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measure µfis almost surely the semi-circular law for any f∈L2[0,1]. /square Acknowledgements: I am grateful to Prof. Rajendra Bhatia for his interest and comme nts. I also thank Prof. Krishna Maddaly for discussions at the earlier stag e of this work. References [AGZ10] Greg W. Anderson, Alice Guionnet, and Ofer Zeitou... | https://arxiv.org/abs/2503.23940v1 |
Multivariate Species Sampling Models Beatrice Franzolini, Antonio Lijoi, Igor Pr¨ unster Bocconi Institute for Data Science and Analytics, Bocconi University, Italy Giovanni Rebaudo ESOMAS Dept., University of Torino and Collegio Carlo Alberto, Italy Abstract Species sampling processes have long served as the framework... | https://arxiv.org/abs/2503.24004v1 |
across observations in distinct groups. Con- versely, the assumption of independence across populations prevents information sharing between experiments, despite this being a key objective in applied multi-sample analyses (see, for instance, Woodcock and LaVange, 2017; Chen and Lee, 2019; Ouma et al. , 2022; Suet al. ,... | https://arxiv.org/abs/2503.24004v1 |
, 2023; Denti et al. , 2023), and many others (e.g. Horiguchi et al. , 2024; Yan and Luo, 2023; Bi and Ji, 2023). Models employing such priors define a dependence among the processes in ( P1, . . . , P J) and among the observations in X, facilitating the desired borrowing of information among different populations. The... | https://arxiv.org/abs/2503.24004v1 |
can be generalized even beyond partial exchangeability. Finally, it is crucial to clarify that while mSSPs indeed generalize SSPs, the essence of mSSPs lies in their multivariate nature and, in particular, in the dependence induced across populations and, consequently, across elements within the vector ( P1, . . . , P ... | https://arxiv.org/abs/2503.24004v1 |
same atoms. However, it is worth noting that Definition 1 is highly general in that the πj,h’s could potentially be null almost surely, resulting in instances where the random probabilities P1, . . . , P Jshare only a handful or none of the species with positive probability. When analyzing dependence and quantifying sh... | https://arxiv.org/abs/2503.24004v1 |
multivariate species sampling processes A notable subclass of mSSPs, which we name regular , arises by imposing a simple indepen- dence condition on the weights associated with non-shared species. First, consider a bivariate 5 mSSP ( P1, P2) and define π(j)= π(j) j,h′ P ℓ≥0π(j) j,ℓ! h′≥0forj= 1,2 where the weights π(j)... | https://arxiv.org/abs/2503.24004v1 |
Green, 1997), and SSP( Lπ, P0) denotes the law of a (univariate) species sampling process with weights distribution defined by Lπand base measure P0(Pitman, 1996). See Section S.1 of the Supplementary Material for more details about (univariate) SSP, related exchangeable prob- ability partition function (EPPF), predict... | https://arxiv.org/abs/2503.24004v1 |
+µj(X), µ 0∼CRM ((1−z)exp{−s} s, c, P 0), µ jind∼CRM (zexp{−s} s, c, P 0) •GM-dependent σ-stable (GM- σ, Lijoi et al. , 2014): Pj=µ0+µj µ0(X) +µj(X), µ 0∼CRM ((1−z)σs−1−σ Γ(1−σ), c, P 0), µ jind∼CRM (zσs−1−σ Γ(1−σ), c, P 0) then, for any couple (Pj, Pk),π(j)⊥π(k). Thus, (P1, . . . , P J)is a rmSPP. If(P1, . . . , P J)i... | https://arxiv.org/abs/2503.24004v1 |
, J , then E[Pj(A)] =P0(A),Var[Pj(A)] =P(Xj,1=Xj,2)P0(A) 1−P0(A) . By marginal exchangeability, the probability of the tie P(Xj,i=Xj,m) between observa- tions extracted from population jdoes not depend on the indexes ( i, m). Moreover, such probability can be written in terms of the weights in the representation in (... | https://arxiv.org/abs/2503.24004v1 |
impossible to construct zero-correlated rmSSPs whose components are not pairwise independent. Theorem 1(Null correlation) .If(P1, . . . , P J)is a rmSSP, then ∀j̸=k∈[J] Cor[Pj(A), Pk(A)] = 0 iffPj⊥Pk. Within the class of rmSSP, it is notable that not only a correlation equal to one does imply exchangeability (as for al... | https://arxiv.org/abs/2503.24004v1 |
1,0(2)⋆ EPPF(2) 1,0(2) EPPF(2) 1,1(2) + EPPF(2) 2,1(1,1)EPPF(2) 1,0(2) EPPF(2) 1,0(2) = 0 EPPF(2) 1,1(2) = 0 NDP1 1 +α1 (1 +α)(1 + β)1 1 +βα→+∞ α→0 NPY1−σα 1 +α(1−σα)(1−σβ) (1 +α)(1 + β)1−σβ 1 +βα→+∞ orσα→1(α, σ α)→ →(0,0) NDM1 +ρα 1 +ραMα(1 +ρα)(1 + ρβ) (1 +ραMα)(1 + ρβMβ)1 +ρβ 1 +ρβMβMα→+∞ Mα→1 NGN2γα γα+ 14γαγβ (γα+... | https://arxiv.org/abs/2503.24004v1 |
, q h, E[Pj(A1)q1···Pj(Ah)qh] =E P0(A1)K(j) 1:q1P0(A2)K(j) q1+1:q2···P0(Ah)K(j) qh−1+1:qh|E̸= P(E̸=), where K(j) a:bis the random number of species in the “block of observations” from the a-th to theb-th observation, in a sample of size q1+···+qhfrom Pj, and E̸=is the event of not observing any shared species across ... | https://arxiv.org/abs/2503.24004v1 |
Moreover, marginal posterior sampling schemes require knowledge of the pEPPF to be derived. To define the pEPPF for mSSM, let Dbe the number of distinct values among the n=PJ j=1Ijobservations in the sample ( Xj,i:i∈[Ij], j∈[J]). Denote the vector of frequency counts in a group jbynj= (nj,1, . . . , n j,D) with nj,dind... | https://arxiv.org/abs/2503.24004v1 |
the restaurant. As in the classical metaphor, each table serves a unique dish and when a customer sits at an empty new table, they order a new dish not yet served at any table. 15 Proposition 10.IfX∼mSSM with pEPPF, then a multivariate Chinese restaurant process (mCRP) (i.e., a sequential sampling scheme that allows sa... | https://arxiv.org/abs/2503.24004v1 |
. . , ℓ ·,D)induced by Q. 16 Example 2 (Continue). If(P1, . . . , P J)is an NSSP, then pEPPF(n) D,aug(n1, . . . ,nJ,ℓ,q) = EPPF(J) R,0(ℓ1, . . . , ℓ R)QR r=1EPPF(I⋆ r) Dr(q1,·, . . . , q Dr,),(7) where EPPF(J) R,0(ℓ1, . . . , ℓ R)is the EPPF induced by Lπ,0that controls the clustering of the group labels j= 1, . . . , ... | https://arxiv.org/abs/2503.24004v1 |
Empirical distribution functions of the four groups (left) and empirical probabilities of ties (right), based on the entire available dataset. In the left panel, species of all groups are ordered based on their frequency in Group 2. to uncover new species, and are also encountered in genomics, where often the goal is t... | https://arxiv.org/abs/2503.24004v1 |
ties an excellent summary of dependence (and thus borrowing of information) across groups for rmSSPs. All details about models, algorithms, and hyperpriors can be found in Section S.3 of the Supplementary Material. Figure 3 showcases the average number of species discovered by two hierarchical species sampling processe... | https://arxiv.org/abs/2503.24004v1 |
prob. of ties Uniform DP PY +DP +PY HDP HPY Oracle Avg. num. 0.2335 0.3317 0.3298 0.3312 0.3262 0.3322 0.3237 0.3467 RMSE NA 0.1563 0.0743 0.1621 0.0655 0.1929 0.0655 0 Table 6: Simulated Scenario: Average number of species discovered per sampling step (Avg. num.) and the root mean squared error of the predictive proba... | https://arxiv.org/abs/2503.24004v1 |
809–838. Battiston, M., Favaro, S., and Teh, Y. W. (2018). Multi-armed bandit for species discovery: a Bayesian nonparametric approach. J. Am. Stat. Assoc. ,113, 455–466. Beraha, M., Guglielmi, A., and Quintana, F. A. (2021). The semi-hierarchical Dirichlet process and its application to clustering homogeneous distribu... | https://arxiv.org/abs/2503.24004v1 |
feature of the Dirichlet process. Scand. J. Stat. ,33, 105–120. Leisen, F. and Lijoi, A. (2011). Vectors of two-parameter Poisson–Dirichlet processes. J. Multivar. Anal. ,102, 482–495. Lijoi, A., Nipoti, B., and Pr¨ unster, I. (2014). Bayesian inference with dependent normalized completely random measures. Bernoulli ,2... | https://arxiv.org/abs/2503.24004v1 |
Analytics, Bocconi University, Italy Giovanni Rebaudo ESOMAS Dept., University of Torino and Collegio Carlo Alberto, Italy S.1 Some basics on (univariate) species sampling In classical species sampling problems, a random sample (X 1, . . . , X n) is extracted from an unknown and typically discrete distribution and each... | https://arxiv.org/abs/2503.24004v1 |
iid the unique values associated with each set in the partition fromP 0. The EPPF and the SSP can also be characterized by a specific sequence of predictive distri- butions (Pitman, 1996) also known as thegeneralized Chinese restaurant process(gCRP). In the culinary metaphor, we can think of observations corresponding ... | https://arxiv.org/abs/2503.24004v1 |
we can derive the well-known gCRP of the PYP from the EPPF in (S.5) applying the definition of conditional probability. P/parenleftbig Xn+1=x|X/parenrightbig = nk−α γ+nifx=X∗ kandk= 1, . . . , K γ+αK γ+nifx=X∗ K+1.(S.6) S.1.2 Dirichlet process (DP) If we considerP∼PYP(α, γ;P 0) as in the previous section and we r... | https://arxiv.org/abs/2503.24004v1 |
MATERIALS S.6 Let us define, forj= 1,2, /parenleftig (θ0,h, π(12) j,h) :h∈ H 0/parenrightig := ((θ h, πj,h) :h∈ H 0) and/parenleftig π(j) j,h:h∈ ¯H0/parenrightig :=/parenleftbig πj,h:h∈ ¯H0/parenrightbig , andθ j,hiid∼P 0, forj= 1,2 andh∈ ¯H0, independent from all the previous random variables, i.e.,/parenleftig (... | https://arxiv.org/abs/2503.24004v1 |
Definition 3 we have thatP/parenleftbig ω(j,k) j¯π(j,k) j,1ω(j,k) k¯π(j,k) k,1>0/parenrightbig >0 that entails P(X j,1=X k,1)≥E/bracketleftig ω(j,k) j¯π(j,k) j,1ω(j,k) k¯π(j,k) k,1/bracketrightig >0 that contradictsP(X 1,j=X 1,k) = 0. Sinceω(j,k) ja.s.=ω(j,k) ka.s.= 0 we can rewrite Pja.s.= ¯π(j) j,0P0+/summationdisp... | https://arxiv.org/abs/2503.24004v1 |
q1···P 0(AJ)K(J) qJ|Πq∈ A q1,...,qJ/bracketrightig P(Π q∈ A q1,...,qJ). S.2.12 Proof of Proposition 9 Proof. pEPPF(n) D(n1, . . . ,n J) =E/bracketleftbigg/integraldisplay XD∗D/productdisplay d=1P1(dxd)n1,d. . . P J(dxd)nJ,d/bracketrightbigg =E/braceleftbigg/integraldisplay XD∗D/productdisplay d=1J/productdisplay j=1/b... | https://arxiv.org/abs/2503.24004v1 |
probabilities of ties as a function of the hyperparameters and approximately match their expected values and variances. This selection procedure ensures a fair performance comparison, as the probabilities of ties provide an excellent summary of dependence for rmSSPs. The resulting expected probability of ties and corre... | https://arxiv.org/abs/2503.24004v1 |
Sci.,28, 209–222. Lijoi, A., Mena, R. H., and Pr¨ unster, I. (2005). Hierarchical mixture modeling with normal- ized inverse-Gaussian priors.J. Am. Stat. Assoc.,100, 1278–1291. Lijoi, A., Mena, R. H., and Pr¨ unster, I. (2007). Controlling the reinforcement in Bayesian non-parametric mixture models.J. R. Stat. Soc. Ser... | https://arxiv.org/abs/2503.24004v1 |
Wasserstein KL-divergence for Gaussian distributions Adwait Datar1[0000 −0002−4085−9675]and Nihat Ay1,2[0000 −0002−8527−2579] 1Institute for Data Science Foundations, Hamburg University of Technology, 21073 Hamburg, Germany 2Santa Fe Institute, Santa Fe, NM 87501, USA {adwait.datar,nihat.ay}@tuhh.de Abstract. We introd... | https://arxiv.org/abs/2503.24022v1 |
a point xis the vector grad f(x)∈Rn. We use A≻0(A⪰0) to indicate that Ais symmetric positive definite (semi-definite). The matrix exponential is denoted by eAand for any A≻0, the matrix logarithm and symmetric positive definite square root ofAis denoted by log(A)and√ A(orA1 2), respectively. Finally, the Frobenius norm... | https://arxiv.org/abs/2503.24022v1 |
0X=Σ1. Under the constraints that Σ0≻0andΣ1≻0, uniqueness of the positive definite solution follows from [7, Theorem 5] proving the uniqueness of the solution Ato (7). Uniqueness of bis obtained immediately since (eA−I)A†+A⊥ is non-singular. This proves the final statement. We now present the main result of the paper... | https://arxiv.org/abs/2503.24022v1 |
(σ1−σ0)2+ (µ1−µ0)2 +1 2log(R)⊥(µ1−µ0)2 which gives the desired result. The final limit can be shown via successive appli- cations of L’Hopital’s rule. Letµ=N(µ0, σ2)andν=N(µ1, σ2)be two univariate Gaussian distribu- tions with equal variance σ2and possibly different means µ0andµ1. Let us now compare the WKL-divergence... | https://arxiv.org/abs/2503.24022v1 |
have no competing interests to declare that are relevant to the content of this article. References 1. Amari, S.i., Nagaoka, H.: Methods of information geometry, vol. 191. American Mathematical Soc. (2000) 2. Ay, N.: Information geometry of the otto metric. Information Geometry pp. 1–24 (2024) 3. Ay, N., Amari, S.i.: A... | https://arxiv.org/abs/2503.24022v1 |
Smooth and rough paths in mean derivative estimation for functional data Max Berger and Hajo Holzmann∗ Department of Mathematics and Computer Science Philipps-Universit¨ at Marburg {mberger, holzmann }@mathematik.uni-marburg.de April 1, 2025 Abstract In this paper, in a multivariate setting we derive near optimal rates... | https://arxiv.org/abs/2503.24066v1 |
fixed synchronous design with sufficiently many design points spline estimators may achieve the parametric√n-rate of convergence even under the supremum norm. In this paper, in a multivariate setting we derive near optimal rates of convergence in the minimax sense for estimating partial derivatives of the mean function... | https://arxiv.org/abs/2503.24066v1 |
1, . . . , d . We set ab..=ab1 1···abd d,|a|=a1+. . .+ad,a!..=a1!···ad!, amin= min 1≤r≤dar. If the coordinates of aare non-negative integers we denote the partial derivative operator by ∂a..=∂a1 1. . . ∂ad d. Set1..= (1, . . . , 1)⊤∈Nd. Given p..= (p1, . . . , p d)⊤we denote by {j∈Nd|1≤j≤p}the set of j∈Ndthat are compo... | https://arxiv.org/abs/2503.24066v1 |
to order ≤ ⌊β⌋, ∃random variable Ms.th.∥Z∥H,β≤Ma.s. and E[M2]≤CZ (4) Thus processes Z∈ P(β, C Z) have Lipschitz-continuous derivatives of order <⌊β⌋, and H¨ older- continuous derivatives of orders = ⌊β⌋with exponent β− ⌊β⌋a.s., and moreover, all random Lipschitz - and H¨ older constants can be upper-bounded by a square... | https://arxiv.org/abs/2503.24066v1 |
h∨1 ,x,y∈[0,1]d. Assumption 3 (Sub-Gaussian errors) .The random variables {εi,j|1≤i≤n,1≤j≤p}are independent and independent of the processes Z1, . . . , Z n. Further we assume that the distribu- tion of εi,jis sub-Gaussian, and setting σ2 ij..=E[ε2 i,j] we have that σ2..= supnmax1≤j≤pσ2 j<∞ and that there exists ζ >0 ... | https://arxiv.org/abs/2503.24066v1 |
the paths themselves are smooth of order <|s|, then on the one hand consistent estimation is still possible, but on the other hand the parametric rate can no longer be achieved, as we show below by providing lower bounds. Next we turn to lower bounds. For the first bound we use the following design assumption, which we... | https://arxiv.org/abs/2503.24066v1 |
for the triangular array of processes ( Xni)n∈N,1≤i≤ngiven by (27). While this is broadly analogous to the case of the mean function µin Berger et al. (2023), some extensions are required and we provide the proof in Section 5. 2.3 Local polynomial estimators We briefly discuss local polynomial estimators of partial der... | https://arxiv.org/abs/2503.24066v1 |
effect has been previously observed in nonparametric derivative estimation with local polynomials e.g. in De Brabanter et al. (2013, Figures 2, 4) or in Newell and Einbeck (2007). Therefore in the following we compute the sup-norm error only over the interval [ h,1−h], where hdenotes the bandwidth used in the local pol... | https://arxiv.org/abs/2503.24066v1 |
Irg n(x) =1 npX j=1w(1) j(x;hn)nX i=1Zi(xj) and Ism n(x)1 npX j=1w(1) j(x;hn)nX i=1˜Zi(xj). Our theory, in particular Lemma 7 implies that ∥Irg n∥∞should have a rate slower than 1 /√n hn, while∥Ism n∥∞should converge at the 1 /√nrate. Table 1 contains the results, where we used the optimal bandwitdhs for estimation as ... | https://arxiv.org/abs/2503.24066v1 |
5 shows observations for each of the runners. Pairwise differences together with the averaged observations and a local polynomial estimate of the mean curve are given in Figure 7. The bandwidth for the local polynomial estimator is selected by leave-one-curve-out cross validation, resulting in h= 5.5. Estimating the de... | https://arxiv.org/abs/2503.24066v1 |
the paths are actually not smooth. The conclusion can be drawn from Figure 16 in the Appendix B for all months during the year. 12 Figure 13: Estimates ∂(0,1) uΓ(x, x) and ∂(1,0) uΓ(x, x) of the covariance kernel Γ for temperatures in April. 4 Concluding remarks In this paper we discussed derivative estimation for the ... | https://arxiv.org/abs/2503.24066v1 |
term, given xfor eachxjthere exists θj∈[0,1],1≤j≤p,such that for τj..=x+θj(xj−x)∈[0,1]dit holds that µ(xj) =⌊α⌋−1X |k|=0∂kµ(x)(xj−x)k k!+X |k|=⌊α⌋∂kµ(τj)(xj−x)k k!. Therefore pX j=1w(s) j(x;h)µ(xj)−∂sµ(x) = pX j=1w(s) j(x;h) µ(xj)−(xj−x)s s!∂sµ(x) (by (M1)) = pX j=1⌊α⌋−1X |k|=0∂kµ(x)(xj−x)k k!+X |k|=⌊α⌋∂kµ(τj)(xj−x)... | https://arxiv.org/abs/2503.24066v1 |
(28)) ≤CdMiϵ . (ϵ≤h) Then the bound in i)follows from Lemmas 9 and 8. ii).We proceed similarly as for i), but using the envelope Φn=Ce(M1, . . . , M n)⊤ √n h|s|−(β−δ). To show that this is indeed an envelope for Xnfor appropriate choice of Ce>0, again we use a Taylor expansion, this time of order ⌊β⌋. Then for intermed... | https://arxiv.org/abs/2503.24066v1 |
adjusted to obtain a H¨ older norm ≤L. Finally, ∥∂sµ0−∂sµ1∥∞=∥∂sµ1∥∞=˜L(c p)−α+|s|2|s|dY r=1sup x∈Tl,r g(sr)(x) ≥c1p−α+|s|. forc1>0, with Tl,r= (xl,r+xl+1,r)/2−1/(2cp),(xl,r+xl+1,r)/2 + 1 /(2cp) . ii) Lower bound of order (log(np1)/(np1))(α−|s|)/(2α+1).This is derived in analogy to the case for estimating the mean fu... | https://arxiv.org/abs/2503.24066v1 |
associated with the Riemann-Liouville β1-fractional Brownian motion, which is computed in Van Der Vaart and Van Zanten (2008, Lemma 10.2). Therefore, it follows that for the Kullback- Leibler divergence, KL(Y0, Y1,n) =1 2∥µ1,h∥2 H =1 2h2(α−β1)h−1 Γ(β1+ 1/2)2Z1 0g2(t/h) dt (using (32)) =R1 0g2(t) dt Γ(β1+ 1/2)2n−1. as r... | https://arxiv.org/abs/2503.24066v1 |
calculate similarly as for ii) ρ2 n(x,x′) =E"pX j=1w(s) j(x;h)Z1(xj)pX k=1w(s) k(x;h)Z1(xk) −2pX j=1w(s) j(x′;h)Z1(xj)pX k=1w(s) k(x;h)Z1(xk) +pX j=1w(s) j(x′;h)Z1(xj)pX k=1w(s) k(x′;h)Z1(xk)# n→∞−→∂(s,s)⊤Γ(x,x)−2∂(s,s)⊤Γ(x,x′) +∂(s,s)⊤Γ(x′,x′) =..ρ(x,x′) uniformly for all x,x′∈[0,1]dby (M1). Now let ( xn,yn)n∈N⊂[0,1]d... | https://arxiv.org/abs/2503.24066v1 |
(2011). Optimal estimation of the mean function based on discretely sampled functional data: Phase transition. Ann. Statist. 39 (5), 2330–2355. Cao, G. (2014). Simultaneous confidence bands for derivatives of dependent functional data. Electron. J. Stat. 8 (2), 2639–2663. Da Costa, N., M. Pf¨ ortner, L. Da Costa, and P... | https://arxiv.org/abs/2503.24066v1 |
(2005). Functional data analysis for sparse longitudinal data. J. Amer. Statist. Assoc. 100 (470), 577–590. Zhang, X. and J.-L. Wang (2016). From sparse to dense functional data and beyond. Ann. Statist. 44 (5), 2281–2321. A Further proofs Proof of Lemma 6, ii). We apply Dudley’s entropy bound (van der Vaart and Wellne... | https://arxiv.org/abs/2503.24066v1 |
being generated with the mean function µl. This is done by 1 Ndn,pX lKL(Pµl,P0) =1 Ndn,pn 2σ2 0X lX 1≤j≤pµ2 l(xj) (normal distr.) ≤n 2σ2 0˜L2 e2dh2α+d n,pX 1≤j≤pX ldY r=11|2(xjr−zlr)|<hn,p (Def. µl) ≤n 2σ2 0˜L2 e2dh2α+d n,pp1(disjoint supp.) ≤const. log( np1). (choice of hn,p) Proof of Lemma 5. Analogously to (Berger e... | https://arxiv.org/abs/2503.24066v1 |
ASYMPTOTICALLY DISTRIBUTION-FREE GOODNESS-OF-FIT TESTING FOR POINT PROCESSES JUSTIN BAARS, S. UMUT CAN, AND ROGER J. A. LAEVEN Abstract. Consider an observation of a multivariate temporal point process 𝑁with lawPon the time interval [0,𝑇]. To test the null hypothesis that Pbelongs to a given parametric family, we con... | https://arxiv.org/abs/2503.24197v1 |
ignored; see [3, 10, 11, 19, 31, 32, 34, 49, 50]. In recent decades, various goodness-of-fit testing procedures for temporal point processes have been pro- posed, including random thinning [38], random superposition [7], super-thinning [12], and random-time- change-based methods [13, 45]. However, those procedures rely... | https://arxiv.org/abs/2503.24197v1 |
as we formally establish, converges to a standard Wiener process, as 𝑇→∞ . The remainder of this paper is organized as follows. In Section 2, we introduce our setup and state the assumptions on point processes. Section 3 is dedicated to deriving a functional central limit theorem (FCLT) for the compensated empirical c... | https://arxiv.org/abs/2503.24197v1 |
probabilistic behavior of 𝑁𝜃can be characterized by a conditional intensity function 𝜆𝜃:R+→R𝑑 +:𝑡↦→𝜆𝜃(𝑡); see [14], Proposition 7.3.IV. Denote its components by 𝜆(𝑘) 𝜃,𝑘∈[𝑑]. Let H𝑡:=𝜎(𝑁𝜃(𝑠):𝑠∈(−∞,𝑡]). 4 JUSTIN BAARS, S. UMUT CAN, AND ROGER J. A. LAEVEN Then𝜆𝜃can be taken to be any (H𝑡)𝑡∈R-pred... | https://arxiv.org/abs/2503.24197v1 |
test process In this section, we derive a distribution-free limit process from a transformation of the process 𝑁𝜃0, meaning that the limit is in fact independent of the true parameter 𝜃0, as well as of the model FΘ. As indicated in the Introduction, this allows us to overcome a crucial shortcoming of goodness-of-fit... | https://arxiv.org/abs/2503.24197v1 |
the projection operator onto the 𝑖th coordinate. Since 𝜋−1 𝑖{𝑥𝑖∈𝐷([0,1],R):∥𝑥𝑖−𝑦𝑖∥∞<𝜖}=Ø 𝑧∈𝐷([0,1],R𝑑)𝑧𝑖=𝑦𝑖 𝑥∈𝐷([0,1],R𝑑):∥𝑥−𝑧∥∞<𝜖 lies in the uniform topology on 𝐷([0,1],R𝑑), it follows that 𝑊𝑈⊂𝑆𝑈. Next, note that SU on 𝐷([0,1],R𝑑) is generated by the sets 𝑥∈𝐷([0,1],R𝑑):∥𝑥−𝑦∥∞<�... | https://arxiv.org/abs/2503.24197v1 |
𝑢𝑇∫𝑢𝑇 0𝐷𝜆𝜃0(𝑠)d𝑠√ 𝑇(𝜃0−ˆ𝜃𝑇)+𝑜P(1). (12) We prove the result by showing that the 𝑖th coordinate of the right-hand side (12) converges uniformly in probability to the 𝑖th coordinate of the right-hand side of (11). From this, convergence in the Skorokhod topology follows. Denote the𝑘th row of𝛼𝜃0by𝛼(𝑘)... | https://arxiv.org/abs/2503.24197v1 |
(7). Corollary 1. Grant Assumptions A–C. Let 𝑊𝜃0be a𝑑-variate Wiener process with covariance 𝑢·diag E[𝑁(1) 𝜃0[0,1]],...,E[𝑁(𝑑) 𝜃0[0,1]] . Then it holds under the law P𝜃0of𝑁𝜃0∈FΘthat 1√ 𝑇 𝑁𝜃0(𝑢𝑇)−∫𝑢𝑇 0𝜆ˆ𝜃𝑇(𝑠)d𝑠 𝑢∈[0,1]𝑑−→(𝑊𝜃0(𝑢)+𝑢𝛼𝜃0𝑍)𝑢∈[0,1]=:(ˆ𝜂𝜃0(𝑢))𝑢∈[0,1], (14) as𝑇→∞ , w... | https://arxiv.org/abs/2503.24197v1 |
(16) is a Wiener process with variance 𝜇(𝑘) 𝜃0𝑢, implying that T𝜃0(ˆ𝜂𝜃0)is a standard 𝑑-dimensional Wiener process. Indeed, T𝜃0(ˆ𝜂𝜃0)is a𝑑-dimensional Gaussian process with zero mean, hence its probabilistic behavior is characterized by its covariance structure 𝐶𝑖𝑗(𝑢):=Cov T𝜃0(ˆ𝜂𝜃0))𝑖(𝑢),(T𝜃0(ˆ𝜂�... | https://arxiv.org/abs/2503.24197v1 |
after which one asks whether FΘ={𝑁𝜃:𝜃∈Θ}is indeed a suitable model for the data. To answer this question, we use Theorem 4 to construct a goodness-of-fit test for testing the null hypothesis 𝐻0given in (19). The ideas of this subsection will be implemented in Sections 6–7. In Section 5, we consider consistency unde... | https://arxiv.org/abs/2503.24197v1 |
statistics based on a functional of the empirical process — such as the Kolmogorov-Smirnov, Anderson-Darling and Cram ´er-von Mises tests — converge at rate√︁ 𝑛(𝑇)≍√ 𝑇, the bias of the ‘naive’ testing procedure cannot be mitigated through a choice of the hyperparameter 𝑛. ASYMPTOTICALLY DISTRIBUTION-FREE GOF TESTIN... | https://arxiv.org/abs/2503.24197v1 |
in (7). This transformed process converges to a process independent of the true parameter 𝜃0, as well as of the model FΘ, hence leads to asymptotically correct goodness-of-fit tests. However, we notice that due to the transformation information may be lost, see [24]. In other words, it may be possible that a test base... | https://arxiv.org/abs/2503.24197v1 |
process with covariance 𝑢↦→𝑢·lim 𝑇→∞© «2∫𝜏𝑇 0E𝑔𝑖(0)−¯𝑔𝑇 𝑖(0) 𝜇(𝑖) ˆ𝜃𝑇𝑔𝑗(𝑡)−¯𝑔𝑇 𝑗(𝑡) 𝜇(𝑗) ˆ𝜃𝑇d𝑡ª® ¬𝑖,𝑗∈[𝑑]. Note that ¯𝑍+𝑊2in a non-zero stochastic process if there is some 𝑘∈[𝑑]such that Var(𝑔𝑘(0))≠0, hence if𝜆is not equal (in 𝐿1(P)) to𝜆𝜃+𝑐for some𝜃∈Θand some𝑐∈R𝑑.... | https://arxiv.org/abs/2503.24197v1 |
but nonstationary deviation 𝑁from the null, but where 𝜆→𝜆𝜃a.s., for some𝜃∈Θ. However, one might argue that any such model is in fact ‘close’ to the null, since it converges to a null model as it converges to stationarity. 6.Simulations for parametric Hawkes null hypotheses In this section, we investigate the behav... | https://arxiv.org/abs/2503.24197v1 |
right rejection rates for various choices of 𝑁Exp 𝜇,𝛼,𝛽. However, one should be careful with selecting models with𝛼close to𝛽, since, for fixed 𝑇, for such parameters ˆ𝑊(𝑇)might behave more erratically than a Brownian motion. This is a property inherent to the Hawkes process, not to our testing procedure; see, ... | https://arxiv.org/abs/2503.24197v1 |
need not hold in general. Suppose that in step (v) of the testing procedure described in Algorithm 1, we apply the Kolmogorov-Smirnov normality test. We can then compare the simulated Kolmogorov-Smirnov test statistics for 𝑁ExpH∈𝐻Exp 0 with their theoretical null distribution given by the Kolmogorov distribution, usi... | https://arxiv.org/abs/2503.24197v1 |
observed ‘clusters’. However, the regularity of these clusters should reveal that the process is not an overdispersed one. Finally, since the self-correcting process 𝑁SCmodels behavior opposite to that of a Hawkes process, rejecting the null under this alternative should be quite easy. Although Theorem 5 ensures that ... | https://arxiv.org/abs/2503.24197v1 |
43; 269; 481 87; 378; 496 204; 468; 498 𝑁Periodic431; 498; 500 487; 500; 500 499; 500; 500 409; 499; 500 493; 500; 500 499; 500; 500 𝑁SC500; 500; 500 500; 500; 500 500; 500; 500 500; 500; 500 500; 500; 500 500; 500; 500 We repeat our experiments for 𝐻PL 0∋𝑁PLHconsisting of univariate power-law Hawkes processes. We ... | https://arxiv.org/abs/2503.24197v1 |
procedure. 7.Data analysis In this section, we illustrate the testing procedure outlined in Algorithm 1 by applications to real-world data. In particular, we compare the goodness-of-fit tests conducted in [39] and [46] to our asymptotically exact test. 7.1. Temporal ETAS model. First, we apply the testing procedure fro... | https://arxiv.org/abs/2503.24197v1 |
distributed under the null. Following [39], we perform a Kolmogorov-Smirnov test for the null hypothesis stating that (𝑥𝑖)𝑖∈[482]is a sample from a standard exponential distribution. This yields a 𝑝-value of.5756. Hence, for any reasonable significance level, the null is not rejected, indicating a seemingly good fi... | https://arxiv.org/abs/2503.24197v1 |
model is specified through its conditional intensity 𝜆(𝑡)=𝜇+∑︁ 𝑡𝑖<𝑡𝐻(𝜆(𝑡𝑖))𝑔(𝑡−𝑡𝑖), (38) with𝜇>0, where𝐻:(0,∞)→[ 0,∞)is assumed to be a non-increasing function and where 𝑔:[0,∞)→ [0,∞)is a density function. Note that this model is self-exciting, and can be seen as a variant of the classical Hawkes proc... | https://arxiv.org/abs/2503.24197v1 |
is very few. For the large sample, we use 𝑛=ceil√︁ 𝑁[0,𝑇]/4 =13. Following the procedure, we apply an Anderson-Darling test for normality in step (v), as suggested in Section 6. For the small sample, the 𝑝-value is 0.2506 (compared to 0 .5948 for the ‘naive’ testing procedure). For the large sample, the 𝑝-value ... | https://arxiv.org/abs/2503.24197v1 |
dynamic is accounted for by the recursive model but is not captured by the traditional self-exciting model. 8.Concluding remarks In this work, we introduced an asymptotically distribution-free test process for point processes, which can be used to construct asymptotically distribution-free goodness-of-fit tests. Such t... | https://arxiv.org/abs/2503.24197v1 |
pp. 3415–3426. [11] F. Chen andW. H. Tan (2018). Marked self-exciting point process modelling of information diffusion on Twitter. The Annals of Applied Statistics 12, pp. 2175–2196. [12] R. A. Clements ,F. P. Schoenberg , and A. Veen (2013). Evaluation of space–time point process models using super-thinning. Environme... | https://arxiv.org/abs/2503.24197v1 |
Krishnamurthy andE. Blasch (2022). Hawkes process modeling of block arrivals in Bitcoin Blockchain. Preprint. Available at https://arxiv.org/abs/2203.16666 . [35] I. W. McKeague ,A. M. Nikabadze , and Y. Q. Sun (1995). An omnibus test for independence of a survival time from a covariate. Annals of Statistics 23, pp. 45... | https://arxiv.org/abs/2503.24197v1 |
processes. Journal of Applied Probability 50, pp. 760–771. Appendix For the following set of assumptions, let H0,𝑡:=𝜎(𝑁𝜃(𝑠):𝑠∈ ([ 0,𝑡]),and let𝜆∗ 𝜃be any(H𝑡)𝑡∈R- predictable function such that 𝜆∗ 𝜃(𝑡)=lim Δ𝑡↓01 Δ𝑡P 𝑁𝜃[𝑡,𝑡+Δ𝑡)>0|H0,𝑡. Assumptions C.1(v). (a)E[sup𝛿∈(0,1]𝛿−1(𝑁[0,𝛿])2]<∞. (b) For... | https://arxiv.org/abs/2503.24197v1 |
arXiv:2503.24209v1 [math.ST] 31 Mar 2025Optimal low-rank approximations for linear Gaussian inver se problems on Hilbert spaces Part II: posterior mean approximation Giuseppe Carere∗and Han Cheng Lie† Institut f¨ ur Mathematik, Universit¨ at Potsdam, Potsdam O T Golm 14476, Germany Abstract In this work, we construct o... | https://arxiv.org/abs/2503.24209v1 |
is nondegenerate observation noise with known covariance Cobsand zero mean, and Y takes values in Rn. For a given realisation y∈RnofY, the posterior distribution then is N(mpos,Cpos), where mpos=mpr+CposG∗C−1 obs(y−Gmpr),Cpos=Cpr−CprG∗(Cobs+GCprG∗)−1GCpr, see [34, Example 6.23]. The posterior covariance Cposis independ... | https://arxiv.org/abs/2503.24209v1 |
surjective, and neither is Cprsince ranCpr⊂ranC1/2 pr. Furthermore, ifCprandC1/2 prare injective, then we can define the inverses as unbounded opera tors which are only defined on a dense subspace, c.f. Lemma A.12(ii). This is in con trast with the finite-dimensional setting, in which all the operators involved are bounde... | https://arxiv.org/abs/2503.24209v1 |
reduced-rank mat rix approximation by [31] and [16] to infinite dimensions, which can be found in [5]. The solutions and the corr esponding minimal losses are identified in Theorems 5.10 and 5.11 and Corollary 5.12. The resulting op timal approximations share the property with mposthat they lie in ran Cposwith probabilit... | https://arxiv.org/abs/2503.24209v1 |
on the relevant 3 operators for the analysis of the Bayesian update. Certain aspec ts of low-rank posterior covariance approximation are briefly recalled in Section 4. In this section we also in terpret the prior-to-posterior update in terms of variance reduction. Optimal low-rank posterior mean approximation is conside... | https://arxiv.org/abs/2503.24209v1 |
as follows. If T:H→K,S:H→K andU:K→Z for some separable Hilbert space Z, thenT+S: domT+S⊂H→K with dom T+S:= domT∩domSand UT: domUT⊂H→Z with dom UT:=T−1(domU). IfT∈B(H) is positive and self-adjoint, then the norm /bardbl·/bardblT−1on ranTis defined by/bardblh/bardblT−1= /bardblT−1/2h/bardbl, forh∈ranT. HereT−1/2: ranT1/2⊂... | https://arxiv.org/abs/2503.24209v1 |
mean depends on yand lies in ranCpos, by (3a). The posterior covariance is independent ofy, as (3b) shows. Equation(3c)requiressomeinterpretation. Since µprisnondegeneratebyAssumption2.1, supp µpr= H, c.f. [3, Definition 3.6.2] and Cpris positive, hence injective, c.f. Lemmas A.2 and A.12. Therefore, we caninvertCpronit... | https://arxiv.org/abs/2503.24209v1 |
the posterior mean approximations ofthe form (4a) which areoptimal, in the sense specified in Section 5, do in fact correspondto self-adjoint updates−B. By (3a), it follows that there exists r0≤nsuch that mpos∈M(1) r∩M(2) rfor allr≥r0. Indeed, ifr≥rank(G∗) = rank( G), then (Cpr−B)G∗C−1 obs∈B00,r(Rn,H) for every B∈B00,r(... | https://arxiv.org/abs/2503.24209v1 |
divergences can be expre ssed explicitly in terms of the means and covariances of ν1andν2usingR(·/bardbl·) defined in (7). These formulations rely on a generalisation of the determinant to infinite-dimensional Hilbert sp aces. For A∈L1(H), the so- called ‘Fredholm determinant’ det( I+A) can be defined, and if only A∈L2(H)... | https://arxiv.org/abs/2503.24209v1 |
subspace. T he directions (C−1/2 prwi)i≤rank(H) are orthogonal with respect to the Cpr-weighted inner product /an}bracketle{th1,h2/an}bracketri}htCpr:=/an}bracketle{tCprh1,h2/an}bracketri}ht, and not the unweighted inner product of H. The equation (11) can be interpreted as follows. Given an r-dimensional subspace Vr⊂r... | https://arxiv.org/abs/2503.24209v1 |
of the range condition ran K⊂ranCpr. Furthermore, the assumption K∈ B(Rr,H) implies the rank restriction rank( K)≤r. Thus, for rsmall compared to n,Cpr−KK∗can be interpreted as a low-rank update of Cpr.Therefore, the class Crprovides an extension to infinite dimensions of the finite-dimensional updates considered in [33]... | https://arxiv.org/abs/2503.24209v1 |
es one operator, which can then can be applied in the subsequent ‘online’ stage to any realisation of the da ta. This is in analogy to the finite- dimensional case studied in [33, Section 4.1] and its generalisation to c ertain nonlinear forward models and to losses with respect to the average Amari α-divergences as stu... | https://arxiv.org/abs/2503.24209v1 |
compact when His infinite-dimensional, the series/summationtextm i=1(1+−λi 1+λi)wi⊗widoes not converge as m→∞. However, there is pointwise convergence: for h∈H, we may compute, /parenleftBigg I+/summationdisplay i−λi 1+λiwi⊗wi/parenrightBigg h=/summationdisplay i/parenleftbigg 1+−λi 1+λi/parenrightbigg /an}bracketle{th,... | https://arxiv.org/abs/2503.24209v1 |
to (27). For example, if ranS⊥/ne}ationslash={0}, then one can modify Non ranS⊥without changing the operator TNS. The condition (28) ensures that a unique solution of (27) can be obtained. Furthermore, (28) also has a natural inter pretation as giving minimal solutions of (27). Indeed, any N∈L2(H2,H3) satisfies N=PkerT⊥... | https://arxiv.org/abs/2503.24209v1 |
is /summationtext i>r/parenleftBig −λi 1+λi/parenrightBig3 and the solution Aopt,(1) ris unique if and only if the following holds: λr+1= 0or λr< λr+1. By (16),Copt r=Cpr−/summationtext i>r−λi(Cprwi)⊗(Cprwi). We thus see that the optimal operator Aopt,(1) r in Theorem 5.11 is of the form ( Cpr−B)G∗C−1 obs, whereBsatisfi... | https://arxiv.org/abs/2503.24209v1 |
2for every i∈N, then this decay does not occur, and one can obtain a smaller loss by ignoring the structure. In the following, we interpret the optimal low-rank poserior mean ap proximations in terms of projec- tions of the prior and the posterior means. Lemma 5.13. Letr≤nandAopt,(i)fori= 1,2be defined in Theorems 5.10 ... | https://arxiv.org/abs/2503.24209v1 |
differs from Section 2 in the replacement of the forward map GbyGP. As before, we denote by yan arbitrary realisation of Y. LetµP,pos(y)∼ N(mP,pos(y),CP,pos) be the posterior distribution corresponding to (29) and µpr=N(0,Cpr). Because GPis continuous, it follows from [34, Theorem 6.31] that µP,pos(y)∼µpr∼µpos(y), where... | https://arxiv.org/abs/2503.24209v1 |
describe the prior-preconditioned Hessian C1/2 prG∗C−1 obsGC1/2 prand its square root C1/2 prG∗C−1/2 obsin (20). The eigendecomposition of the prior-preconditioned Hessian can be used in the construction of the optimal projector in Section 7, and the SVD of (20) can be used to form the optimal posterior mean approximat... | https://arxiv.org/abs/2503.24209v1 |
Thus, the temperature field ( x,t)/ma√sto→u(x,t) on (0,1)×[0,T] solves, ∂tu−∂xxu= 0, in (0,1)×(0,T), u(·,0) =x†, on (0,1), u(0,·) =u(1,·) = 0, on (0,T], where the true initial condition x†is unknown and where we impose a homogenous Dirichlet spatial boundary condition. We assume that the data consists of a noisy ob serv... | https://arxiv.org/abs/2503.24209v1 |
0((0,1))), because /an}bracketle{texp(t∆)h,k/an}bracketri}ht1=/summationdisplay i(1+ai)/an}bracketle{texp(t∆)h,ei/an}bracketri}ht/an}bracketle{tei,k/an}bracketri}ht=/summationdisplay i(1+ai)exp(−tai)/an}bracketle{th,ei/an}bracketri}ht/an}bracketle{tei,k/an}bracketri}ht, is symmetric in h,k∈H1 0((0,1)). By[4, Theorem10.... | https://arxiv.org/abs/2503.24209v1 |
Gaussian posterior and the approximate Gaussian posterior given by an approximate posterior mean and the exact posterior co variance, after averaging over the data distribution. The chosen divergences are the Hellinger distanc e and the Renyi, Amari, and forward and reverse KL divergences. These loss classes form a nat... | https://arxiv.org/abs/2503.24209v1 |
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