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affine varieties. Let ϕ:V−→Wbe a surjective morphism, and let f∈C[W]be a regular function. Then z∈Vandp=ϕ(z)are critical points of f′=f◦ϕandf, respectively, if the differential dϕz:Tz(V)−→Tp(W)is surjective. 11 Proof. The differential dϕzis, in coordinates, equal to the Jacobian of ϕrestricted to Tz(V). By the chain ru... | https://arxiv.org/abs/2505.15969v1 |
standard unit vectors of Cknin the last k(n−k) columns indexed by the entries of ˜Zk+1, . . . , ˜Zn. Hence, multiplying JacZΦ with the vectors eijfori=k+ 1, . . . , n andj= 1, . . . , k that are tangent vectors to Vk,natZgives k(n−k) linearly independent vectors in the tangent space to pGr( k, n) at ZZT. Since the dime... | https://arxiv.org/abs/2505.15969v1 |
=RTΛR where Λ is the diagonal matrix with eigenvalues of Aon the diagonal. Since Λ is diagonal andRhas the desired block structure, the points in the disjoint union are critical points. Conversely, suppose QTAQhas the desired block structure. Since Ais diagonalizable, every block of QTAQis diagonalizable. Assembling th... | https://arxiv.org/abs/2505.15969v1 |
j=1j2. 4.1 Diagonal case Our computational experiments indicate that the number of critical points of the heteroge- neous quadratics minimization problem stays stable if we take the input matrices A1, . . . , A k to be generic diagonal matrices. While we do not have a general proof for this observation, we present a re... | https://arxiv.org/abs/2505.15969v1 |
computation shows that the images of the k(n−k) + k 2 vectors under the Jacobian of the parametrization map stay linearly independent. Hence, this image has dimension dim(pFl(1 ,2, . . . , k ;n)). Therefore, the differential of the parametrization map is surjective. Lemma 3.2 implies that the critical points on Vk,nar... | https://arxiv.org/abs/2505.15969v1 |
is a critical point of (15). We proceed by induction on i. When i= 1, the optimization problem simplifies to max uTu=vTv=1vTAu.By computing the transpose of the Jacobian, we find that the critical points of this problem are characterized by the leftmost column of the matrix Av u 0 ATu0v being in the span of the rest ... | https://arxiv.org/abs/2505.15969v1 |
only it satisfies UTU=VTV= Id kand there exist symmetric matrices M, N such that AV=UM and ATU=V N. We must have M=N, since UTAV=UTUM =MandVTATU=VTV N= NandM, N are symmetric. Thus ( U, V) is a critical point if and only if UTU=VTV= Id k and there exists a single symmetric matrix Mwith AV=UMandATU=V M. The matrix Msati... | https://arxiv.org/abs/2505.15969v1 |
arXiv:2505.16124v1 [stat.ME] 22 May 2025Controlling the False Discovery Rate in High-Dimensional Linear Models Using Model-X Knockoffs and p-values Jinyuan Chang1,2, Chenlong Li3, Cheng Yong Tang∗4, and Zhengtian Zhu2 1Joint Laboratory of Data Science and Business Intelligence, Southwestern University of Finance and Ec... | https://arxiv.org/abs/2505.16124v1 |
control the FDR regardless of the dependence structure among the test statistics. The BH procedure is known to control the false discovery rate at its target level under the assumption of independence among test statistics. However, when test statistics are dependent, the situation becomes more complex. Under the condi... | https://arxiv.org/abs/2505.16124v1 |
the target FDR level is low or the signal strength is weak. However, because the method requires evaluating ordinary least squares (OLS) estimators and their associated p-values, it is not applicable in high-dimensional settings where the number of predictors is much larger than the sample size. Our aim in this study i... | https://arxiv.org/abs/2505.16124v1 |
extreme value distributions, leveraging asymptotic results as the number of hypotheses diverges. As a result, the performance of these approaches depends on the accuracy of distribu- tional approximations in both regimes. Moreover, an implicit rationale behind the use of extreme value approximations is that the tails o... | https://arxiv.org/abs/2505.16124v1 |
copies of a d-dimensional random vector x. Our investigation takes place in the context of the linear model: y=Xβ+ε, (1) where y= (y1, . . . , y n)⊤∈Rnis the response vector, ε= (ε1, . . . , ε n)⊤∈Rnis the error term with E(εi) = 0 and Var( εi) =σ2, and β∈Rdis the parameter vector. The primary objective of interest is ... | https://arxiv.org/abs/2505.16124v1 |
A new set of parameter estimates is obtained by fitting the linear model (1) with yand the new design matrix ( x+ ·,j,x− ·,j,X−j), where X−jdenotes the matrix Xwithout the j-th column. Let ˆβ+ j and ˆβ− jbe the components of the estimator corresponding to x+ ·,jandx− ·,j, respectively. Based 7 on these estimates, the s... | https://arxiv.org/abs/2505.16124v1 |
difficulty lies in constructing appropriate knockoff variables. In the high-dimensional linear model setting, Cand` es et al. (2018) propose the model-X knockoff framework. Using model-X knockoff variables, an augmented design matrix can be constructed similarly to that in Barber and Cand` es (2015). To estimate the hi... | https://arxiv.org/abs/2505.16124v1 |
covariance matrix of ( x⊤,˜x⊤)⊤is structured as Γ= Cov {(x⊤,˜x⊤)⊤}= Σ Σ −D Σ−D Σ , (8) where D= diag( s), and sis a hyperparameter chosen to ensure that the covariance matrix Γis positive semidefinite. We refer to Cand` es et al. (2018) for a comprehensive discussion of the model- X knockoff methodology and its c... | https://arxiv.org/abs/2505.16124v1 |
by {ˆΛj,j}2d j=1. Let ˆ σbe an estimator of the standard deviation σof the model error εi, which will be specified in Section 2.3. Write ˆβ(bc) 1= (ˆβ(bc) 1,1, . . . , ˆβ(bc) 1,d)⊤and ˆβ(bc) 2= (ˆβ(bc) 2,1, . . . , ˆβ(bc) 2,d)⊤. Based on (10) and (11), for j∈[d], we define a set of paired test statistics ( t1,j, t2,j) ... | https://arxiv.org/abs/2505.16124v1 |
recommended balanced choice. While it is feasible within our framework to develop adaptive multiple testing procedures by incorporating an estimate of the proportion of true null hypotheses – e.g., following the approach of Storey (2002) – we omit this extension to maintain focus on our primary objective. Moreover, in ... | https://arxiv.org/abs/2505.16124v1 |
introduces complex dependence structures, making parameter es- timation more challenging and requiring careful theoretical treatment to ensure proper control of these dependencies. Further challenges arise from analyzing and ensuring the properties of the inter- mediate parameter estimates, including the high-dimension... | https://arxiv.org/abs/2505.16124v1 |
sub-Gaussian rows, with zero mean and sub-Gaussian norm ∥x∥ψ2=κ1, then the model-X knockoff variate ˜Xalso has independent sub-Gaussian rows, with zero mean and sub-Gaussian norm ∥˜x∥ψ2=κ1. Furthermore, the augmented covariate Zhas independent sub-Gaussian rows, with zero mean and sub-Gaussian norm κ1≤ ∥z∥ψ2≤2κ1, and s... | https://arxiv.org/abs/2505.16124v1 |
statistical methods under consideration. The matrix containing the knockoff varibles is constructed following the second-order model-X knockoff procedure of Cand` es et al. (2018). We explore combinations of n∈ {200,500}andd∈ {n,1.5n,2n}. For each setting, we randomly assign kcomponents of β0to be nonzero, with the rem... | https://arxiv.org/abs/2505.16124v1 |
pre-specified FDR levels, weaker signals, and smaller sample sizes, demonstrating their effectiveness under more challenging conditions. A likely explanation is the variability intro- duced by the ratio estimators used in the empirical FDP calculations of knockoff-based methods (Cand` es et al., 2018), which can substa... | https://arxiv.org/abs/2505.16124v1 |
demonstrate the benefits of using the debiased estimator, particularly in settings where low-dimensional approaches, such as that of Sarkar and Tang (2022), are applicable. Figure 4 presents results highlighting these advantages in cases where both our proposed methods and the method in Sarkar and Tang (2022) are appli... | https://arxiv.org/abs/2505.16124v1 |
j-th 23 0.000.100.200.300.400.500.60 0.100.150.200.250.300.350.400.450.50 AmplitudeSimulated FDRn = 200, d = 200 0.000.100.200.300.400.500.60 0.100.150.200.250.300.350.400.450.50 AmplitudeSimulated FDRn = 200, d = 300 0.000.100.200.300.400.500.60 0.100.150.200.250.300.350.400.450.50 AmplitudeSimulated FDRn = 200, d = 4... | https://arxiv.org/abs/2505.16124v1 |
we apply our proposed methods to identify mutations in HIV-1 associated with resistance to Protease Inhibitors (PIs). Specifically, we focus on six of the seven drugs: APV, ATV, IDV, LPV, NFV, and RTV. As an example with comparable nandd, analyzing this real dataset provides valuable insights into the advantages of our... | https://arxiv.org/abs/2505.16124v1 |
in high-dimensional settings. In summary, our proposed p-value-based multiple testing methods demonstrate competitive performance in addressing multiple testing in practical settings. We recommend our approach, particularly in cases where the target FDR level is lower or the signals from contributing variables are weak... | https://arxiv.org/abs/2505.16124v1 |
The control of the false discovery rate in multiple testing under dependency. The Annals of Statistics , 29, 1165–1188. B¨ uhlmann, P. and van de Geer, S. (2011). Statistics for High-Dimensional Data: Methods, Theory and Applications . Springer, Heidelberg. Cai, T., Liu, W., and Luo, X. (2011). A constrained ℓ1minimiza... | https://arxiv.org/abs/2505.16124v1 |
therapy in subtype B isolates and implications for drug-resistance surveillance. The Journal of Infectious Diseases , 192, 456–465. 29 Rhee, S.-Y., Taylor, J., Wadhera, G., Ben-Hur, A., Brutlag, D. L., and Shafer, R. W. (2006). Genotypic predictors of human immunodeficiency virus type 1 drug resistance. Proceedings of ... | https://arxiv.org/abs/2505.16124v1 |
reported here. Figures S3 and S4 present the results with relatively small n S1 0.000.050.100.150.200.25 200220240260280300320340360380400 dSimulated FDRSetting 1, n = 200 0.000.050.100.150.200.25 200220240260280300320340360380400 dSimulated FDRSetting 2, n = 200 0.000.050.100.150.200.25 2002202402602803003203403603804... | https://arxiv.org/abs/2505.16124v1 |
matrix were generated from setting 2. The sparsity level is k= 0.04dand the FDR level is α= 0.2. The methods compared are Algorithm 1 (squares and red solid line), Algorithm 2 (circles and yellow solid line), the knockoff-based method of Cand` es et al. (2018) (triangles and green dotted line), the Gaussian Mirror meth... | https://arxiv.org/abs/2505.16124v1 |
level is α= 0.2. The methods compared are Algorithm 1 (squares and red solid line), Algorithm 2 (circles and yellow solid line), the knockoff-based method of Cand` es et al. (2018) (triangles and green dotted line), the Gaussian Mirror method of Xing et al. (2021) (diamonds and blue dashed line), and the Gaussian Mirro... | https://arxiv.org/abs/2505.16124v1 |
and the FDR level is α= 0.05. The methods compared are Algorithm 1 (squares and red solid line), Algorithm 2 (circles and yellow solid line), the knockoff-based method of Cand` es et al. (2018) (triangles and green dotted line), the Gaussian Mirror method of Xing et al. (2021) (diamonds and blue dashed line), and the G... | https://arxiv.org/abs/2505.16124v1 |
d= 200 (middle column), n= 700 and d= 300 (right column). The rows of the design matrix were generated from setting 2. The sparsity level is k= 0.1dand the FDR level is α= 0.05. The methods compared are Algorithm 1 (squares and red solid line), Algorithm 2 (circles and green solid line), and the Bonferroni-Benjamini-Ho... | https://arxiv.org/abs/2505.16124v1 |
arXiv:2505.16275v1 [math.ST] 22 May 2025Semiparametric Bernstein–von Mises theorems for reversible diffusions Matteo Giordano and Kolyan Ray University of Turin and Imperial College London Abstract We establish a general semiparametric Bernstein–von Mises theorem for Bayesian nonparametric priors based on continuous ob... | https://arxiv.org/abs/2505.16275v1 |
the position of a particle diffusing in a potential energy field that exerts a force directed towards its local extrema, see Figure 1. By a classic result of Kolmogorov, the drift taking the form of a gradient vector field ∇Bis equivalent to time reversibility of the process X(e.g. [7], p. 46). Reversible systems are w... | https://arxiv.org/abs/2505.16275v1 |
statistical inference in the T→ ∞regime, where one can use the average behaviour of the particle trajectory for inference. This requires a suitable notion of statistical ergodicity to ensure the particle exhibits enough recurrence to use long-time averages. We ensure this by following [43, 47, 68, 1, 20, 40, 27] in res... | https://arxiv.org/abs/2505.16275v1 |
can be seen via the semiparametric information bounds in Section 2.2, which involve the solution to an elliptic PDE, see [25] for a similar situation in inverse problems. In contrast, in the non-reversible case studied in [40], such bounds are simpler weighted L2-norms. Notation. LetTd= (0,1]ddenotethe d-dimensionaltor... | https://arxiv.org/abs/2505.16275v1 |
consider the unique equivalence class withR TdB(x)dx= 0, namely B∈˙L2(Td). In the present model, the process lives on all of Rd, but will notbe globally recurrent in this space. However, the periodicity of Bmeans that the values of (Xt)tmodulo Zd encode all the relevant statistical information about ∇Bcontained in the ... | https://arxiv.org/abs/2505.16275v1 |
the information or local asymptotic normality (LAN) norm induced by the statistical model(e.g.Chapter25of[67]). Inthepresentreversiblediffusionsetting, theLANinner product is ⟨B,¯B⟩L:=⟨∇B,∇¯B⟩µ0(see Lemma 2), giving corresponding functional expansion Ψ(B) = Ψ( B0) +⟨ψL, B−B0⟩L+r(B, B 0). Define the second order ellipti... | https://arxiv.org/abs/2505.16275v1 |
r(B, B 0) =o(1/√ T)means the functional Ψ(B)is approximately linear with expansion (4), which nonetheless allows to cover several interesting nonlinear functionals. Condition (9) requires invariance of the prior for the full parameter Bunder a shift Bu=B−uγT/√ Tin the approximate least favourable direction γT, which sh... | https://arxiv.org/abs/2505.16275v1 |
e.g. [49]. Taking µBbounded away from zero rules out such situations, where statistical estimation problems can behave qualitatively differently [48, 44, 50] and it is unclear if√ T-rates are attained. Remark 2 (Functional regularity) .Employing a standard Bayesian nonparametric prior for Band considering the induced m... | https://arxiv.org/abs/2505.16275v1 |
terms in the likelihood (e.g. Lemma 3). The rescaling in (11)ensures such conditions are satisfied. Deriving posterior contraction rates, let alone BvM results, for non-rescaled Gaussian priors in such nonlinear settings is currently an open problem. Theorem 2. LetΠbe the rescaled Gaussian prior from (11)withW∼ΠWsatisf... | https://arxiv.org/abs/2505.16275v1 |
(10). Since B0∈˙Hs+1(Td)and s > 3d/2, we also have µ0∝e2B0∈Hs+1(Td)⊂Cd+1+κ(Td), the last inclusion holding by the Sobolev embedding. Hence by Lemma 9, there exists a unique ele- ment A−1 µ0ψ∈˙H(d/2+1+ κ)∨2(Td)such that Aµ0A−1 µ0ψ=ψalmost everywhere, and ∥A−1 µ0ψ∥H(d/2+1+ κ)∨2≲∥ψ∥H(d/2−1+κ)+for any κ > 0small enough. De... | https://arxiv.org/abs/2505.16275v1 |
prescribe random wavelet coefficients with tail behaviour between the Laplace (corresponding to p= 1) and the Gaussian distribution ( p= 2). Besov-Laplace priors have recently enjoyed significant popularity within the inverse problems and imaging communities [35, 17, 33, 31, 4, 36] since they exhibit attractive sparsit... | https://arxiv.org/abs/2505.16275v1 |
1.25−(7.5x−5.5)2−(5y−1.25)2+e−(7.5x−2)2−(7.5y−2)2.(19) For each given B, we simulate the continuous trajectory (Xt:t≥0)via the Euler- Maruyama scheme, iterating xr+1=xr+∇B(xr)δt+δtWr, W riid∼N(0,1), r ≥0. Across all the experiments, we set the time stepsize to δt= 10−4, resulting in realistic approximations of the cont... | https://arxiv.org/abs/2505.16275v1 |
q= 4) and Ψ3(B) =R µB(x) logµB(x)dxto be the entropy of the invariant measure (Example 5). For each combination of ground truth Band functional Ψ, the obtained coverages are higher at the larger time horizon; in particular, for T= 100, they are very close to the nominal level 95%predicted by Theorem 2. The average leng... | https://arxiv.org/abs/2505.16275v1 |
transform satisfies EΠDTh eu√ T(Ψ(B)−eΨT) XTi =eu2 2TRT 0∥∇γT(Xt)∥2dtR DTeℓT(Bu)dΠ(B)R DTeℓT(B)dΠ(B)(1 +oP0(1)) asT→ ∞, with centering eΨT= Ψ( B0) +1 TZT 0∇γT(Xt).dWt. (22) Proof.Using Bayes formula and setting ZT=R DTeℓT(B)dΠ(B)to be the normalizing constant, IT(u) :=EΠDT[eu√ T(Ψ(B)−eΨT)|XT] =e−u√ TRT 0∇γT(Xt).dWt1 ZT... | https://arxiv.org/abs/2505.16275v1 |
we obtain the following lemma controlling the remainder term in the LAN expansion, uniformly over a function class. The constant η >0below can be arbitrarily small and does not affect the required regularity in a significant way. Lemma 3. For some η >0, suppose B0∈C(d/2+η)∨2(Td)and for M > 0andζT→0, let DT⊆ {B∈˙C2(Td) ... | https://arxiv.org/abs/2505.16275v1 |
Further let (γT:T >0)⊂Hd/2+1+ κ(Td)be a sequence of functions such that Kγ:= lim supT→∞∥γT∥Hd/2+1+ κ<∞and∥∇A−1 µ0ψ− ∇γT∥µ0→0asT→ ∞. Then forˆΨTandeΨTdefined in (6)and(22), respectively, we have ˆΨT−eΨT=oP0(1/√ T). Furthermore, as T→ ∞, 1 TZT 0∥∇γT(Xt)∥2dtP0−→ ∥∇ A−1 µ0ψ∥2 µ0. Proof.Using the definitions (6) and (22), ˆ... | https://arxiv.org/abs/2505.16275v1 |
B 0)|=O(ε2 T) =o(1/√ T) since s > d. We conclude that, for all sufficiently large M > 0, the set DTsatisfies the condition (7) of Theorem 1 with the choices ζT=εd/(2d+2κ) T andξT=√ Tε2 T. (ii) It remains to verify the asymptotic invariance property (9). For Bu=B− uγT/√ TandΠu:=L(Bu), using the Cameron-Martin theorem (e... | https://arxiv.org/abs/2505.16275v1 |
0, define the sets DT=DT(M) := B∈VJ:∥∇B− ∇B0,T∥1≤MεT,∥B∥Hd+1+κ≤M ,(30) where κ > 0is an arbitrarily small constant. For Mlarge enough, Lemma 7 below implies that Π(DT|XT)P0−→1asT→ ∞. Using (31), ∥∇B−∇B0∥1≲εTforall B∈ DT, whichverifiesthefirstrequirementin (7). Next, in view of the continuous embedding W1,1(Td)⊂B1 1∞(T... | https://arxiv.org/abs/2505.16275v1 |
one in P0-probability as T→ ∞. Starting with DT,1, note that since B0∈Hs+1(Td), ∥∇B0− ∇B0,T∥1≲∥B0−B0,T∥H1≤2−Js∥B0∥Hs+1≲εT. (31) Hence, provided that Mis sufficiently large, DT,1⊇ {B∈˙C2(Td) :∥∇B− ∇B0∥1≤MεT/2} By Theorem 2.4 of [27] (with the choice p= 1), whose assumptions are precisely recov- ered under the conditions... | https://arxiv.org/abs/2505.16275v1 |
last display, Ψ(B)−Ψ(B0) = 2Z (B−B0)µB0(φ−Ψ(B0)) +O(ε2 T). Many interesting functionals are approximately linear in the invariant measure µ, and Lemma 8 allows us to perform a further linearization in Bon the linearizations in µ. This is the approach we take for several examples. Proof of Example 5. For the entropy fun... | https://arxiv.org/abs/2505.16275v1 |
C > 0. The case q=∞follows identically. Using the last display, the definition of Aµand the multiplication inequality for Besov norms, ∥u∥Btpq≲∥(1/µ)Aµu∥Bt−2 pq+∥(1/µ)∇µ.∇u∥Bt−2 pq ≲∥1/µ∥B|t−2| ∞∞∥Aµu∥Bt−2 pq+∥1/µ∥B|t−2| ∞∞∥µ∥B|t−2|+1 ∞∞∥u∥Bt−1 pq(37) for all u∈ H, with constants depending on t, p, q, d. We will deduce... | https://arxiv.org/abs/2505.16275v1 |
URL https://doi.org/10.3150/22-bej1563 . [5] S. Agapiou, M. Dashti, and T. Helin. Rates of contraction of posterior distributions based on p-exponential priors. Bernoulli , 27(3):1616–1642, 2021. ISSN 1350-7265. doi: 10.3150/20-bej1285. URL https://doi.org/10.3150/20-bej1285 . [6] S. Agapiou, M. Dashti, and T. Helin. R... | https://arxiv.org/abs/2505.16275v1 |
University Press, Cambridge, 2017. ISBN 978-0-521-87826-5. doi: 10.1017/9781139029834. URL https://doi.org/10.1017/9781139029834 . [23] D. Gilbarg, N. S. Trudinger, D. Gilbarg, and N. Trudinger. Elliptic partial differ- ential equations of second order , volume 224. Springer, 1977. [24] E. Giné and R. Nickl. Mathematic... | https://arxiv.org/abs/2505.16275v1 |
, 45(4):1664–1693, 2017. [43] O. Papaspiliopoulos, Y. Pokern, G. O. Roberts, and A. M. Stuart. Nonparametric estimation of diffusions: a differential equations approach. Biometrika , 99(3):511– 531, 2012. [44] T. Patschkowski and A. Rohde. Adaptation to lowest density regions with appli- cation to support recovery. Ann... | https://arxiv.org/abs/2505.16275v1 |
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arXiv:2505.16302v1 [math.ST] 22 May 2025Covariance matrix estimation in the singular case using regularized Cholesky factor Olivier Besson∗ May 23, 2025 Abstract We consider estimating the population covariance matrix when the number of available samples is less than the size of the observations. The sample covariance ... | https://arxiv.org/abs/2505.16302v1 |
is hypothetical since it depends on Σ. Tables 1, 2 and 3 of [18] provide the expressions of the FSOPT estimator for a large number of loss functions. The next step is to derive the asymptotic -as p, n→ ∞ with p/n→c- limit of L(ˆΣ,Σ), say L∞(ˆΣ,Σ). Then the Oracle estimator is defined as the one which minimizes this lim... | https://arxiv.org/abs/2505.16302v1 |
tail of the Cholesky factor of the population covariance matrix is not identifiable. Therefore, there is a need to fill this unknown part and a method is proposed to fulfill this need. 2 Covariance matrix estimation 2.1 Outline LetX= x1x2···xn be the p×ndata matrix whose columns xkare independent and drawn from a mul... | https://arxiv.org/abs/2505.16302v1 |
and covariance matrix Σ, which we denote as Xd=Np,n(0,Σ,In), it follows that Xd=L¯Xwhere ¯Xd=Np,n(0,Ip,In) whered= means “has the same distribution as ”. Consequently Sd=L¯WLT where ¯Wfollows a singular Wishart distribution [29]. We let ¯W=¯G¯GTwith ¯Ga lower triangular matrix with positive diagonal elements. Then all ... | https://arxiv.org/abs/2505.16302v1 |
we observed that the FSOPT and the Oracle estimators perform the same so we only consider the latter. For comparison purposes we also evaluate Stein’s risks of the benchmark linear shrinkage estimator of [9] (LW-LS in the figures), the nonlinear shrinkage estimator of [11] (LW-NLS) and the methods of [27] (BL) and [28]... | https://arxiv.org/abs/2505.16302v1 |
when nis small and the large eigenvalues occupy at least 30% of the spectrum. References [1] M. Pourahmadi. High dimensional covariance estimation . Wiley series in probability and statistics. John Wiley & Sons, Hoboken, NJ, 2013. [2] A. Zagidullina. High-dimensional covariance matrix estimation - An introduction to ra... | https://arxiv.org/abs/2505.16302v1 |
mean of a multivariate normal distribution. The Annals of Statistics , 9(6):1135–1151, November 1981. [18] O. Ledoit and M. Wolf. Shrinkage estimation of large covariance matrices: Keep it simple, statistician? Journal of Multivariate Analysis , 186:104796, November 2021. [19] T. Bodnar, A. K. Gupta, and N. Parolya. On... | https://arxiv.org/abs/2505.16302v1 |
arXiv:2505.16428v1 [math.ST] 22 May 2025Sharp Asymptotic Minimaxity for One-Group Priors in Sparse Normal Means Problem Sayantan Paul, Prasenjit Ghosh and Arijit chakrabarti Abstract In this paper, we consider the asymptotic properties of the B ayesian multiple testing rules when the mean parameter of the sparse normal... | https://arxiv.org/abs/2505.16428v1 |
of generality, a typica l choice ofσ2isσ2= 1. This is the famous normal means model orGaussian sequence model . In the normal means problem as given in (1), one is interested in testing simultaneously H0i:θi= 0 vsH1i:θi∝ne}ationslash= 0 fori= 1,...,n. A multiple testing procedure, defined in terms of a measurable functi... | https://arxiv.org/abs/2505.16428v1 |
for all iimplies the corresponding minimax risk is exactly 1. Hence, it is of practitioners’ importance to consider a non-trivial t esting procedure such that the resultant R(Θ) is strictly less than 1. Intuition suggests that it is necessary to ass ume some conditions on the least possible value of signals for the sep... | https://arxiv.org/abs/2505.16428v1 |
an≤cbn. Similarly, an∝bnimplies there exists some constant C >0 such that an=Cbnfor alln. Finally,an=o(bn) denotes lim n→∞an bn= 0.φ(·) denotes the density of a standard normal distribution. 2 Existing results on minimax risk The most recent and widely studied work on the minimax risk for the Ga ussian sequence model i... | https://arxiv.org/abs/2505.16428v1 |
Abraham et al. (2024) assumed the slab density to b e quasi-Cauchy, such that the convolution ofφ∗f(fbeing the density of Fgiven in (9)), say ˜fis such that ˜f(x) =1√ 2πx−2/parenleftbigg 1−exp/parenleftbigg −x2 2/parenrightbigg/parenrightbigg ,x∈R. Now, given any t∈(0,1), they considered a multiple testing rule which d... | https://arxiv.org/abs/2505.16428v1 |
which is of the form 1 qninf ψsup θ∈ΘbEL(θ,ψ) = 1−Φ(b)+o(1), (17) whereΘbis already defined in (7) and bmay be either finite or b=bn→ ±∞. This result also holds for the BH andl−value-based procedures under the assumption of polynomial spar sity stated before. To the best of our knowledge, the only work based on one-group... | https://arxiv.org/abs/2505.16428v1 |
ques tions raised in the last part of this Section based on one-group priors. 3 Results based on one-group priors Inspired by the works of Abraham et al. (2024) and Salomond (2017 ), we were interested in studying the multiple testing problem H1i:θi∝ne}ationslash= 0 simultaneously, when each θiis modeled as a scale mix... | https://arxiv.org/abs/2505.16428v1 |
θi∝ne}ationslash= 0|X)) corresponding to the spike and slab prior. They proposed the following decision rule intuitive ly, RejectH0iifE(1−κi|Xi,τ)>1 2,i= 1,2,···,n. (22) Later Datta and Ghosh (2013) and Ghosh and Chakrabarti (2017 ) established optimality of the decision rule (22) in the context of minimizing the Bayes... | https://arxiv.org/abs/2505.16428v1 |
as a good choice for the broad class of one-group priors considered here. Proposition 1. SupposeXiind∼ N(θi,1),i= 1,2,···,n. We want to test hypotheses H0i:θi= 0vsH1i:θi∝ne}ationslash= 0 simultaneously based on the decision rule (23)using one-group priors. Assume that τ→0asn→ ∞such that nτ qn→C∈(0,∞). Also assume that ... | https://arxiv.org/abs/2505.16428v1 |
the minimax risk corresponding to the classificatio n or Hamming loss. To the best of our knowledge, it is the first result in the literature of global-local priors i n this context. In order to establish this result, the only condition one needs on τis thatτis of the same order asqn n, i.e., the proportion of non-zero m... | https://arxiv.org/abs/2505.16428v1 |
arguments used in Theorem 1 for controlling type I error in th at part. On the complementary part of the range, we need to provide nontrivial arguments by exploiting the st ructure of /hatwideτ. For controlling type II error, we apply the procedure on E(κi|xi,τ), using the fact that for any for any fixed xi∈R,E(κi|xi,τ)... | https://arxiv.org/abs/2505.16428v1 |
FDR and FNR. As stated earlier, the only work in the context of obtaining an upper bound to the minimax risk based on this loss induced by one-group priors was considere d by Salomond (2017). However, their choice of the minimum value of the non-zero means exceeds the universa l threshold. Not only this, they did not p... | https://arxiv.org/abs/2505.16428v1 |
ri sk can be made arbitrarily small by choosing b=bn→ ∞, even for an unknown sparse situation. In this way, their res ult becomes a special case of ours under our chosen class of priors under study. Theorem 6. Consider the setup of Theorem 1. Suppose we want to test hypot hesesH0i:θi= 0vsH1i:θi∝ne}ationslash= 0 simulta... | https://arxiv.org/abs/2505.16428v1 |
et al. (2016). For that class of priors, first, it is needed to prove that, assuming the level of sparsity to be known, whether any multiple testing rule can attain the minimax risk, asympto tically, under the beta-min condition of Abraham et al. (2024). The question regarding the optimality of a te sting procedure for ... | https://arxiv.org/abs/2505.16428v1 |
some 0 <δ1<1. Later we may choose a more specific choice of g(τ), if required. Now define BnandCnasBn={g(Xi,τ,η,δ)>1 2}and Cn={|Xi|>/radicalBig ρlog/parenleftbig1 τ/parenrightbig }withρas defined before. Clearly, t2i=Pθi(E(κi|Xi,τ)>1 2) ≤Pθi(Bn)≤Pθi(Bn∩Cn)+Pθi(Cc n). (39) 13 Observe that, for any θi Pθi(g(Xi,τ,η,δ)>1 2,|X... | https://arxiv.org/abs/2505.16428v1 |
the empirical Bayes approac h corresponding to ith hypothesis, respectively. For any αn>0, we have tEB 1i=Pθi=0(E(1−κi|Xi,/hatwideτ)>1 2) =Pθi=0(E(1−κi|Xi,/hatwideτ)>1 2,/hatwideτ >2αn)+Pθi=0(E(1−κi|Xi,/hatwideτ)>1 2,/hatwideτ≤2αn) =I+IIsay, (56) Recall that, by definition, for any given x∈R,E(1−κ|x,τ) is increasing in ... | https://arxiv.org/abs/2505.16428v1 |
≤Pθi/negationslash=0(1 c2nn/summationdisplay j=1(/negationslash=i)1(|Xj|>/radicalbig c1logn)≤γζn) ≤P(|1 n−1n/summationdisplay j=1(/negationslash=i)1(|Xj|>/radicalbig c1logn)−νn| ≥νn−c2γn n−1ζn) ≤νn(1−νn) (n−1)(νn−c2γn n−1ζn)2, (70) whereνn=P(|Xi|>√c1logn). Since,νn(1−νn)≤1 4for any 0< νn<1, it is enough to show that th... | https://arxiv.org/abs/2505.16428v1 |
have for a=1 2, asn→ ∞ E(1−κi|Xi,αn)≤K1eX2 i 2αn(1+o(1)). (86) 20 Hereo(1) depends only on nsuch that lim n→∞o(1) = 0 and is independent of iandK1is a constant depending onM. Hence, we have the following : tFB 1i=PH0i(E(1−κi|X)>1 2) ≤PH0i/parenleftbiggX2 i 2>log/parenleftbigg1 αn/parenrightbigg −logK1−log(1+o(1))/paren... | https://arxiv.org/abs/2505.16428v1 |
the choice of λimpliest>0 for sufficiently large n. In order to obtain (96), we use a uniform upper bound ont2i,i:θi∝ne}ationslash= 0 as derived in (53). Using (96), we obtain, for all |θi| ≥/radicalbigg 2log/parenleftBig n qn/parenrightBig +b,i= 1,2,···,qn Pθ(/summationdisplay i:θi/negationslash=0ψi≤λqn)≤exp/parenleftbi... | https://arxiv.org/abs/2505.16428v1 |
ality confirms, for sufficiently large n sup θ∈ΘbVI=o(1),asn→ ∞. (111) Combining (109) and (111) ensures sup θ∈ΘbFDR(θ,ψ) =o(1),asn→ ∞. (112) ForFNR(θ,ψ), from definition, we obtain FNR(θ,ψ) =Eθ(/summationtext i:θi/negationslash=0(1−ψi)) qn =1 qn/summationdisplay i:θi/negationslash=0tFB 2i. (113) Recall that, in Theorem 3,... | https://arxiv.org/abs/2505.16428v1 |
and Chakrabarti, A. (2016). Asymp totic properties of bayes risk of a general class of shrinkage priors in multiple hypothesis testing under sparsit y.Bayesian Analysis , 11(3):753– 796. Gin´ e, E., and Nickl, R. (2021). Mathematical foundations of infinite -dimensional statistical models. Cambridge university press . G... | https://arxiv.org/abs/2505.16428v1 |
arXiv:2505.16651v1 [math.OC] 22 May 2025Risk-averse formulations of Stochastic Optimal Control and Markov Decision Processes Alexander Shapiro∗Yan Li† May 23, 2025 Abstract The aim of this paper is to investigate risk-averse and distributionally robust modeling of Stochastic Optimal Control (SOC) and Markov Decision Pr... | https://arxiv.org/abs/2505.16651v1 |
in [6] and [10] in a sense is static since the ambiguity sets of transition kernels are defined before realizations of the decision process. As a consequence, in order to derive the corresponding dynamic equations there is a need to introduce the so-called rectangularity conditions on the ambiguity sets of transition k... | https://arxiv.org/abs/2505.16651v1 |
of G-measurable random variables (called versions) which are equal to each other P-almost surely. By P|G(A) =E|G[1A],A∈ F, is denoted the respective conditional probability. For a random variable Y, we denote by E|Y[Z] the conditional expectation and by P|Y(A) =E|Y[1A],A∈ F, the respective conditional probability. 2 Ri... | https://arxiv.org/abs/2505.16651v1 |
measures. Then the robust counterpart of RPis defined as R(Z) := sup P∈MRP(Z), (2.4) assuming that supP∈MRP(Z)<∞for all Z∈ Z. IfRPsatisfies any of axioms (A2) - (A4) for allP∈M, then so is its robust counterpart R. The monotonicity axiom is more involved, this is because the inequality Z≥Z′is defined P-almost surely wi... | https://arxiv.org/abs/2505.16651v1 |
then V@RPNαmay not converge, in probability, toV@RP α(Z) asN→ ∞ . That is, in general there is no guarantee that V@RPNαis a consistent estimator of V@RP α(Z). Recall the Dvoretzky-Kiefer-Wolfowitz (DKW) inequality (e.g., [7, Theorem 11.6]): for Z∼P, FP=FP Zandε >0 the following inequality holds P sup z∈R FPN(z)−FP(z) ... | https://arxiv.org/abs/2505.16651v1 |
α(Zx′) ≤L x−x′ , x, x′∈ X. Indeed, let νx α:=V@RP α(Zx). It is clear that for any ϵ >0, we have P(Zx≤νx α+ϵ)≥1−α,and hence P(Zx′− ∥Zx′−Zx∥ ≤νx α+ϵ)≥1−α. 6 This implies νx′ α≤νx α+∥Zx′−Zx∥+ϵ. Since this holds for any ϵ >0, it follows that νx′ α≤ νx α+∥Zx′−Zx∥. Interchanging νx′ αandνx αyields νx′ α−νx α ≤ ∥Zx′−Zx∥ ≤L x−... | https://arxiv.org/abs/2505.16651v1 |
replacing V@RP α(Z) with its conditional counterpart. For the Value-at-Risk its conditional counterpart is V@RP α|G(Z) = inf z:P|G(Z≤z)≥1−α , α∈(0,1). (2.26) In particular we consider the following construction where we follow [21, Appendix A1 - A2]. Assume that the measurable space (Ω ,F) is given as the product of m... | https://arxiv.org/abs/2505.16651v1 |
be arbitrary forω1∈Ω1\supp( P1). 2.2.1 Multistage setting. This can be extended in a straightforward way to the setting where Ω = Ω 1× ··· × ΩTand F=F1⊗···⊗F T, with T≥2. For P∈Pandt∈ {2, ..., T}we can define conditional probabilities P|ω[t−1]determined by the respective Regular Probability Kernel KP t:Ft×Ω[t−1]→[0,1],... | https://arxiv.org/abs/2505.16651v1 |
|ω[t−1](Zt) follows by the construction. In general, the considered variables Ztmay be restricted to an appropriate linear space of measurable functions (such lin- ear space is explicitly mentioned in the definition of axioms (A1)-(A4)). For the Value-at-Risk functionals, the corresponding linear space consists of all ... | https://arxiv.org/abs/2505.16651v1 |
ξt), ξ[t]) , (3.6) which can be viewed as the counterpart of dynamic equations (3.3). A sufficient condition for policy ¯ut= ¯πt(xt, ξ[t−1]),t= 1, ..., T , to be optimal is that ¯ut∈arg min ut∈UtRP |ξ[t−1] ct(xt, ut, ξt) +Vt+1(Φt(xt, ut, ξt), ξ[t]) . (3.7) Proof of the above dynamic equations is based on an intercha... | https://arxiv.org/abs/2505.16651v1 |
theorem, equation (3.13) has a unique solution. That is, consider the space Bof bounded measurable functions g:X →R, equipped with the sup-norm ∥g∥∞= supx∈X|g(x)|, and Bellman operator T:B→B, T(g)(·) := inf u∈Usup P∈MRP c(·, u, ξ) +βg(Φ(·, u, ξ)) , g∈B. (3.14) We have that if |Z| ≤κ,P-almost surely for some κ∈R, then... | https://arxiv.org/abs/2505.16651v1 |
u)−(x′, u′)∥, for all x, x′∈ X,u, u′∈ U,ξ∈Ξ. Unfortunately the above assumption does not guarantee Lipschitz continuity of the value func- tionV(·). Nevertheless we can proceed as follows. Let ˜Vdenote an approximation of Vsuch that ˜Vis˜L-Lipschitz continuous. Then ∥˜V−VN∥∞=∥T˜V+˜V− T˜V− TNVN∥∞ ≤ ∥T ˜V− TNVN∥∞+∥˜V− T˜... | https://arxiv.org/abs/2505.16651v1 |
and ˜L-Lipschitz continuity of ˜V, we have ∥˜Zx,u−˜Zx′,u′∥∞≤L∥(x, u)−(x′, u′)∥,L=L(1 +β˜L). (3.22) For a fixed x, u∈ X × U , from Proposition 2.1 and the definition of κα, forN≥1 2κ−2 αlog(2/δ), with probability 1 −δwe have T(˜V)(x, u) =TN(˜V)(x, u). Consider an η-net of X ×U , denoted by Nη. That is, for any ( x, u)∈ ... | https://arxiv.org/abs/2505.16651v1 |
we assume that RQt×Pt(Zt(ut, ξt)) :=EQt RPt(Zt(ut, ξt)) , (3.26) where ut∼QtandRPtis a risk functional with respect to ξt∼Pt. In particular if RPt=EPt, equation (3.25) becomes eVt(xt) = inf Qt∈Stsup Pt∈MtEQt×Pt ct(xt, ut, ξt) +eVt+1(Φt(xt, ut, ξt) . (3.27) We say that there exists a non-randomized optimal policy if... | https://arxiv.org/abs/2505.16651v1 |
concave in P. Another example of concave inPfunctional is the Value-at-Risk measure RP=V@RP α. The following result about existence of the non-randomized optimal policies is an extension of [20, Theorem 4.1]. Proposition 3.1. Suppose that for t= 1, ..., T , there exists saddle point (u∗ t, P∗ t)satisfying condi- tions ... | https://arxiv.org/abs/2505.16651v1 |
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