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t o c onsider a far br oader c lass of low coor dinate degr ee (L CD ) alg orithms . In fact w e w ill c onsider a slightly br oader c lass of 𝐑𝑁- v alued L CD alg orithms , supplement ed b y an additional r ounding sc heme int o Σ𝑁 . The r andomized r ounding ̃ 𝒜 of 𝒜 is de- fined b y ̃ 𝒜 ≔ 𝗋𝗈𝗎𝗇𝖽 ⃗ 𝑈( 𝒜 )...	https://arxiv.org/abs/2505.20607v1
is essentially best -possible under the lo w degr ee heuristic. In particular , this heuristic sugg est s fr om the ener gy - degr ee tr adeoff of 𝐷 ≤ ̃ 𝑜 ( 𝐸 ) that finding solutions w ith ener gy 𝐸 r equir es time 𝑒̃ Ω ( 𝐸 ). This tr adeoff is att ainable along the full r ang e 1 ≪ 𝐸 ≤ 𝑁 v ia the f ollo w ing...	https://arxiv.org/abs/2505.20607v1
em 1. 4 is not able and highly sugg estiv e . See also [L S24] f or r elat ed discussion. W e also mention that c onjectur ally optimal tr ade - off s of a similar fla v or bet w een running time and signal-t o -noise r atio ha v e been est ablished f or spar se PC A in [ A WZ23] , [Din+24] and f or t ensor PC A in [K ...	https://arxiv.org/abs/2505.20607v1
f or the E uc lidean ball of r adius 𝑟 ar ound 𝑥 . In addition, w e w rit e 𝐵Σ ( 𝑥 , 𝑟 ) ≔ 𝐵 ( 𝑥 , 𝑟 ) ∩ Σ𝑁 = { 𝑦 ∈ Σ𝑁 : ‖ 𝑦 − 𝑥 ‖ < 𝑟 } , t o denot e point s on Σ𝑁 w ithin dist anc e 𝑟 of 𝑥 . W e use 𝒩 ( 𝜇 , 𝜎2) t o denot e the scalar N ormal distribution w ith giv en mean and v arianc e , and the ...	https://arxiv.org/abs/2505.20607v1
( 𝑔 ) depends only on 𝑔𝑆 } , 𝑉≤ 𝐷 ≔ span ( { 𝑉𝐽 : 𝐽 ⊆ [ 𝑁 ] , \| 𝐽 \| ≤ 𝐷 } ) These subset s describe functions whic h only depend on some subset of c oor dinat es , or on some bounded number of c oor dinat es . N ot e that 𝑉[ 𝑁 ] = 𝑉≤ 𝑁 = 𝐿2( 𝐑𝑁, 𝜋⊗ 𝑁) . The coor dinate degr ee of a function 𝑓 ∈ 𝐿2...	https://arxiv.org/abs/2505.20607v1
, w e kno w that f or eac h 𝒜𝑖 ∈ 𝑉≤ 𝐷 w e ha v e ( 1 − 𝜀 )𝐷𝐄 \| 𝒜𝑖 ( 𝑔 ) \|2≤ 𝐄 [ 𝒜𝑖 ( 𝑔 ) 𝒜𝑖 ( 𝑔′) ] ≤ 𝐄 \| 𝒜𝑖 ( 𝑔 ) \|2. Summing this o v er 𝑖 giv es ( 1 − 𝜀 )𝐷≤ 𝐄 ⟨ 𝒜 ( 𝑔 ) , 𝒜 ( 𝑔′) ⟩ ≤ 1 . C ombining this w ith the abo v e , and using 1 − ( 1 − 𝜀 )𝐷≤ 𝜀 𝐷 , y ields ( 1. 4) . □ R emar k ...	https://arxiv.org/abs/2505.20607v1
uppose 𝒜 : 𝐑𝑁→ 𝐑𝑁 is a deterministic alg orithm with pol ynomial degr ee (r esp . coor di- nate degr ee ) 𝐷 and norm 𝐄 ‖ 𝒜 ( 𝑔 ) ‖2≤ 𝐶 𝑁 . Then, f or 𝑟 = 𝑂 ( 1 ) and standar d N ormal r . v .s 𝑔 and 𝑔′ whic h ar e ( 1 − 𝜀 ) - corr elated (r esp . ( 1 − 𝜀 ) -r esampled), ̂ 𝒜𝑟 as in Definition 1. 8 sat...	https://arxiv.org/abs/2505.20607v1
same ar gument pr o v es har dness when 𝒜 is a lo w degr ee poly nomial alg orithm; this is omitt ed f or br e v it y .) 2 Har dness f or Lo w Degr ee Alg orithms In this section, w e pr o v e Theor em 1.3 and Theor em 1. 4 – that is , w e e xhibit str ong lo w degr ee har dness f or both lo w poly nomial degr ee and ...	https://arxiv.org/abs/2505.20607v1
, sugg esting that a sharp analy sis v ia global OGP ar gument s ma y be c hallenging t o implement. Let us giv e a mor e det ailed outline of our str at egy , in the case of lo w c oor dinat e degr ee . Let 𝐸 be an ener gy le v el, and assume 𝒜 is a Σ𝑁 - v alued alg orithm w ith c oor dinat e degr ee at most 𝐷 ≤ ̃...	https://arxiv.org/abs/2505.20607v1
degr ee alg orithms w ithout the r andomized r ounding st ep . T o handle the latt er , w e obser v e that solutions t o a giv en NPP inst anc e ar e isolat ed w ith high pr obabilit y . This implies that r andomized r ounding either c hang es only 𝑂 ( 1 ) c oor dinat es ( whic h pr eser v es st abilit y ), or else in...	https://arxiv.org/abs/2505.20607v1
1 − 𝑥 ) log2 ( 𝑥 ) be the binar y entr opy func tion. Then, f or 𝑝 ≔ 𝐾 / 𝑁 , ∑ 𝑘 ≤ 𝐾(𝑁 𝑘) ≤ exp2 ( 𝑁 ℎ ( 𝑝 ) ) ≤ exp2 ( 2 𝑁 𝑝 log2 (1 𝑝) ) . Pr oof : F or a Bin ( 𝑁 , 𝑝 ) r andom v ariable 𝑆 , w e ha v e: 1 ≥ 𝐏 ( 𝑆 ≤ 𝐾 ) = ∑ 𝑘 ≤ 𝐾(𝑁 𝑘) 𝑝𝑘( 1 − 𝑝 )𝑁 − 𝑘≥ ∑ 𝑘 ≤ 𝐾(𝑁 𝑘) 𝑝𝐾( 1 − 𝑝 )𝑁 − �...	https://arxiv.org/abs/2505.20607v1
t o see that 𝜂 ∈ ( 0 , 1 / 2 ) and (2.3 ) holds f or all 𝐸 ≤ 𝑁 . The r esulting as y mpt otics f ollo w immediat ely . □ 2.2 C onditional Landscape Obstruction W e turn no w t o est ablishing the c entr al c onditional landscape obstruction of our ar gument. The idea is that f or an 𝑥 ∈ Σ𝑁 depending on 𝑔 and a r ...	https://arxiv.org/abs/2505.20607v1
that ‖ 𝑥 − 𝑥′‖2= 4 𝑘 . Thus ‖ 𝑥 − 𝑥′‖ ≤ 2√𝜂 𝑁 if and only if 𝑘 ≤ 𝑁 𝜂 < 𝑁 / 2 , so b y Lemma 2.2 , \| 𝐵Σ ( 𝑥 , 𝜂𝑁 ) \| = ∑ 𝑘 ≤ 𝑁 𝜂(𝑁 𝑘) ≤ exp2 ( 2 𝜂 log2 ( 1 / 𝜂 ) 𝑁 ) . (2.7) T o bound the inner pr obabilit y under 𝑔′∼1 − 𝜀 𝑔 , fix an y 𝑥′∈ Σ𝑁 and w rit e 𝑔′= 𝑝 𝑔 + √ 1 − 𝑝2 ̃ 𝑔 f or 𝑝 ≔ ...	https://arxiv.org/abs/2505.20607v1
exp2 ( − 𝐸 + 𝑂 ( 1 ) ) , and w e c onc lude (2.5 ) as in the pr e v ious case . □ The f ollo w ing lemma sho w s a positiv e c orr elation pr opert y that enables us t o a v oid union-bound- ing o v er 𝒜 finding solutions t o c orr elat ed or r esampled pair s of inst anc es . Belo w , the set 𝑆 pla y s the r ole o...	https://arxiv.org/abs/2505.20607v1
) ∈ 𝑆 ) , 𝑄 = 𝐏 ( ( 𝑔 , 𝜔 ) ∈ 𝑆 , ( 𝑔′, 𝜔 ) ∈ 𝑆 ) , 𝑞 ( 𝜔 ) = 𝐏 ( ( 𝑔 , 𝜔 ) ∈ 𝑆 \| 𝜔 ) , 𝑄 ( 𝜔 ) = 𝐏 ( ( 𝑔 , 𝜔 ) ∈ 𝑆 , ( 𝑔′, 𝜔 ) ∈ 𝑆 \| 𝜔 ) . Lemma 2.5 sho w s that f or an y 𝜔 ∈ Ω𝑁 , 𝑄 ( 𝜔 ) ≥ 𝑞 ( 𝜔 )2. Then, b y J ensen ’ s inequalit y , 𝑄 = 𝐄 [ 𝑞 ( 𝜔 ) ] ≥ 𝐄 [ 𝑞 ( 𝜔 )2] ≥ 𝐄 [ 𝑞...	https://arxiv.org/abs/2505.20607v1
. □ Pr oof of Theor em 1.3 : Let 𝑝solve ≔ 𝐏 ( 𝒜 ( 𝑔 ) ∈ 𝑆 ( 𝐸 ; 𝑔 ) ) be the pr obabilit y that 𝒜 solv es one inst anc e . By Lemma 2.5 , 𝐏𝑔′ ∼1 − 𝜀 𝑔 ( 𝑆solve ) ≥ 𝑝2 solve . In addition, let 𝑝cond ≔ max 𝑥 ∈ Σ𝑁1 − 𝐏𝑔′ ∼1 − 𝜀 𝑔 ( 𝑆cond ( 𝑥 ) ) . 𝑝unstable ≔ 1 − 𝐏𝑔′ ∼1 − 𝜀 𝑔 ( 𝑆stable ) , Set...	https://arxiv.org/abs/2505.20607v1
addition, define 𝑆diff ≔ { 𝑔 ≠ 𝑔′} . Lemma 2.8 . F or 𝑥 ≔ 𝒜 ( 𝑔 ) , we have 𝑆diff ∩ 𝑆solve ∩ 𝑆stable ∩ 𝑆cond ( 𝑥 ) = ∅ . Pr oof : This f ollo w s fr om Lemma 2. 7 , noting that the pr oof did not use that 𝑔 ≠ 𝑔′ almost sur ely . □ As bef or e , our pr oof f ollo w s fr om sho w ing that f or appr opriat e ...	https://arxiv.org/abs/2505.20607v1
𝐷 𝜀 =𝐷 𝑁log2 (𝑁 𝐷) = 𝑜 ( 1 ) . Lik e w ise , f or (b ) log2𝑁 ≪ 𝐸 ≪ 𝑁 and 𝐷 = 𝑜 ( 𝐸 / log2( 𝐶 𝑁 / 𝐸 ) ) , w e g et 16 𝑝unstable ≲𝐷 log2 ( 𝑁 / 𝐷 ) log2 ( 𝐶 𝑁 / 𝐸 ) 𝐸 As 𝐷 is an upper bound on the maximum possible alg orithm degr ee , w e ma y incr ease 𝐷 w ithout loss of g ener alit y in the ana...	https://arxiv.org/abs/2505.20607v1
appear , f ocusing on the L CD case . In the ne x t st ep t o w ar ds Theor em 1. 4 , w e sho w that our pr ec eding analy sis e x t ends t o 𝑂 ( 1 ) - dist anc e pertur bations of an L CD alg orithm, thank s t o the pr eser v ation of st abilit y . Thr oughout the belo w , fix a dist anc e 𝑟 = 𝑂 ( 1 ) . W e c onsid...	https://arxiv.org/abs/2505.20607v1
of 𝑝unstable and 𝑝cond as in (2.9 ) , ag ain r eplacing 𝒜 w ith ̂ 𝒜𝑟 . Thus , w e c hoose 𝜀 ≔ log2 ( 𝑁 / 𝐷 ) / 𝑁 , so that 𝐏 ( 𝑆diff ) → 1 . Additionally , c hoose 𝜂 as Lemma 2.3 , so that 𝑝cond = 𝑜 ( 1 ) . It then suffic es t o sho w that 𝑝unstable = 𝑜 ( 1 ) . T o see this , r ecall that when 𝐸 = 𝛿 �...	https://arxiv.org/abs/2505.20607v1
abo v e Theor em 1. 4 .) Lemma 2.10 . F ix 𝑥 ∈ 𝐑𝑁. Dr aw 𝑁 coin flips 𝐼𝑥 , 𝑖 ∼ Bern ( 2 𝑝𝑖 ( 𝑥 ) ) as well as 𝑁 signs 𝑆𝑖 ∼ Unif { ± 1 } , all mutuall y independent; define the r andom variable ̃ 𝑥 ∈ Σ𝑁 by ̃ 𝑥𝑖 ≔ 𝑆𝑖 𝐼𝑥 , 𝑖 + ( 1 − 𝐼𝑥 , 𝑖 ) 𝗌𝗂𝗀𝗇 ( 𝑥 )𝑖 . Then ̃ 𝑥 ∼ 𝗋𝗈𝗎𝗇𝖽 ( 𝑥 ) . Pr o...	https://arxiv.org/abs/2505.20607v1
, the number of coor dinates ̃ 𝑥 is r esampled in di ver g es in pr obabilit y (i. e . e x ceeds an y fix ed 𝑚 < ∞ with pr obabilit y 1 − 𝑜 ( 1 ) ). Pr oof : R ecall that f or 𝑥 ∈ [ 0 , 1 / 2 ] , w e ha v e log2 ( 1 − 𝑥 ) = Θ ( 𝑥 ) . Thus , as eac h c oor dinat e of 𝑥 is r ounded independently , w e can c omput ...	https://arxiv.org/abs/2505.20607v1
≤ 2− 𝐸, − 2− 𝐸≤ ⟨ 𝑔 , 𝑥𝐽 ⟩ − ⟨ 𝑔 , 𝑥𝐽 ⟩ ≤ 2− 𝐸. Multiply ing the lo w er equation b y − 1 and adding the r esulting inequalities giv es \| ⟨ 𝑔 , 𝑥𝐽 ⟩ \| ≤ 2− 𝐸. 19 Thus , finding pair s of distinct solutions w ithin dist anc e 2√ 𝑘 implies finding a subset 𝐽 ⊆ [ 𝑁 ] of at most 𝑘 c oor dinat es and \| 𝐽 \|...	https://arxiv.org/abs/2505.20607v1
f or an y possible 𝑥 = 𝒜 ( 𝑔 ) . N o w , let 𝐽 ⊆ [ 𝑁 ] denot e the set of the fir st 𝐾 c oor dinat es t o be r esampled, so that 𝐾 = \| 𝐽 \| , and c onsider 𝐏 ( ̃ 𝑥 ∈ 𝑆 ( 𝐸 ; 𝑔 ) \| ̃ 𝑥𝐽 ) , wher e w e fix the c oor dinat es out side of 𝐽 and let ̃ 𝑥 be unif ormly sampled fr om a 𝐾 - dimensional subcube ...	https://arxiv.org/abs/2505.20607v1
fr om Phase Tr ansitions , ” in 2008 4 9th A nnual IEEE S ymposium on F oundations of C omputer Science , Oct. 2008 , pp . 79 3–80 2. doi: 10 .1109 /F OCS .2008 .11 . [ A CR11] D . Ac hliopt as , A . C oja- O ghlan, and F . Ric ci- T er senghi, “ On the solution-spac e g eometr y of r andom c onstr aint satisfaction pr...	https://arxiv.org/abs/2505.20607v1
C ommunications on Pur e and A pplied Mathematics , v ol. 7 6 , no . 10 , pp . 2410–24 7 3 , 20 23 . [Bar+19] B . Bar ak , S . Hopkins , J . K elner , P . K. K othari, A . Moitr a, and A . P ot ec hin, “ A near ly tight sum- of-squar es lo w er bound f or the plant ed c lique pr oblem, ” SIAM J ournal on C omputing , v...	https://arxiv.org/abs/2505.20607v1
. 2, pp . 217–240 , 2009 , doi: 10 .100 2 / r sa.20 25 5 . [Bor+09b] C. Bor g s , J . Cha y es , S . Mert ens , and C. N air , “Pr oof of the Local REM C onjectur e f or N umber P artitioning . II. Gr o w ing Ener gy Scales , ” R andom S truc tur es & Alg orithms , v ol. 34, no . 2, pp . 241–284, 2009 , doi: 10 .100 2 ...	https://arxiv.org/abs/2505.20607v1
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arXiv:2505.20668v1 [math.ST] 27 May 2025Eigenstructure inference for high-dimensional covariance with generalized shrinkage inverse-Wishart prior Seongmin Kim1, Kwangmin Lee2, Sewon Park3, and Jaeyong Lee1 1Department of Statistics, Seoul National University 2Department of Big Data Convergence, Chonnam National Univers...	https://arxiv.org/abs/2505.20668v1
suggested sparse covariance estimation using thresholding methods. In the Bayesian lit erature,Banerjee and Ghosal (2015) suggested sparse precision estimation using the Gaussian graphical model. Lee et al.(2022) introduced a beta-mixture shrinkage prior for sparse cova riance and showed the posterior has nearly optima...	https://arxiv.org/abs/2505.20668v1
we directly compute the posterior expectation. Commonly, to investigate the asymptotic properties of the p osterior, researchers resort to general theorems on posterior convergence rates (e.g., Ghosal et al. (2000);Ghosal and van der Vaart (2007)). However, in this paper, we directly evaluate the posterio r expectation...	https://arxiv.org/abs/2505.20668v1
in high-dimensional settings, be comes even more severe. On the other hand, when b >1, the prior forces the eigenvalues to become nearly identic al. Accordingly, as proposed in Berger et al. (2020), we restrict bto the range [0 ,1] to avoid these undesirable behaviors. ConsiderthefollowingspectraldecompositionofΣ = UΛU...	https://arxiv.org/abs/2505.20668v1
Ã(¹\|Γ12,Γ−12,Λ;H0)∝etr(−1 2H1R¹Γ12Λ−1ΓT 12RT ¹). 9 Let the spectral decomposition of Γ 12Λ−1ΓT 12be Γ12Λ−1ΓT 12= cosÉ−sinÉ sinÉcosÉ  s10 0s2  cosÉsinÉ −sinÉcosÉ , withs1> s2andÉ∈(−Ã 2,Ã 2]. Then, the conditional density of ¹simpliﬁes to Ã(¹\|Γ12,Γ−12,Λ;H0)∝exp/parenleftbig ccos2(¹+É)/parenrightbig , wherec=−...	https://arxiv.org/abs/2505.20668v1
posterior convergence rates for eigenstruct ure, we assume the following conditions: A1. We consider high-dimensional settings where n/p→0. A2. There exist positive constants c0andC0such that, for all n, C0> ¼0,k+1>···> ¼0,p> c0. A3. The klargest eigenvalues are suﬃciently separated by a constant value¶0>0: ¼0,j−¼0,j+1...	https://arxiv.org/abs/2505.20668v1
may depend on nandp. Letϵ >0, and deﬁneÄ= min l<klog/parenleftbigg¼0,l ¼0,l+1/parenrightbigg . Suppose thatnp anz¼0,k n¼0,1Ä 2nϵ2andϵ{/radicalbiggnp an((nÄ)−1/2. Then, /integraldisplay (/uniontextL l=1Dϵ,l)cÃ(Γ\|Xn)(dΓ) =O/parenleftBig exp/parenleftBig −Ä 2nϵ2/parenrightBig ·/parenleftBig2¼0,1 ¼0,k/parenrightBigkq/paren...	https://arxiv.org/abs/2505.20668v1
the convergence rate of the eigenvalues, while the second and third terms are attribute d to the convergence of the eigenvectors. Comparing the rate in ( 7) with the sample covariance rate in ( 8), we observe that the ﬁrst term in the sample covariance rate is larger. 17 4 Simulation Studies To evaluate the performance...	https://arxiv.org/abs/2505.20668v1
0.221 0.206 À2 0.419 0.443 0.466 0.427 0.673 0.428 À3 0.639 0.642 0.658 0.679 0.921 0.620 Table 2: Average errors (Err À) for estimated eigenvectors. Table1presents the average errors (Err ¼), coverage probabilities (CP), and credible (or conﬁdence) interval length (IL) for the estimated spike d eigenvalues across diﬀe...	https://arxiv.org/abs/2505.20668v1
are relatively weak. Table3represents the average errors (Err ¼), coverage probabilities (CP), and credible (or conﬁdence) interval length (IL) for the estimated spike d eigenvalues, across diﬀerent methods and values of n. The gSIW, SIW, and S-POET methods consistently outperform the sample covariance. In contrast, th...	https://arxiv.org/abs/2505.20668v1
k, the gSIW prior is applied, and the WAIC and IC p3are computed. WAIC and IC p3selectk= 7 and k= 9, respectively, while the GR method selects k= 1 as the optimal number of spiked eigenvalues. (a) Dimensionality reduction with k= 7 (WAIC). (b) Dimensionality reduction with k= 9 (ICp3). Figure 2: The images after dimens...	https://arxiv.org/abs/2505.20668v1
improving asymptotic bounds for sam ple eigenstructures in more general settings. Acknowledgements This work was supported by the National Research Foundation o f Korea (NRF) grant funded by the Korea government(MSIT) (No. NRF-2023R1A2C100305 0). 25 References Ahn, S. C. and Horenstein, A. R. (2013). Eigenvalue ratio t...	https://arxiv.org/abs/2505.20668v1
C. (2016). Nonparametric eigenvalue-regularized prec ision or covariance matrix estimator, AnnalsofStatistics 44(3): 928–953. Ledoit, O. and Wolf, M. (2004). A well-conditioned estimato r for large-dimensional co- variance matrices, Journalofmultivariate analysis88(2): 365–411. Ledoit, O. and Wolf, M. (2012). Nonlinear...	https://arxiv.org/abs/2505.20668v1
sum of two matrices AandB, whereA=1 nk/summationdisplay i=1¼0,iZiZT iandB=1 np/summationdisplay i=k+1¼0,iZiZT i. Lemma S1.2 (Asymptotic properties of eigenvalues of sample covariance) .Under model (1), the eigen- values of sample covariance satisfy the following properties for all suﬃciently large n, ˆ¼j ¼0,j=  ...	https://arxiv.org/abs/2505.20668v1
is given by D¿= 2/radicalbig min(n,p−n). Lemma S1.5 (Probabilityofsubsetonorthogonalgroup) .Consider the following subset of the orthogonal groupO(p), deﬁned as Bϵ=/braceleftbigg Γ∈O(p) : inf Q1∈O(n), Q2∈O(p−n)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsing...	https://arxiv.org/abs/2505.20668v1
on Cϵ p/productdisplay i=n+1ci nf/parenleftBigh n/parenrightBigp−n/bracketleftbigg 1+2ϵ2n (p−n)h¼0,1/bracketrightbiggp−n , where´max= max 1fjfk´j, and for all suﬃciently large n. Proof.For Γ∈Cϵ, the following inequality holds: (Γii−1)2+p/summationdisplay j=1 j̸=iΓ2 ji< ϵ2, fori= 1,...,n. Sincep/summationdisplay j=1Γ2 j...	https://arxiv.org/abs/2505.20668v1
the equality holds for all Γ ∈B¸: inf Q∈O(p−n)\|\|Γ22−Q\|\|2 F= inf Q∈O(p−n)\|\|D−Q\|\|2 F = inf Q∈O(p−n)p−n/summationdisplay i=1[(/radicalBig ˜¼i−qi)2+(1−q2 i)] =p−n/summationdisplay i=1(/radicalBig ˜¼i−1)2 11 fmax ifp−n/vextendsingle/vextendsingle/vextendsingle˜¼i−1/vextendsingle/vextendsingle/vextendsingle (/radicalbig˜¼i+1...	https://arxiv.org/abs/2505.20668v1
ji(ap−an) k+1×M(p−n−p/summationtext i=n+1p/summationtext j=n+1Γ2 ji)(ap+n 2−1) Mk(a1+n 2−1)+n/summationtext i=k+1p/summationtext j=n+1Γ2 ji(an+n 2−1)/bracketrightBigg = sup Acϵ/bracketleftBigg 1 Kk/summationtext j=1p/summationtext i=k+1Γ2 ji(an−ak)+k/summationtext j=1p/summationtext i=n+1Γ2 ji(ap−ak) k×Ln/summationtext...	https://arxiv.org/abs/2505.20668v1
ii)+p/summationdisplay j=1 j̸=iΓ2 ji< ¸2holds and it implies Γ2 iig1−¸2. Therefore, we obtain the upper bound of ( S4) as follows (S4)fsup D¸k/productdisplay i=1/bracketleftbiggh n+Γ2 iiˆ¼i+/summationdisplay j=1 j̸=iΓ2 jiˆ¼j/bracketrightbiggai+n/2−1 17 fsup D¸k/productdisplay i=1/bracketleftbiggh n+Γ2 iiˆ¼i+(1−Γ2 ii)ˆ¼...	https://arxiv.org/abs/2505.20668v1
non-negative diagonal elements, and \|\|P−S\|\|Fgϵfor all permutation matrix P. Ifa1=···=ak, then the inequality holds sup D¸k/producttext i=1/parenleftBigci n/parenrightBigai+n/2−1 inf E¸k/producttext i=1/parenleftBigci n/parenrightBigai+n/2−1≼exp/parenleftBigg n¸2ˆ¼1 ˆ¼k/parenrightBigg exp/parenleftBigg nϵ2min i<klog/par...	https://arxiv.org/abs/2505.20668v1
ii−1)2+(1−Γ2 ii)< ϵ2, it follows that Γ2 ii>1−ϵ2. Givenci n=h n+n/summationdisplay j=1Γ2 jiˆ¼j,we obtain the bounds: ci n∈/bracketleftbigg (1−ϵ2)ˆ¼i+h n,(1−ϵ2)ˆ¼i+ϵ2ˆ¼1+h n/bracketrightbigg ,fori= 1,...,n, ci n∈/bracketleftbiggh n, ϵ2ˆ¼1+h n/bracketrightbigg ,fori=n+1,...,p. Under Γ ∈Dϵ, each¼ifollows an inverse gamma ...	https://arxiv.org/abs/2505.20668v1
Ã(Λ,Γ\|Xn)(dΛ)(dΓ) = 1+ O/parenleftBig exp/parenleftBig −Ä 2nϵ2/parenrightBig ·/parenleftBig2¼0,1 ¼0,k/parenrightBigkq/parenrightBig +O/parenleftBig/parenleftBign¼0,k+p p/parenrightBig−ϵ2an/parenrightBig , whereÄ= min l<klog/parenleftbigg¼0,l ¼0,l+1/parenrightbigg . Following the same bounding steps as in Lemma S1.16, w...	https://arxiv.org/abs/2505.20668v1
(dΓ). The last equality holds due to /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle Sl0 0Q2 −Γ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/v...	https://arxiv.org/abs/2505.20668v1
( S1.14), the upper bound of ( S18) is given by (S18)≼exp/parenleftBigg n(ϵ+ϵ5)2ˆ¼1 ˆ¼k/parenrightBigg exp/parenleftBigg n(ϵ4−(ϵ+ϵ5))2min i<klog/parenleftBiggˆ¼i ˆ¼i+1/parenrightBigg/parenrightBigg×/parenleftBig2¼0,1 ¼0,k/parenrightBigkq ≼exp/parenleftbigg nϵ2¼0,1 ¼0,k/parenrightbigg exp/parenleftbigg nϵ2 4min i<klog/p...	https://arxiv.org/abs/2505.20668v1
1−/integraldisplay (/uniontextL l=1Dϵ,l)c/parenleftBig/productdisplay ci/parenrightBig−ai−n 2+1 (dΓ) /integraldisplay/parenleftBig/productdisplay ci/parenrightBig−ai−n 2+1 (dΓ)/parenrightBigg ginf Dϵf(c)×/parenleftBig 1+O/parenleftBig exp/parenleftBig −Ä 2nϵ2/parenrightBig ·/parenleftBig2¼0,1 ¼0,k/parenrightBigkq/paren...	https://arxiv.org/abs/2505.20668v1
for some constant ³0∈[−C,C], we have t=¯dp n+³0/radicalbiggp n. Therefore, the posterior expectation simpliﬁes to: E/bracketleftBig¼i−¼0,i ¼0,i/vextendsingle/vextendsingleXn/bracketrightBig =1 ¼0,i(³i−³0)/radicalbiggp n+´i+1 ¼0,ih n +O/parenleftBig ϵ2¼0,1 ¼0,i+¼0,1 ¼0,iexp/parenleftBig −» 2ϵ/parenrightBig +¼0,1 ¼0,i/pa...	https://arxiv.org/abs/2505.20668v1
diﬀerence of eigenvectors) .For the spiked eigenvector matrix Γ1, the following inequality holds on the set Dϵ: \|\|Γ0,1−Γ1\|\|2=O(dk)+Op(·k), where·k=1 ¼0,k/radicalbiggp n+p n3/2¼0,k+1 n. Proof. \|\|Γ0,1−Γ1\|\|2f \|\|Γ0,1−Γ1\|\|2 F fk/summationdisplay i=1\|\|À0,i−Ài\|\|2 2 = 2k/summationdisplay i=1(1−/vextendsingle/vextendsingleÀT 0,...	https://arxiv.org/abs/2505.20668v1
0,i)·/parenleftbig O(dk)+Op(·k)/parenrightbig +k/summationdisplay i=1(¼0,i−ˆ¼i)2/bracketrightBig +6/bracketleftBig 8/summationdisplay i>k¼2 0,i+/summationdisplay i>k(¼0,i−ˆ¼i)2/bracketrightBig . (S21) By Lemma S1.2, we obtain (¼0,i−ˆ¼i)2=  O(p2 n2+¼2 0,i n1−2¶) fori= 1,...,k O(p2 n2) for i=k+1,...,n O(1) f...	https://arxiv.org/abs/2505.20668v1
arXiv:2505.20681v1 [stat.ME] 27 May 2025HYBRID BAYESIAN ESTIMATION IN THE ADDITIVE HAZARDS MODEL ´Alvarez Enrique Ernesto1, Riddick Maximiliano Luis2 1Instituto de C´ alculo, Universidad de Buenos Aires - CONICET Universidad Nacional de Luj´ an, Argentina mail: enriqueealvarez@fibertel.com.ar 2NUCOMPA and Departamendo ...	https://arxiv.org/abs/2505.20681v1
the previous ones by proposing a rescaling for the duration times themselves. I.e., T∗=T∗ 0exp(Z′β),where T∗ 0∼F0(·) is the baseline cumu- lative distribution function. Interestingly, for the corresponding hazard functions, this entails that λ(t,β) =λ0 texp(Z′β) exp(Z′β), (3) which is neither multiplicative nor addit...	https://arxiv.org/abs/2505.20681v1
hazard function itself, but not necessarily all the coefficients. In our context, there are two ways to accomplish that goal: (i) by adding the specific constraint that the estimated ˆλ(t) =ˆλ0(t) +β′zbe nonnegative and performing constrained inference; or ( ii) by using only positive covariates zk, either as given by ...	https://arxiv.org/abs/2505.20681v1
observations that belong to each grid interval, and calling ( zkj, δkj) to their corresponding covariate and censoring indicator, this yields Ln(β, λ0) = exp( −β′nX i=1tizi) exp( −nX i=1Λ0(ti))mY j=1njY kj=1 aj+β′zkjδkj(6) (7) Notice that the formula in Equation (6) above does not correspond to any stan- dard density...	https://arxiv.org/abs/2505.20681v1
Being a mixture of Gammas, the display above provides a tractable expression which we extensively exploit in the sequel. 2.1 Uninformative Priors 7 Without this convenient expression, several algorithms were presented in lit- erature with innovative adaptations of acceptance rejection sampling, Metropolis- Hastings and...	https://arxiv.org/abs/2505.20681v1
are currently under research. 3 The Hybrid Bayesian Method We attempt to develop a Bayesian method that achieves two goals, ( i) it disen- tangles estimation of βfrom the baseline hazard function λ0(·), as the latter is often treated as a nuisance in many applications, and ( ii) it generates estimators in closed form. ...	https://arxiv.org/abs/2505.20681v1
the mean or the mode of a multivariate normal distribution with mean vector m= (V−1 2V1) and covariance matrix is D=n−1(V−1 2V3V−1 2). This entails that in a Bayesian context we could consider LY estimators as belonging to a posterior normal distribution for the Euclidean parameter with a flat (improper) prior. In this...	https://arxiv.org/abs/2505.20681v1
of (1 −α) coverage. The inter- val is of the form [ bl, bu], where 0 ≤bl< bu<∞. It is noteworthy that in Lin and Ying’s formulation, since the confidence intervals are based on the approximate normal distribution, they take the form ˆβLY±z1−α/2ˆσ(ˆβLY), which allows for a possibly negative lower endpoint. Here for the ...	https://arxiv.org/abs/2505.20681v1
order now to simplify notation, let us call d(j) k:=dk (sj−sj−1)k×Γ(c αj), cj:=Pnj kj=1(tkj−sj−1) +mj(sj−sj−1) (sj−sj−1)+c. With that notation we express the posterior fΛ+ 0j\|X,β(a+ 0) =PNj k=0d(j) k(a+ 0j)(k+c αj−1)e−a+ 0jcj R∞ −∞PNj k=0d(j) k(a+ 0j)(k+c.αj−1)e−a+ 0j.cjda+ 0 =PNj k=0d(j) k ck jΓ(k+c αj)fG(k+c αj,c...	https://arxiv.org/abs/2505.20681v1
a single covariate with regression parameter β= 0.5, (ii) Censoring variable Cwith an Exponential distribution with mean 2, ( iii) Covariate Z∼χ2 1, (iv) sample sizes n= 100 or 500, and ( v)R= 1000 replicates. Simulations for the regression parameter β.The results are presented in Tables (1) and (2). We observe that in...	https://arxiv.org/abs/2505.20681v1
) (0.1024 ) (0.1030 ) (0.1033 ) (0.1035 ) (0.1035 ) (0.1036 ) 2 0.6543 0.5419 0.5264 0.5186 0.5138 0.5122 0.5106 (0.0982 ) (0.1024 ) (0.1030 ) (0.1033 ) (0.1035 ) (0.1035 ) (0.1036 ) 10 1.4375 0.7128 0.6128 0.5620 0.5312 0.5209 0.5107 (0.0982 ) (0.1024 ) (0.1030 ) (0.1033 ) (0.1035 ) (0.1035 ) (0.1036 ) Table 2: Simula...	https://arxiv.org/abs/2505.20681v1
2.6005 log(EXP+1) 1.7172 1.7081 1.7172 Table 5: (Results are multiplied by 103) The results are presented in Table (5), where we visualize the impact of the choice of the hyperparameters on the final estimates. Those results show the versatility of our proposed Hybrid Bayesian method, which can be calibrated to give th...	https://arxiv.org/abs/2505.20681v1
& Sons. Klein, J. P. and Moeschberger, M. L. (2006). Survival analysis: techniques for censored and truncated data . Springer Science & Business Media. 20 Lawless, J. F. (2003). Event history analysis and longitudinal surveys. Analysis of Survey data , 221-243. Lin, D. Y. and Ying, Z. (1994). Semiparametric Analysis of...	https://arxiv.org/abs/2505.20681v1
arXiv:2505.20708v1 [econ.TH] 27 May 2025Berk-Nash Rationalizability∗ Ignacio Esponda Demian Pouzo UC Santa Barbara UC Berkeley May 28, 2025 Abstract We introduce Berk–Nash rationalizability , a new solution concept for misspeciﬁed learning environments. It parallels rationalizability in games and captures all actions t...	https://arxiv.org/abs/2505.20708v1
environments or rely on paramet ric assumptions such as Gaussian signals or binary states. Examples include Nyarko(1991),Fudenberg, Romanyuk and Strack (2017), Heidhues, K˝ oszegi and Strack (2018),Heidhues, K˝ oszegi and Strack (2021),Bohren and Hauser (2021), and He(2022) among many others. 3Limit actions are deﬁned ...	https://arxiv.org/abs/2505.20708v1
space A, a space of observable consequences Y, and a consequence function that maps actions to probability distributions over consequences, denoted by Q:A→∆Y. The agent does not necessarily knowQbut possesses a parametric model of it, represented as Qθ:A→∆Y, where the model parameter θbelongs to a parameter space Θ. Th...	https://arxiv.org/abs/2505.20708v1
distribution Q(· \|a) corresponds to a normal distribution with mean ( α∗+a)θ∗and the same variance. These distributions admit densities with respect to the Lebesgue measure that vary co ntinuously in aandθ, and they satisfy the uniform integrability condition with a quadratic e nvelope. The agent’s payoﬀ function is π(...	https://arxiv.org/abs/2505.20708v1
about consequences arises f rom actions drawn from the setA, then the models that provide the best ﬁt are those that minimize KL divergence for some action distribution supported on A. Consequently, the agent will follow actions that are optimal for beliefs that assign probability one to these best-ﬁt models. Example (...	https://arxiv.org/abs/2505.20708v1
ytdrawn from the distribution Q(· \|at), •And updates beliefs via Bayes’ rule: µt+1=Bay(at,yt,µt), 6EP also considered mixed actions. Every action in the support of an e quilibrium mixed action is ratio- nalizable. 7EP consider forward-looking agents under a condition—weak identiﬁ cation—that guarantees incentives to ex...	https://arxiv.org/abs/2505.20708v1
\|a). The marginal distribution of PoverA∞is denoted by PA∞. An action a∈Ais called a limit action of the sequence a∞= (a1,a2,...) if there exists a subsequence ( atk)ksuch that atk→aask→ ∞. Equivalently, ais a limit action if, for every open neighborhood U⊆Aofa, there are inﬁnitely many times t∈Nsuch that at∈U. WhenAis...	https://arxiv.org/abs/2505.20708v1
and Γ is nonempty-valued and upper hemicontinuous, eac h set Bkis nonempty and compact. The inﬁnite intersection of nested, none mpty compact sets is nonempty, so Bis nonempty. This typeofcharacterizationisstandardwhen rationalizabilityisdeﬁ nedasclosureunder a set-valued operator. Just as in other familiar settings—su...	https://arxiv.org/abs/2505.20708v1
rationalizable actions. The limits also satisfy T(a∞ min) =a∞ max, T(a∞ max) =a∞ min, 12 so they form a 2-cycle of Tand are ﬁxed points of T2. IfT2has a unique ﬁxed point, then it must also be a ﬁxed point of T, and the limit is a singleton. In that case, rationalizability coincides with equilibrium. If T2has multiple ...	https://arxiv.org/abs/2505.20708v1
actions, denoted aSandaL, respectively. These actions are also the smallest and largest Berk-Nash equilibrium actions. They can be cons tructed as the limits of the monotone sequences: ak+1 S=a(θ(ak S)), ak+1 L= ¯a(¯θ(ak L)), starting from a0 S= minAanda0 L= maxA. The limits aS= limk→∞ak SandaL= limk→∞ak L are ﬁxed poi...	https://arxiv.org/abs/2505.20708v1
where˜B0=Aand˜Bk+1=T(˜Bk), whereT(·) = ˜a◦˜θ(·). This characterization is convenient because the operator Γ is deﬁn ed over mixed actions and mixed beliefs, which can be diﬃcult to work with directly. In contra st, the operator 15 Tin Proposition 3allows us to focus entirely on degenerate actions and beliefs, while st ...	https://arxiv.org/abs/2505.20708v1
the fact that there is a function ga∈L2(Y,R,Q(· \|a)) such that supθ′∈O(θ,ε)(g(θ′,y,x))2≤(ga(y))2, whereg(θ,y,x) := log( q(y\|a)/qθ(y\|a)) and O(θ,ε) :={θ′:\|\|θ′−θ\|\|< ε}. In our case, we need a function Gthat does not depend on a; hence our assumption (v). In Step 2 of their proof, they conclude that, for each θ∈Θ, Λ δ(a) ...	https://arxiv.org/abs/2505.20708v1
Consider θ/\e}atio\slash≤¯θ(ah). Since¯θ(xh) is the largest minimizer of K(·,xh) andθ∨¯θ(ah)> ¯θ(ah), thenK(¯θ(ah),ah)< K(θ∨¯θ(ah),ah). Since−Kis single crossing in ( θ,a), this implies K(¯θ(ah),a)< K(θ∨¯θ(ah),a) for all a∈[al,ah]. Since −Kis quasi-supermodular in θ, this implies K(θ∧¯θ(xh),a)< K(θ,a) for all a∈[al,ah]...	https://arxiv.org/abs/2505.20708v1
similar if a < a). Strict quasiconcavity of U(·,θ) implies that, for all θ∈˜θ(A), U(a,θ)< U(¯a,θ)≤U(˜a(θ),θ). Therefore, for any µ∈∆˜A, /integraldisplay U(a,θ)µ(dθ)</integraldisplay U(¯a,θ)µ(dθ), implying that a /∈F(∆˜θ(A)) =F(∆(∪σ∈∆AΘm(σ))), where the equality follows from equa- tion (16). SinceF(∆(∪σ∈∆AΘm(σ)))⊇Γ(A), ...	https://arxiv.org/abs/2505.20708v1
arXiv:2505.20946v1 [math.ST] 27 May 2025Almost Unbiased Liu Estimator in Bell Regression Model: Theory and Application Caner Tanı¸ s†and Yasin Asar‡ †Department of Statistics, C ¸ankırı Karatekin University e-mail: canertanis@karatekin.edu.tr ‡Department of Mathematics and Computer Sciences, Necmettin Erbakan Universit...	https://arxiv.org/abs/2505.20946v1
devel- oped to ﬁt the speciﬁc needs of these regression models. More recent studies have proposed a novel Liu estimator for Bell regression, with performance e valuations conducted through simula- tion studies. Comparative analyses between ridge andLiu es timators have also beenundertaken, particularly in the context o...	https://arxiv.org/abs/2505.20946v1
distribution. •The Bell distribution is a member of the one-parameter expon ential distributions. •The Bell distribution is unimodal. •The Poisson distribution does not follow the Bell family of d istributions. But if the pa- rameter has a small value, the Bell distribution approximat es to the Poisson distribution. •T...	https://arxiv.org/abs/2505.20946v1
solution of U/parenleftBig /hatwideβ/parenrightBig =0p, where 0prefers a p-dimensional vector of zeros. Regrettably, the maximum likelihood estimator /hatwideβlacks a closed-form solution, necessitating its numerical computation. For instance, the Newton–Raphs on iterative method is one possible approach. Alternatively...	https://arxiv.org/abs/2505.20946v1
d in the Bell regression model. 6 In the Bell regression, the AULE is /hatwideβAULE=/parenleftBig I−(1−d)2(F+I)−2/parenrightBig /hatwideβMLE. The covariance matrix and bias vector of the AULE are Cov/parenleftBig /hatwideβAULE/parenrightBig =Cov/parenleftBig I−(1−d)2(F+I)−2/hatwideβMLE/parenrightBig =/parenleftBig I−(1...	https://arxiv.org/abs/2505.20946v1
λj+d/parenrightbig2/parenleftbig λj+2−d/parenrightbig2 λj/parenleftbig λj+1/parenrightbig4 =p/summationdisplay j=1/parenleftbig λj+1/parenrightbig4−/parenleftbig λj+d/parenrightbig2/parenleftbig λj+2−d/parenrightbig2 λj/parenleftbig λj+1/parenrightbig4. The diﬀerence between the variances of MLE and AULE is positiv e f...	https://arxiv.org/abs/2505.20946v1
5.2482 5.1354 5.3143 200 0.9 15.4715 4.7503 4.7207 4.5908 4.7601 400 0.9 11.7297 3.0397 3.0136 2.9411 3.0421 100 0.95 17.0654 5.5808 5.5327 5.5002 5.5956 200 0.95 15.6238 4.8431 4.8150 4.7800 4.8511 400 0.95 10.3919 2.5382 2.5024 2.4961 2.5321 Table 5: Simulated MSE values when p= 8 nρMLE LE AULE(d1)AULE(d2)AULE(d3) 10...	https://arxiv.org/abs/2505.20946v1
data y(response variable) the number of defects per laminated pla stic plywood area x1 volumetric shrinkage x2 assembly time x3 wood density x4 drying temperature The design matrix is centered and standardized so that X⊤Xis in the correlation form before obtaining the estimators. A Bell regression model without i nterc...	https://arxiv.org/abs/2505.20946v1
1.0 dSBEstimators AULE LE Figure 4: MSEs and SBs of the estimators for −1< d <1 200400600 0.00 0.25 0.50 0.75 1.00 dMSEEstimators AULE LE MLE 0204060 0.00 0.25 0.50 0.75 1.00 dSBEstimators AULE LE Figure 5: MSEs and SBs of the estimators for 0 < d <1 6 Conclusion In this paper, we introduced a new biased estimator call...	https://arxiv.org/abs/2505.20946v1
Amin, M., Qasim, M., Afzal, S., Naveed, K. (2022). New ridgee stimators in theinverse Gaussian regression: Monte Carlo simulation and application to chem ical data. Communications in Statistics–Simulation and Computation, 51(10), 6170–618 7. Amin, M., Akram, M. N., Majid, A. (2023). On the estimation of Bell regression...	https://arxiv.org/abs/2505.20946v1

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