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are introduced in Definition C.2, below. Definition C.2. Forδ >0, letη(δ), τ(δ)>0be given by η(δ) =γa2δr 2 πlog 4e η(δ), (53) and τ(δ)≜η(δ) c4log 2e η(δ), (54) where c4>0is given in Definition C.1. To condense notation, the explicit parameterization by δwill in general be dropped and left implicit in this manuscrip... | https://arxiv.org/abs/2502.18393v1 |
max{α, δ}. Fixθ∗∈Θ. LetJ ⊆ 2[d]andC ⊂Θbe finite sets, and let J′′≜{supp( θ)∪J:θ∈ C, J∈ J } . Define k′′ 0≜min{k+ max J′′∈J′′|J′′|, d}. If n≥max64α0 γ2c2δ2max 3 log6 ρ3|J ||C| ,2(k′′ 0−1) ,4 α0log6 ρ3|J ||C| , (64) then with probability at least 1−ρ3, uniformly for all J′′∈ J′′, 2∥hf;J′′(θ∗,θ∗)−E[hf;J′′(θ∗,θ∗)]∥... | https://arxiv.org/abs/2502.18393v1 |
and third, sup J∈J, θ∈C\B τ(θ∗)2∥hf;supp( θ)∪J(θ∗,θ∗)−E[hf;supp( θ)∪J(θ∗,θ∗)]∥2 ∥E[θ+hf(θ∗,θ)]∥2≤sup J′′∈J′′2∥hf;J′′(θ∗,θ∗)−E[hf;J′′(θ∗,θ∗)]∥2 ∥E[θ+hf(θ∗,θ)]∥2≤cδ, where J′≜{supp( θ∗)∪J:J∈ J } andJ′′≜{supp( θ)∪J:θ∈ C, J∈ J } . It follows that under the stated condition on n, with probability at least 1−ρ, for all ˆθ∈Θa... | https://arxiv.org/abs/2502.18393v1 |
1 2,1 2−r 2 π 1−β2 6β 4,r 2 π1 β) . (71) 28 Using the above bound on αin Equation (71), an upper bound on α0can also be obtained. Noting that δ≤1 2, and letting b2≜3√ 2π(5+√ 21), ifβ <b2 ϵ=q 2 π1 δ, then α0= max {α, δ} ≤min{1 2,q 2 π1 β}, whereas if β≥b2 ϵ=q 2 π1 δ, thenα0= max {α, δ}=δ. Next, an explicit form for an... | https://arxiv.org/abs/2502.18393v1 |
z=0z2e−1 2z2p(z)p(−z)dz ≤Zz=∞ z=0e−1 2z2p(−z)dzZz=∞ z=0e−1 2z2p(z)p(−z)dz Rz=∞ z=0ze−1 2z2p(z)p(−z)dzRz=∞ z=0e−1 2z2p(z)p(−z)dz!2 −Rz=∞ z=0z2e−1 2z2p(z)p(−z)dzRz=∞ z=0e−1 2z2p(z)p(−z)dz . (77) Under the assumed correctness of Claim C.8, the proof of Corollary 5.2 can be completed. The right-hand-side of the ... | https://arxiv.org/abs/2502.18393v1 |
of η(δ)andτ(δ). Take nto be at least n≥max{n1, n2, n3, n4, n5}= ˜Ok β2ϵ2 ,ifβ∈(0, b1), ˜Ok βϵ2 ,ifβ∈[b1,b2 ϵ), ˜Ok ϵ , ifβ∈[b2 ϵ,∞), where n1= 432π3 1−b2 1 62β2δlog 24 ρkX ℓ=0d ℓkX ℓ′=0d ℓ′b τℓ′! ,ifβ∈(0, b1), 216π2 b2 0δlog 24 ρkX ℓ=0d ℓkX ℓ′=0d ℓ′b τℓ′! , ifβ... | https://arxiv.org/abs/2502.18393v1 |
1−p(−z),ifv= 0, p(−z), ifv= 1,(86) foru, v∈ {0,1}andz≥0. Applying, in order twice, the law of total probability, the definition of conditional probabilities, and Equations (84)-(86) obtains: fU(1) =Zz=∞ z=−∞fU,|Z|(1, z)dz=Zz=∞ z=0fU||Z|(1|z)f|Z|(z)dz=r 2 πZz=∞ z=0e−1 2z2p(z)p(−z)dz, fV(1) =Zz=∞ z=−∞fV,|Z|(1, z)dz=Zz=∞ ... | https://arxiv.org/abs/2502.18393v1 |
is equivalently stated as: ∂ ∂z1−p(z+w) +p(−(z+w)) 1−p(z) +p(−z)=∂ ∂zRu=∞ u=0e−1 2(u+z+w)2duRu=∞ u=0e−1 2(u+z)2du≤0. To evaluate this partial derivative, observe: ∂ ∂z1−p(z+w) +p(−(z+w)) 1−p(z) +p(−z) =∂ ∂zRu=∞ u=0e−1 2(u+z+w)2duRu=∞ u=0e−1 2(u+z)2du =Ru=∞ u=0e−1 2(u+z)2du ∂ ∂zRu=∞ u=0e−1 2(u+z+w)2du Ru=∞ u=0e−1 2... | https://arxiv.org/abs/2502.18393v1 |
satisfied for probit regression, Step (a) is completed. 38 Moving ahead with Step (b), recall that the aim here is to derive αandγ. For α, observe: α=1√ 2πZz=∞ z=0e−1 2z2(1−p(z) +p(−z))dz =2√ 2πZz=∞ z=0e−1 2z2p(−z)dz =2√ 2πZz=∞ z=0e−1 2z21√ 2πZu=−βz u=−∞e−1 2u2dudz =2√ 2πZz=∞ z=0e−1 2z21√ 2πZu=0 u=−∞e−1 2u2du−1√ 2πZu=... | https://arxiv.org/abs/2502.18393v1 |
|| ˜C|e−1 18π(1+s)nt2arccos( ⟨θ∗,θ⟩)− |˜C|e−1 3πns2arccos( ⟨θ∗,θ⟩), (96) P ∀J′′∈ J′′ hf;J′′(θ∗,θ∗)√ 2π−Ehf;J′′(θ∗,θ∗)√ 2π 2≤r α0(1 +s′)(k′′ 0−1) n+α0t′! ≥1−2|J′′|e−1 12α0nt′2− |J′′|e−1 8α0nt′2 1+s′−e−1 3α0ns′2, (97) where in expectation, for any J⊆[d]andθ∈Θ, E[hJ(θ∗,θ)] =θ∗−θ, (98) E[hf;J(θ∗,θ∗)] =E[⟨hf;J(θ∗,θ∗),θ∗⟩]... | https://arxiv.org/abs/2502.18393v1 |
=2∥hJ(θ∗,θ)−E[hJ(θ∗,θ)]∥2 ∥E[ˆθ+hf;J(θ∗,ˆθ)]∥2+2∥hsupp( θ∗)∪J(θ,ˆθ)−E[hsupp( θ∗)∪J(θ,ˆθ)]∥2 ∥E[ˆθ+hf;J(θ∗,ˆθ)]∥2 +2∥hf;supp( θ)∪J(θ∗,θ∗)−E[hf;supp( θ)∪J(θ∗,θ∗)]∥2 ∥E[ˆθ+hf;J(θ∗,ˆθ)]∥2 43 ▶by the linearity of expectation and the triangle inequality =2∥hJ(θ∗,θ)−E[hJ(θ∗,θ)]∥2 ∥E[θ+hf(θ∗,θ)]∥2+2∥hsupp( θ∗)∪J(θ,ˆθ)−E[hsupp(... | https://arxiv.org/abs/2502.18393v1 |
finitely many values that can be taken by the function hJ′(θ,·) is determined by the number of values that can be taken by I(sign( ˜Xθ)̸= sign( ˜Xˆθ))∈ {0,1}nover all choices of ˆθ∈ B′ 2τ(θ). As such, write W(θ)≜{I(sign( ˜Xθ)̸= sign( ˜Xˆθ)) :ˆθ∈ B′ 2τ(θ)}forθ∈Θ. In addition, for θ,ˆθ∈Θ, define L(θ,ˆθ)≜∥I(sign( Xθ)̸= si... | https://arxiv.org/abs/2502.18393v1 |
hJ′—in this case: hJ′(θ,ˆθ) =* hJ′(θ,ˆθ),θ−ˆθ ∥θ−ˆθ∥2+ θ−ˆθ ∥θ−ˆθ∥2+* hJ′(θ,ˆθ),θ+ˆθ ∥θ+ˆθ∥2+ θ+ˆθ ∥θ+ˆθ∥2+¯hJ′(θ,ˆθ), (121) where ¯hJ′(θ,ˆθ) =hJ′(θ,ˆθ)−* hJ′(θ,ˆθ),θ−ˆθ ∥θ−ˆθ∥2+ θ−ˆθ ∥θ−ˆθ∥2−* hJ′(θ,ˆθ),θ+ˆθ ∥θ+ˆθ∥2+ θ+ˆθ ∥θ+ˆθ∥2 per Equation (47). Note that similar orthogonal decompositions appear in, e.g., Plan et a... | https://arxiv.org/abs/2502.18393v1 |
arbitrary choice of J′∈ J′,θ∈˜C,ˆθ∈ D(θ); ▶by a union bound and an earlier discussion about the cardinality of D(·) ≤5|J′||˜C|qe−1 8ηnt2+d k e−1 64ηn ▶by Equation (129) and Lemma C.11 ≤5|J ||C| qe−1 8ηnt2+d k e−1 64ηn. ▶ ∵|J′| ≤ |J | and|˜C| ≤ |C| 48 Note that the last line follows from recalling the definition of ... | https://arxiv.org/abs/2502.18393v1 |
+s′)(k′′ 0−1) n+√ 2πα0t′≤r 2πα0(1 +s′)(k′′ 0−1) n+vuut6πα0log 6 ρ3|J′′| n ≤1 2·rπ 8γcδ+1 2·rπ 8γcδ =rπ 8γcδ, (138) where c > 0is a constant as per Definition C.1. Together with Equation (115) from the proof of Lemma C.3, Equation (138) gives way to the following bound: P ∀J′′∈ J′′∥hJ′′(θ∗,θ∗)−E[hJ′′(θ∗,θ∗)]∥2≤rπ 8γc... | https://arxiv.org/abs/2502.18393v1 |
θ∗+θ ∥θ∗+θ∥2 + ¯hJ(θ∗,θ)−E[¯hJ(θ∗,θ)] (152) due to Equation (150) and the linearity of expectation. Applying the triangle inequality to the ℓ2-norm of the orthogonal decomposition in Equation (152) and scaling it by a factor of1√ 2πyields: 1√ 2πhJ(θ∗,θ)−E1√ 2πhJ(θ∗,θ) 2 ≤ 1√ 2πhJ(θ∗,θ),θ∗−θ ∥θ∗−θ∥2 −E 1√ 2πhJ(θ∗,... | https://arxiv.org/abs/2502.18393v1 |
2π 2 ≤α0t′ 2+r α0(1 +s′)(k′′ 0−1) n+α0t′ 2 =r α0(1 +s′)(k′′ 0−1) n+α0t′. Thus, Equation (97) holds. D.2.2 Proof of Equations (98)–(100) Next, the four expectations, Equations (98)–(100), in Lemma D.2 are verified. Let J⊆[d]be an arbitrary coordinate subset. Note that it suffices to establish the results for hJas those... | https://arxiv.org/abs/2502.18393v1 |
X|L hJ(θ∗,θ),θ∗−θ ∥θ∗−θ∥2 L=ℓ =πℓ∥θ∗−θ∥2 narccos( ⟨θ∗,θ⟩), (167) E X|L hJ(θ∗,θ),θ∗+θ ∥θ∗+θ∥2 L=ℓ = 0, (168) E X|L[¯hJ(θ∗,θ)|L=ℓ] =0. (169) Taking θ∗,θ∈Θarbitrarily, via the law of total expectation, Equations (144) and (145) follow from Equa- tions (168) and (169), respectively: E X hJ(θ∗,θ),θ∗+θ ∥θ∗+θ∥2 =E L E... | https://arxiv.org/abs/2502.18393v1 |
xi,θ⟩) (173) ▶see, justification below =1 nnX i=1 ˜ xi,θ∗−θ ∥θ∗−θ∥2 I sign (⟨˜ xi,θ∗⟩)̸= sign ( ⟨˜ xi,θ⟩) , (174) ▶usign ( u) =|u|for any u∈R where the second to last equality, (173), follows from the observation that either the indicator term takes the value 0, or otherwise, if sign(⟨˜ xi,θ∗⟩)̸= sign( ⟨˜ xi,θ⟩), th... | https://arxiv.org/abs/2502.18393v1 |
=Eh es(Ui−E[Ui])i =fRi(1)Eh es(Ui−E[Ui]) Ri= 1i +fRi(0)Eh es(Ui−E[Ui]) Ri= 0i ▶by the law of total expectation =fRi(1)ψ(Ui|Ri=1)−E[Ui|Ri=1](s) +fRi(0)ψ(Ui|Ri=0)−E[Ui|Ri=0](s) ▶by the definition of mgfs =1 πarccos( ⟨θ∗,θ⟩)ψ(Ui|Ri=1)−E[Ui|Ri=1](s) + 1−1 πarccos( ⟨θ∗,θ⟩) ψ(Ui|Ri=0)−E[Ui|Ri=0](s) ▶by Equation (175) ≤1 πa... | https://arxiv.org/abs/2502.18393v1 |
1 is standard Gaussian, i.e., (Vi|Ri= 1)∼ N (0,1), and thus, the density function of Vi|Riis given for z∈Randr∈ {0,1}by: fVi|Ri(z|r) = 0, ifr= 0, z̸= 0, 1, ifr= 0, z= 0, 1√ 2πe−1 2z2,ifr= 1. Since Vi|Ri= 1is standard Gaussian, its expectation is E[Vi|Ri= 1] = 0 , while also, E[Vi|Ri= 0] =Zz=∞ z=−∞zfVi|Ri(z|0)dz= ... | https://arxiv.org/abs/2502.18393v1 |
basis is given and subsequently rewritten as follows: ¯hJ(θ∗,θ) =k′X j=1 ¯hJ(θ∗,θ)√ 2π,vj vj =k′X j=1 hJ(θ∗,θ)√ 2π− hJ(θ∗,θ)√ 2π,vk′−1 vk′−1− hJ(θ∗,θ)√ 2π,vk′ vk′,vj vj ▶by the choice of vk′−1=θ∗−θ ∥θ∗−θ∥2,vk′=θ∗+θ ∥θ∗+θ∥2 =k′X j=1 hJ(θ∗,θ)√ 2π,vj vj− hJ(θ∗,θ)√ 2π,vk′−1 vk′−1− hJ(θ∗,θ)√ 2π,vk′ vk′ ▶due to the orthogona... | https://arxiv.org/abs/2502.18393v1 |
>(1 +s)µL)≤e−1 3πns2arccos( ⟨u,v⟩). (199) Lemma D.5. Fixt′′, σ > 0and0< m≤d. LetJ′′⊆[d],|J′′|=m, andX∼ N(0, σ2P j∈J′′ejeT j). Then, P ∥X−E[X]∥2>√mσ+t′′ ≤P(∥X−E[X]∥2>E[∥X∥2] +t′′)≤e−1 2σ2t′′2. (200) Proof (Proof of Lemma D.5). Note that ∥X−E[X]∥2=∥X∥2due to the lemma’s condition that Xis zero-mean. By standard propert... | https://arxiv.org/abs/2502.18393v1 |
some of the steps taken above can be obtained by extending those appearing in the proof of Lemma D.1. The first step towards deriving Equations (146) and (148) is characterizing the distribution of each ithsummand, i∈[n], in Equation (211). Let Zi∼ N (0,1)andRi≜I(f(Zi)̸= sign( Zi)),i∈[n]. Then, each ithsummand, i∈[n], ... | https://arxiv.org/abs/2502.18393v1 |
denoted by ψ(Wi|Ri)−E[Wi|Ri]andψ(−Wi|Ri)−E[−Wi|Ri], respectively. Write µ0≜E[Wi|Ri= 0] = 0 andµ1≜E[Wi|Ri= 1] =√ 2/π−γ 2α, where these expectations were calculated previously in Equations (217) and (218). Conditioned on Ri= 1, the mgfs are given at s∈[0,∞)by ψ(Wi|Ri=1)−E[Wi|Ri=1](s) =Eh es(Wi−E[Wi]) Ri= 1i 70 =Zz=∞ z=01... | https://arxiv.org/abs/2502.18393v1 |
W, P hf;J′′(θ∗,θ∗)√ 2π,θ∗ −E hf;J′′(θ∗,θ∗)√ 2π,θ∗ > αt ≤e−1 3αnt2. Moreover, by a nearly identical argument (omitted here), the other side of the bound is obtained: P(W−E[W]<−αt) =P(−W−E[−W]> αt)≤e−1 3αnt2, and thus, P hf;J′′(θ∗,θ∗)√ 2π,θ∗ −E hf;J′′(θ∗,θ∗)√ 2π,θ∗ <−αt ≤e−1 3αnt2. Combining the above inequalit... | https://arxiv.org/abs/2502.18393v1 |
{0,1}by fUi,j|Ri(z|r) = 0, ifz̸= 0, r= 0, 1, ifz= 0, r= 0, fZi,j(z)fYi|Ri(1|1) +f−Zi,j(z)fYi|Ri(−1|1),ifr= 1, = 0, ifz̸= 0, r= 0, 1, ifz= 0, r= 0, 1 2fZi,j(z) +1 2f−Zi,j(z),ifr= 1, = 0, ifz̸= 0, r= 0, 1, ifz= 0, r= 0, 1√ 2πe−1 2z2,ifr= 1,(234) where the third case on the right-hand-side of the first e... | https://arxiv.org/abs/2502.18393v1 |
that (U|L=ℓ) = k′−1X j=1Ujej L=ℓ ∼ N 0,ℓ n2k′−1X j=1ejeT j , and hence, U|L≤ℓis at most√ ℓ n-subgaussian with mean E[U|L≤ℓ] =0and support of cardinality ∥U∥0=k′−1. Therefore, by Lemma D.7 and standard properties of Gaussians, P ∥U−E[U]∥2>p (k′−1)ℓ n+α0t L≤ℓ! (237) 76 ≤P ∥U−E[U]∥2>p (k′−1)ℓ n+α0t L=ℓ! (238) ≤e−n... | https://arxiv.org/abs/2502.18393v1 |
dsf1(s)≤0, is true: d dsf1(0) =1√ 2πZz=∞ z=0(z−µ1)e−1 2z2ν(z)dz =1√ 2πZz=∞ z=0ze−1 2z2ν(z)dz−µ11√ 2πZz=∞ z=0e−1 2z2ν(z)dz =p 2/π−γ 2−p 2/π−γ 2αα 78 ▶by the definitions of α, γ in Equations (6) and (7), respectively, and an earlier remark in Equation (218) that µ1=√ 2/π−γ 2α = 0. On the other hand, the case when s >0wil... | https://arxiv.org/abs/2502.18393v1 |
calculus, this implies that sup s≥0f1(s) =f1(0), verifying Equation (221). Verification of Equation (222) .Equation (222) can be derived through an analogous approach. As such, most of the analysis to upper bound f2falls onto showing thatd dsf2(s)≤0for all s≥0, from which it will directly follow that f2(s)≤f2(0)fors≥0.... | https://arxiv.org/abs/2502.18393v1 |
arXiv:2502.18634v1 [math.ST] 25 Feb 2025Kernel Estimation for Nonlinear Dynamics∗†‡ Marie-Christine D¨ uker FAU Erlangen-N¨ urnbergAdam Waterbury Denison University February 27, 2025 Abstract Many scientific problems involve data exhibiting both temporal and cr oss-sectional dependen- cies. While linear dependencies hav... | https://arxiv.org/abs/2502.18634v1 |
(1.1), we employ a regularized least squares estimator also know n as Kernel Ridge Regression (KRR). KRR is a popular technique in supervised learning and has been used to avoid overfitting in regression problems. However, to the best of our knowledg e, there is no literature on the use of KRR estimators for temporal mo... | https://arxiv.org/abs/2502.18634v1 |
Adamczak and Bednorz (2015),Adamczak and Wolff (2015),Paulin (2015),Chen and Wu (2017),Alquier, Doukhan, and Fan (2019),Fan, Jiang, and Sun (2021). Given the kernel-based estimation approach, our work natur ally connects to the literature on U-statistics, particularly concentration inequalities f or U-statistics. The pr... | https://arxiv.org/abs/2502.18634v1 |
one and treat it as a Mark ov chain. This Xp-valued process tYtuis defined by Yt.“` X1 t,...,X1 t´p`1˘1, t “p,...,T, (2.1) 3 and, fort“p,...,T, we define ¨ ˚˝Xt ... Xt´p`1˛ ‹‚“¨ ˚˚˚˝gpXt´1,...,X t´pq Xt´1 ... Xt´p`1˛ ‹‹‹‚`¨ ˚˚˚˝εt 0 ... 0˛ ‹‹‹‚orYt“GYpYt´1q `ξt. For notational convenience, we sometimes write Yt“ pY1 t,1,... | https://arxiv.org/abs/2502.18634v1 |
q PRpT´pqˆpT´pq, withKipX,X q.“ rpKipX,X qqs,tss,t“p`1,...,Tdefined as pKipX,X qqs,t“KipXps´pq:ps´1q,Xpt´pq:pt´1qq, s,t “p`1,...,T. 5 We also define an empirical kernel matrix KpX,X q “diagpK1pX,X q,...,K dpX,X qq PRdpT´pqˆdpT´pq. (2.11) Then, vectorizing the estimated counterpart of GXpgqin (2.4), we have vec¨ ˚˝pgTpXp,... | https://arxiv.org/abs/2502.18634v1 |
density ψsuch that for each compact set AĎsupp pεi,tq, infxPAψpxq ą0. 7 WerefertoProposition2.5.2in Vershynin (2018) fordifferentcharacterizations ofsub-Gaussianity. We also state two slightly more restrictive assumptions for the discussion of our convergence rates below. Both are special cases of Assumption N.1. Assump... | https://arxiv.org/abs/2502.18634v1 |
if δěc0c logpTq TLb2 2γ, (3.6) and γěa 4σ2logpdTq (3.7) then, with probability at least 1´c1T´c2, η1KpX,X qηďδ2. (3.8) Constants: Note that the constant b2in (3.3) and (3.6) stems from Assumption K.4and is kernel-dependent. Similarly, the last summand in ( 3.5) depends on β1,β2which are also kernel- dependent and stem ... | https://arxiv.org/abs/2502.18634v1 |
from Lemma 6.1, Lemma 6.5, and Lemma 6.7. /squaresolid The proof of the results used in the proof of Theorem 3.4are quite involved and can be found in Section 6. We present two more examples but, for simplicity, consider the cased“p“1. 10 Periodic kernel: Suppose the kernel K1is the periodic Sobolev kernel, i.e., Kpx,y... | https://arxiv.org/abs/2502.18634v1 |
Hbe the Hilbert space defined as H.“ąd i“1Hi“ thPBpXp:Rdq:h“ ph1,...,h dq1, hiPHi, i“1,...,d u,(4.1) with inner product xh,rhyH“ÿd i“1xhi,rhiyHi. (4.2) For a kernel Kisatisfying Assumption K.2, we have, for f,gPHi, xf,gyHi“ÿM k“1λ´1 i,kÿNpkq j“1xf,φi,j,kyL2pXp,πqxg,φi,j,kyL2pXp,πq, where x¨,¨yL2pXp,πqis the standard inn... | https://arxiv.org/abs/2502.18634v1 |
where Pnpy,Aq.“PypYnPAq, A PBpXpq, yPXp, (4.10) and } ¨ }TVdenotes the total variation norm on PpXpq. Remark 4.9.We require the somewhat nonstandard assumption that Jis integrable. In view of (1.1) and Assumption G.1, this is satisfied, for instance, when Assumption N.1holds. Then, one can infer that the chain is aperio... | https://arxiv.org/abs/2502.18634v1 |
qZX,Tq ď pλmaxpZX,Tqq21 T2trpη1KpX,X qηq ďλ´21 T2η1KpX,X qη, (5.6) where the first identity follows from ( 5.5) and Lemma 4.6. For the first inequality, note that the matricesηη1,ZX,TandKpX,X qareall positivesemidefinite, sotheinequality follows fro mTheorem 1 ofFang, Loparo, and Feng (1994). The final inequality follows u... | https://arxiv.org/abs/2502.18634v1 |
inequality and the s econd inequality uses that the maximum of a sum is bounded above by the sum of the maxima. We consider t he diagonal and cross terms of the sum in the last line of ( 5.18) separately. Define ΦpYt´1,Ys´1q.“max k“1,...,Mmax j“1,...,N pkqβ´2 i,j,kλi,kφi,j,kpYt´1qφi,j,kpYs´1q, and note, due to the defini... | https://arxiv.org/abs/2502.18634v1 |
2d˙ ď2d δ2EpBi,Mq. (5.31) Applying the Fubini-Tonelli Theorem, we have EpBi,Mq “E¨ ˝8ÿ k“Mλi,kNpkqÿ j“1˜ 1 TTÿ t“p`1εi,tφi,j,kpYt´1q¸2˛ ‚ 21 ďσ2 T28ÿ k“Mλi,kNpkqÿ j“1Tÿ t“p`1Eppφi,j,kpYt´1qq2q, (5.32) where (5.32) follows upon using the law of total expectation and noting t hatεtis independent of Yt´1, thatEpεi,tq “0, ... | https://arxiv.org/abs/2502.18634v1 |
ď››››1 TÿT t“p`1fpryt´1qKpryt´1,¨q ´1 TÿT t“p`1fp¯yt´1qKp¯yt´1,¨q›››› H “1 T}fprzqKprz,¨q ´fp¯zqKp¯z,¨q}H ď1 Tsup w,rwPXp}fpwqKpw,¨q ´fprwqKprw,¨q}Hď1 TCF.(5.39) The final inequality in ( 5.39) is verified in the calculations below. Forw,rwPXpwe have }fpwqKpw,¨q ´fprwqKprw,¨q}H ď }fpwqKpw,¨q ´fpwqKprw,¨q}H` }fpwqKprw,¨q ... | https://arxiv.org/abs/2502.18634v1 |
(5.43) and the fact that πphq “LKpfq, and (5.55) uses (5.51) and (5.52). /squaresolid The next lemma is also used in the proof of Theorem 3.2. 26 Lemma 5.5. LetFi,M:pXpqT´p`1Ñ r0,8qdefined by Fi,Mpyp,...,yTq “max k“1,...,Mmax j“1,...,N pkq1 TˇˇˇˇˇTÿ t“p`1ˆ pyt,1´fpyt´1qqiβ´1 i,j,kλ1 2 i,kφi,j,kpyt´1q1t|pyt,1´fpyt´1qqi|ď... | https://arxiv.org/abs/2502.18634v1 |
k!pk`1qpd. Proof:Recall the quantity nk,jintroduced in Lemma 6.1. Sincekandjare fixed, we, with a slide abuse of notation, write nin place of nk,j, wheren.“ pn1,1,...,n 1,p,...,n d,1,...,n d,pq. Let Zα,i,r „Npαi,σ2 iq, α “ pα1,...,α dq1PRd, i“1,...,d, r “1,...,p, and note that Lemma 6.3ensures that there is some bP p0,8... | https://arxiv.org/abs/2502.18634v1 |
01a pk´xqxdx“π, which shows that ( 6.15) holds ford“2. Now, suppose that ( 6.15) holds for some dě2. Using the inductive hypothesis, we see that ÿ n1`¨¨¨`nd`1“kd`1ź i“1n´1{2 i “k´dÿ nd`1“1ÿ n1`¨¨¨`nd“k´nd`1n´1{2 d`1dź i“1n´1{2 i “k´dÿ nd`1“1n´1{2 d`1¨ ˝ÿ n1`¨¨¨`nd“k´nd`1dź i“1n´1{2 i˛ ‚ ďπd{2k´dÿ nd`1“1n´1{2 d`1pk´nd`1... | https://arxiv.org/abs/2502.18634v1 |
A. M., Fieguth, P. W., and Chen, H. H. A new Mercer sigmo id kernel for clinical data classifi- cation. In 2014 36th Annual International Conference of the IEEE Engin eering in Medicine and Biology Society, pages 6397–6401. IEEE, 2014. Chakrabortty, A. and Kuchibhotla, A. K. Tail bounds for canonica l U-statistics and U... | https://arxiv.org/abs/2502.18634v1 |
R., and Smola, A. J. A generalized represe nter theorem. In International conference on computational learning theory , pages 416–426. Springer, 2001. Sharma, S., Sharma, S., and Athaiya, A. Activation functions in neur al networks. Towards Data Sci , 6(12): 310–316, 2017. Shen, Y., Han, F., and Witten, D. Exponential ... | https://arxiv.org/abs/2502.18634v1 |
A Matsuoka-Based GARMA Model for Hydrological Forecasting: Theory, Estimation, and Applications Guilherme Pumia,∗, Danilo Hiroshi Matsuokaa, Taiane Schaedler Prassaand Bruna Gregory Palmb Abstract Time series in natural sciences, such as hydrology and climatology, and other envi- ronmental applications, often consist o... | https://arxiv.org/abs/2502.18645v1 |
supports informed decision-making and is essential for successfully planning and implementing water management strategies. Forecasting plays a crucial role in this context; however, obtaining prediction confidence intervals for the forecasts is equally important. These intervals provide valuable information to quantify... | https://arxiv.org/abs/2502.18645v1 |
random component follows a Matsuoka’s distribution in (0 ,1), with G. Pumi, D.H. Matsuoka, T.S. Prass and B.G. Palm 3 systematic component models the process’ conditional mean through an ARMA-like structure with possible exogenous covariates, called MARMA (Matsuoka autoregressive moving average) models. We propose a pa... | https://arxiv.org/abs/2502.18645v1 |
and general class of distributions introduced in Reis et al. (2024), called the Unit Gamma-G class. Despite all these works, little was known about the distribution until the work of Matsuoka et al. (2024), where the authors study a semiparametric three-step approach to production frontier estimation. Although the pape... | https://arxiv.org/abs/2502.18645v1 |
Fokianos and Kedem (2004), but it is slightly different from the approach commonly used in GARMA-like models for which the distribution is not a member of the canonical exponential family, like the KARMA, UWARMA e βARMA. In these cases, it is G. Pumi, D.H. Matsuoka, T.S. Prass and B.G. Palm 5 usually simpler to paramet... | https://arxiv.org/abs/2502.18645v1 |
understood to hold almost surely. Let Ht(γ) :=−∂2ℓt(γ) ∂γ∂γ′,and H(γ) :=−∂2ℓ(γ) ∂γ∂γ′=−nX t=1∂2ℓt(γ) ∂γ∂γ′=nX t=1Ht(γ). Notice that H(γ) and ℓ(γ) both depend on n. However, for simplicity and since no confusion will arise, we shall drop the dependence on non the notation. Let In(γ) :=E(H(γ)) be the information matrix c... | https://arxiv.org/abs/2502.18645v1 |
almost surely in the domain of g−1, for allγ∈Ω and all t. 3. There is a probability measure λonRp+q+1such thatZ Rp+q+1vv′λ(dv) is positive defi- nite and such that 1 nnX t=1I(Zt−1∈A)P−→λ(A), asn→ ∞ , atγ0. A detailed discussion on these assumptions and their implications can be found on section 5 of Fokianos and Kedem ... | https://arxiv.org/abs/2502.18645v1 |
based on the so-called quantile residuals, defined as e(q) t:= Φ−1(F(Yt|Ft−1)), (9) where F(·|Ft−1) denotes the cumulative distribution function associated with the model’s random component and Φ−1denotes the standard normal quantile function. In the present setting, when the model is correctly specified, the quantile ... | https://arxiv.org/abs/2502.18645v1 |
parameter ˆ µ(b) n+kand update ˆr(b) n+k=g(ˆY(b) n+k)−g(ˆµ(b) n+k). From these steps, we obtain a collectionˆY(b) n+1,···,ˆY(b) n+h m b=1of bootstrap samples. For each k∈ {1,···, h}, a level δconfidence interval for Yn+kis obtained from the (1 −δ/2)th and δ/2th sample quantiles calculated from ˆY(1) n+k,···,ˆY(m) n+k.... | https://arxiv.org/abs/2502.18645v1 |
in italics), and standard deviations (in parentheses) calculated from the 1,000 replicas. The table shows that parameter βis remarkably well estimated in all cases. Parameter αis also well estimated, especially when n= 500. A comparison between the mean and median estimates shows that for n= 100, the estimates are slig... | https://arxiv.org/abs/2502.18645v1 |
1 β=−0.5 ϕ=−0.4 θ=−0.2 100 0.943 0.949 (0.117) -0.500 -0.500 (0.031) -0.319 -0.329 (0.174) -0.292 -0.294 (0.196) 200 0.970 0.971 (0.074) -0.500 -0.501 (0.021) -0.358 -0.363 (0.111) -0.246 -0.250 (0.130) 500 0.988 0.988 (0.043) -0.500 -0.500 (0.014) -0.383 -0.384 (0.065) -0.218 -0.220 (0.078) n α = 1 β=−0.5 ϕ= 0.4 θ= 0.... | https://arxiv.org/abs/2502.18645v1 |
Solid lines in the scatter plot represent the true values. G. Pumi, D.H. Matsuoka, T.S. Prass and B.G. Palm 15 5.2 Goodness-of-fit tests This section examines the finite-sample performance of goodness-of-fit tests based on the simple and quantile residuals discussed in Section 4.2. For the simple residuals, ˆ et=Yt−ˆµt... | https://arxiv.org/abs/2502.18645v1 |
0.2 θ=−0.40.03 0.06 0.06 0.05 0.06 0.03 0.04 0.05 0.05 0.04 200 0.01 0.05 0.05 0.06 0.05 0.02 0.06 0.06 0.05 0.06 500 0.00 0.05 0.06 0.05 0.06 0.01 0.05 0.05 0.05 0.04 100ϕ=−0.8 θ= 0.20.04 0.06 0.05 0.06 0.08 0.06 0.09 0.08 0.06 0.17 200 0.06 0.06 0.05 0.05 0.09 0.06 0.07 0.07 0.06 0.15 500 0.06 0.04 0.04 0.03 0.05 0.0... | https://arxiv.org/abs/2502.18645v1 |
Reservoir, located at the border between Itapecerica da Serra and Embu-Gua¸ cu, SP, Brazil. Accurately modeling the useful water volume of a reservoir is crucial for effective water resource management. This estimation is key to informed decision-making, and for water management plans to be successful, they must rely o... | https://arxiv.org/abs/2502.18645v1 |
effects. G. Pumi, D.H. Matsuoka, T.S. Prass and B.G. Palm 19 To evaluate the out-of-sample forecasting performance, we compared the proposed MARMA model with the KARMA model (Bayer et al., 2017) and the Holt-Winters additive method (Holt, 2004; Winters, 1960). The KARMA model is widely applied in hydrological modeling,... | https://arxiv.org/abs/2502.18645v1 |
normality for the logit and loglog link functions, which suggests model misspecification. Model parsimony is a crucial consideration in selecting an appropriate forecasting model, as it balances goodness of fit with complexity to enhance interpretability and generalizability. Table 3 shows that while the KARMA models a... | https://arxiv.org/abs/2502.18645v1 |
KARMA model (loglog) (a) In-sample forecast TimeGuarapiranga UV 2022.8 2023.0 2023.2 2023.4 2023.6 2023.8 2024.00.20.40.60.81.0 Observed data Predicted Values (MARMA) Bootstrap Prediction (MARMA) Bootstrap PI (MARMA) Predicted Values (KARMA) Bootstrap Prediction (KARMA) Bootstrap PI (KARMA) (b) Out-of-sample forecast F... | https://arxiv.org/abs/2502.18645v1 |
0.1272 0.1549 0.1492 0.1372 HW 0.0328 0.0316 0.0551 0.0884 0.1347 0.1911 0.2588 0.2715 0.2801 0.2984 0.2850 0.2664KARMAlogit 0.0163 0.0373 0.0339 0.0277 0.0330 0.0454 0.0717 0.0648 0.0669 0.0898 0.0835 0.0800 loglog 0.0083 0.0288 0.0270 0.0216 0.0282 0.0385 0.0630 0.0591 0.0576 0.0798 0.0738 0.0712 cloglog 0.0264 0.044... | https://arxiv.org/abs/2502.18645v1 |
performed better for long-term horizons. Turning to the in-sample performance, the KARMA model showed slightly better accuracy measures than MARMA. Overall, MARMA models exhibited a competitive per- formance with fewer parameters and significantly lower computational costs. These findings reinforce the practitioner to ... | https://arxiv.org/abs/2502.18645v1 |
. John Wiley & Sons. Holt, C. C. (2004). Forecasting seasonals and trends by exponentially weighted moving aver- ages. International Journal of Forecasting , 20(1):5–10. Kalliovirta, L. (2012). Misspecification tests based on quantile residuals. The Econometrics Journal , 15(2):358–393. Kim, J. H. (2014). vrtest: Varia... | https://arxiv.org/abs/2502.18645v1 |
perform estimation via PMLE are available in R package BTSR (Prass and Pumi, 2022). Below we examine pairwise scatter plots and marginal behavior (histograms and boxplots). From the scatter plots, we observe the convergence of the points to a familiar Gaussian be- havior asnincreases. The marginal behavior (histograms ... | https://arxiv.org/abs/2502.18645v1 |
−0.6−0.5−0.4−0.3−0.2 0.2 0.3 0.4 0.5 0.6 α^ β^ −0.6−0.5−0.4−0.3−0.2 0.2 0.3 0.4 0.5 0.6 Sample size (n): 100 200 500 0510 0.10.20.30.40.50.6density −0.75−0.50−0.250.000.25 −0.75−0.50−0.250.000.25 0.20.30.40.50.6 α^ φ^ −0.75−0.50−0.250.000.25 0.20.30.40.50.6 Sample size (n): 100 200 500 051015 −0.6 −0.5 −0.4 −0.3 −0.2de... | https://arxiv.org/abs/2502.18645v1 |
0.8 1.0density −1.00−0.75−0.50−0.250.00 −1.00−0.75−0.50−0.250.00 0.6 0.8 1.0 α^ φ^ −1.00−0.75−0.50−0.250.00 0.6 0.8 1.0 Sample size (n): 100 200 500 01020 −0.6 −0.5 −0.4 −0.3density −1.00−0.75−0.50−0.250.00 −1.00−0.75−0.50−0.250.00 −0.6 −0.5 −0.4 −0.3 β^ φ^ −1.00−0.75−0.50−0.250.00 −0.6 −0.5 −0.4 −0.3 Sample size (n): ... | https://arxiv.org/abs/2502.18645v1 |
arXiv:2502.19254v1 [cs.LG] 26 Feb 2025Set and functional prediction: randomness, exchangeability, and conformal Vladimir Vovk February 27, 2025 Abstract This papercontinues thestudyofthe efficiencyof conformal p rediction as compared with more general randomness prediction and exc hangeabil- ity prediction. It does not r... | https://arxiv.org/abs/2502.19254v1 |
opposite to w hat we did for p-values). However, the property of validity of conformal e-predictors is slightly more difficult to state in terms of prediction sets: now validity m eans that the integral of the probability of error for Γαoverα∈(0,∞) does not ex- ceed 1 [9, end of Appendix B]. This implies that the probabi... | https://arxiv.org/abs/2502.19254v1 |
P(zσ(1),...,z σ(n),zn+1) =P(z1,...,z n,zn+1) (1) for each data sequence z1,...,z n+1and each permutation σof{1,...,n} (training-invariant functions were called simply invariant in [8]). We will sometimes refer to the values taken by p-variables as p-values, and our notation for the classes of all randomness and conform... | https://arxiv.org/abs/2502.19254v1 |
measurable functions E:Z→[0,∞is denoted by E. The class EtXofconformal e-variables consists of all functions E∈ EXthat are training-invariant: E(zσ(1),...,z σ(n),zn+1) =E(z1,...,z n,zn+1) (4) for each data sequence z1,...,z n+1and each permutation σof{1,...,n}. We often regardthe randomnesse-variables E∈ ERasrandomness... | https://arxiv.org/abs/2502.19254v1 |
projections, we cannot claim that t hese ways of moving between different function classes are always optimal. Lemma 2 lists the only two cases where the combination of two of our t hree basic operators (i,X, andt) gives something interesting. The other four cases are: (EX)i= (Ei)X= 1,(Ei)t= (Et)i=Ei. 3 Main results Let... | https://arxiv.org/abs/2502.19254v1 |
e−1in Theorem 3 cannot be replaced by a larger one. Proof.In this proof we follow the example in [8, Sect. B.1] (the example in [8] is informal, and here we formalize it). Without loss of generality we ass ume |X|= 1 (so that the objects become uninformative and we can omit them from ournotation)and Y={0,1}(withthedisc... | https://arxiv.org/abs/2502.19254v1 |
n,xn+1,y) EtX(z1,...,z n,xn+1,y)B(dy|zn+1) =/radicalbig G1(z1,...,z n+1)G2(z1,...,z n+1)≤G(z1,...,z n+1) (the existence of G1andG2follows from Theorems 3 and 6, respectively). It is known that, for any δ∈(0,1), the function p/mapsto→δpδ−1transforms p- values to e-values and that the function e/mapsto→e−1transforms e-va... | https://arxiv.org/abs/2502.19254v1 |
n,zn+1) :=e−1/2 |Y|−1/summationdisplay y∈Y\{yn+1}/radicalBigg E(z1,...,z n,xn+1,y) EtX(z1,...,z n,xn+1,y).(17) The interpretation of (17) is that the conformal e-predictor EtXis almost as efficientastheoriginalrandomnesse-predictor Eonaverage;asbefore,efficiency is measured by the degree to which we reject the false labels... | https://arxiv.org/abs/2502.19254v1 |
a predic tion func- tion of the form f(y) =D1{y≥b}, whereb∈Ris the upper prediction limit and D >0 reflects the confidence in this prediction. Proposition 10. LetB:Z֒→Ybe a Markov kernel. Suppose a monotonic randomness e-predictor E, given a training sequence z1,...,z nand test object xn+1, outputs a set prediction f(y) ... | https://arxiv.org/abs/2502.19254v1 |
S. Miettinen. Theoretical Epidemiology: Principles of Occurrence Re- search in Medicine . Wiley, New York, 1985. [7] Glenn Shafer. The language of betting as a strategy for statist ical and scientific communication (with discussion). Journal of the Royal Statistical Society A , 184:407–478, 2021. [8] Vladimir Vovk. Rand... | https://arxiv.org/abs/2502.19254v1 |
Modeling Extreme Events in the Presence of Inlier: A Mixture Approach Shivshankar Nilaa, Ishapathik Dasa,∗, N. Balakrishnaa aDepartment of Mathematics and Statistics Indian Institute of Technology Tirupati India. Abstract In many random phenomena, such as life-testing experiments and environmental data (like rainfall d... | https://arxiv.org/abs/2502.19793v1 |
appropriate threshold—beyond which EVT models can accurately approximate the tail behaviour of the population as given in Scarrott and MacDonald (2012) and Hu and Scarrott (2018). A low threshold leads to a poor approximation of the generalized Pareto distribution (GPD) and biased return level estimates, while a high t... | https://arxiv.org/abs/2502.19793v1 |
for data below a threshold and utilized a GPD to model the tail. Slightly earlier, Do Nascimento et al. (2012) proposed a semi-parametric Bayesian method by employing a mixture of gamma distributions below the threshold. Dirichlet process mixture of gamma densities for bulk proposed by F´ uquene Pati˜ no (2015). The fi... | https://arxiv.org/abs/2502.19793v1 |
factors are often overlooked in single-tail models with positive support or are not thoroughly studied when considering the entire support range. Our study aims to address the following key questions: 1. How does the inclusion of inliers in modeling affect the extreme values in the data when modeling them (i.e., the ex... | https://arxiv.org/abs/2502.19793v1 |
Weibull families can be combined into a single family of models, represented by a distribution function denoted as Hξ(x), and can be written in the following form, Hξ(x) = exp( − 1 +ξx−µ σ−1 ξ) (1) where 1 + ξ x−µ σ >0,µ∈R,σ >0, and ξ∈R. The shape parameter ξgoverns the tail behaviour: ξ= 0 corresponds to the Gumb... | https://arxiv.org/abs/2502.19793v1 |
for extreme values beyond the threshold, called as the “tail model”. The distribution function Fof an EVMM can be generally defined as F(x|φ, θ) = H(x|φ), ifx≤u, H(u|φ) + [1−H(u|φ)]G(x|θ),ifx > u,(3) where handHdenote the density and distribution functions, respectively, parameterized by φ, representing the bul... | https://arxiv.org/abs/2502.19793v1 |
EVIMM framework offers a key advantage by accounting for inliers concentrated at zero, which improves the modeling of extreme values and enhances the overall accuracy and comprehensive- ness of the data analysis. The density plot of the EVIMM for different parameter sets is shown in Figure 2. Figures 2 and 3 show that ... | https://arxiv.org/abs/2502.19793v1 |
likelihood function and then develop the corresponding MLE based on a random sample of size n. Maximum Likelihood Function Letx= (x1, x2, . . . , x n) be a random sample from the population with the CDF Fas defined in equation (4). The likelihood function L(x; Θ) is given by: L(x; Θ) =nY i=1f(xi;α, η, β, u, ξ, σ ), 11 ... | https://arxiv.org/abs/2502.19793v1 |
the confidence interval includes the respective true parameter values. The bias and MSE are calculated using the formula Bias (ˆθ) =1 NNX i=1ˆθ(i)−θ, MSE (ˆθ) =1 NNX i=1 ˆθ(i)−θ2(8) where ˆθdenotes the estimator of the parameter θandˆθ(i)denotes the estimate of θforith data set, where Nis the number of data generated... | https://arxiv.org/abs/2502.19793v1 |
) 94.0 σ 5 5.2111 ( 5.9488 ) 1.2364 (2.2095, 6.9792) 3.0297 ( 9.1987 ) 0.2111 ( 0.9488 ) 92.6 750α 0.2 0.1991 (NA) 0.0159 (0.1747, 0.2358) 2e-04 (NA) -9e-04 (NA) 97.6 η 1 1.017 ( 1.0072 ) 0.075 (1.0228, 1.316) 0.0042 ( 0.0032 ) 0.017 ( 0.0072 ) 95.5 β 5 4.9227 ( 4.9729 ) 0.361 (3.4107, 4.817) 0.1767 ( 0.1558 ) -0.0773 ... | https://arxiv.org/abs/2502.19793v1 |
1.0092 ) 0.091 (0.8827, 1.2401) 0.0078 ( 0.0066 ) 0.0216 ( 0.0092 ) 96.2 β 5 4.8767 ( 4.9928 ) 0.584 (3.7188, 6) 0.3778 ( 0.3369 ) -0.1233 ( -0.0072 ) 92.9 u 8.9587 8.7006 ( 11.443 ) 1.1931 (6.0534, 10.8587) 1.0995 ( 10.6307 ) -0.2582 ( 2.4842 ) 94.0 ξ 0.2 0.1511 ( 0.1364 ) 0.2406 (-0.3053, 0.6644) 0.0491 ( 0.2063 ) -0... | https://arxiv.org/abs/2502.19793v1 |
11.1679 ) 0.9994 (8.3798, 12.3087) 1.1074 ( 3.5818 ) 0.1096 ( 0.7706 ) 96.0 ξ -0.2 -0.2699 ( -0.2719 ) 0.2269 (-0.9052, 0.043) 0.0606 ( 0.1034 ) -0.0699 ( -0.0719 ) 94.3 σ 5 5.3348 ( 5.3483 ) 1.7983 (3.1906, 10.4964) 2.9058 ( 5.1131 ) 0.3348 ( 0.3483 ) 94.2 500α 0.2 0.1988 (NA) 0.0192 (0.1691, 0.2441) 4e-04 (NA) -0.001... | https://arxiv.org/abs/2502.19793v1 |
5.1836 ) 1.593 (1.6459, 7.612) 4.3261 ( 8.1771 ) 0.4877 ( 0.1836 ) 92.9 400α 0.4 0.3999 (NA) 0.0257 (0.3832, 0.4845) 6e-04 (NA) -1e-04 (NA) 96.8 η 1 1.0257 ( 1.0147 ) 0.1087 (0.9316, 1.3575) 0.0096 ( 0.0082 ) 0.0257 ( 0.0147 ) 96.0 β 5 4.8655 ( 4.9593 ) 0.6017 (3.2613, 5.6381) 0.4265 ( 0.3881 ) -0.1345 ( -0.0407 ) 90.7... | https://arxiv.org/abs/2502.19793v1 |
10.3972 10.3537 ( 11.2879 ) 1.0186 (7.9238, 12.0314) 1.15 ( 4.8305 ) -0.0435 ( 0.8907 ) 96.3 ξ 0 -0.084 ( -0.1079 ) 0.2906 (-0.8132, 0.3704) 0.1044 ( 0.2356 ) -0.084 ( -0.1079 ) 93.5 σ 5 5.5427 ( 6.0494 ) 2.2795 (2.5864, 11.5353) 5.2195 ( 11.512 ) 0.5427 ( 1.0494 ) 93.4 400α 0.2 0.1997 (NA) 0.0211 (0.1648, 0.246) 4e-04... | https://arxiv.org/abs/2502.19793v1 |
and convergence probability (CP). 19 Sample Size Parameters True Value Sample Mean BSE BCI MSE Bias CP 300α 0.4 0.3997 (NA) 0.0293 (0.3584, 0.4739) 8e-04 (NA) -3e-04 (NA) 97.3 η 1 1.0313 ( 1.0193 ) 0.1161 (0.8539, 1.309) 0.0129 ( 0.0116 ) 0.0313 ( 0.0193 ) 96.8 β 5 4.836 ( 4.9564 ) 0.7138 (3.4024, 6.2097) 0.5518 ( 0.57... | https://arxiv.org/abs/2502.19793v1 |
( 5.4092 ) 0.9218 (3.0288, 6.4961) 1.0254 ( 3.7208 ) 0.1712 ( 0.4092 ) 94.5 Table 6: Simulation results based on N= 2000 replications for sample sizes n= 300 ,400,500,750,and 1000, with an inlier proportion of 40% and exponential-tailed generalized Pareto distributions (GPD). The table presents the sample mean of estim... | https://arxiv.org/abs/2502.19793v1 |
9.7295, σ= 5 and ξ= 0.2,−0.2 with sample size n= 2000. 22 0 20 40 60 80 10010 20 30 40 50 60Return Level Plot Return Period (years)Return LevelTrue EVIMM EVMM 95% CI(a) Return level estimates for a heavy-tailed GPD ( ξ= 0.2). 0 20 40 60 80 10012 14 16 18 20 22 24 26Return Level Plot Return Period (years)Return LevelTru... | https://arxiv.org/abs/2502.19793v1 |
Plot N/A N/A N/A 220 76.000 -0.0720 (N/A) (N/A) EVM Model (Bulk) N/A 1.0106 118.4097 173.9996 67.4980 0.0040 (0.0017) (0.2490) (0.6272) (0.4114) (0.0045) EVM Model (Para.) N/A 0.9816 145.6462 258.7006 72.1900 -0.0599 (0.1387) (1.0025) (0.3989) (0.4510) (0.0184) EVIM Model 0.2130 1.0426 110.9591 218.8999 58.6405 0.1588 ... | https://arxiv.org/abs/2502.19793v1 |
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