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δ)o , the conditions 3, 7 and 10 reduce to the 58 following: ∆1 2θ N,M log(d−1 θ∆−1 2θ N,M)≳d−1 θ×log(2 δ) l, (102) ∆N,Mr log(d−1 θ∆−1 2θ N,M)≳h log(log(N+M) ˜α)i4 δ2×1 (N+M)(103) and ∆1+2θ 2θ N,Mr log(d−1 θ∆−1 2θ N,M)≳d−1 θ× log(log(N+M) ˜α)2 δ(N+M)2. (104) (102) holds true if e−1≥λ≳[log(4 δ)+log(8 α)]logl land l≥2¯... | https://arxiv.org/abs/2502.20755v1 |
which depends on the unknown θ. Note that the constanth c(˜α,δ,θ) 16c2 1i1 2θcan be expressed as maxn 1 16c2 1,1o ×maxn (1 2θ∗)1 4θ∗,√ 2o ×[log(8 ˜α)+log(4 δ)]2 θ∗ δand therefore, it is a constant that depends only ˜α,δandθ∗, since θ≥θ∗. Now,log(N+M) N+M≤max log(N+M) N+M,[log(N+M)]1 4θ×[log log( N+M)]1 2θ (N+M)1 2θ ≤... | https://arxiv.org/abs/2502.20755v1 |
the regularization parameter λas stated in Proposition 1, we have that4κlog(2 δ) λl≤1 40andq 2κN1(λ) log(2 δ) √ λl≤1 40. Therefore, using Bernstein’s inequality in L2(L2(R))(See A.1), we have that, for any 0< δ < 1, P θ1:l:t=∥M2∥L2(L2(R))≤4κlog(2 δ) λl+q 2κN1(λ) log(2 δ) √ λl≤1 20 ≥1−δ. Fix any δ∈(0,1). Therefo... | https://arxiv.org/abs/2502.20755v1 |
2,6400N1(λ) ×κlog(2 δ) λ, then, for any 0< δ < 1, we have ηλ,l≥C4 2∥u∥2 L2(R) with probability at least 1−δ. Remark 1. Under Assumption A0, the covariance operator ΣPQand the integral operator TPQ corresponding to the kernel Kand distribution R=P+Q 2is trace-class with the same eigenvalues (λi)i∈Ias defined in (20). A... | https://arxiv.org/abs/2502.20755v1 |
provided the event Eoccurs, we must have, 140κ slog32κs 1−√ 1−δ≤λ≤ ∥AlAl∗∥L∞(L2(R)). Therefore, using Lemma B.2(ii) of Sriperumbudur and Sterge (2022), we have that, conditional on the occurrence of the event E, P( Z1:s:r 2 3≤ ∥M l∥L∞(Hl)≤√ 2|E) ≥√ 1−δ. (125) 67 Using (124) and (125), we have, by the law of total proba... | https://arxiv.org/abs/2502.20755v1 |
the kernel Kby covariance operators corresponding to the kernel Kl. Lemma 3(ii) is also required for the derivation. Lemma 5. LetB:Hl→ H lbe a bounded operator, G∈ H lbe an arbitrary function and (Xi)n i=1i.i.d∼Q. Define I=2 nnX i=1D a(Xi),BΣ−1/2 PQ,λ,l(G−µQ,l)E Hl where a(x) =BΣ−1/2 PQ,λ,l(Kl(·, x)−µQ,l)andµQ,l=R XKl(... | https://arxiv.org/abs/2502.20755v1 |
of Lemma A.8 in (Hagrass et al., 2024b) upon replacing Hby Hl, inclusion operator corresponding to the kernel Kby approximation operator corresponding to the kernel Kl, and covariance and integral operators corresponding to the kernel Kby covariance and integral operators corresponding to the kernel Kl. Lemma 9. Letu=d... | https://arxiv.org/abs/2502.20755v1 |
A > 0, and α >1, be−τi≤λi≤Be−τi. Then, we have that, N2 2(λ) =X i≥1λi λi+λ2 ≤X i≥1A2i−2α (ai−α+λ)2 =A2 a2X i≥1 i−α i−α+λ a!2 ≤A2 a2Z∞ 0 x−α x−α+λ a!2 dx =A2 a2a λ1 αZ∞ 01 1 +xα2 dx. Due to the finiteness of the integral when α >1, we thus obtain that N2 2(λ)≲λ−1 αor equivalently, N2(λ)≲λ−1 2α. The proof of the lo... | https://arxiv.org/abs/2502.20755v1 |
with probability at least 1−δ 2, N2 2,l(λ)≤4 Tr B−1 λAB−1 λA . (133) Now, we observe that Tr B−1 λAB−1 λA = Trh B−1/2 λAB−1/2 λB−1/2 λAB−1/2 λi = Trh B−1/2 λAB−1/2 λ∗ B−1/2 λAB−1/2 λi = B−1/2 λAB−1/2 λ 2 L2(L2(R)) ≤2 B−1/2 λ(A−B)B−1/2 λ 2 L2(L2(R))+ 2 B−1/2 λBB−1/2 λ 2 L2(L2(R)) = 2 B−1/2 λ(A−B)B−1/2 λ 2 L2(L2(R)... | https://arxiv.org/abs/2502.20755v1 |
for both the above bounds to hold true. However, the conditions imposed on λandlin the statement of Lemma 13 imply that λ≳logl l, which is an even stronger condition. Therefore, N2 2,l(λ)≲N2 2(λ)holds if λ≳logl l. Next, we assume exponential decay of eigenvalues of ΣPQ. 1.1st term: N2 2(λ)≳log4 δ λlholds if e−1≥λ≳[log(... | https://arxiv.org/abs/2502.20755v1 |
al., 2024b) by replacing the kernel Kwith the RFF-based kernel Kland probabilities with conditional probabilities, we obtain that PH1(qλ,l 1−α> C∗γ4,l|θ1:l)≤δ. If the event Eoccurs, we must have that γ4,l≤γ3,land therefore, we obtain PH1(qλ,l 1−α> C∗γ3,l|E)≤δ. Lemma 17. LetXbe a random variable, λbe a deterministic par... | https://arxiv.org/abs/2502.20755v1 |
2lwith 1at the i-th position and 0elsewhere. To compute Φ(X), the cosine and sine functions are first applied elementwise to the n×lmatrix MX, yielding the matrices cos(MX)andsin(MX). These two matrices are then concatenated horizontally to form the n×2lmatrix cos(MX)|sin(MX) . Next, the columns of this matrix are pe... | https://arxiv.org/abs/2502.20755v1 |
Location Characteristics of Conditional Selective Confidence Intervals via Polyhedral Methods∗ Andreas Dzemski†, Ryo Okui‡, and Wenjie Wang§ March 3, 2025 Abstract We examine the location characteristics of a conditional selective con- fidence interval based on the polyhedral method. This interval is con- structed from... | https://arxiv.org/abs/2502.20917v1 |
with notable exceptions by Kivaranovic & Leeb (2021, 2024). This note 2 addresses such a gap in the literature by examining the characteristics of condi- tional selective confidence intervals constructed using the polyhedral approach described by Lee et al. (2016). We are particularly interested in the behavior of the ... | https://arxiv.org/abs/2502.20917v1 |
cluster close to common critical values, as documented by Gerber & Malho- tra (2008), Brodeur et al. (2016), and Brodeur et al. (2020). This clustering has raised concerns about potential p-hacking and led to marginally significant results being viewed with skepticism. Selection-robust inference methods may be able to ... | https://arxiv.org/abs/2502.20917v1 |
that the estimator ˆθis unbiased and Gaussian: ˆθ∼N(θ, σ2). For simplicity, we assume that the variance σ2is known. In this case, it is without loss of generality to normalize σ2= 1 and ˆθis identical to its t-statistic X= X/σ. Let xobsdenote the observed value of X. First, we discuss the one-sided significance, where ... | https://arxiv.org/abs/2502.20917v1 |
conventional inference that ignores the selection problem. In this case, the median-unbiased estimator is close to the conventional parameter estimate ( ˆθMU≈ˆθ) and the conditional confidence interval approximates the traditional equal-tailed interval ( θ(0.975)≈xobs−1.96 and θ(0.025)≈xobs+ 1.96). When the observed t-... | https://arxiv.org/abs/2502.20917v1 |
the observed t-statistic xobs. For the latter, we consider the behavior of θ(p) when xobsis close to the critical value ¯ x. We first consider the case of marginal significance. We show that the bounds of the conditional selective confidence intervals and the median-unbiased esti- mator can take arbitrarily large negat... | https://arxiv.org/abs/2502.20917v1 |
= 0 otherwise. Let θ2-sided (p) denote the value of θthat solves p=F2-sided (xobs, θ,¯x). The median-unbiased estimator and conditional confidence interval are de- fined analogously to the one-sided significance case. In particular, the median- unbiased estimator is given by ˆθMU,2=θ2-sided (0.5) and a (1 −α)-level con... | https://arxiv.org/abs/2502.20917v1 |
the narrowest. Here, the length of the unconditional confidence interval is 3 .92(= 1 .96×2), and that of the conditional one is 1 .77; the difference is substantial. 5 Conclusion In this paper, we explore the properties of selective inference methods. Our main finding is that the results of these inference methods can... | https://arxiv.org/abs/2502.20917v1 |
ican Statistical Association 116(534), 845–857. URL: https://doi.org/10.1080/01621459.2020.1732989 Kivaranovic, D. & Leeb, H. (2024), ‘A (tight) upper bound for the length of con- fidence intervals with conditional coverage’, Electronic Journal of Statistics 18, 1677–1701. 18 Lee, J. D., Sun, D. L., Sun, Y. & Taylor, J... | https://arxiv.org/abs/2502.20917v1 |
Modeling discrete common-shock risks through matrix distributions Martin Bladt, Eric C. K. Cheung, Oscar Peralta and Jae-Kyung Woo Abstract We introduce a novel class of bivariate common-shock discrete phase-type (CDPH) distributions to describe dependencies in loss modeling, with an emphasis on those induced by common... | https://arxiv.org/abs/2502.21172v1 |
insurance mathematics, there is increasing need for re- searchers and practitioners to accurately model the dependencies among multiple risk fac- tors or risk events, as this is crucial for effective risk management. Such dependence mod- eling is particularly important in scenarios where different risks are influenced ... | https://arxiv.org/abs/2502.21172v1 |
does not admit closed-form expressions for the joint probability mass function (pmf). A more recent development was presented by Bladt and Yslas (2023b) which is a discrete version of Bladt (2023)’s work, where the authors investigated the termination time of discrete-time Markov chains that begin in the same initial s... | https://arxiv.org/abs/2502.21172v1 |
4, we address the estimation of the CDPH distributions from data, proposing an EM algorithm tailored to our models. Section 5 presents simulation studies that demonstrate the implementation and performance of our estimation procedures, as well as an application to a dataset comprising bivariate insurance claim frequenc... | https://arxiv.org/abs/2502.21172v1 |
starting in the same initial state) before proceeding independently. The exact details behind our novel construction are as follows. Consider two jointly evolving terminating processes, M1={M1(n)}n∈NandM2={M2(n)}n∈N, each taking values in the state space E ∪ S , where E ∩ S =∅and|E| ∧ |S| ≥ 1. We construct the joint pr... | https://arxiv.org/abs/2502.21172v1 |
Proof. Equations (2.1) and (2.2) follow by straightforward application of the law of total probability. Next, P(τ1,2=m, τ 1−τ1,2=z1, τ2−τ1,2=z2) =X j∈SP(τ1,2=m, M 1(m) =M2(m) =j) ×P(τ1−τ1,2=z1, τ2−τ1,2=z2|τ1,2=m, M 1(m) =M2(m) =j) =X j∈SP(τ1,2=m, M 1(m) =M2(m) =j) ×P(τ1−τ1,2=z1|τ1,2=m, M 1(m) =j)P(τ2−τ1,2=z2|τ1,2=m, M ... | https://arxiv.org/abs/2502.21172v1 |
and param- eters ( α,P,U,Q1,Q2). Employing Proposition 2.1, we obtain the following immediate result. Corollary 2.4. Let(N1, N2)follow a bivariate CDPH distribution with support (c1N+ k1)×(c2N+k2)and parameters (α,P,U,Q1,Q2). Then, the joint pmf of (N1, N2)for x1∈c1N+k1andx2∈c2N+k2, namely fN1,N2(x1, x2) :=P(N1=x1, N2=... | https://arxiv.org/abs/2502.21172v1 |
mimicking the common shock component τ1,2, eventually jumping from a state i∈ Eto a state ( j, j)∈ S × S with probability uij. •It then evolves in S×S according to Q1⊗Q2such that a transition from ( i1, i2)∈ S×S to (j1, j2)∈ S × S occurs with probability q1,i1j1q2,i2j2(where, for k= 1,2, we use qk,ijto denote the ( i, ... | https://arxiv.org/abs/2502.21172v1 |
We have the following proposition. Proposition 2.7. Given a CDPH( ααα,P,U,Q1,Q2)random vector (τ1, τ2), the sum τ1+τ2 follows a DPH distribution. Proof. With the representation (2.12) for the random variable τ1+τ2, it follows from Theorem 5.2 in Navarro (2019) that the distribution of τ1+τ2is a mixture of a probability... | https://arxiv.org/abs/2502.21172v1 |
The state space corresponding to the transitions specified by PsumisEsum:=E(1)∪(S(1)× E(2)), and that for the transitions specified by Q1,sumandQ2,sumis(S(1)× S(2))∪ S(2). Here, with the obvious notation, E(i)∪ S(i)is the state space of the underlying terminating Markov chains that define the vector (τ1(i), τ2(i)). Pro... | https://arxiv.org/abs/2502.21172v1 |
using Proposition 2.9 repeatedly. 3 Compound sums under bivariate CDPH claim count Having defined the class of bivariate CDPH distributions, we now turn our attention to a class of bivariate random variables ( Y1, Y2) with a random sum representation Y1:=τ1X ℓ=1X1,ℓ, Y 2:=τ2X ℓ=1X2,ℓ, (3.1) where, for each coordinate k... | https://arxiv.org/abs/2502.21172v1 |
random vector (Y1, Y2)defined in (3.1) belongs to the class of MME distributions introduced in Bladt and Nielsen (2010). In the following subsection, we establish a related result by showing that if ( X1, X2) be- longs to the subclass of multivariate phase-type distributions (MPH∗) proposed in Kulkarni (1989) then so d... | https://arxiv.org/abs/2502.21172v1 |
∂2}). For each ℓ∈N+, suppose that Jℓis the process underlying the MPH∗pair ( X1,ℓ, X2,ℓ). Then, we concatenate independent realizations of Jℓ’s to match the number of steps of γ prior to termination. Thus, we construct a process that keeps track of the state of γwhile simultaneously evolving within C. After starting γ×... | https://arxiv.org/abs/2502.21172v1 |
1}n m=1from state i∈ Etoj∈ E. 19 3.NT ij: the total number of jumps of {M(m) 1}n m=1from state i∈ Etoj∈ S. 4.NB,k ij: the total number of jumps of {M(m) k}n m=1from state i∈ Stoj∈ S, where k∈ {1,2}. 5.NB,k i: the total number of jumps of {M(m) k}n m=1from state i∈ Sto the absorbing state, where k∈ {1,2}. Having specifi... | https://arxiv.org/abs/2502.21172v1 |
simulations involving bivariate Poisson and Poisson- Lindley distributions, followed by an application to real-world insurance claims data. Our focus is on assessing how well CDPH can capture or mimic these distributions. 21 5.1 Bivariate Poisson distribution with common shocks We begin by demonstrating that our propos... | https://arxiv.org/abs/2502.21172v1 |
random variables), suitably transformed to comply with the matrix constraints. 22 Figure 5.1: Bivariate Poisson distribution with common shock intensity λZ= 1. In the top panels we show the empirical pmf (black circles) versus the fitted pmf (by increasing matrix dimension: red triangles, blue crosses, green diamonds),... | https://arxiv.org/abs/2502.21172v1 |
be found in Table 1 of Vernic (2000), and was originally modeled using the bivariate generalized Poisson distribution (BGPD). It consists of claim frequencies from two related insurance categories, with a total of 708 claims. We fit this dataset to our proposed CDPH distribution, applying the same shift trans- formatio... | https://arxiv.org/abs/2502.21172v1 |
CDPH class as well. In particular, min {τ1, τ2}, max{τ1, τ2}and τ1+τ2are shown to be DPH whereas mixtures and sums of independent and identically distributed copies of ( τ1, τ2) belong to the CDPH class. Further properties of compound sums with CDPH claim counts are also discussed, and an EM algorithm for the CDPH clas... | https://arxiv.org/abs/2502.21172v1 |
in risk theory. Insurance: Mathematics and Eco- nomics , 106:364–389. Cossette, H., Mailhot, M., Marceau, E., and Mesfioui, M. (2016). Vector-valued Tail Value-at-Risk and capital allocation. Methodology and Computing in Applied Probability , 18(3):653–674. Drekic, S., Dickson, D. C. M., Stanford, D. A., and Willmot, G... | https://arxiv.org/abs/2502.21172v1 |
HALFSPACE REPRESENTATIONS OF PATH POLYTOPES OF TREES AMER GOEL, AIDA MARAJ, AND ´ALV ARO RIBOT ABSTRACT . Given a tree T, its path polytope is the convex hull of the edge indicator vectors for the paths between any two distinct leaves in T. These polytopes arise naturally in polyhedral geometry and applications, such a... | https://arxiv.org/abs/2502.21204v1 |
of observational and interventional data. In fact, the path Page 1arXiv:2502.21204v1 [math.CO] 28 Feb 2025 Halfspace Representations of Path Polytopes of Trees A.Goel, A.Maraj, ´A.Ribot parametrization has already shown essential for all the progress related to the MLE of Brownian motion tree models [1, 3]. Therefore, ... | https://arxiv.org/abs/2502.21204v1 |
Section 2 we review literature on polytopes and toric fiber products that is relevant for this paper. In Section 3 we show path polytopes are inductively constructed as toric fiber products of pyramids over path polytopes on star trees. Then, in Section 4, we describe the facets and their respective halfspaces of path ... | https://arxiv.org/abs/2502.21204v1 |
together with a set of halfspaces which describe the facets of Pis called an H-representation of P. Given two positive integers k, n, the(n, k)-hypersimplex is ∆n,k=( (x1, . . . , x n)∈Rn|nX i=1xi=k,0≤x1, . . . , x n≤1) . When k= 1,∆n,1is the standard simplex ∆n. Consider a tree T= (V, E). Recall that for two vertices ... | https://arxiv.org/abs/2502.21204v1 |
PRODUCTS We use toric fiber products to construct the path polytope of a tree using path polytopes of its subtrees. More precisely, just as a Tcan be formed using gluing on stars Sn1, . . . , S nk, we show that the polytope PTcan be obtained by toric fiber products on PSn1⃝ ∨ {0}, . . . , P Snk⃝ ∨ {0}. This is done by ... | https://arxiv.org/abs/2502.21204v1 |
k2}where ki∈Lv(Ti)fori= 1,2. Let{ae|e∈E(T1)∪E(T2)}be the standard basis of RE(T1)×RE(T2)∼=RE(T1)∪E(T2), and let {be|e∈E(T)}be the standard basis of RE(T). Consider the affine map given by ϕ(ae1) =ϕ(ae2) =1 2b{u1,u2}andϕ(ae) =beife̸=e1, e2. The vertices of Qcan be divided in three classes, one for each vertex of ∆3. Fir... | https://arxiv.org/abs/2502.21204v1 |
independent, so the dimension of the polytope is at most |E| − |{ v∈V|deg(v) = 2}| −1. Page 7 Halfspace Representations of Path Polytopes of Trees A.Goel, A.Maraj, ´A.Ribot We show that it is equal to this quantity. Consider a gluing of two trees T=T1∗e1,e2T2. Let πi:PTi⃝ ∨ {0} →∆3(i= 1,2)be a pair of gluing integral p... | https://arxiv.org/abs/2502.21204v1 |
T2) , which is isomorphic to F{u1,u2} T . Note that F{u1,k1} T1× {0} ⊂L1:= x∈RE(T1)∪E(T2)|xe= 0for all e∈E(T2),X e∈Eleaf(T1)\{e1}xe= 2 , {0} ×F{u2,k2} T2⊂L2:= x∈RE(T1)∪E(T2)|xe= 0for all e∈E(T1),X e∈Eleaf(T2)\{e2}xe= 2 . Since L1andL2are skew linear spaces, we have the free join (F{u1,k1} T1×{0})⃝ ∨({0... | https://arxiv.org/abs/2502.21204v1 |
T⃝ ∨ci↔j T). Therefore, dim(G(u,v) T) = dim( PT)−1, so G(u,v) T=PT∩ {x∈RE(T)|x{u,v}=P w∈N(v)\{v}x{u,w}}is a facet of PT. Case 3: Consider sets of the form PT1×∆3ˆPT2. We have that PT1×∆3ˆPT2= conv {(cT1,i↔k1,cT2,k2↔j)|i∈Lv(T1)\ {k1}, j∈Lv(T2)\ {k2}} ∪ {(cT1,i↔j,0)|i, j∈Lv(T1)\ {k1}} is included in ˆPT1×∆3(G(u2,k2) T2... | https://arxiv.org/abs/2502.21204v1 |
defining probability tree models. Journal of Symbolic Compu- tation , 99:127–146, 2020. [7] Eliana Duarte, Benjamin Hollering, and Maximilian Wiesmann. Toric fiber products in geometric modeling. In International Conference on Geometric Science of Information , pages 494–503. Springer, 2023. [8] Stephen E Fienberg and ... | https://arxiv.org/abs/2502.21204v1 |
LSD OF THE COMMUTATOR OF TWO DATA MATRICES JAVED HAZARIKA AND DEBASHIS PAUL Abstract. We study the spectral properties of a class of random matrices of the form S− n=n−1(X1X∗ 2− X2X∗ 1) where Xk= Σ1/2 kZk,Zk’s are independent p×ncomplex-valued random matrices, and Σ karep×p positive semi-definite matrices that commute ... | https://arxiv.org/abs/2503.00014v1 |
dimension grows to infinity. Partially motivated by these, we look at a different class of ”commutator/ anti-commutator matrices”, namely that of two independent rectangular data matrices under certain regularity conditions. In this paper, we study the asymptotic behavior of the spectra of random commutator matrices un... | https://arxiv.org/abs/2503.00014v1 |
two samples as X1= Σ1 2ZandX2= Σ1 2Wwhere W= (Wij), Z= (Zij)∈Rp×n having independent entries with zero mean and unit variance. The experimenter suspects an element-wise dependence, i.e., Corr( Zij, Wij) =ρ, and would like to test the hypothesis H0:ρ= 0 against H1:ρ̸= 0. We can characterize this dependence in terms of a... | https://arxiv.org/abs/2503.00014v1 |
0, v∈R}and CR:={u+ iv:u >0, v∈R}denote the left and right halves of the complex plane respectively excluding the imaginary axis. ℜ(z) and ℑ(z) denote the real and imaginary parts respectively of z. The norm of a vector xwill be denoted as ||x||and the operator norm of a matrix Ais denoted by ||A||op. Definition 2.1. Fo... | https://arxiv.org/abs/2503.00014v1 |
distribution function of − iX, the real counterpart of X. Then, Fis defined as follows. F( ix) :=F(x) for x∈R (3.1) It is clear that Fis the clockwise rotated version of F. The analogous Levy metric between distribution functions F, G on the imaginary axis can be defined as Lim(F, G) :=L(F,G) (3.2) where L(F,G) is the ... | https://arxiv.org/abs/2503.00014v1 |
z∈C+implies that iz∈CL. Thus, we have sF(z) =ZdF(x) x−z=ZdF( ix) − i( ix− iz)= iZdF(y) y− iz= isF( iz) (3.13) □ The following is an analogue of a result linking convergence of Stieltjes transforms to the weak convergence of measures on the real axis. Proposition 3.5. Forn∈N, letsn(·)be the Stieltjes transform of Fn, a ... | https://arxiv.org/abs/2503.00014v1 |
(4.3) Then, FSnd− →Fwhere the Stieltjes Transform of Fatz∈CLis characterized by the below set of equations. sF(z) =Z R2 +dH(λ) −z+λTρ(ch(z))(4.4) where h(z) = (h1(z), h2(z))∈C2 Rare unique numbers such that h(z) =Z R2 +λdH(λ) −z+λTρ(ch(z)), andλ= (λ1, λ2)T(4.5) Moreover, h1, h2themselves are Stieltjes Transforms of mea... | https://arxiv.org/abs/2503.00014v1 |
as a stepping stone to prove the result under general conditions mentioned in Theorem 4.3. The assumptions are as follows. 8 LSD OF THE COMMUTATOR OF TWO DATA MATRICES 4.1.1. Assumptions. •A1: There exists a constant τ >0 such that max k=1,2 sup n∈N||Σkn||op ≤τ •A2: For k∈ {1,2},Ez(k) ij= 0,|z(k) ij| ≤Bn, where Bn:=n... | https://arxiv.org/abs/2503.00014v1 |
¯Q(z) is invertible for sufficiently large ndepending on z. The proof is given in Section C. At this point, we define a few additional deterministic quantities that will serve as approximations to the random quantity hkn(z) for z∈CLandk= 1,2. Definition 4.16. ˜hn(z) := ( ˜h1n(z),˜h2n(z)) where ˜hkn(z) =1 ptrace{Σkn¯Q(z... | https://arxiv.org/abs/2503.00014v1 |
Tn:=1 n(Λτ 1nZ1Z∗ 2Λτ 2n−Λτ 2nZ2Z∗ 1Λτ 1n) (4.15) Step5: Recall that for k= 1,2, we have Zk= (z(k) ij)∈Cp×n. With Bn=naas inA2of Assumptions 4.1.1, define ˆZk:= (ˆz(k) ij) with ˆ z(k) ij=z(k) ij 1{|z(k) ij|≤Bn}. Now, let Un:=1 n(Λτ 1nˆZ1ˆZ2∗Λτ 2n−Λτ 2nˆZ2ˆZ1∗Λτ 1n) (4.16) Step6: Fork= 1,2, let ˜Zk=ˆZk−EˆZk. Then, defin... | https://arxiv.org/abs/2503.00014v1 |
to 1−2/c. The proof is given in Section E.3. Figure 1. Illustration of the result of Theorem 4.23 as cvaries when β= 0.7 Theorem 4.24. Suppose L(Hn, H)→0where Hn, Hare bi-variate distributions over R2 +andL(·,·)denotes the Levy distance. If ∃K > 0such that max k=1,2 lim sup n→∞Z R2 +λ2 kdHn(λ1, λ2) < K (4.19) then, l... | https://arxiv.org/abs/2503.00014v1 |
fixed z∈CL, denote the unique solutions to (5.5) corresponding to HnandHash(z, H n)andh(z, H)respectively. Then h(z, H n)− →h(z, H). The proof is given in Section F.2. LSD OF THE COMMUTATOR OF TWO DATA MATRICES 13 6.LSD when the common covariance is the Identity Matrix When Σ n=Ipa.s., we have FΣn=δ1for all n∈Nand thus... | https://arxiv.org/abs/2503.00014v1 |
( sFc(·)) as a function of cin Proposition G.5 and regarding the location of the Stieltjes Transform in Proposition G.6. 6.2.Deriving the density of the L.S.D. Certain properties of the LSD such as symmetry about 0 and existence and value of point mass at 0 have already been established in Proposition 4.21 and Theorem ... | https://arxiv.org/abs/2503.00014v1 |
iz) (7.3) where h( iz) = (h1( iz), h2( iz))T∈C2 Rare unique numbers such that ih( iz) =Z R2 +λdH(λ) −z− iλTρ(ch( iz))(7.4) Moreover, h1, h2themselves are Stieltjes Transforms of measures (not necessarily probability measures) over the imaginary axis and continuous as functions of H. Proof. The proof is immediate from ... | https://arxiv.org/abs/2503.00014v1 |
variables. arXiv:2101.09444 , 2021. [20] Joel L. Schiff. Normal Families . Springer-Verlag, New York, 1993. [21] Jack W. Silverstein and Zhidong Bai. On the empirical distribution of eigenvalues of a class of large dimensionsal random matrices. Journal of Multivariate Analysis , 1995. [22] Elias M. Stein and Rami Shaka... | https://arxiv.org/abs/2503.00014v1 |
= 1 = P(Ay). Then ∀ω∈Ax∩Ay, we have 0 ≤ |Xjn(ω)+Yjn(ω)| ≤ |Xjn(ω)|+|Yjn(ω)|. Hence, lim n→∞max 1≤j≤n|Xjn(ω)+ Yjn(ω)|= 0. But, P(Ax∩Ay) = 1. Therefore, the result follows. □ Lemma A.3. Let{Ajn, Bjn, Cjn, Djn: 1≤j≤n}∞ n=1be triangular arrays of random variables. Suppose max 1≤j≤n|Ajn−Cjn|a.s.− − →0and max 1≤j≤n|Bjn−Djn|a... | https://arxiv.org/abs/2503.00014v1 |
1≤j≤n 1 nu∗ jnAjvjn a.s.− − →0 Proof. LetQj(u, v) :=1 nu∗ jnAjvjn. Define Qj(v, v), Qj(u, u), Qj(v, u) similarly. Let xjn=1√ 2(ujn+vjn). Now applying Lemma A.6, we get max 1≤j≤n 1 nx∗ jnAjxjn−1 ntrace( Aj) a.s.− − →0(A.4) =⇒max 1≤j≤n 1 2(Qj(u, u)−Tj) +1 2(Qj(v, v)−Tj) +1 2(Qj(u, v) +Qj(v, u)) a.s.− − →0, where Tj:=1 nt... | https://arxiv.org/abs/2503.00014v1 |
u≤C0 u(B.6) For arbitrary ϵ >0, there exist δ(ϵ)>0 such that |θ1|,|θ2|< δ(ϵ) =⇒ |ρk(θ1, θ2)|< ϵ. Without loss of any generality, we can choose δ(ϵ)<1. By choosing u > cC 0/δ(ϵ), we can ensure that |chk(z)|< δ(ϵ). Then for such z, we have for k= 1,2 |ρk(ch)|=|ρk(ch1, ch2)|< ϵ (B.7) Now by (B.1), we have ℜ(h1(z)) = uI1,0... | https://arxiv.org/abs/2503.00014v1 |
cg2, ch1, ch2∈ S(cC0/u) and ρ1, ρ2are Lipschitz continuous with constant K0= 1. Now using ¨Holder’s Inequality, we get |g1−h1| ≤Z λ2 1|ρ1(ch)−ρ1(cg)|+λ1λ2|ρ2(ch)−ρ2(cg)| dH(λ) | −z+λTρ(cg)| × | − z+λTρ(ch)| ≤K0||ch−cg||1Zλ1λ1dH(λ) | −z+λTρ(cg)| × | − z+λTρ(ch)|+Zλ1λ2dH(λ) | −z+λTρ(cg)| × | − z+λTρ(ch)| ≤cK0||h−g||1... | https://arxiv.org/abs/2503.00014v1 |
Assumptions 4.1.1, for 1≤j≤n,z∈CLand sufficiently large n, we have max 1≤j≤n|trace{MnQ(z)} −trace{MnQ−j(z)}| ≤4cC0B ℜ2(z)a.s. Consequently, max 1≤j≤n|1 ptrace{Mn(Q(z)−Q−j(z))}|a.s.− − →0 Proof. Fixz∈CLand denote Q(z) as Q. By R0andR4of (A.1), for any 1 ≤j≤n, |trace{MnQ} −trace{MnQ−j}| (B.24) =|trace{Mn(Sn−zIp)−1} −trac... | https://arxiv.org/abs/2503.00014v1 |
Letz∈CL. Then there exists s, tindependent of nsuch that 0< s≤tand for sufficiently large nand under A1of Assumptions 4.1.1, we have (1)cnhn(z) = (cnh1n(z), cnh2n(z))∈ H2 s,t (2)cnEhn(z) = (cnEh1n(z), cnEh2n(z))∈ H2 s,t (3)cn˜hn(z) = (cn˜h1n(z), cn˜h2n(z))∈ H2 s,t Proof. Under A1of Assumptions 4.1.1, we have ||Σ1n||op,... | https://arxiv.org/abs/2503.00014v1 |
results for z∈CL. 1:|ρk(cnhn(z))−ρk(cnEhn(z))|a.s.− − →0 2:|ρk(cn˜hn(z))−ρk(cnEhn(z))| − →0 Proof. Fixz∈CL. By Lemma B.7, for sufficiently large n, we must have . The first result follows from Lemma B.5 and Lemma B.9. The second result follows from (4.13) and Lemma B.9. □ Lemma B.11. Under Assumptions 4.1.1, the operat... | https://arxiv.org/abs/2503.00014v1 |
show max 1≤j≤n|c1j−vn|a.s.− − →0, define for 1 ≤j≤n, 1:Ajn= 1−Ej(1,2) 2:Bjn=Den(j) , defined in (C.3) 3:Cjn= 1 and Djn=vn By Remark B.17, we see that Ajn, Bjn, Cjn, Djnsatisfy the conditions of Lemma A.3. Therefore, we have the result associated with c1j. The results for c2j, d1j, d2jfollow from similar arguments. □ Ap... | https://arxiv.org/abs/2503.00014v1 |
D.Proof of Theorem 4.19 Proof. By Theorem 4.12, every sub-sequence of {hn(·)}∞ n=1has a further sub-sequence that converges uni- formly in each compact subset of CL. Let h∞(·) = ( h∞ 1(·), h∞ 2(·)) be one such subsequential limit corre- sponding to the sub-sequence {hnm(·)}∞ m=1. Additionally, due to (3.11) and (4.7), ... | https://arxiv.org/abs/2503.00014v1 |
This completes the proof of Theorem 4.3 under Assumptions 4.1.1. □ Appendix E.Proof of Theorem 4.20 E.1.Proof of Step8 and Step9. Proof. Since Theorem 4.3 holds for ˜Un, we have F˜Und− →Fτfor some LSD Fτand for z∈CL, there exists functions sτ(z) and hτ(z) satisfying (4.4) and (4.5) with Hτreplacing Hand mapping CLtoCRa... | https://arxiv.org/abs/2503.00014v1 |
By Theorem 4.11, all these subsequential limits coincide which we will denote by h∞(z) = (h∞ 1(z), h∞ 2(z)). Now we will show that sτ(z)→sF(z) asτ→ ∞ where sF(·) is given by (4.4). Note that, |sτ(z)−sF(z)| (E.6) = ZdHτ(λ) −z+λTρ(chτ(z))−ZdH(λ) −z+λTρ(ch∞(z)) ≤ Zd{Hτ(λ)−dH(λ)} −z+λTρ(chτ(z)) +Z 1 −z+λTρ(chτ(z))−1 −z+λTρ... | https://arxiv.org/abs/2503.00014v1 |
Inequality to get the following bound. P1 pX i,jI(k) ij> ϵ ≤PX i,j(I(k) ij−P(I(k) ij= 1)) >pϵ 2 ≤2 exp −p2ϵ2/4 2(pϵ/2 +P i,jVarI(k) ij) ≤2 exp −p2ϵ2/4 2(pϵ/2 +pϵ/2) = 2 exp −pϵ 8 By Borel Cantelli lemma,1 pP ijI(k) ija.s.− − →0 and thus1 prank( Zk−ˆZk)a.s.− − →0. Combining this with (E.9), we have||FTn−FUn||i... | https://arxiv.org/abs/2503.00014v1 |
E.1, all we need is show that hk(−ϵ) is bounded. For k= 1,2 and ϵ >0,t= (x, y)∈R2 +, define the functions Gk(ϵ,t) :R3 +→R+as follows. Gk(ϵ,t) :=ZλkdH(λ) ϵ+λ1ρ2(ct) +λ2ρ1(ct)=ZβλkdH1(λ) ϵ+λTρ(ct)(E.19) First of all, as an implication of D.C.T, we have lim ϵ↓0Gk(θ, ϵ) =θkfork= 1,2. This is clear from the arguments presen... | https://arxiv.org/abs/2503.00014v1 |
0), we can make T1andT3arbitrarily small. Now let’s look at T2. We have cg1, cg2, ch1, ch2∈ S(cC0/u) and due to Remark B.3, ρ1, ρ2are Lipschitz continuous with constant K0= 1. Using H¨ older’s Inequality, we have T2= Zλ2 1(ρ1(cg)−ρ1(ch)) +λ1λ2(ρ2(cg)−ρ2(ch)) (−z+λTρ(cg))(−z+λTρ(ch))dG(λ) (E.23) ≤K0||cg−ch||1q I2,0(g, ... | https://arxiv.org/abs/2503.00014v1 |
< ϵ+c| i+ch|−2I2(h, H) for some arbitrarily small ϵ > 0. The last inequality follows since the integrand in K1is bounded by |ℜ(σ(ch))|−2, we can arbitrarily control the first term by taking¯Hsufficiently close to Hin the Levy metric. The argument for bounding |G2|is exactly the same. Therefore we have |G1|<p ϵ+c| i+ch|... | https://arxiv.org/abs/2503.00014v1 |
with a convex shape. When c >2, we have 0 < R−< R +. In this case, g(x) = 0 when x2=R±andg(x)<0 when x2∈(R−, R+). Thus∀x∈(−√R+,−√R−)∪(√R−,√R+) =Sc, we have g(x)<0. Similarly, for 0 < c≤2,g(x)<0 ∀x∈(−√R+,0)∪(0,√R+) =Sc. Therefore, for any c >0, we have g(x)<0 on the set Sc.g(x)≥0 on Sc c\{0}follows from the convexity of... | https://arxiv.org/abs/2503.00014v1 |
Therefore, we get t3 0=−q3(x) s3 0=V+(x)V−(x) iV+(x)=− iV−(x) Finally we observe that s0, t0satisfy the below relationship. •s3 0+t3 0= 2r(x) = lim ϵ↓02R(−ϵ+ ix) = lim ϵ↓0 S3 0(−ϵ+ ix) +T3 0(−ϵ+ ix) •s0t0=−q(x) =−lim ϵ↓0Q(−ϵ+ ix) = lim ϵ↓0 S0(−ϵ+ ix)T0(−ϵ+ ix) From the above it turns out that lim ϵ↓0S3 0(−ϵ+ ix),... | https://arxiv.org/abs/2503.00014v1 |
shrinking to 0 as ϵ↓0. Therefore, by (3.8) and the symmetry about 0 fc(x) =−1 πlim ϵ↓0ℜ(sF(−ϵ+ ix)) = 0 So, the density is positive on Scand zero on Sc c\{0}. Finally we check if the density can exist at x= 0 for 0 < c < 2. For this we evaluate L:= lim ϵ↓0ℜ(sF(−ϵ)). 1 sF(−ϵ)=−(−ϵ) +1 i+csF(−ϵ)+1 − i+sF(−ϵ) =⇒1 L=1 i+cL... | https://arxiv.org/abs/2503.00014v1 |
the Stieltjes transform s(z, c) is continuous in c >0. Proof. From Theorem 4.3, one of the roots of (6.3) is a Stieltjes transform and hence analytic. Let us denote this functional root as s(z, c). Let z0∈CLbe fixed and {cn}∞ n=1⊂R+be such that lim n→∞cn=c0∈(0,∞). Letsn:=s(z0, cn) be the corresponding functional root (... | https://arxiv.org/abs/2503.00014v1 |
3 LSD OF THE COMMUTATOR OF TWO DATA MATRICES 49 Since θ+∈(0, π/2), we have cos(2π+θ+ 3)<0 and sin(2π+θ+ 3)>0. Thus we have m2∈Q2. Third Root: By (6.5), the last root is m3=ω2s0+ω1t0 (G.16) =|V+|1 3ei(4π+θ+) 3+|V−|1 3ei(2π−θ+) 3 = (|V+|1 3− |V−|1 3)ei(4π+θ+) 3 Since θ+∈(0, π/2), we have cos(4π+θ+ 3)<0 and sin(4π+θ+ 3)<0... | https://arxiv.org/abs/2503.00014v1 |
arXiv:2503.00178v1 [math.ST] 28 Feb 2025Aspects of a Generalized Theory of Sparsity based Inference in Linear Inverse Problems Ryan O’Dowd∗†Raghu G. Raj∗Hrushikesh N. Mhaskar† ∗U.S. Naval Research Laboratory, Radar Division, Washingto n D.C. †Institute of Mathematical Sciences, Claremont Graduate Un iversity, Claremont... | https://arxiv.org/abs/2503.00178v1 |
in tomographic imaging and CS and which, using statistical arguments, were shown to reduce to ℓ1-based linear inverse problems under limiting conditions. Furthe r- more, a statistical learning theory for CG-based neural net - works that theoretically confirms the numerical experiment al results obtained in tomographic i... | https://arxiv.org/abs/2503.00178v1 |
are equal in distribution. TABLE I DISTRIBUTIONS USED IN THIS PAPER ,INCLUDING THEIR NAME ,SYMBOL , AND PROBABILITY DISTRIBUTION FUNCTIONS . Name Symbol Pdf Normal distributionN(µ,σ2)1√ 2πσe−(x−µ)2/2σ2 Multivariate Normal distributionN(µ,Σ)1 (2π)n/2Σ1/2e−(x−µ)TΣ−1(x−µ)/2 Laplace distributionL(µ,λ)λ 2e−λ|x−µ| Rayleigh d... | https://arxiv.org/abs/2503.00178v1 |
Theorem 4.5]. IV. C ONNECTION TO THE CG PRIOR In this section, we investigate a subset of CG priors, which we refer to as Compound Laplacian (CL), and outline some properties of their resulting regularizers. We start with a propo- sition showing how the Laplacian distribution is a specific C G distribution with a Raylei... | https://arxiv.org/abs/2503.00178v1 |
to set wk j= Rj/parenleftBig/radicalBig (ck j)2+ǫ2/parenrightBig /((ck j)2+ǫ2)for eachj∈[n]so that the first term of Lapproaches1 2R(c). This gives us some further restrictions on the class of regularizers we consider in thi s section. In addition to Rbeing subadditive, even, and R(0) = 0, we would also like Rto be cont... | https://arxiv.org/abs/2503.00178v1 |
0< κ < K−4+6γ 1−γ, then ǫ≤2(1+γ)δ (1−γ)(K−κ)−4−6γ. (34) Proof : Part 1) Since ǫ >0by assumption, there exists some C >0such that wk j≥Cfor allj∈[n],k∈N. As a consequence of the same argument as in [5, Lemma 5.1], we have ∞/summationdisplay k=1/vextendsingle/vextendsingle/vextendsingle/vextendsingleck−ck+1/vextendsingle... | https://arxiv.org/abs/2503.00178v1 |
questions of interest are: 1) How can the theory in this paper be extended to consider CG distributions more generally? 2) Need we restrict to the space of Gyin Theorem 5.1? Our definition for the weak null space property may allow us to consider more broad sets of solutions. 3) Detailed calculations of rates of converg... | https://arxiv.org/abs/2503.00178v1 |
A Few Observations on Sample-Conditional Coverage in Conformal Prediction John Duchi∗ Stanford University March 4, 2025 Abstract We revisit the problem of constructing predictive confidence sets for which we wish to obtain some type of conditional validity. We provide new arguments showing how “split conformal” methods... | https://arxiv.org/abs/2503.00220v1 |
measure Leb(bC(x)) is almost always infinite (see also Barber et al. [3]): Corollary 1.1. LetXbe a metric space and assume that X∈ X has continuous distribution. If bCprovides distribution free (1−α)conditional coverage, then for P-almost all x∈ X, P(Leb(bC(x)) = + ∞)≥1−α. Similar results apply when the marginals over ... | https://arxiv.org/abs/2503.00220v1 |
costly optimization. This suggests split-conformal approaches that provide adaptive confidence sets of the form bCn(x):=n y∈ Y | s(x, y)≤bhn(x)o , where bhnis chosen based only on the sample ( Xi, Yi)n i=1—hence the name “split conformal”— making the set bCneasy to compute [25, 8]. In spite of their ease of computation... | https://arxiv.org/abs/2503.00220v1 |
behavior, without any as- sumptions on the underlying distribution, allowing analogues of the guarantee (7) in approximate conditional senses. Bian and Barber [6] also consider such sample-conditional coverage, showing that it is impossible to achieve without stronger assumptions for many predictive methods. 4 2 Sample... | https://arxiv.org/abs/2503.00220v1 |
>bτn). Then because we choose bτnso that Pn(S≤bτn)≥1−α, we obtain P(Sn+1>bτn|Pn)≤α+ sup τ∈R|P(S > τ )−Pn(S > τ )|. (10a) If the values Siare distinct, then Pn(S≤bτn)≤1−α+1 n, and so a completely similar calculation yields P(Sn+1>bτn|Pn)≥α−1 n−sup τ∈R|P(S > τ )−Pn(S > τ )|. (10b) In either case, if we can control the de... | https://arxiv.org/abs/2503.00220v1 |
2(1−α)α+8 3t! . Notably, t7→nt2 2(1−α)α+8 3tis increasing in t, so that P(bτn< q⋆(1−α−γ))≤exp −nγ2 2(1−α)α+8 3γ! . If the scores Shave a density, P(S≤q⋆(β)) =βfor any β∈(0,1). Then we may also consider the event that bτn> q⋆(1−α+γ). For this to occur, we require Pn(S < q⋆(1−α+γ))≤1−α, and defining Bi= 1{Si< q⋆(1−α+γ)},... | https://arxiv.org/abs/2503.00220v1 |
confidence set The population-level confidence set Cθ⋆(x) ={y|s(x, y)≤ ⟨θ⋆, ϕ(x)⟩}immediately suggests devel- oping an empirical analogue [11, 16]. Thus, we turn to an analysis of the empirical confidence set, considering the estimator bθn∈argmin θEPn[ℓα(⟨θ, ϕ(X)⟩ −S)], (11) which Jung et al. [16] consider for the spec... | https://arxiv.org/abs/2503.00220v1 |
t≥0, with probability at least 1−e−nt2, the randomized confidence set bCn(·, U)achieves ((1−α), ϵn)-weighted coverage (Definition 1.1) for the class W:={w(x) =⟨u, ϕ(x)⟩}u∈B2with ϵn≤cbϕ r dlogn d n+d n+t! . As another corollary to Theorem 1, let us assume that G={G1, . . . , G d}consists of sets Gi partitioning X, and d... | https://arxiv.org/abs/2503.00220v1 |
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