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on ΣandΘ. As demonstrated in our proof, Equation (3.11) is a crucial condition for the concentration inequalities in Lemma B.26 of Bai and Silverstein (2010) to hold, thereby making the limiting behavior of the estimators traceable. For con- crete estimators bβR(θ) andbβM(Θ), we have shown that such condition holds for... | https://arxiv.org/abs/2503.12155v1 |
developing advanced methods to im- prove the prediction of complex traits and diseases in non-European populations (Zhang et al., 2023, 2024; Jin et al., 2024). Many proposed methods improve predictive power by integrating summary- level data from diverse populations to develop ancestry-specific models tailored to each... | https://arxiv.org/abs/2503.12155v1 |
data XT jyj∈Rp,WT jWj, for 1≤j≤K, and hyperparameter Θ ={ωj,Θj}K j=1. forj←1 toKdo hj←N (0,Ip), //Sample p-dimension standard Gaussian random variable. s(tr) j←n(tr) nXT jyj+q n(tr)(n−n(tr)) n2 Cov( XT jyj)1/2hj, //Construct s(tr) jfor training. end for s(v) 1←XT 1y1−s(tr) 1, //Construct s(v) 1for validation in the pop... | https://arxiv.org/abs/2503.12155v1 |
data from the target population. These findings highlight that cross-ancestry genetic correlation (Brown et al., 2016; Xue and Zhao, 2023) plays a crucial role in determining whether incorporating multi-ancestry data, ei- ther through resampling-based self-training or individual-level data training, can improve predict... | https://arxiv.org/abs/2503.12155v1 |
hyperparameter values, respectively. Importantly, the besting-performing tuning parameters, θ∗ sum,Randθ∗ ind,R, which maximize R2 sum,R(θ) and R2 ind,R(θ), respectively, are well-aligned. A similar phenomenon is observed between θ∗ sum,Mandθ∗ ind,M. In addition, the right panels of Figures 1 and 2 compare prediction a... | https://arxiv.org/abs/2503.12155v1 |
approach by using Algorithm 2, denoted as Ensemble-pseudo, which combines PRS models trained using di fferent methods via a linear combination strategy (Jin et al., 2025). The hyperparameter for these resampling-based methods, implemented through Algorithms 1 and 2, is selected using the resampling-based pseudo-validat... | https://arxiv.org/abs/2503.12155v1 |
validation samples, respectively, while LDpred2-pseudo has a prediction accuracy of 2 .8%. Overall, our real data analysis provides valuable insights into the relative prediction accuracy of different estimators and algorithms for DXA imaging data. Among 71 DXA traits, LDpred2-pseudo outperforms Lassosum2-pseudo, while... | https://arxiv.org/abs/2503.12155v1 |
in fields that utilize shared summary data. As the first statistical framework to examine the properties of resampling-based self-training, our study has a few limitations. First, we focus on linear estimators and provide two concrete examples commonly used in genetic and dense-signal predictions (Choi et al., 2020; Ge... | https://arxiv.org/abs/2503.12155v1 |
dimensional random matrices , volume 20. Springer. Bonomi, L., Huang, Y ., and Ohno-Machado, L. (2020). Privacy challenges and research opportuni- ties for genomic data sharing. Nature Genetics , 52(7):646–654. Boyle, E. A., Li, Y . I., and Pritchard, J. K. (2017). An expanded view of complex traits: from polygenic to ... | https://arxiv.org/abs/2503.12155v1 |
Auton, A., Babalola, E., Bell, R. K., et al. (2024). Mussel: Enhanced bayesian polygenic risk prediction leveraging information across multiple ancestry groups. Cell Genomics , 4(4). Kachuri, L., Chatterjee, N., Hirbo, J., Schaid, D. J., Martin, I., Kullo, I. J., Kenny, E. E., Pasaniuc, B., in Diverse Populations (PRIM... | https://arxiv.org/abs/2503.12155v1 |
Visscher, M., O’Donovan, C., Sullivan, F., Sklar, P., Ruderfer, M., McQuillin, A., Morris, W., et al. (2009). Common polygenic variation contributes to risk of schizophrenia and bipolar disorder. Nature , 460(7256):748–752. Qian, J., Tanigawa, Y ., Du, W., Aguirre, M., Chang, C., Tibshirani, R., Rivas, M. A., and Hasti... | https://arxiv.org/abs/2503.12155v1 |
trans-ancestry genetic correlation with unbal- anced data resources. Journal of the American Statistical Association , 119(546):839–850. Zhao, B., Zheng, S., and Zhu, H. (2024b). On blockwise and reference panel-based estimators for genetic data prediction in high dimensions. The Annals of Statistics , 52(3):948–965. Z... | https://arxiv.org/abs/2503.12155v1 |
a random matrix satisfying Conditions 1 and 2 and Abe a p×pnonnegative definite matrix. Then, with probability one, for eachθ∈R+, asn,p→∞ proportionally, we have A+bΣn+θIp−1≍ A+τn(θ)Σ+θIp−1, (S.2.1) whereτn(θ) is defined as the solution in C+to the fixed point equation τn(θ)−1=1+1 ntraceh Σ(A+τn(θ)Σ+θIp)−1i . When... | https://arxiv.org/abs/2503.12155v1 |
S.9 in Fu et al. (2024), which computes the closed-form of a more general trace functional. Lemma S.2.4 (Lemma S.9 in Fu et al. (2024)) .For any deterministic symmetric matrices P,Q,R∈ Rp×p, for each entry i,j, we have E PΣ−1/2bΣnΣ−1/2QΣ−1/2bΣnΣ−1/2R i,j =1 nn E x4 0−3 Pdiag( Q)R i,j+(n+1) P Q R i,j+Tr(Q) P R i,... | https://arxiv.org/abs/2503.12155v1 |
nw·σ2 ϵ·1 ntrace τnw(θ)Σ+θIp−1bΣn +op(1). It follows that Ω(1) sum= Ω(2) sum+ Ω(3) sum−Ω(4) sum+op(1) =n(tr) nw·κσ2 β p·trace τnw(θ)Σ+θIp−1bΣ2 n +n(tr) nw·σ2 ϵ·1 ntrace τnw(θ)Σ+θIp−1bΣn −n(tr) nw·κσ2 β p·trace τnw(θ)Σ+θIp−1bΣ2 n +2·n(tr) nw·κσ2 β p·trace τnw(θ)Σ+θIp−1bΣnΣ −n(tr) nw·κσ2 β p·trace τ... | https://arxiv.org/abs/2503.12155v1 |
n(tr) τnw(θ)Σ+θIp−1X(v)TX(v) n(v)# +op(1). Based on the fact that bΣn(tr)=X(tr)TX(tr)/n(tr)≍ΣandbΣn(v)=X(v)TX(v)/n(v)≍Σ, we have Ω(1) ind=n(tr) nw·κσ2 β p·trace τnw(θ)Σ+θIp−1Σ2 +op(1). This demonstrates the limit of Ω(1) ind. For Ω(2) ind, we have Ω(2) ind= X(tr)Ty(tr)T WTW+θnwIp−1Σ WTW+θnwIp−1X(tr)Ty(tr) =... | https://arxiv.org/abs/2503.12155v1 |
s(v),D(Θ)s(tr)E +op(1)=n(tr) nw·κσ2 β p·trace D(Θ)Σ2 +op(1). 40 ForΨ(2) sum, we have Ψ(2) sum=n(tr) nXTy+r n(tr)(n−n(tr)) n2(Cov( XTy))1/2hT A WTW,ΘΣA WTW,Θn(tr) nXTy+r n(tr)(n−n(tr)) n2(Cov( XTy))1/2h =1 n2w·n(tr) nXTy+r n(tr)(n−n(tr)) n2(Cov( XTy))1/2hT E(Θ)n(tr) n... | https://arxiv.org/abs/2503.12155v1 |
sumas follows Λ(4) sum=KX j=2ωjn(tr) nw·κ1κjσ2 1,j p·trace Σ1DjΣj . It follows that Λ(1) sum= Λ(3) sum+ Λ(4) sum =ω1·n(tr) nw·κ1σ2 1,1 p·trace D1Σ2 1 +KX j=2ωjn(tr) nw·κ1κjσ2 1,j p·trace Σ1DjΣj . ForΛ(2) sum, we have Λ(2) sum= KX j=1ωjAj(WT jWj,Θj)s(tr) j 2 Σ1 =X 1≤i,j≤Kωiωjn(tr) nXT iyi+r n(tr)(n−n(tr)) n... | https://arxiv.org/abs/2503.12155v1 |
EjΣ2 j +n(tr) n2w·σ2 ϵ·trace EjΣj . Therefore, we have Λ(2) ind=X 1≤i<j≤K2ωiωj n(tr) nw!2κiκjσ2 i,j p·trace ΣiDiΣ1DjΣj +X 1≤j≤Kω2 j·n(tr) n2w·κjσ2 j,j p·1 h2 jtrace Σj ·trace EjΣj + n(tr) nw!2 ·κjσ2 j,j ptrace EjΣ2 j+op(1). It follows that R2 ind,MA(θ)=n(v) ∥y(v)∥2 2· Λ(1) ind2 Λ(2) ind, wh... | https://arxiv.org/abs/2503.12155v1 |
0.0012 0.0007 0.0005 21133 0.1647 0.0025 0.0013 0.0010 0.0005 0.0005 0.0004 21122 0.3780 0.0128 0.0213 0.0178 0.0179 0.0146 0.0099 21134 0.3471 0.0099 0.0187 0.0145 0.0149 0.0129 0.0082 21135 0.0778 0.0011 0.0005 0.0026 0.0013 0.0012 0.0013 23244 0.3277 0.0153 0.0316 0.0277 0.0271 0.0252 0.0166 23245 0.1669 0.0006 0.00... | https://arxiv.org/abs/2503.12155v1 |
ASYMPTOTIC EXPANSIONS OF GAUSSIAN AND LAGUERRE ENSEMBLES AT THE SOFT EDGE II: LEVEL DENSITIES FOLKMAR BORNEMANN Abstract. We continue our work [ 8] on asymptotic expansions at the soft edge for the classical n-dimensional Gaussian and Laguerre random matrix ensembles. By revisiting the construction of the associated sk... | https://arxiv.org/abs/2503.12644v1 |
dimension with accordingly adjusted indices. Still,we presentedithere in the form (1.2)to show the striking duality between the orthogonal and symplectic cases; a probabilistic proof of that relation is given in Appendix A. 1.2.Scaling limits at the soft edge. It is known that the largest level (the so-called soft edge... | https://arxiv.org/abs/2503.12644v1 |
1, n, β = 2, 2n+1 2, β = 4, noting that the left hand sides of (1.2) are then subject to exactly the same modification, (1.4b) (2n+ 1)′ β=1=n′ β=4. In Section 4, we infer from the kernel expansions established in [ 8, Lemma 2.1/3.1] an asymptotic expansion of the form (1.5a) ρ2,n(x)dx ds x=µn+σns=mX j=0ω2,j(s;τn)hj n+h... | https://arxiv.org/abs/2503.12644v1 |
P β,1,2(β= 1,2,4), Pβ,2,1, P β,2,2, P β,2,3, P β,2,4(β= 1,4), from ˜p1,˜q1,˜r1andpj, qj, rj, uj, vj(j= 1,2). The agreementwiththose polynomials calculated from the more general theory provides further compelling evidence supporting the conjectures underlying the cases β= 1,4in [7], [8]: only the final form (1.7)of the ... | https://arxiv.org/abs/2503.12644v1 |
related to β= 4. Hence, the hidden parameter in (2.3)isq= 2(p−n). According to (2.2), here and in what follows, the Laguerre weight functions associated with a hidden parameter qare given by5 (2.4a) w1(x) =x(q−1)/2e−x/2,p w2(x) =xq/2e−x/2,p w4(x) =x(q+1)/2e−x/2. In particular, we have√w4=w2/w1, which is exactly the con... | https://arxiv.org/abs/2503.12644v1 |
real parameter q. By construction, the functions ϕn/√w2are the classical orthonormal polynomials of degree nassociated with the weight w2displayed in (2.4). Remark 2.1.Rescaling the three-term-recurrences [ 5, Eqs. (6.1.10/6.2.5)] of the Hermite and Laguerre polynomials yields, for n= 0,1,2, . . ., that xϕn,∞=rn 2ϕn−1,... | https://arxiv.org/abs/2503.12644v1 |
in the Hermite caseZ Ωψ0(x)dx=1√ 2πZ∞ −∞e−x2/2dx= 1, and, in the Laguerre case with hidden parameter q >−1,Z Ωψ0(x)dx=1 Γ((q+ 1)/2)Z∞ 0e−tt(q+1)/2−1dt= 1, where the Euler integral is obtained by substituting x= 2t. □ Remark 2.2.Alternatively, the integrals in Corollary 2.1 can be established head-on. Using symmetry, th... | https://arxiv.org/abs/2503.12644v1 |
even and odd dimension for the orthogonal ensembles; in fact, the expression of the level density can be brought to a form that holds independently of the parity of n. For the bi-othonormal wave functions ψndefined in (2.11), we consider the particular antiderivative Ψnwith constant of integration fixed by taking a val... | https://arxiv.org/abs/2503.12644v1 |
(nodd). Even though the formulae for the even cases (3.4a)and the odd ones (3.4b)look superficially different, they can easily be brought to the same form: by paying attention to (3.1), we obtain, for each parity of n, (3.5) ρ1,n=ρ2,n+ψn−1· 1 + Ψ♯ n =ρ2,n−1+ψn−1· 1 + Ψ♯ n−2 . 10Note that there is a weight factor mi... | https://arxiv.org/abs/2503.12644v1 |
gives an expansion of 1 2 ρ2,n−1(x) +ρ2,n(x)dx ds x=µn′+σn′s in powers of hn′since the odd powers of h1/2 n′that appear when individually re-expanding ρ2,n−1(x)dx/dsandρ2,n(x)dx/dsmust cancel by symmetry. Similarly, by Theorem B.1, Ψ♯ n−2 expands using the parameters indexed by n−3/2andΨ♯ nusing those indexed by n+ 1... | https://arxiv.org/abs/2503.12644v1 |
for Theorem 4.2. □ Remark 4.1.Comparing the expansion coefficients in Theorems 4.1 and 4.2 with those in Theorems 4.3 and 4.4, we note that17 ˜pj(t) = ˜pj(t; 0),˜qj(t) = ˜qj(t; 0),˜rj(t) = ˜rj(t; 0), pj(t) =pj(t; 0), q j(t) =qj(t; 0), r j(t) =rj(t; 0), u j(t) =uj(t; 0), v j(t) =vj(t; 0). This is consistent with the sca... | https://arxiv.org/abs/2503.12644v1 |
, ω′′ 0=1 2 sAi AI−Ai Ai′ . Thus, the equation becomes (5.3) pAi2+qAi′2+rAi Ai′+uAi AI + vAi′AI =−1 2 P1+ 2sP′ 1+P′ 2 Ai2+P′ 1Ai′2−1 2P2Ai Ai′+1 2 P′ 1+sP2 Ai AI +1 2 P1+P′ 2 Ai′AI. By using tools from the Siegel–Shidlovskii theory of transcendental numbers and the differ- ential Galois theory, we give in [ 6, ... | https://arxiv.org/abs/2503.12644v1 |
x n) =1 n! n det j,k=1ψj−1(xk) . Here, the factor 1/n!comes from rescaling the Selberg integrals; see, e.g., [ 2, Eqs. (2.5.10/11)]. Theorem A.1. The system (2.11)of wave functions can be represented in the form d dxE1 2sgn(x−x1)···sgn(x−xn) =ψn−1(x), where x1, . . . , x nare the levels of the orthogonal ensemble. Pr... | https://arxiv.org/abs/2503.12644v1 |
uniformly for sbounded from below, with certain pk,qk∈Q[s]; the first of which are p1(s) =−s 5,q1(s) =s2 5,p2(s) =s5 50+9s2 70,q2(s) =−3s3 35−9 35. If we denote the antiderivatives of ψ♯ n,∞andAithat vanish at x=∞byΨ♯ n,∞andAI0, the asymptotic expansion integrates to (B.3) Ψ♯ n,∞(x) x=µ+σs= AI 0(s) +mX k=1 Pk(s) Ai(s)... | https://arxiv.org/abs/2503.12644v1 |
τbeing bounded away from zero (that is, for τ0⩽τ⩽1given any fixed τ0>0), applying the scaling limits, which are for fixed n, falls short of a proof. Appendix C.The Laguerre-to-Hermite Scaling Limit Using the scaling and expansion parameters (4.1)and(4.4), we easily get from (B.1)that lim p→∞σ1/2 n,p γpµn,p+σn,ps 2−1/... | https://arxiv.org/abs/2503.12644v1 |
symplectic-unitary transitions at the hard and soft edges. Nuclear Phys. B 553(3), 601–643 (1999) 19For the LOEwith integer pthis transition law is implied by the multivariate central limit theorem: namely, if a matrix Xn,pis drawn from the standard n-variate Wishart distribution with pdegrees of freedom, one gets in d... | https://arxiv.org/abs/2503.12644v1 |
arXiv:2503.12808v2 [stat.ML] 18 Mar 2025Estimating stationary mass, frequency by frequency Milind Nakul⋆, Vidya Muthukumar†,⋆, Ashwin Pananjady⋆,† H. Milton Stewart School of Industrial and Systems Engineer ing⋆ School of Electrical and Computer Engineering† Georgia Institute of Technology March 19, 2025 Abstract Suppo... | https://arxiv.org/abs/2503.12808v2 |
small ζ(i.e., to estimate the probability mass placed on small-frequency symbols) and the plug-in or empirical esti mator to approximate Mπ ζfor large ζ (i.e., to estimate the probability mass placed on large-fre quency symbols). In particular, the prob- lem of estimating the probability vector ( Mπ ζ)n ζ=0has been stu... | https://arxiv.org/abs/2503.12808v2 |
shows that estimation can always be performed consistently in n. We show that such an estimator then naturally yields an esti mator for the stationary distribution πdefined over the sample space X(see Lemma 1). Contributions and organization Our main contributions are summarized below: •Weproposeasimpleandefficientestimat... | https://arxiv.org/abs/2503.12808v2 |
that f(u)≤C·g(u) for all uin the domain offandg. We use the notation f(u)/greaterorsimilarg(u) wheng(u)/lessorsimilarf(u). We write f(u)≍g(u) if both relations f(u)/greaterorsimilarg(u) andg(u)/lessorsimilarf(u) hold. Logarithms are taken to the base e. We use ( c,C) to denote universal positive constants that could be... | https://arxiv.org/abs/2503.12808v2 |
which for each input x, outputs the duplicated sequence ( x,...,x/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright ktimes) with probability α, andxotherwise. The output process is ergodic and satisfies exponential α-mixing, and the mixing time satisfies tmix(ǫ) =kfor allǫ >0. 2.2 Frequency-by-frequen... | https://arxiv.org/abs/2503.12808v2 |
As an aside, we note that such a com petitive relation was proved for the KL-divergence by Orlitsky and Suresh (2015), who showed an approximation constant of 1 instead ofas above. AcorrespondingresultfortheTVdistancehasno t appearedbefore(to ourknowledge). 3 Methodology We now turn our attention to the estimation prob... | https://arxiv.org/abs/2503.12808v2 |
and we make a specificchoice of ζin stating our theorem to follow. When ζandτare clear from context, we often use the shorthand /hatwiderMζ:=/hatwiderMζ(τ;ζ) to denote the scalar estimate and /hatwiderM:= (/hatwiderMζ(τ;ζ))n ζ=0to denote its vector counterpart. 4 Main results Our main result is a bound on the TV error a... | https://arxiv.org/abs/2503.12808v2 |
a sequence of random variables from an ergodic, stochastic process that is exponentially α-mixing with parameters µandρ, with each Uj∈[0,B]almost surely. Recall the definition of mixing time from Eq. (2). (a) For each fixed τ≥tmix(ǫ/n), the following Bernstein-type inequality holds: /vextendsingle/vextendsingle/vextendsi... | https://arxiv.org/abs/2503.12808v2 |
If the window size is chosen such that τ≥tmix(n−2), then we have E/bracketleftBig/vextendsingle/vextendsingle/vextendsingleMπ ζ(Xn)−/hatwiderMWingIt,ζ/vextendsingle/vextendsingle/vextendsingle/bracketrightBig ≤C/radicalbiggτ n/parenleftBigg/radicalBig E[Mπ ζ]+/radicalBig ζlog(2τ)E[Mπ ζ]+/radicaltp/radicalvertex/radical... | https://arxiv.org/abs/2503.12808v2 |
which in turn are proved in Sections 5.4and5.5. Denote/hatwiderMuas theunnormalized estimator, i.e. /hatwiderMu:=/hatwiderM(τ;ζ)·ν, with /hatwiderMu ζ=/braceleftBigg/hatwiderMWingIt,ζ(τ) ifζ≤ζ /hatwiderMPI,ζ ifζ >ζ, We have dTV(Mπ(Xn),/hatwiderM(τ;ζ))(i) ≤ /⌊ard⌊lMπ−/hatwiderMu/⌊ard⌊l1 =ζ/summationdisplay ζ=0/vextendsi... | https://arxiv.org/abs/2503.12808v2 |
claim (16):First, note that n=/summationtextn ζ=0ζϕζ. Part (a) of the inequality then follows by noting that n≥n/summationdisplay ζ=ζ+1ζϕζ≥(ζ+1)n/summationdisplay ζ=ζ+1ϕζ. (18) To prove part (b), we use the Cauchy–Schwarz inequality to ob tain n/summationdisplay ζ+1ϕζ/radicalbig ζ=n/summationdisplay ζ=ζ+1√ϕζ/radicalbig... | https://arxiv.org/abs/2503.12808v2 |
By construction of Yn, the random variables {Jk}n0 k=1 are also independent and identically distributed. Moreove r, because the process Ynis initialized at the stationary distribution π, we have E[Y1] =E[J1]. Finally, recall that we defined v2:= 1 τ2var(/summationtextτ i=1Ui) in the statement of the lemma. It is easy to... | https://arxiv.org/abs/2503.12808v2 |
sequence4Yn, we have /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleE[Y1]−1 nn0/summationdisplay k=1/summationdisplay j∈SkYj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤2/radicalBig τBlog(2/δ)/summationtextn0 k=1/summationtext j∈SkYj n−2τ+7τBl... | https://arxiv.org/abs/2503.12808v2 |
for completeness. Working now on the intersection of the high-probability eve nts in (30) and (31), which in turn occurs with probability at least 1 −11δ−3ǫ, and considering ζ≥max/braceleftbigg 36τ0log(2n/δ),1+/radicalbig (4+8τ0)log(n/δ)+4τ0log(n/δ) 3/bracerightbigg , 20 we have /summationdisplay x∈X≥ζI{Nx(Xn) =ζ}/vext... | https://arxiv.org/abs/2503.12808v2 |
Proof of Proposition 2 In this section we bound the error of the WingIt estimator for small frequencies. We first derive bounds on the mean squared error ( MSE) for each ζ, and then use these to obtain bounds on the ℓ1 error. Throughout this proof, note that τshould be thought of as an arbitrary integer that divides n, ... | https://arxiv.org/abs/2503.12808v2 |
j+1 n2 0n0/summationdisplay j=1n0/summationdisplay k=1 k/\e}atio\slash=jZjZk. 24 We first bound the cross terms in T2. We define the following random variables for each j,k∈[n0] withj/\e}atio\slash=kand conditioned on a random variable Y∼π: Qj,k:=I/braceleftbig NY(XHj∩Hk) =ζ/bracerightbig −I{NY(XHk) =ζ}. (41) Observe tha... | https://arxiv.org/abs/2503.12808v2 |
It remains to prove Lemmas 4,5and6, which we do next. 5.5.1 Proof of Lemma 4 The proof of this lemma is analogous to the proof of Lemma 1 in Pananjady et al. (2024). Define, for convenience, Qj,k:=EY[Qj,k] =EY/bracketleftbig I/braceleftbig NY(XHj∩Hk) =ζ/bracerightbig −I{NY(XHk) =ζ}/bracketrightbig . Note that by auxilia... | https://arxiv.org/abs/2503.12808v2 |
we have RY=n0/summationdisplay j=1n0/summationdisplay k=1 k/\e}atio\slash=jI/braceleftbig Y∈XBj/bracerightbig ·I{NY(XHk) =ζ} /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright A1+n0/summationdisplay j=1n0/summationdisplay k=1 k/\e}atio\slash=jI/braceleftbig Y∈XBj/bracerightbig ·I/braceleftbig NY(X... | https://arxiv.org/abs/2503.12808v2 |
NY(XBj∪XBk) =u/bracerightbig , 31 where step ( i) follows from the union bound and step ( ii) follows by introducing the summation overuwhich counts the number of occurrences of YinXBj∪XBk. Consequently, from the above statement, we have A2≤2n0/summationdisplay j=1n0/summationdisplay k=1 k/\e}atio\slash=j4τ−2/summation... | https://arxiv.org/abs/2503.12808v2 |
long-range depe ndence. Statistical science , pages 404–416, 1992. P. Chandra and A. Thangaraj. Missing mass under random dupli cations. In 2024 IEEE Interna- tional Symposium on Information Theory (ISIT) , pages 522–526. IEEE, 2024. P. Chandra, A. Thangaraj, and N. Rajaraman. How good is Good– Turing for Markov sample... | https://arxiv.org/abs/2503.12808v2 |
, 7(1):41–153, 2024. H. Ney, U. Essen, and R. Kneser. On structuring probabilisti c dependences in stochastic language modelling. Computer Speech & Language , 8(1):1–38, 1994. M. I. Ohannessian and M. A. Dahleh. Rare probability estimat ion under regularly varying heavy tails. In Conference on Learning Theory , pages 2... | https://arxiv.org/abs/2503.12808v2 |
{−1,0,1} andI/braceleftbig x∈XBj/bracerightbig ·I/braceleftbig Nx(XHj∩Hk) =ζ/bracerightbig ∈ {0,1}. Let us thus analyze the case when the equality is violated and show that this cannot happen. Note that Qj,k= 1 implies Nx(XHj∩Hk) =ζandNx(XHk)/\e}atio\slash=ζ. We can write the set XHkas the union of the disjoint sets XH... | https://arxiv.org/abs/2503.12808v2 |
dTV(Zi1,i2,Z′ i1,i2)≤dTV(Zi1,i2,/tildewideZ)+dTV(/tildewideZ,Z′ i1,i2)(i) ≤6µ·ρτ, where step ( i) follows from Lemma 11. This completes the proof of the lemma. Lemma 13. Letn0=n/τ. For each j∈[n0], define the stochastic process (X′ k)k∈Djτas a|Djτ|- length sequence and initial state sampled from the distribu tionπand in... | https://arxiv.org/abs/2503.12808v2 |
arXiv:2503.12863v1 [math.PR] 17 Mar 2025Parameter Estimation for Generalized Mixed Fractional Stochastic Heat Equation B.L.S. PRAKASA RAO CR Rao Advanced Institute of Mathematics, Statistics and Computer Science, Hyderabad, India Abstract: We study the properties of a stochastic heat equation with a g eneralized mixed ... | https://arxiv.org/abs/2503.12863v1 |
the multivariate process {(u(t1,x),...,u(tn,x)),x∈R} is a centered stationary Gaussian process. (iii) The variance of u(t,x) is given by Var(u(t,x)) =E[u(t,x)]2=σ2 1vt(H1)+σ2 2vt(H2),t >0,x∈R (2. 3) where vt(H) =cHtH+1andcH=2H+1(2H−1)Γ(H+1 2)√π(H+1). (2. 4) (iv)Fort,s∈[0,T] andx >0,the covariance function admits the fo... | https://arxiv.org/abs/2503.12863v1 |
can be shown that the function f:R→(0,∞) is strictly increasing function in Hfollowing arguments in Avetisian and Ralchenko (2023) and hence has an inverse f−1.We define the estimator ˆH1Nof the parameter H1by the equation ˆH1N=f−1(t−(H2+1) 2VN(t2)−t−(H2+1) 1VN(t1) t−(H2+1) 3VN(t3)−t−(H2+1) 1VN(t1)) (3. 5) which will be... | https://arxiv.org/abs/2503.12863v1 |
Computer Science, Hyder- abad 500046, India. References: Arcones, M. (1994) Limit theorems for nonlinear functional s of a stationary Gaussian se- quences of vectors, Ann. Probab. ,22, 2242-2274. Avetisian, D. and Ralchenko, K. (2020) Ergodic properties o f the solution to a fractional stochastic heat equation with an ... | https://arxiv.org/abs/2503.12863v1 |
ON A CONJECTURE OF ROVERATO REGARDING G-WISHART NORMALISING CONSTANTS CHING WONG, GIUSI MOFFA, AND JACK KUIPERS Abstract. The evaluation of G-Wishart normalising constants is a core component for Bayesian analyses for Gaussian graphical models, but remains a computationally intensive task in general. Based on empirical... | https://arxiv.org/abs/2503.13046v1 |
and updat- ing the precision matrix adds considerable computational overhead. It would therefore be desirable to derive a formula for CG(δ, D) based on CG(δ, In), and in 2002, Roverato [1] studied the function hG(δ, D) := 2n 2det(Iss G(DG))−1 2det(DG)−δ−2 2CG(δ, D)−1, where DGis the PD-completion of Dwith respect to G,... | https://arxiv.org/abs/2503.13046v1 |
is Iss G(DG). Then, the determinant of Iss( DG) is 2ndet(DG)n+1= det(Iss( DG)) = det(Iss G(DG)) det (Iss(DG))−1 n+m+ 1, n+n 2−1 = det(Iss G(DG)) det Iss((DG)−1) n+m+ 1, n+n 2−1 . 2 where the first step follows from the block determinant involving the Schur complement, which is expressed in terms of a block ... | https://arxiv.org/abs/2503.13046v1 |
cycle of length 4 and Gbe the path of length 3. For both graphs, exact formulae for the normalising constants (when Dis the identity matrix) are known [7]: CG(δ, I4) = 22δ+3π3 2Γδ+ 1 23 Γδ 2 , CG∗(δ, I4) = 22δ+4π2Γ δ 2 Γ δ+1 2 Γ δ+2 23 Γ δ+3 2 . Therefore, the correct ratio of the normalising constants is (4)CG... | https://arxiv.org/abs/2503.13046v1 |
does not hold for all graphs. Instead we showed that the conjecture implies an approximation that had previously been employed in Bayesian samplers to speed up inference when computing the ratio of normalising constants with identity scale matrices [6]. In Bayesian inference the parameter of the normalising constant is... | https://arxiv.org/abs/2503.13046v1 |
Spearman’s rho for bivariate zero-inflated data Jasper Arendsa, Elisa Perrone1a aDepartment of Mathematics and Computer Science, Eindhoven University of Technology, Groene Loper 5, 5612 AZ, Eindhoven, The Netherlands Abstract Quantifying the association between two random variables is crucial in applications. Tradition... | https://arxiv.org/abs/2503.13148v1 |
bias. In [6], the authors derived the lower and upper bounds of the newly introduced estimator, making its interpretation possible. A further step has been taken in [24] where the work of [27] was extended to zero-inflated count data, for 1Corresponding author, email e.perrone@tue.nl Preprint submitted to arXiv March 1... | https://arxiv.org/abs/2503.13148v1 |
cases where XandYare zero or strictly positive. To that end, we consider the probabilities p1=P[X= 0], p2=P[Y= 0], p11=P[X > 0, Y > 0],p10=P[X > 0, Y= 0], p01=P[X= 0, Y > 0] and p00=P[X=Y= 0]. Moreover, let X10(X11) be a positive random variable with the distribution of X, given that Y= 0 ( Y > 0). Similarly, let Y01(Y... | https://arxiv.org/abs/2503.13148v1 |
ρSPis Spearman’s rho away from zero. The estimator amounts to substituting each component by its relative frequency and ρSPby the standard estimator for ρSon the observations that are strictly positive. However, we believe that the derivations that led to Eq. (4) in [26] are incorrect resulting in a substantial bias of... | https://arxiv.org/abs/2503.13148v1 |
respectively, a simple rewrite shows that ρS11=ρS10=ρS01=ρS00= 0 and therefore ρA= 0. An estimator of ρAcan be derived by replacing the probabilities in Eq. (5) with their respective relative frequencies. Moreover, ρS11, that is, ρSaway from zero, is estimated using the standard rank-based estimator ˆρS, where tied obs... | https://arxiv.org/abs/2503.13148v1 |
any joint distribution can be expressed as a copula Cthat takes as arguments the marginal distributions of the random vector, that is P[X≤x, Y≤y] =C(F(x), G(y)). Going back to the most (least) concordance random vectors, we can construct them through the copulas MandWthat satisfy W(F(x), G(y))≤P[X≤x, Y≤y]≤M(F(x), G(y))... | https://arxiv.org/abs/2503.13148v1 |
that ˜ sis the point that is attainable by X10andX11. However, in the case where F(˜s) = 1, and therefore there is no such ˜ u∈VG, we simply set G(˜u) =G(˜u−) = 1. Similarly for p1≥p2, let˜t∈VG 5 Figure 1: Comparison of our derived bounds (red dashed line) to those in Eq. (7) (black solid line) for πG= 0.5 fixed. and ˜... | https://arxiv.org/abs/2503.13148v1 |
the estimators of ρS, i.e., ˆ ρMand ˆρA, for Npairs generated from two random variables joined through the Fr´ echet copula Cα(u, v) = (1 −α)uv+αmin(u, v), where u, v, α ∈[0,1] [22]. We compare the derived estimates ˆ ρMand ˆρAwith the true value of ρScalculated as proposed by Safari-Katesari et al. [29] where ρSwas ex... | https://arxiv.org/abs/2503.13148v1 |
0.20 2.31 0.10 0.24 0.09 0.19 0.5 0.22 0.51 9.30 0.24 0.44 0.22 0.29 0.8 0.35 0.80 20.8 0.38 0.59 0.34 0.35 0.8 0.2 0.19 0.20 0.68 0.27 1.49 0.19 0.60 0.5 0.47 0.50 0.67 0.61 2.51 0.48 0.56 0.8 0.76 0.80 0.46 0.94 3.73 0.77 0.28 8 0.2 0.2 0.10 0.20 2.02 0.11 0.26 0.10 0.23 0.5 0.24 0.50 7.45 0.26 0.37 0.24 0.31 0.8 0.3... | https://arxiv.org/abs/2503.13148v1 |
size increases. We now further compare our bounds to those derived by Mesfioui and Trufin [19] in Table 2. As discussed in Section 4, our bounds are generally sharper than those derived in [19], and this is also captured by the proposed estimators. We conclude the analysis of the bound estimators by considering the zer... | https://arxiv.org/abs/2503.13148v1 |
rank estimator ˆ ρSwith the tie adjustment. The second data set concerns the frequencies at which the Dutch residents travel by train and bus, tram, and/or metro. A natural question that arises is whether people who travel by train also frequently use other forms of public transport, for example to reach the train stat... | https://arxiv.org/abs/2503.13148v1 |
bivariate zero-inflated Poisson models with application to studies of abundance for multiple species. Environmetrics , 23(2):183–196, 2012. [2] N. Blomqvist. On a measure of dependence between two random variables. Annals of Mathematical Statistics , 21(4):593– 600, 1950. [3] CBS. Onderweg in nederland. www.cbs.nl/nl-n... | https://arxiv.org/abs/2503.13148v1 |
Ronald S. Pimentel. Kendall’s tau and Spearman’s rho for zero-inflated data. PhD Thesis. Western Michigan University, Michigan. , 2009. [27] Ronald S. Pimentel, Magdalena Niewiadomska-Bugaj, and Jung-Chao Wang. Association of zero-inflated continuous variables. Statistics & Probability Letters , 96:61–67, 2015. [28] R ... | https://arxiv.org/abs/2503.13148v1 |
steps for AXyields A=p11h p10 1−2p∗ 1−p† 1 +p01 1−2p∗ 2−p† 1i . Finally, we focus on p1+p+1ρS∗. Conditioning on Sabfor all possible combinations of a, b∈ {0,1}results in p1+p+1ρS∗=p2 11ρS11+p11p10ρS10+p11p01ρS01+p10p01ρS00, which completes the proof. Appendix B. Proof of Proposition 4.1 Here, we formulate a more de... | https://arxiv.org/abs/2503.13148v1 |
and in fact equal to one, hence ρS00= 3. Using ρS11=−1, we conclude that ρA,min=−p3 11+ 3p11p10p01−3p11(p10+p01)−3p10p01 =−(1−p1−p2)3+ 3(1−p1−p2)p1p2−3(1−p1−p2)(p1+p2)−3p1p2 =− 1−3p1−3p2+ 6p1p2+ 3p2 1+ 3p2 2−3p2 1p2−3p1p2 2−p3 1−p3 2 + + 3p1p2−3p2 1p2−3p1p2 2−3p1−3p2+ 3p2 1+ 6p1p2+ 3p2 2−3p1p2 =p3 1+p3 2−1. Appendix ... | https://arxiv.org/abs/2503.13148v1 |
= 0] + P[Y= 0, X= 0]) =1 1−p2(G(y)−p2−p1+p1). Hence, the probability mass function of Y11is P[Y11=y] =1 1−p2( G(y)−G(y−1) if y >0, 0 otherwise. We can now derive ρS10. We start by considering the probability of concordance and discordance separately. We note that, under S10,X1is distributed as X11andX2is distributed as... | https://arxiv.org/abs/2503.13148v1 |
section. This case involves many terms of the same form, therefore we start by recalling some notation. With some abuse of notation, we define W(x, y) = 1−F(x)−G(y) for all ( x, y)∈N2. In fact, we only need this for a few choices of ( x, y). Since we consider the lower Fr´ echet-Hoeffding bound, we still have p11= 1−p1... | https://arxiv.org/abs/2503.13148v1 |
2|S01] +P[X1> X 2, Y1≥Y3|S01] =−P[Y1≥Y3|S01] + 2P[X1≥X2, Y1≥Y3|S01]−P[X1=X2, Y1≥Y3|S01] =−P[Y11=˜t′]P[Y01=˜t′] +P[Y01=˜t′] 2P[X1≥X2, Y1=˜t′|S01]−P[X1=X2, Y1=˜t′|S01] . The last equality follows from P[Y11≥Y01] =P[Y11=Y01=˜t′]. We now analyze the last terms in parenthesis. We sum over the support of the random variabl... | https://arxiv.org/abs/2503.13148v1 |
margins. Theoretically, the margins do not impact the estimates and the parameters are therefore fixed to one. The true value of ρSfor the Fr´ echet copula Cαin the zero-inflated continuous case can be derived directly based on its formula. In particular, one can obtain ρA(Cα) = (1 −α)ρA(Π) + αρA(M), where ρA(Π) = 0 co... | https://arxiv.org/abs/2503.13148v1 |
Stein’s method of moment estimators for local dependency exponential random graph models Adrian Fischer∗, Gesine Reinert†, Wenkai Xu‡ March 18, 2025 Abstract Providing theoretical guarantees for parameter estimation in exponential random graph models is a largely open problem. While maximum likelihood estimation has th... | https://arxiv.org/abs/2503.13191v1 |
for example [29]). Moreover, a practical problem that can arise in parameter estimation for small networks is that the maximum likelihood estimators may lie on, or very close to, the boundary of the parameter space; in such a situation, [32] report that a common approach is to pool small networks into a larger block-di... | https://arxiv.org/abs/2503.13191v1 |
results are illustrated by a set of simulations. We find that Stein estimation achieves similar accuracy as MLE when the latter is available, and that Stein estimation calculation is orders of magnitude faster than the MLE, MPLE, and contrastive divergence algorithms in the ergm package. In the simulation, Stein estima... | https://arxiv.org/abs/2503.13191v1 |
model was first introduced in [24]; consistency as well as non-asymptotic error bounds and normal approximation for the maximum likelihood estimator were developed in [26] and [28]. We assume that the vertex set Acan be partitioned into Kneighbourhoods, or blocks, A1, . . . , A K such that A=∪K k=1Ak. Moreover we defin... | https://arxiv.org/abs/2503.13191v1 |
[12] for which maximum likelihood estimation is not straightforward. In this paper we develop Stein estimation for the LERGM. The first step is to find a suitable Stein operator; in principle, many choices are possible, see for example [19]. In [23] the authors pro- pose a Glauber dynamics Stein operator for the expone... | https://arxiv.org/abs/2503.13191v1 |
< l≤K. (9) Next we sum (9) over all k, lto obtain the two equations X 1≤k≤KX m∈Ek,k σ(⟨βW,∆msk,k(Xk,k)⟩)∆msk,k(Xk,k) +sk,k(♢0 mXk,k)−sk,k(Xk,k) = 0; X 1≤k<l≤KX m∈Ek,l σ(⟨βB,∆msk,l(Xk,l)⟩)∆msk,l(Xk,l) +sk,l(♢0 mXk,l)−sk,l(Xk,l) = 0.(10) Definition 3.2 (LERGM-Stein estimator) .For a LERGM model with statistics s( x)a... | https://arxiv.org/abs/2503.13191v1 |
x) =E( x)satisfies (i)-(v) in Assumption 3.4. Moreover unless Xis the full or the empty graph, (vi) (and hence (vii)) are satisfied. For α̸= 0,1, the probability of obtaining the empty or the complete graph is strictly smaller than 1. 8 The first assumption, (i), ensures that the number of parameters does not exceed th... | https://arxiv.org/abs/2503.13191v1 |
0anda, bare positive integers, and (i)b=i(i+ 1). . .(i+b−1). Taking inspiration from the approach in [20] we show the following result. Lemma 3.9. A LERGM with d1∈ {0,1}, d2∈ {0,1}, such that d1+d2≥1,with statistics of the type (15) foro(i)a strictly decreasing function in isatisfies (i)-(iii) in Assumption 3.4, and (i... | https://arxiv.org/abs/2503.13191v1 |
H(2) i( xk,l)for the degree distribution of the vertices in block l, in the subgraph xk,l. The corresponding statistics are given by sGwd,j ( xk,l) =|A•|−1X i=0o(i)H(j) i( xk,l), 1≤k < l≤K, j = 1,2, where •=lifj= 1 and•=kifj= 2. As a concrete example we take α= 1; we denote byβW= (β(1) W, β(2) W)the within-block parame... | https://arxiv.org/abs/2503.13191v1 |
the other methods tested. Moreover, the computational runtime for the Stein estimator is about two orders of magnitude smaller that that for maximum likelihood estimation. 4 Convergence analysis In order to be rigorous about the dependency structure of the constants and to derive asymptotic results from our error bound... | https://arxiv.org/abs/2503.13191v1 |
the implicit dependence that RWandRBare of the order√d1and√d2,respectively. This assumption is used to bound the second derivatives G(n) WandG(n) B from (12) and (13) away from 0. In [28] this restriction is foregone at the cost of of introducing an assumption on the asymptotic relation between the number of vertices a... | https://arxiv.org/abs/2503.13191v1 |
expansion 16 around β∗ Wyields that for βW∈SW,j E[G(n) W(X(n), βW)]−E[G(n) W(X(n), β⋆ W)] = (βW−β⋆ W)⊤EX 1≤k≤KnX m∈E(n) k,kZ1 0(1−t)σ′(⟨β⋆ W+t(βW−β⋆ W),∆ms(n) k,k(X(n) k,k)⟩) ×∆ms(n) k,k(X(n) k,k)∆ms(n) k,k(X(n) k,k)⊤dt (βW−β⋆ W) ≥1 2X 1≤k≤KnX m∈E(n) k,kσ′(RWLWMCWn)(βW−β⋆ W)⊤E ∆ms(n) k,k(X(n) k,k)∆ms(n) k,k(X(n) k,k... | https://arxiv.org/abs/2503.13191v1 |
normality For standardisation to obtain asymptotic normality, we introduce a matrix for the within-block pa- rameters, E[g(n) W(X(n), β⋆ W)g(n) W(X(n), β⋆ W)⊤(19) and an analogous matrix for the between-block parameters. Define the deterministic quantities Q(n) W=E[g(n) W(X(n), β⋆ W)g(n) W(X(n), β⋆ W)⊤]−1/2E[G(n) W(X(n... | https://arxiv.org/abs/2503.13191v1 |
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