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(i,j)∈E(H)Qxi,xj/parenrightBigg . Now, we make the following observation. Suppose that His not a tree. Then, one can remove some edge ( i,j)∈E(H) to obtain a graph H′that is still connected. Clearly, ΦSBM(p,Q)(H′) =/summationdisplay x1,x2,...,xh∈[k]/parenleftBiggh/productdisplay i=1pxi×/productdisplay (i,j)∈E(H′)Qxi,xj... | https://arxiv.org/abs/2504.17202v1 |
by Lemma 3.2 , /summationdisplay x1,x2,...,xh−2px1px2···pxh−2/productdisplay (ij)∈E(H)\{(par(h−1),h−1),(par(h),h)}|Qxi,xj| ≤/summationdisplay x1,x2,...,xh−2px1px2...pxh−2(max u,v|Qu,v|)|E(H)|−2 = (max u,v|Qu,v|)|V(H)|−3 /lessorsimilarc,D|ΦSBM(p,Q)(Cyc4)||V(H)|−3 4. Altogether, we obtain that /vextendsingle/vextendsingl... | https://arxiv.org/abs/2504.17202v1 |
cancellations in the Fourier coeffi- cients of stars. ByCorollary 2.2 , ΦSBM(p,Q)(StarT) =p1(p1Q11+p2Q1,2)T+p2(p1Q12+p2Q2,2)T. Now, suppose that ( 29) does not hold. Then, for some large absolute constant C,4it must be the case that (4C2)T/vextendsingle/vextendsingle/vextendsinglep1(p1Q11+p2Q1,2)T+p2(p1Q12+p2Q2,2)T/vexte... | https://arxiv.org/abs/2504.17202v1 |
≤pa+t 1×|Q1,1|t×|Q1,2|a+b−1×|Q2,2|s×|Qi,j| /lessorsimilarpa+t+a+b−1 1 ×|Q1,1|a+b+t−1×|Q2,2|s×|Qi,j|, where we used Lemma 3.5 in the second inequality. If (i,j)∈ {(1,1),(1,2)},thena≥1 (as label 1 exists) and |Qi,j|=O(|Q1,1|).Thus, the above is expression is bounded by pa+b+t 1×|Q1,1|a+b+t×|Q2,2|s ≤(p1|Q1,1|)a+b+t×|Q2,2|... | https://arxiv.org/abs/2504.17202v1 |
3.8. Suppose that (30)holds,|Q1,2| ≤p1,andT>1.Then, ΨSBM(p,Q)(H)/lessorsimilarDΨSBM(p,Q)(StarT−1) Proof sketch. The proof is exactly the same as that of Lemma 3.8 . The only difference is that this time|µ|/lessorsimilarDp1,which implies ( p1µT−1)1/T/greaterorsimilarD(p1|µ|T)1/(T+1). Hence, what is left is the case T= 1.... | https://arxiv.org/abs/2504.17202v1 |
≥a+r.Now consider H|V(H)\L(H).Thisisaconnected graph since His connected and L(H) is the set of leaves. Hence, each connected component in H|V(H)\(K∪L(H))has at least one edge to a vertex in K.If, furthermore, this connected component is one of the vertices in S,it must have at least 2 edges to K.At least one edge due ... | https://arxiv.org/abs/2504.17202v1 |
we bound the expression in ( 43) by pa+t+s 1|Q1,1|t|Q1,2|a+b−1+s|p1Q1,2+p2Q2,2||L2(K)||p1Q1,1+p2Q1,2||L1(K)| (UsingEqs. (39) and(41)) /lessorsimilarDpa+t+s+|L2(K)| 1 |Q1,1|t|Q1,2|a+b−1+s+|L2(K)||p1Q1,1+p2Q1,2||L1(K)| /lessorsimilarDpa+t+s+|L2(K)|−1 1 |Q1,1|t×|Q1,2|a+b−1+s+|L2(K)|×p1|p1Q1,1+p2Q1,2||L1(K)|−ξ|×|Q1,2|ξ (Us... | https://arxiv.org/abs/2504.17202v1 |
that Star2is not an approximate maximizer, so ΨSBM(p,Q)(Star3) =o(ΨSBM(p,Q)(Start)). (49) Without loss of generality, let p1≤p2.Let also λi:=p1Q1,i+p2Q2,i,so ΦSBM(p,Q)(Starr) = p1λr 1+p2λr 2.Now, (49) implies that (p1λ2 1+p2λ2 2)1/3=o/parenleftBig (p1λt 1+p2λt 2)1/(t+1)/parenrightBig =⇒ (p1λ2 1+p2λ2 2)(t+1)=o/parenleft... | https://arxiv.org/abs/2504.17202v1 |
Inequalities We note that all arguments in the current section apply more g enerally to any graphon instead of stochastic block model, provided no measurability issues o ccur. 4.1 Cycle Comparisons: A Spectral Approach We prove the following theorem, which explains why signed tr iangles and 4-cycles are used for detect... | https://arxiv.org/abs/2504.17202v1 |
degree d.Then, for any SBM(p,Q)distribution, |ΦSBM(p,Q)(H)| ≤ |ΦSBM(p,Q)(K2,d)|1/2. Proof.Let the vertex set of Hbe{1,2,...,h}such that hhas degree d.Then, |ΦSBM(p,Q)(H)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleIE/bracketleftBigg/productdisplay (i,j)∈E(H)Qxixj/bracketrightBigg/vextendsingl... | https://arxiv.org/abs/2504.17202v1 |
a degree- Dpolynomial test with success probability 1 −ok(1). With a small blow-up in the sample-complexity, we can show th e following theorem. The proofs are identical, observing that the hidden factors are on the o rder of 2O(DlogD)everywhere. Hence, ifD=o(logn/loglogn),the hidden factors are of order no(1). Proposi... | https://arxiv.org/abs/2504.17202v1 |
theorem due to Alon [ Alo81] shows that the maximal number of graphs isomorphic to Hin a graph onMedges is Θ H(M|V(H)|+δ(H) 2),where δ(H):= max S⊆V(H)|S|−|{j∈V(H) :∃i∈Ss.t. (j,i)∈E(H)}|. Reasoning as in Section 1.3 , instead of finding the approximate maximizers of H−→ |ΦSBM(p,Q)(H)|1 |V(H)|,one should look for the appr... | https://arxiv.org/abs/2504.17202v1 |
struct ure. In S´ ebastien Bubeck, Vianney Perchet, and Philippe Rigollet, editors, Proceedings of the 31st Conference On Learning Theory , volume 75 of Proceedings of Machine Learning Research , pages 48–166. PMLR, 06–09 Jul 2018. [BBH+21] Matthew S Brennan, Guy Bresler, Sam Hopkins, Jerry Li, an d Tselil Schramm. Sta... | https://arxiv.org/abs/2504.17202v1 |
random ind ependent sets in the discrete hypercube. Comb. Probab. Comput. , 20(1):27–51, January 2011. [HLL83] Paul W. Holland, Kathryn Blackmond Laskey, and Samu el Leinhardt. Stochastic blockmodels: First steps. Social Networks , 5(2):109–137, 1983. [Hop18] Samuel Hopkins. Statistical inference and the sum o f square... | https://arxiv.org/abs/2504.17202v1 |
Sixth Conference on Learning Theory , volume 195 ofProceedings of Machine Learning Research , pages 5573–5577. PMLR, 12–15 Jul 2023. [MVW24] Jay Mardia, Kabir Aladin Verchand, and Alexander S. Wein. Low-degree phase transitions for detecting a planted clique in sublinear tim e. In Shipra Agrawal and Aaron Roth, editors... | https://arxiv.org/abs/2504.17202v1 |
Theorem A.3 (Balanced 2-SBMs in which 1-Stars Dominate) .There exists a stochastic block modelSBM(n;p,Q)on 2 balanced communities with the following property. It can be distinguished fromG(n,1/2)with high probability via the signed count of 1-stars, but no t via the signed counts of any other connected graph of constan... | https://arxiv.org/abs/2504.17202v1 |
high probability. Construction. Theconstructionisinspiredbythe“quietplanting”distri butionin[ KVWX23 ]. Take someq∈Nsuch that q=ω(n5/8),q=o(n2/3) and consider an SBM on k=q2communities labeled by [q]×[q],wherep(a,b)= 1/q2∀(a,b)∈[q]×[q] and Q(a1,b1),(a2,b2)= 1, a1=b1anda2/\e}a⊔io\slash=b2, −1, a1/\e}a⊔io\slash=b1a... | https://arxiv.org/abs/2504.17202v1 |
arXiv:2504.17451v1 [math.ST] 24 Apr 2025Functional /u1D472Sample Problem via Multivariate Optimal Measure Transport-Based Permutation Test ˇS´arka Hudecov ´a, Daniel Hlubinka and Zden ˇek Hl ´avka Abstract The null hypothesis of equality of distributions of functio nal data coming from/u1D43Esamples is considered. The ... | https://arxiv.org/abs/2504.17451v1 |
which is referred to in the following as the OMT permutation test. Let/u1D44B/u1D457,1,...,/u1D44B/u1D457,/u1D45B/u1D457be independent and identically distributed functional obs er- vations coming from a distribution with a characteristic fu nctional/u1D711/u1D457,/u1D457=1,...,/u1D43E , and let the/u1D43Esamples be in... | https://arxiv.org/abs/2504.17451v1 |
two sample test statistic defined in (3) for samples /u1D457and/u1D459. It follows from (3) that large values of /u1D446/u1D457,/u1D459indicate that the distribution in samples /u1D457and/u1D459may differ. Hence, define the test statistic /u1D47B=(/u1D4461,2,/u1D4461,3,...,/u1D446/u1D43E−1,/u1D43E)⊤=(/u1D4471,...,/u1D447/... | https://arxiv.org/abs/2504.17451v1 |
/u1D47Btakes values inR+with large values indicating violation of the corresponding partial pairwis e hypothesis. Therefore, in view of Section 3.2 in [11], a suitable grid set Gis a subset of {/u1D499:/u1D499∈ [0,1]/u1D451,/bardbl/u1D499/bardbl ≤ 1}. In view of [11], it is beneficial to specify /u1D435such that/u1D435+... | https://arxiv.org/abs/2504.17451v1 |
reasonable results under the null as well as against various alternatives. Foll owing this recommenda- tion, the following test statistics were considered: −/u1D47Binvfor/u1D47Dbeing the approximate inverse to the sample covariance matr ix com- puted from all data (without distinguishing the samples) an d −/u1D47Binv.p... | https://arxiv.org/abs/2504.17451v1 |
for equality of m ean functions for the con- sidered three samples. FP is permutation test based on basis function representation from [6], CS is/u1D43F2-norm-based parametric bootstrap test for heteroscedasti c samples from [3]. L2B is /u1D43F2-norm- based test with bias-reduced method of estimation, while L2 b is/u1D... | https://arxiv.org/abs/2504.17451v1 |
Concentration inequalities and cut-off phenomena for penalized model selection within a basic Rademacher framework Pascal Massart Institut de Math´ ematique d’Orsay Bˆ atiment 307 Universit´ e Paris-Saclay 91405 Orsay-Cedex, France Vincent Rivoirard CEREMADE Universit´ e Paris Dauphine Place du Mar´ echal de Lattre de ... | https://arxiv.org/abs/2504.17559v1 |
assumption is merely a convenience to lighten the presentation, but it is clear that everything we present here extends immediately to the case where the noise variables are bounded in absolute value by a constant M(greater than 1, of course, since the variables have a variance equal to 1). This a priori parametric est... | https://arxiv.org/abs/2504.17559v1 |
to connect the issue of understanding the behavior of this quantity with Talagrand’s works on concentration of product measures is to consider ∥ˆfD∥rather than its square, just because of the formula ∥ˆfD∥= sup b∈B⟨b, ϵ⟩ where B={P 1≤j≤Dθjϕj|P 1≤j≤Dθ2 j≤1}. This formula allows to interprete ∥ˆfD∥as the supremum of a Ra... | https://arxiv.org/abs/2504.17559v1 |
of suprema of Rademacher processes, which, as announced above, will be our target example here. But the fact that no structure is required is important because it also allows to study many examples of functions of independent random variables from random combinatorics. It was in this field that Talagrand’s early work h... | https://arxiv.org/abs/2504.17559v1 |
function ζthat we have used above. Given some function ζsatisfying to condition ( Cv) and combin- ing Talagrand’s convex distance inequality with inequality (4) (used for ζ/√v instead of ζ) leads to the following immediate consequence. Corollary 2 Letζsatisfying regularity condition ( Cv), and X1, X2, . . . , X nbe ind... | https://arxiv.org/abs/2504.17559v1 |
very simple but powerful engine: the variational formula for entropy. This formula is also well known in statistical mechanics, which is another domain of 6 interest of Patrick Cattiaux. Let us briefly recall what this formula says for a random variable ξon some probability space (Ω ,A, P) logEP eξ = sup Q≪P(EQ(ξ)−D(... | https://arxiv.org/abs/2504.17559v1 |
if Bis a subset ofRn, a Rademacher process is nothing else that b→ ⟨b, ϵ⟩, where ⟨., .⟩denotes the canonical scalar product. The quantity of interest here is the supremum of such a process: Z= supb∈B⟨b, ϵ⟩. One knows from Hoeffding’s inequality (see [13]) that for each given vector bwith Euclidean norm and all positive... | https://arxiv.org/abs/2504.17559v1 |
Our aim is to show how some fairly general ideas (as those developed in [6], [8] or [23] for instance) work in a very simple context where the technical aspects are deliberately reduced. The statistical framework we have chosen is that of regression with Rademacher errors which can be described as follows. One observes... | https://arxiv.org/abs/2504.17559v1 |
penalty function pen: M → R+and take ˆ mminimizing ∥Y−ˆfm∥2+ pen ( m) over M. Since by Pythagora’s identity, ∥Y−ˆfm∥2=∥Y∥2− ∥ˆfm∥2, 11 we can equivalently consider ˆ mminimizing − ∥ˆfm∥2+ pen ( m) overM. Then, we can define the selected model Sˆmand the corresponding selected least squares estimator ˆfˆm. Penalized cri... | https://arxiv.org/abs/2504.17559v1 |
introduce some notation. For all m, m′∈ M we define χm,m′= sup g∈Sm′⟨g−fm, ϵ⟩ ∥fm−f∥+∥g−f∥. (22) The role of this supremum of a Rademacher process in the proof of Theorem 6 is elucidated by the following statement. Claim 7 Ifˆmminimizes the penalized least-squares criterion (20), then for every m∈ M and all η∈]0,1[ η∥ˆ... | https://arxiv.org/abs/2504.17559v1 |
on the expected risk, we first prove an expo- nential probability bound and then integrate it. Towards this aim we introduce some positive real number ξ(this is the variable that we shall use at the end of the proof to integrate the tail bound that we shall obtain) and we fix some model m∈ M . Using a union bound, Clai... | https://arxiv.org/abs/2504.17559v1 |
a penalty of the form pen (m) =κσ2Dm, the value κ= 1 is indeed critical in the sense that, below this value the selection method becomes inconsistent. To enlighten this cut- off phenomenon, the lower tails probability bounds established in the section devoted to concentration will play a crucial role. 16 3.3 Cut-off fo... | https://arxiv.org/abs/2504.17559v1 |
we do so, and if we use a union bound, we derive that for all models such that Dm≤N/2 q χ2mN−χ2m≥p (N−Dm−2)+−2√ 2x while simultaneously by Hoeffding’s inequality (10) ⟨gm, ϵ⟩+≤√ 2x except on a set with probability less than 2# MNexp(−x). We choose x= log(2/δ) + log # MNin order to warrant that the above inequalities si... | https://arxiv.org/abs/2504.17559v1 |
is to estimate the function fon the interval [0 ,1] in the model Yk=f(tk) +σϵk, k = 1, . . . , n, (30) where tk=k/nand we assume in the sequel that the ϵk’s are independent Rademacher random variables but our results remain valid if they are only centered i.i.d. bounded random variables; the noise level, σ >0, is assum... | https://arxiv.org/abs/2504.17559v1 |
for any g∈ Sn−1, ∥ˆfˆm−f∥L2≤ ∥ˆfˆm−g∥L2+∥f−g∥L2 ≤ ∥ˆfˆm−g∥n+∥f−g∥L∞ ≤ ∥ˆfˆm−f∥n+ 2∥f−g∥L∞ and ∥fm−f∥n≤ ∥fm−g∥n+∥f−g∥n ≤ ∥fm−g∥L2+∥f−g∥L∞ ≤ ∥fm−f∥L2+ 2∥f−g∥L∞. 21 We have used that ∥f−g∥n≤ ∥f−g∥L∞and∥f−g∥L2≤ ∥f−g∥L∞. To prove optimality of our procedure, we consider the minimax setting and establish rates of our estimat... | https://arxiv.org/abs/2504.17559v1 |
term Dmn−2α+1is smaller than D−2α m∨Dmσ2 n, up to a constant. We end this section by deriving the cut-off phenomenon for the penalty in the functional setting. Even if the analogous general results of Section 3.3 can be obtained, we only consider the case where the collection of models Mis the following: a model m∈ M i... | https://arxiv.org/abs/2504.17559v1 |
with minimal penalties. In Proceedings of the 23rd International Conference on Neural Information Processing Systems , NIPS’09, page 46-54, Red Hook, NY, USA, Curran Associates Inc. (2009). [4]Arlot , S. and Bach , F.. Data-driven calibration of linear estimators with minimal penalties. arXiv:0909.1884v2. (2011). [5]Ar... | https://arxiv.org/abs/2504.17559v1 |
arXiv:2504.17611v1 [math.ST] 24 Apr 2025Some Results on Generalized Familywise Error Rate Controlling Procedures under Dependence Monitirtha Dey∗1and Subir Kumar Bhandari†2 1Institute for Statistics, University of Bremen, Bremen, Ger many 2Interdisciplinary Statistical Research Unit, Indian Stat istical Institute, Kolk... | https://arxiv.org/abs/2504.17611v1 |
are weakly correlated because of this largely loc al presence of correlationbetween SNPs. Storey and Tibshirani (2003)define weak dependence as “any form of dependence whose effect becomes negligible as the num ber of features increases to infinity” and remark that weak dependence generally h olds in genome- wide scans. G... | https://arxiv.org/abs/2504.17611v1 |
one has PΣn/parenleftbig Xi>Φ−1(1−kα⋆/n) for at least k i’s|H0/parenrightbig ≤α with some α⋆/greaterorequalslantDn,k·α1/k? If this is answered in affirmative, then we would have sharper upper b ounds on generalized FWERs and consequently, improved multiple testing proc edures. Otherwise, can one devise improved multiple ... | https://arxiv.org/abs/2504.17611v1 |
We have previously observed that k-FWER is P(at least kout ofn Ai’s occur) for suit- ably defined events Ai, 1≤i≤n. Naturally one wonders whether the theory of probabil- ity inequalities helps to findimproved upper boundson P(at least kout ofn Ai’s occur). Accurate computation of this probability requires knowing the com... | https://arxiv.org/abs/2504.17611v1 |
with correlatio nρ >0. Then, the sequence rmis increasing in 1≤m≤k−1. Proof.Consider the block equicorrelation structure as mentioned in Appen dix. Suppose k= (m+1,m−1,0,...,0) andk⋆= (m,m,0,...,0), where 2 m−2 zeros are there in each of these two vectors. So k>k⋆. Applying Theorem A.1, we obtain am+1am−1≥a2 m. This me... | https://arxiv.org/abs/2504.17611v1 |
bounds are never worse than t he existing ones. We now elucidate the practical utility of our proposed bounds throu gh aprostate cancer dataset (Singh et al. ,2002). This dataset involves expression levels of n= 6033 genes for N= 102 individuals: among them 52 are prostate cancer patients and t he rest are healthy pers... | https://arxiv.org/abs/2504.17611v1 |
paper revisits the classic al testing problem of normal means in different correlated frameworks. We establish u pper bounds on the generalized familywise error rates under each dependence, conse quently giving rise to improved testing procedures. Towards this, we also present impro ved inequalities on the probability t... | https://arxiv.org/abs/2504.17611v1 |
. Springer Series in Statistics. Springer- Verlag, New York, 1990. 14 A Appendix Letk1,...,k nbenonnegativeintegerssuchthat/summationtextn i=1ki=n,anddenote k= (k1,...,k n)′. Without loss of generality, it may be assumed that k1≥ ··· ≥kr>0, kr+1=···=kn= 0 for some r≤n.Tong(1990) defines rsquare matrices Σ11,...,Σrrsuch ... | https://arxiv.org/abs/2504.17611v1 |
Bernstein Polynomial Processes for Continuous Time Change Detection Dan Cunha1, Mark Friedl2, and Luis Carvalho1 1Department of Mathematics and Statistics, Boston University 2Department of Earth and Environment, Boston University Abstract There is a lack of methodological results for continuous time change detection du... | https://arxiv.org/abs/2504.17876v1 |
function of segment length parameters ζor change point locations τj=Pj l=1ζl(Scott and Knott, 1974a; Auger and Lawrence, 1989; Killick et al., 2012; Fearn- head, 2006; Fearnhead and Liu, 2007; Adams and MacKay, 2007). While the state space and segment length models are equivalent in terms of their likelihood distributi... | https://arxiv.org/abs/2504.17876v1 |
continuous time Markov chain πk(zt=h|zs=j) for 0 ≤s < t ≤1 and h≥jfor which exact posterior inference procedures are analytically available without MCMC nor approximation. 1.4 Modeling environmental changes using satellite imagery Change detection is an important and challenging problem in remote sensing data. Appli- c... | https://arxiv.org/abs/2504.17876v1 |
number of ways to choose k−jchange points from n−itime points, Proposition 1 (Marginal noninformative prior in discrete time) .The marginal noninfor- mative prior on the state space {zi}n i=0in discrete time is hypergeometric distributed, πk(zi=j) = n−i k−j i j−1 n k−1 4 An example of these discrete time marginals ... | https://arxiv.org/abs/2504.17876v1 |
time state marginal are the corresponding solid lines. ( Right ) Discrete time noninformative transition probabilities for n= 25 and k= 10 for illustration. The colors range from π10(zi+1= 1|zi= 1)(purple) to π10(zi+1= 10|zi= 10)(pink). asn→ ∞ , and after normalizing the index i(t)/n=⌊tn⌋/n, converges to the Bernstein ... | https://arxiv.org/abs/2504.17876v1 |
Self-transitions π5(zti+1=j|zti=j). Pink is j= 1, with the remaining in gray, following in increasing order of probability. ( Right ) Transitions π5(zti+1=h|zti= 1) forh= 1, . . . , k −1. Pink is h= 1, followed by green, yellow, orange, and brown. Note, one important difference between change point modeling in discrete... | https://arxiv.org/abs/2504.17876v1 |
θj⊥ ⊥θlforj̸=l, as well as conditional independence of the likelihood observations given the model yi⊥ ⊥yj|z,Θk. The choice of likelihood distribution falso does not affect our main results; the EM and simulation procedures for the posterior distribution of zare analytically tractable regardless of the likelihood distr... | https://arxiv.org/abs/2504.17876v1 |
lead to a combinatorially increasing prior probability with respect to the number of segments, which may not be believable. For example, in the discrete time offline setting, this would lead to π(k)∝ n k−1 , that is, the normalizing constant found in Proposition 1. In the continuous time setting, note from Theorem 3, ... | https://arxiv.org/abs/2504.17876v1 |
in pink, having maximum probability at k= 1. We compare these priors from Equations 3 and 4 in Figure 3 across 2000 samples of time from a uniform distribution for an intercept only model. Furthermore, we examine the performance of both priors in the case study and find the noninformative prior on number of segments fr... | https://arxiv.org/abs/2504.17876v1 |
5.5 Marginal posterior distribution on number of segments A major benefit of reparameterizing the change point problem using state variables z∼BPP within a hidden Markov model is the marginal likelihood can be computed using results from the forward backward algorithm. The forward recursions for this model represent th... | https://arxiv.org/abs/2504.17876v1 |
component-wise medians. Furthermore, the Bayes estimator {ˆzti}n i=1is a change point process. This result holds in both the discrete and con- tinuous time settings. Using our EM inference procedure, we estimate this Bayes estimator as follows. The π(k|y) term is estimated using Equation 5 and the π(zti=l|k,y) terms ar... | https://arxiv.org/abs/2504.17876v1 |
model : This is our noninformative discrete time model from Propositions 1 and 2 with location-scale t-distributed likelihood from sub- section 5.2 and noninformative prior on number of segments from Equation 4. •BPPE : This is the BPP model but with prior on number of segments that assumes equally likely sequences acr... | https://arxiv.org/abs/2504.17876v1 |
have much lower compu- tational cost. In our simulated study, the BPP model runs in 3e −1 seconds per dataset, whereas the PELT andBinSeg models run in 7e −4 and 9e −4 seconds per dataset, respec- tively. This computational disparity lies in the difference of the model being assumed. PELT andBinSeg do not allow the pen... | https://arxiv.org/abs/2504.17876v1 |
(Zhu and Woodcock, 2014). 7.2 Interannually varying harmonics One limitation of the above model is it assumes the mean function follows the same season- ality pattern each year. Change point detection algorithms that use the above model may exhibit higher false positive rates, since seasonal anomalies are not captured ... | https://arxiv.org/abs/2504.17876v1 |
Rondonia region of the Amazon rainforest. The second study applies our method to the problem of detecting changes in land management in an agricultural field in the San Joaquin Valley of California, and the third study applies our method to detecting responses of semi-arid vegetation in Texas to drought and year-to-yea... | https://arxiv.org/abs/2504.17876v1 |
are shown in Figure 6. 8.3 Crop rotation in the San Joaquin Valley Identifying and monitoring land management in agricultural regions from remote sensing data is an important task for a wide array of applications such as harvest and food sup- ply projections (Li et al., 2024; Boryan et al., 2011). We chose an agricultu... | https://arxiv.org/abs/2504.17876v1 |
data are provided by the National Drought Mitigation Center, University of Nebraska-Lincoln (Owens, 2007). We use these data to compare drought events to changes detected in NDVI at the same location. Specifically, they provide a No Drought measurement which scales from 0 (i.e. full drought) to 100 (i.e. no drought). W... | https://arxiv.org/abs/2504.17876v1 |
regarding random par- titions that did not fall within the scope of this work. Finally, it would be interesting to see how the continuous time transition probabilities can be parameterized to accommodate additional prior information or for use within an empirical Bayes approach. References Adams, R. P. and MacKay, D. J... | https://arxiv.org/abs/2504.17876v1 |
A. (2014). changepoint: An r package for changepoint analysis. Journal of statistical software , 58:1–19. Konishi, S. and Kitagawa, G. (2008). Information criteria and statistical modeling . Springer Science & Business Media. Li, H., Di, L., Zhang, C., Lin, L., Guo, L., Yu, E. G., and Yang, Z. (2024). Automated in-seas... | https://arxiv.org/abs/2504.17876v1 |
Statistics , pages 1434–1447. Yu, L. and Gong, P. (2012). Google earth as a virtual globe tool for earth science applications at the global scale: progress and perspectives. International Journal of Remote Sensing , 33(12):3966–3986. Zhang, N. R. and Siegmund, D. O. (2007). A modified bayes information criterion with a... | https://arxiv.org/abs/2504.17876v1 |
· (1−t−k n+j n+1 n) (every term isn−i+ const. n) = (1−t)k−j lim n→∞1 nj−1i! (i−j+ 1)!= lim n→∞i· ··· · (i−j+ 2) nj−1 = lim n→∞i n· ··· ·(i−j+ 2) n(there are j−1 terms) = lim n→∞t· ··· · (t−j n+2 n) (every term isi+ const. n) =tj−1 30 lim n→∞nk−1(n−k+ 1)! n!= lim n→∞nk−1 n· ··· · (n−k+ 2) = lim n→∞n n· ··· ·n n−k+ 2(the... | https://arxiv.org/abs/2504.17876v1 |
Using Theorem 2, note that the probability p(zt=h|zs=j) is equiva- lent to the probability p(Ph−1 l=1ζl≤t <Ph l=1ζl|Pj−1 l=1ζl≤s <Pj l=1ζl). As such we evaluate the joint probability of these events, and then divide by the marginal. Case 1: h=j 33 p(j−1X l=1ζl≤s, t <jX l=1ζl) =Zs 0Zs−ζ1 0···Zs−Pj−2 l=1ζl 0Z1−Pj−1 l=1ζl... | https://arxiv.org/abs/2504.17876v1 |
1−1−t 1−r!h−l 1−t 1−r!k−h = 1−t 1−s!k−hhX l=jk−j l−jk−l h−l r−s 1−s!l−j 1−r 1−s!h−l t−r 1−r!h−l = 1−t 1−s!k−hhX l=jk−j l−jk−l h−l r−s 1−s!l−j t−r 1−s!h−l = 1−t 1−s!k−h 1 1−s!h−jhX l=jk−j l−jk−l h−l (r−s)l−j(t−r)h−l = 1−t 1−s!k−h 1 1−s!h−jhX l=j(k−j)! (l−j)!(k−l)!(k−l)! (h−l)!(k−h)!(r−s)l−j(t−r)h−l =k−j h−j... | https://arxiv.org/abs/2504.17876v1 |
Thus, in all cases, the median ˆ zti+1is either ˆ ztior ˆzti+ 1 with probability 1, and the resulting estimator is the Bayes estimator for the weighted Hamming loss in the constrained space of change point processes in discrete time. Show this estimator is a change point process: continuous time case In continuous time... | https://arxiv.org/abs/2504.17876v1 |
hypergeometric distribution, the number of samples until success is con- sidered fixed and the number of successes at that time is considered random, we wish to relate this distribution to one where the segment lengths are random– that is, the number of samples until a specified number of successes is random. With this... | https://arxiv.org/abs/2504.17876v1 |
2)))−1. Finally, the term in the middle can be rewritten in terms of j, in order to understand how it changes during recursion, (n−i1−i2)! (n−i1−k−i2+ 3)!=(n−Pj l=1il)! (n−(Pj l=1il)−k+ (j+ 1))! Using this equation, after recursing through k−2 conditional probabilities, we arrive at, =(k−1)! n(n−1). . .(n−(k−2))·(n−Pk−... | https://arxiv.org/abs/2504.17876v1 |
and change points simulated fromBPP . 45 FPRTPR 0 10 1BPP FPRTPR 0 10 1PELT FPRTPR 0 10 1BinSeg FPRTPR 0 10 1BPP FPRTPR 0 10 1PELT FPRTPR 0 10 1BinSeg FPRTPR 0 10 1BPP FPRTPR 0 10 1PELT FPRTPR 0 10 1BinSegFigure 10: Row 1 is a subset of the data generated with error variance 0 .1. Row 2 is a subset of the data generate... | https://arxiv.org/abs/2504.17876v1 |
and 0 precision except forβ. Since we do not want short periods of change to be captured by sharp slopes, we set the precision of βto be 5 to help regularize and avoid spurious changes. Denote corresponding precision matrix as Λ θ. Now, consider the prior distribution on the annual harmonic contrasts ϕ= ({γh,l}H,J h=1,... | https://arxiv.org/abs/2504.17876v1 |
10 1BPP: Normal Likelihood FPRTPR 0 10 1Noninf Discrete FPRTPR 0 10 1BPPE FPRTPR 0 10 1BPP: Normal Likelihood FPRTPR 0 10 1Noninf Discrete FPRTPR 0 10 1BPPEFigure 18: Study is broken down by number of changes. First row is 0 changes, to the fourth row of 3 changes. 54 0.1 0.2 0.3 0.4Rondonia Deforestation: Case Study P... | https://arxiv.org/abs/2504.17876v1 |
this prior incurs extra falsely detected changes compared to the prior in Equation 4. 14 Appendix E: Supplementary Results for Methodol- ogy: EM and Simulation 14.1 Expectation Maximization Expectation maximization will be used to obtain posterior expectations of the robustness variables {qi}n i=0as well as the state v... | https://arxiv.org/abs/2504.17876v1 |
distribution of the mean vectors θjfollow a Gaussian distribu- tion since their prior is Gaussian. Let W(j) ii=qi1{zi=j}be diagonal, p(θj|y,z,q, σ2 j)∝exp{−1 2(θj−µj)TΛj(θj−µj)} Where µ= (XTW(j)X+Φ−1)−1XTW(j)yand Λ j= (XTW(j)X+Φ−1)/σ2are the mean and precision matrix of the Gaussian posterior for θj. The posterior cond... | https://arxiv.org/abs/2504.17876v1 |
arXiv:2504.17885v1 [math.PR] 24 Apr 2025Maximal Inequalities for Independent Random Vectors Supratik Basu1Arun Kumar Kuchibhotla2 1supratik.basu@duke.edu 2arunku@cmu.edu 1Department of Statistical Science, Duke University 2Department of Statistics & Data Science, Carnegie Mellon University April 28, 2025 Abstract Maxim... | https://arxiv.org/abs/2504.17885v1 |
sup f∈Fj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 nn/summationdisplay i=1f(Wi)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketrightBigg , which is also an expected value of a finite maximum of averages of rand om variables. Hence, the bounding En(F) can be solv... | https://arxiv.org/abs/2504.17885v1 |
first claim, we use Pro position 1 of Pruss(1997) and for the second claim, we use Theorem 1.4.4 of de la Pe˜ na and Gin´ e (1999). Define the sub-class of mean zero probability distributions PonRdas Pq(σ,B) :=/braceleftbigg Pa probability distribution : EP[X] = 0,max 1≤j≤pEP[X2(j)]≤σ2,(EP[/bar⌈blX/bar⌈blq ∞])1/q≤B/brace... | https://arxiv.org/abs/2504.17885v1 |
qBenn(.) denote the upper bound provided to the tail probability of the ran dom variable /bar⌈bl¯X/bar⌈bl∞obtained using an application of the Bennett’s Inequality. We theref ore have qBenn(t) = 2pexp/parenleftbigg −nσ2 B2Ψ/parenleftbiggtB σ2/parenrightbigg/parenrightbigg fort≥0 4 and further define pBenn(t) = min{1,qBe... | https://arxiv.org/abs/2504.17885v1 |
random vectors X1,X2,···,Xnsuch that for all i= 1,2,···,n max 1≤j≤p|Xi(j)| ≤Bandmax 1≤j≤pV[Xi(j)]≤σ2. Then there exist independent random vectors Y1,Y2,···Ynsuch thatYi(1),Yi(2),···,Yi(p)are jointly independent for all i= 1,2,···,nsuch that max 1≤j≤p|Yi(j)| ≤Bandmax 1≤j≤pV[Yi(j)]≤σ2 and E/bracketleftbig /bar⌈blX/bar⌈bl... | https://arxiv.org/abs/2504.17885v1 |
3.1. WhiletheiranalysisisrestrictedtoBernoullirandomvariables,ouran alysisimpliesthatBernoullirandom variables, in some sense, result in the worst case bound. In the follo wing section, we also consider the case of unbounded random vectors. 4 Bounds under Integrable Envelope This section focuses on acquiring upper boun... | https://arxiv.org/abs/2504.17885v1 |
4.2becomes the Bennett bound. To elaborate on the behavior of the s econd term, note that from Theorem 2.3 of Hoorfar and Hassani (2008), we have W(x)≥ln(x)−ln/parenleftbigg ln/parenleftbiggx(1+1/e) lnx/parenrightbigg/parenrightbigg ,for allx>0. This follows by taking y=x/ein Theorem 2.3 of Hoorfar and Hassani (2008) a... | https://arxiv.org/abs/2504.17885v1 |
is clear that the collection is decreasing in qi.e.Sq′⊂ Sq. Hence the corresponding sets Ω \ Sqare increasing in q. Further, observe that for any b≥a≥1/3, we have /parenleftbigg 1−1 q/parenrightbigg ln(b/a)<1 6(b/a−1)+1≤1+1 2/parenleftbigg 1−2 q/parenrightbigg (b/a−1). Now, take a= (B2/σ2)(ln(2p)/n)1−2/q′ andb= (B2/σ2)... | https://arxiv.org/abs/2504.17885v1 |
future research. Finally, our results can be applied to study or refine maximal inequalitie s for the supremum of empirical processes. For instance, some results of Kuchibhotla and Patra (2022) can be improved by replacing their Proposition B.1 with our Theorem 4.3. We leave these important applications to future resear... | https://arxiv.org/abs/2504.17885v1 |
volume6of Cambridge Series in Statistical and Probabilistic Mathematics . Cambridge University Press, Cambridge. Van der Vaart, A. W. (1998). Asymptotic statistics , volume 3 of Cambridge Series in Statistical and Proba- bilistic Mathematics . Cambridge University Press, Cambridge. vanderVaart, A. W. andWellner, J.A. (... | https://arxiv.org/abs/2504.17885v1 |
q(σ,B)≥max/braceleftbigg E∗ q,∞(σ,B,τ q(σ,B)),1 2nMq(σ,B)/bracerightbigg ≥1 2E∗ q,∞(σ,B,τ q(σ,B))+1 4nMq(σ,B). This completes the proof of lower bound. To prove that the final statement that ( 2) holds even when Mq(σ,B) is replaced with M′ q(σ,B), first note that M′ q(σ,B)≤ Mq(σ,B). This implies that the lower bound with... | https://arxiv.org/abs/2504.17885v1 |
≥((2qn(logn)−2)1/q−e)/parenleftbigg 1−exp/parenleftbigg −q 2(logn)2 (logn+log(2q))2/parenrightbigg/parenrightbigg /greaterorsimilar(n(logn)−2)1/q. The penultimate inequality is a simple application of the fact that 1 −x≤e−xforx≥0 which leads to the fact that 1 −(1−x)n≥1−e−nx. The last inequality holds because the multi... | https://arxiv.org/abs/2504.17885v1 |
∃β∈(0,α) satisfying ( E.10) by Lagrange’s Mean Value Theorem. Also, from ( E.9) and (E.11), we haveB n(f(A)−f(B))≤2a−1 2ln2(B−A). Consequently, we must have for B <ℓ2, /integraldisplayB ApBenn(t)dt≥B−A 2≥ln2 2a−1B n(f(A)−f(B)). (E.12) Combining the results ( E.8) and (E.12) for the cases B≥ℓ2andB <ℓ2respectively, we fin... | https://arxiv.org/abs/2504.17885v1 |
parts. In the first part, alongside B2ln(2p)>neσ2, we havet∗B≥4(σ2+B2)/(5n) where t∗=4σ2 9B/parenleftbigg(1+B2/σ2)(4ln(2p)/5n) W((1+B2/σ2)(4ln(2p)/5n))−1/parenrightbigg . Now plugging in x=B2/σ2andy= ln(2p)/nin Lemma S.4.14, we get Ψ−1(B2ln(2p)/(nσ2))≤e e−1B2ln(2p)/(nσ2)−1 ln(1+(B2ln(2p)/(nσ2)−1)/e)≤e2 e−1(1+B2/σ2)(ln(2... | https://arxiv.org/abs/2504.17885v1 |
As a consequence, P/parenleftbigg max 1≤j≤p/vextendsingle/vextendsingleWj/vextendsingle/vextendsingle≥t∗/parenrightbigg = 1−(1−pσ,B)np=≥1−/parenleftbigg 1−1 2np/parenrightbiggnp ≥1−e−1/2. Hence by Markov’s inequality, E[/bar⌈blW/bar⌈bl∞]≥t∗P/parenleftbigg max 1≤j≤p/vextendsingle/vextendsingleWj/vextendsingle/vextendsin... | https://arxiv.org/abs/2504.17885v1 |
2 p/summationdisplay j=1P/parenleftBig/vextendsingle/vextendsingle/vextendsingleY(j)/vextendsingle/vextendsingle/vextendsingle≥t/parenrightBig 2 +1 2p/summationdisplay j=1P/parenleftBig/vextendsingle/vextendsingle/vextendsingleY(j)/vextendsingle/vextendsingle/vextendsingle≥t/parenrightBig2 ≥p/summationdisplay j=1P/... | https://arxiv.org/abs/2504.17885v1 |
/tildewideQSn(1/z)≤(T¯ℓ1)(logz)+(Tℓ2)(logz). We now evaluate the quantities on the right-hand side. By Lemma S.4.19, we have (T¯ℓ1)(logz) =n(σ2/B2)q/(q−2)(Bq/σ2)1/(q−2)Ψ−1/parenleftbigglogz n(σ2/B2)q/(q−2)/parenrightbigg 32 =nB/parenleftbiggσ2 B2/parenrightbigg(q−1)/(q−2) Ψ−1/parenleftbigglogz n(σ2/B2)q/(q−2)/parenrigh... | https://arxiv.org/abs/2504.17885v1 |
Inequality, E/bracketleftBigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1 nn/summationdisplay i=1Xi/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble ∞/bracketrightBigg ≤E/bracketleftBigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1 nn/summationdi... | https://arxiv.org/abs/2504.17885v1 |
as follows. K= 2B(ln(2p)/n)−1/q/bracketleftBig ln/parenleftBig (B/σ)(ln(2p)/n)1/2−1/q/parenrightBig/bracketrightBig1/q . It is thus easy to see that both Bq Kq−1/lessorsimilarB/parenleftbiggln(2p) n/parenrightbigg1−1/q/bracketleftBigg ln/parenleftBigg B2 σ2/parenleftbiggln(2p) n/parenrightbigg1−2/q/parenrightBigg/brack... | https://arxiv.org/abs/2504.17885v1 |
by observing that showing the validity of the lower bound boils down to an equivalent problem: Ψ′(Ψ−1(x))≥√ 2ln(1+√x) ⇐⇒ln(1+Ψ−1(x))≥√ 2ln(1+√x) ⇐⇒Ψ−1(x)≥(1+√x)√ 2−1 ⇐⇒√ 2(1+√x)√ 2ln(1+√x)−(1+√x)√ 2+1−x≤0 Takeζ(x) =√ 2(1+√x)√ 2ln(1+√x)−(1+√x)√ 2+1−x. Sinceζ(0) = 0, in order for us to show that ζ(x)≤0 forx∈[0,e], it suffi... | https://arxiv.org/abs/2504.17885v1 |
first inequality while there is strict inequa lity for the second one if p1p2>0. Let the lemma hold for n=mi.e. m/summationdisplay i=1pi−m−1/summationdisplay i=1m/summationdisplay j=i+1pipj≤1−m/productdisplay i=1(1−pi)≤m/summationdisplay i=1pi 42 Now, forn=m+1, 1−m+1/productdisplay i=1(1−pi) = 1−(1−pm+1)m/productdisplay... | https://arxiv.org/abs/2504.17885v1 |
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