand it holds for any t −d4 dt4f(υ+tu) =−/an}b∇acketle{t∇4f(υ),u⊗4/an}b∇acket∇i}ht= 12... | https://arxiv.org/abs/2504.11834v2 |
−F−1 zzFzAΦ−1 K,AAK2A∗ Φ−1 K,AAK2A∗/parenrightigg =/parenleftigg −1 2FzAΦ−1 K,AAK2A∗ Φ−1 K,AAK2A∗/parenrightigg ,(C.17) whereΦ−1 K,AAis theAA-block of F−1 G,ηη: ΦK,AA=FAA+K2−1 2FAzFzA. By (C.4) andz∗ m−A∗ m⊤θ∗≡0, the matrix FAzFzAis block-diagonal with the blocks θ∗θ∗⊤. This implies ΦK,AmAm=µ2/C1p+K2 m−1 2θ∗θ∗⊤, m= ... | https://arxiv.org/abs/2504.11834v2 |
On the Intersection and Composition properties of conditional independence Tobias Boege UiT The Arctic University of Norway, Tromsø, Norway post@taboege.de Abstract Compositional graphoids are fundamental discrete structures which appear in probabilistic reasoning, particularly in the area of graphical models. They are... | https://arxiv.org/abs/2504.11978v1 |
et al. (2023). Intersection classically appears as a technical condition which ensures the equivalence of different Markov properties of graphical models (see (Lauritzen, 1996, Theorem 3.7)). It also guarantees the uniqueness of Markov boundaries by Pearl and Paz (1985) and drives certain identifiability results descri... | https://arxiv.org/abs/2504.11978v1 |
to ijkL and condition on L. Thus, we arrive at the following problem formulation which is addressed in this paper. Problem. For jointly distributed discrete random variables ( X,Y,Z), find sufficient conditions such that Intersection [X⊥ ⊥Y|Z]∧[X⊥ ⊥Z|Y] =⇒[X⊥ ⊥Y]∧[X⊥ ⊥Z], respectively, Composition [X⊥ ⊥Y]∧[X⊥ ⊥Z] =⇒[X⊥... | https://arxiv.org/abs/2504.11978v1 |
Xis a function of both YandZbut non-constant (hence has positive Shannon entropy H(X)), then the conclusion of Intersection is not satisfied. 4 Remark 3.3. Note that the conditions (2)and(3)in Example 3.1 enforce in both cases thatYis a function of Zand vice versa. It is possible to violate Intersection without any fun... | https://arxiv.org/abs/2504.11978v1 |
other components contain distributions violating Intersection. This explains the computational results observed in Example 3.1. Example 3.7 (Incompleteness of the G´ acs–K¨ orner criterion) .The following table defines a joint distribution of four binary random variables in which Gis the G´ acs–K¨ orner common informat... | https://arxiv.org/abs/2504.11978v1 |
Fallat et al. (2017) show that MTP 2implies upward stability (i.e., [I⊥ ⊥J|K]=⇒[I⊥ ⊥J|L]for any L⊇K), which is far stronger than Composition. We again begin the investigation with two example classes in which Composition is violated. Example 4.1 (Matroids) .Matroids provide a class of functional dependence structures w... | https://arxiv.org/abs/2504.11978v1 |
4.3, the Composition property is obtained simultaneously for all conditional distributions given the auxiliary G. This makes the criterion appear to be somewhat harder to work with as it requires a suitable coupling of the conditional distributions through G. The following Examples 4.4 and 4.5 show possible application... | https://arxiv.org/abs/2504.11978v1 |
incompatibility, we believe that the coincidence of the cardinalities of I4andC4is an artifact of the small ground set size rather than a reflection of a deeper connection between the two properties. Relation to Gaussianity. Regular Gaussian distributions satisfy both, Intersection and Composition. For the third implic... | https://arxiv.org/abs/2504.11978v1 |
s (2007), Theorem 3 and Corollary 2, it is possible to construct discrete random variables Y∗,Z∗,G∗ 1,G∗ 2with possibly infinite support such that: •H(G∗ 1|Y∗) =I(X:Y∗|G∗ 1) = 0, •H(G∗ 2|Z∗) =I(X:Z∗|G∗ 2) = 0, and •the entropy profiles of ( X,Y,Z) and ( X,Y∗,Z∗) agree. Then G∗= (G∗ 1,G∗ 2) is a function of ( Y∗,Z∗) and... | https://arxiv.org/abs/2504.11978v1 |
Series . Oxford University Press, 1996. S. Lauritzen and K. Sadeghi. Unifying Markov properties for graphical models. Ann. Statist. , 46(5):2251–2278, 2018. doi: 10.1214/17-AOS1618. R. Lnˇ eniˇ cka and F. Mat´ uˇ s. On Gaussian conditional independence structures. Kybernetika , 43(3):327–342, 2007. F. Mat´ uˇ s. Probab... | https://arxiv.org/abs/2504.11978v1 |
1 New Three Different Generators for Constructing New Three Different bivariate Copulas Iman M. Attia * Imanattiathesis1972@gmail.com ,imanattia1972@gmail.com *Department of Mathematical Statistics, Faculty of Graduate Studies for Statistical Research, Cairo University, Egypt Abstract In this paper, the author introduc... | https://arxiv.org/abs/2504.11993v1 |
bivariate copula is a non-decreasing and a right-continuous bivariate function mapping 𝐶:[0,1]×[0,1]→[0,1] which satisfies the following conditions 1) 𝐶(0,𝑣)=𝐶(𝑢,0)= 0, ∀(𝑢,𝑣)∈[0,1]ଶ ( Grounded) 2) 𝐶(1,𝑣)=𝑣 and 𝐶(𝑢,1)=𝑢 ∀(𝑢,𝑣)∈[0,1]ଶ ( uniform marginal) 3) 𝐶(𝑢ଵ,𝑣ଵ)−𝐶(𝑢ଵ,𝑣ଶ)−𝐶(𝑢ଶ,𝑣ଵ)+𝐶(𝑢ଶ,𝑣ଶ)≥... | https://arxiv.org/abs/2504.11993v1 |
ln(𝑢))ଵ ఈ+(−α ln𝑣)ଵ ఈ൨ఈିଶ ቆ1 𝛼(−α ln(𝑣))ଵ ఈିଵቀ−𝛼 𝑣ቁቇ = 𝑒𝑥𝑝ቆ−1 𝛼(−α ln(𝑢))ଵ ఈ+(−α ln𝑣)ଵ ఈ൨ఈ ቇ(−α ln(𝑢))ଵ ఈ+(−α ln𝑣)ଵ ఈ൨ଶఈିଶ ቆ(−α ln(𝑣))ଵ ఈିଵ൬1 𝑣൰ቇ× ቆ(−α ln(𝑢))ଵ ఈିଵ൬1 𝑢൰ቇ+𝑒𝑥𝑝ቆ−1 𝛼(−α ln(𝑢))ଵ ఈ+(−α ln𝑣)ଵ ఈ൨ఈ ቇቆ(−α ln(𝑢))ଵ ఈିଵ൬1 𝑢൰ቇ× (𝛼 −1)(−α ln(𝑢))ଵ ఈ+(−α ln𝑣)ଵ ఈ൨ఈିଶ ቆ(−α ln(𝑣))ଵ ఈିଵ൬−1... | https://arxiv.org/abs/2504.11993v1 |
𝑧∈(0,1) The generator should fulfill the sufficient conditions: 1) 𝜑(0)=(− ln0)ఈషమ= ∞ 2) 𝜑(1)=(− ln1)ఈషమ= 0 3) 𝜑ᇱ(𝑧)= 𝛼ିଶ (− ln𝑧)ఈషమିଵቀିଵ ௭ቁ < 0 This ensures that the generator is a decreasing function. 4) 𝜑ᇱᇱ(𝑧)= ଵ ఈమቀଵ ఈమ−1ቁ (− ln𝑧)ఈషమିଶቀିଵ ௭ቁቀିଵ ௭ቁ+ 𝛼ିଶ (− ln𝑧)ఈషమିଵቀଵ ௭మቁ> 0 This ensures that the generat... | https://arxiv.org/abs/2504.11993v1 |
න𝜑(𝑢) 𝜑ᇱ(𝑢)ଵ 𝑑𝑢 +1 = 4 ቆ−𝛼ଶ 2ቇ +1 = 1−2 𝛼ଶ If 𝛼 𝑖𝑠 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑖𝑛𝑔 0 𝜏= 1 indicating positive dependency. If 𝛼= 1 𝑠𝑜 𝜏= −1 indicating negative dependency. To be product copula indicating independency, the alpha parameter should be square root of 2 at which the copula is invalid. So this copula ca... | https://arxiv.org/abs/2504.11993v1 |
Attia, 2024), transform this variable to be defined on the interval from zero to infinity as shown below: 6 𝛼ଶ 1−𝑦ଵ ఈమ൨𝑦ቀଶ ఈమିଵቁ , 0 < 𝑦 < 1 , 𝛼 > 0 Let 𝑤 = −ln (𝑦)భ ഀయ → 𝑤 =ି୪୬௬ ఈయ → −𝑤𝛼ଷ= ln𝑦 → 𝑦 = 𝑒ିఈయ௪ 𝑑𝑦 =𝑒−𝛼3𝑤(−𝛼ଷ) 𝑑𝑤 න6 𝛼ଶ ቈ1−ቀ𝑒ି𝛼3 ௪ ቁଵ ఈమቀ𝑒ି𝛼3௪ ቁቀଶ ఈమቁஶ ቀ𝑒ି𝛼3௪ ቁିଵ (𝛼ଷ)𝑒ି𝛼3௪ 𝑑�... | https://arxiv.org/abs/2504.11993v1 |
Proof: when u=1 so : 𝜑(𝑤)= 𝛼 2 ቐ−10+ඨ ൬1+24 1൰ +ඨ ൬1+24 𝑣൰ቑ= = 𝛼 2 ቐ−5 +ඨ ൬1+24 𝑣൰ቑ 𝐶(1,𝑣) =6 𝛼ଶ (𝜑(𝑤)+2𝛼)(𝜑(𝑤)+3𝛼) 𝐶(1,𝑣) =6 𝛼ଶ ቆ 𝛼 2 ቊ−5 +ට ቀ1+24 𝑣ቁቋ+2𝛼ቇቆ 𝛼 2 ቊ−5 +ට ቀ1+24 𝑣ቁቋ+3𝛼ቇ 𝐶(1,𝑣) =6 𝛼ଶ ቆ 𝛼 2 ቊ−5 +ට ቀ1+24 𝑣ቁቋ+4𝛼 2ቇቆ 𝛼 2 ቊ−5 +ට ቀ1+24 𝑣ቁቋ+6𝛼 2ቇ 𝐶(1,𝑣) =6 𝛼ଶ ൭ 𝛼 2 ቆቊ−5 +ට ቀ1+2... | https://arxiv.org/abs/2504.11993v1 |
=1 Fig. 26 shows the joint PDF copula (copula density) at alpha =1 32 Fig. 27 shows the joint CDF copula (copula) at alpha =10 Fig. 28 shows the joint PDF copula (copula density) at alpha =1 To sum up for this copula, the generator and inverse generator fulfill the conditions. The copula density fulfills the criteria b... | https://arxiv.org/abs/2504.11993v1 |
association analysis of systolic and diastolic blood pressure by copula models. BMC Proceedings , 8(S1), S72. https://doi.org/10.1186/1753-6561-8-S1-S72 Kuss, O., Hoyer, A., & Solms, A. (2014). Meta-analysis for diagnostic accuracy studies: A new statistical model using beta-binomial distributions and bivariate copulas... | https://arxiv.org/abs/2504.11993v1 |
arXiv:2504.12190v2 [math.ST] 28 Apr 2025CREATING NON-REVERSIBLE REJECTION-FREE SAMPLERS BY REBALANCING SKEW-BALANCED MARKOV JUMP PROCESSES By Erik Jansson1, Moritz Schauer1 Ruben Seyer1,aand Akash Sharma1 1Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg,arubense@chalm... | https://arxiv.org/abs/2504.12190v2 |
it behaves qualitatively different. The essential difference of the proposed sampler is that it avoids random walk behavior that may be present in HMC [ 30]. This is due to its non-reversibility, which manifests as robustness to tuning and mixing for a wider range of hyperparameters. In practice, the computational cost... | https://arxiv.org/abs/2504.12190v2 |
The law of the process is determined by: 1. The initial distribution Z0∼π0on(S,B(S)). 2.The rate function λ:S→[0,∞). Conditional on Zτi=a, the time until the next jump is exponentially distributed with rate λ(a) = (E[τi+1−τi|Zτi=a])−1. 3.The jump kernel κ:S×B(S)→[0,1], such that P(Zτi+1∈B|Zτi=a) =κ(a,B)for alla,B∈S×B(S... | https://arxiv.org/abs/2504.12190v2 |
with an additional semi-local condition yields the necessary stationarity. Proposition 2.2.Letπbe a probability measure, a π-isometric involution s, and µa rate kernel with bounded rate λ(a) =/integraltext Sµ(a,db)<∞. If a (weaker) skew balance condition (2.7)/integraldisplay S×Sf(b)π(da)µ(a,db) =/integraldisplay S×Sf(... | https://arxiv.org/abs/2504.12190v2 |
is invariant under sandT, and define ˜µ(a,db) =˜λ(a)δT(a)(db), whereδxis the Dirac measure at x. Then the rate kernel ˜µis in skew-detailed balance with respect to sand˜π. SKEW-BALANCED MARKOV JUMP SAMPLERS 7 Proof. For any ˜π-integrable f:S×S→R, a direct calculation with the change of variablesa∝⇕⊣√∫⊔≀→(s◦T)(b′)yields... | https://arxiv.org/abs/2504.12190v2 |
a leapfrog step. Thus, in the context of Section 2.2, T= LF,˜λ= 1and˜µ(a,db) =δLF(a)(db). This process obviously does not sample from the measure π∝e−H(q,p)˜π, where ˜π denotes the Lebesgue measure. However, note that ˜πis preserved under LFby the volume preservation. Therefore, since π≪˜π, we can apply Theorem 2.1 to ... | https://arxiv.org/abs/2504.12190v2 |
of the FFF sampler relies on, reversibility and volume preservation, are the same properties underpinning HMC. It is remarkable that, by simply applying them differently, one can instead obtain a rejection-free, non-reversible, and continuous-time sampler. In the construction of HMC, the basis is detailed balance. By o... | https://arxiv.org/abs/2504.12190v2 |
someR≥2β/α. In our case, we have that both the drift and minorization conditions hold, as detailed in the following theorem: Theorem 3.2.Let Assumption 1 hold. Furthermore, let the leapfrog integration step sizeεsatisfy the bound (3.9) 0<ε<2√κ L∇U. Then, the norm-like function (3.10) V(q,p) =C+H(q,p), whereC∈Ris such t... | https://arxiv.org/abs/2504.12190v2 |
(Zτn)/parenrightigg almost surely, for almost every Z0, whereτ0= 0. Proof. A direct application of the ergodic theorem and the Rao–Blackwell theorem as the total rate is bounded from above and below, see [ 11,39]. The geometric ergodicity proven in Theorem 3.2 implies a stronger version of the above proposition where ... | https://arxiv.org/abs/2504.12190v2 |
the maximal (i.e., worst) mean KS distance across all marginals. OurmetricdiffersfromthepopularchoiceofmetricswithinMarkovchainMonteCarlo based on effective sample size (ESS), which compare (an estimate of) the asymptotic variance to the corresponding estimator for an i.i.d. sequence [ 37]. While ESS-based metrics are ... | https://arxiv.org/abs/2504.12190v2 |
many small steps. This implies that exploration can be performed much more efficiently in such settings using the true non-reversibility of the FFF sampler. Wedefinethetargetpotential U(x) =(∥x∥−R)2 2σ2,andruntheexampleintwodimensions withR= 2.6,σ= 0.0165. Short sample trajectories for different settings of FFF and HMC... | https://arxiv.org/abs/2504.12190v2 |
The model thus shares several features with the previously covered synthetic models. TheresultsofthegridsearchareshowninFig.7,whereabudgetof150000evaluations of∇Uwas used for performance reasons, and the reference Fin the distributional 16 Fig 6. Contours of some marginal pair distributions in the Bayesian PKPD model, ... | https://arxiv.org/abs/2504.12190v2 |
momentum on lines in Rd. Of course, Tbeing less sophisticated compared to FFF, more flips will occur, and hence one explores less efficiently than with Hamiltonian dynamics. 3.The above construction also works on discrete spaces, for example S=Z×{− 1,+1} with ˜πas the counting measure. Refreshments can be avoided in th... | https://arxiv.org/abs/2504.12190v2 |
the function h=U(x)−κ∥x∥2/2is convex outside a ball with radius r≥0. Letℓ∈Rdenote the smallest (possibly negative) eigenvalue of ∇2h inside this ball. WithR2=/integraltext1 0∇2U(q+αδ)δdα, we expand ⟨δ,R 1−R2⟩=−/integraldisplay1 0κδTδαdα−/integraldisplay1 0δT∇2h(q+αδ)δαdα ≤−1 2κ∥δ∥2+/integraldisplay∥δ∥ 0/parenleftbigg −... | https://arxiv.org/abs/2504.12190v2 |
the elementary bound exp(−x+)x≤1/e, AV(q,p)≤1 e+λFRESHd 2−λFRESH∥p∥2/2. OnA2, ∥p∥2≥K2 1(U(q) +C) so AV(q,p)≤/parenleftbigg1 e+λFRESHd 2−λFRESHC 2 +K2 1/parenrightbigg −K2 1 2 +K2 1λFRESHH(q,p) and we get AV≤−α2V+β2. for someα2>0,β2∈RonA2. Combining both bounds proves the proposition. 22 A.2. Minorization condition. We ... | https://arxiv.org/abs/2504.12190v2 |
DKS(ˆF,F) = sup x∈R|ˆF(x)−F(x)| = max/braceleftigg sup n=1,...,N|ˆF(Xn)−F(Xn)|,sup n=1,...,N|ˆF(Xn−1)−F(Xn)|/bracerightigg = max n=1,...,Nmax{Wn−F(Xn), F(Xn)−Wn−1}. Similarly, by splitting the integral at the jump points in the definition2of the Anderson–Darling (AD) distance, we obtain DAD(ˆF,F) =/integraldisplay∞ −... | https://arxiv.org/abs/2504.12190v2 |
=/integraldisplay [(λFLIP◦s)(q,p)−λFLIP(q,p)]f(q,p)e−H(q,p)dqdp =/integraldisplay [λFROG (q,p)−(λFROG◦s)(q,p)]f(q,p)e−H(q,p)dqdp, where the last step used the definition of λFLIPand the elementary identity (x)+− (−x)+=xto recover the FROGrates. With this rewrite, we are in a position to cancel the FROGterm. First, we r... | https://arxiv.org/abs/2504.12190v2 |
Geometric Convergence for MALA under Verifiable Conditions. https://doi.org/10.48550/arXiv.2201.01951 [13]Durmus, A. ,Moulines, É. andSaksman, E. (2020). Irreducibility and Geometric Ergodicity of Hamiltonian Monte Carlo. Ann. Statist. 483545–3564. https://doi.org/10.1214/19-AOS1941 [14]Ethier, S. N. andKurtz, T. G. (1... | https://arxiv.org/abs/2504.12190v2 |
arXiv:2504.12307v1 [stat.ME] 3 Apr 2025On a new PGDUS transformed model using Inverse Weibull distribution Gauthami P . & Chacko V . M. Department of Statistics St. Thomas College (Autonomous), Thrissur, University of C alicut, Kerala, India - 680001 gauthamistat@gmail.com, chackovm@gmail.com Abstract The Power General... | https://arxiv.org/abs/2504.12307v1 |
Chacko (2021), who pro- posed and thoroughly examined the DUS Inverse Weibull (DUS- IW) distribution, which is distin- guished by its upside-down bathtub-shaped hazard rate. Rel ated studies that expanded these find- ings to the DUS-Kumaraswamy (DUS-K) distribution include t hose by Karakaya et al. (2021) and Anakha and... | https://arxiv.org/abs/2504.12307v1 |
the three-parameter Weibull model in early researches (see Harter , 1970, 1971). Due in large pa rt to its adaptability in modeling a vari- ety of experimental data, subsequent research, including a s studies by Mudholkar et al. (1996) and Evans et al. (2019), has made a substantial contribution to t he understanding o... | https://arxiv.org/abs/2504.12307v1 |
and since alternative methods can address potential challenges in parameter estimation, we also expl ore the Maximum Product of Spacings 3 (MPS) method, detailing estimation procedures, all given i n Section 3 and conducting simulation studies given in Section 4 to compare these approaches. Furt hermore, a real-data in... | https://arxiv.org/abs/2504.12307v1 |
∑ m=0(−1)k−1 m!/parenleftbigg2(γ−1) k/parenrightbigg (2γ−k)mΓ/parenleftig 1+1 λ/parenrightig (m+2)2+1 λ. 2.1. Distribution of Order Statistic Taking randomly a sample of size nfrom PGDUS−IW(λ,θ,γ)distribution, the rthorder statistic, where, r=1, 2, ..., nwill have the CDF and PDF as given in (2.7) and (2.8) respectiv... | https://arxiv.org/abs/2504.12307v1 |
the log-PS function lnPS(λ,θ,γ) = ( n+1)−1/braceleftigg γln ee−/parenleftbiggt1:n θ/parenrightbigg−λ −1 +ln/bracketleftigg (e−1)γ−/parenleftigg ee−(tn:n θ)−λ −1/parenrightiggγ/bracketrightigg +n ∑ i=2ln ee−/parenleftbiggti:n θ/parenrightbigg−λ −1 γ − ee−/parenleftbiggti−1:n θ/parenrightbigg−λ −1 γ ... | https://arxiv.org/abs/2504.12307v1 |
MSE ( ˆλ) 50 0.9511 −0.0489 0.0144 100 0.9619 −0.0380 0.0072 150 0.9699 −0.0301 0.0049 350 0.9816 −0.0184 0.0022 n ˆθ bias ( ˆθ) MSE ( ˆθ) 50 0.8376 0.2376 0.7693 100 0.7164 0.1164 0.3414 150 0.6339 0.0339 0.2134 350 0.5874 −0.0126 0.1062 n ˆγ bias ( ˆγ) MSE ( ˆγ) 50 0.9850 0.6850 4.3225 100 0.8876 0.5876 3.0054 150 0.... | https://arxiv.org/abs/2504.12307v1 |
i=1fT1PGDUS(t1i,λ,θ,γ1)M ∏ j=1fT2PGDUS(t2j,λ,θ,γ2). On required substitution, simplification and taking logari thm, we get the log-L function, 10 lnL=Nlnγ1+Mlnγ2+(N+M)(lnλ+λlnθ)−(Nγ1+Mγ2)ln(e−1) −(λ+1)/bracketleftigg N ∑ i=1lnt1i+M ∑ j=1lnt2j/bracketrightigg −N ∑ i=1/parenleftbiggt1i θ/parenrightbigg−λ −M ∑ j=1/parenl... | https://arxiv.org/abs/2504.12307v1 |
Rc,k, we find the estimators ofγ1and γ2. The likelihood function is L(λ,θ,γ1,γ2|tl 1j,t2j) =N ∏ j=1/braceleftigg/bracketleftigg k ∏ l=1fT1PGDUS(tl 1j,λ,θ,γ1)/bracketrightigg fT2PGDUS(t2j,λ,θ,γ2)/bracerightigg . On required substitution, simplification and taking logari thm, we get lnL=Nklnγ1+Nlnγ2+N(k+1)(lnλ+λlnθ)−N(... | https://arxiv.org/abs/2504.12307v1 |
the est imating methods used to fit the two datasets. Having been considered, this thorough invest igation demonstrates its resilience and suitability for use in reliability theory, especially for p arallel systems, offering a useful resource for researchers working in this field. 15 References Abu El Azm, W. S., Almetwa... | https://arxiv.org/abs/2504.12307v1 |
In Stochastic models in reliability engineering , pages 81-100. CRC Press. Drapella, A. (1993). The complementary Weibull distributi on: unknown or just forgotten?. Quality and reliability engineering international, 9 (4), 383-385. Elbiely, M. M., & Yousof, H. M. (2019). A new inverse Weibull d istribution: properties ... | https://arxiv.org/abs/2504.12307v1 |
of entropy. Statistical Science , 40-58. Loganathan, A., & Uma, A. (2017). Comparison of estimation m ethods for inverse weibull parame- ters. Global and Stochastic Analysis, 4 (1), 83-93. Mahmoud, M. A. W., Sultan, K. S., & Amer, S. M. (2003). Order st atistics from inverse Weibull distribution and associated inferenc... | https://arxiv.org/abs/2504.12307v1 |
arXiv:2504.12520v1 [math.ST] 16 Apr 2025Interpreting Network Differential Privacy Jonathan Hehir, Xiaoyue Niu, and Aleksandra Slavkovi´ c Department of Statistics, Pennsylvania State University, University Park, PA April 18, 2025 Abstract How do we interpret the differential privacy (DP) guarantee for network data? We ... | https://arxiv.org/abs/2504.12520v1 |
assumptions are generally unrealistic but frequently left unstated. (See Section 3.1.) A major goal of our work is to determine reasonable conditions under whic h similar claims can be made. Related Work. Tschantz et al. (2020, Appendix A) provides a detailed histo ry of varied per- spectives on the interpretation of D... | https://arxiv.org/abs/2504.12520v1 |
We provide add itional background on Pufferfish and its relationship with DP in Section 4. Outline. We review the classical (non-network) setting of different ial privacy in Section 2, providing discussion on both inferential and causal interp retations. Section 3 offers a review of edge DP and its interpretations, wher... | https://arxiv.org/abs/2504.12520v1 |
neighbors: /u1D4370∼/u1D4371iff|/u1D4370|=|/u1D4371|=/u1D45Band/u1D4370,/u1D4371agree on all but one tuple.2 •Unbounded neighbors: /u1D4370∼/u1D4371iff|/u1D4370|=|/u1D4371| +1and/u1D4371⊂/u1D4370(or vice-versa). Under a bounded-neighbors definition, the size of the databa se (i.e., number of rows, /u1D45B) is fixed and n... | https://arxiv.org/abs/2504.12520v1 |
tuples /u1D4610,/u1D4611at level/u1D6FCwith power greater than /u1D452/u1D700/u1D6FC. After all, conditioned on the remaining /u1D45B−1records, the hypothesis tests for the /u1D456-th record are equivalent to hypothesis tests for the full database. This h ypothetical almost-all-knowing adversary is often referred to as... | https://arxiv.org/abs/2504.12520v1 |
Perhaps the most popular of these is edge differential privacy , which, loosely speaking, defines two networks as neighboring if they differ on a single edge. We explore her e some of the subtleties of edge DP, its interpretations and common misinterpretations, and th e implications of edge DP on adversarial inference. ... | https://arxiv.org/abs/2504.12520v1 |
unnatural because there often lacks a natural setting corresponding to the notion of neighboring databases in edge DP. Recall that in clas- sical DP, neighboring databases capture the distinction be tween an individual contributing true data vs. fabricated data (bounded neighbors) or contributing tr ue data vs. withold... | https://arxiv.org/abs/2504.12520v1 |
count of edges from the graph /u1D43A=(/u1D449,/u1D438) ∈ G/u1D45B, Medges(/u1D43A)=|/u1D438| +Laplace(1//u1D700), satisfies/u1D700–edge DP. Having established a privacy mechanism, we now provide an ex ample in which this mechanism fails to prevent inference of the presence or absence of a spe cific edge. Example 1.Suppo... | https://arxiv.org/abs/2504.12520v1 |
as present or absent. These two possibilities cor respond to two hypothetical, complete databases that could have been used to calculate ˜/u1D451—but only one of these databases is plausible under the constraints of the network data generating proces s. This hypothesis test is essentially meaningless to an adversary kn... | https://arxiv.org/abs/2504.12520v1 |
8 frames Puf ferfish as a proper generaliza- tion of any pure DP variant. In particular, the instantiatio n of Pufferfish in Corollary 8 remains distribution-agnostic. This is a different presentation f rom in Kifer and Machanavajjhala (2014, Section 6), where various results equate specific DP setting s with Pufferfish in... | https://arxiv.org/abs/2504.12520v1 |
that for all P/u1D703∈Θ,/u1D454∈ G/u1D45B, and{/u1D456, /u1D457} ∈ I/u1D449: P/u1D703(/u1D43A=/u1D454∪{/u1D456, /u1D457} | {/u1D456, /u1D457} ∈/u1D438) ≤/u1D452/u1D6FCP/u1D703(/u1D43A=/u1D454\{/u1D456, /u1D457} | {/u1D456, /u1D457}∉/u1D438) (2) P/u1D703(/u1D43A=/u1D454\ {/u1D456, /u1D457} | {/u1D456, /u1D457}∉/u1D438) ... | https://arxiv.org/abs/2504.12520v1 |
edge secrets S,Spairsfrom Theorem 9 and /u1D6FC=sup /u1D703,/u1D454,/u1D456,/u1D457/barex/barex/barex/barex/u1D6FD/u1D447 /u1D703Δ/u1D703(/u1D454,/u1D456, /u1D457)) −log/parenleftbiggP/u1D703({/u1D456, /u1D457} ∈/u1D438) P/u1D703({/u1D456, /u1D457}∉/u1D438)/parenrightbigg/barex/barex/barex/barex≤sup /u1D703,/u1D454,/u1... | https://arxiv.org/abs/2504.12520v1 |
the case of attributed graphs (i.e., networks with node covariates) edge DP alone offers no privacy for node attribu tes, since the privacy definition does not specify how a mechanism must treat differences in node-leve l data. Treatment of attributed graphs in practice varies, sometimes proceeding with edge DP anywa y ... | https://arxiv.org/abs/2504.12520v1 |
a pproximations in circulation, /u1D453- DP (Dong et al., 2022) is a natural starting point, since it is parameterized in terms of a trade-off function /u1D453between the Type I and Type II errors of the hypothesis test fo r neighboring databases /u1D43B0:/u1D437=/u1D4370vs./u1D43B1:/u1D437=/u1D4371. Data Collection an... | https://arxiv.org/abs/2504.12520v1 |
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