text string | source string |
|---|---|
[ ·]idenotes the ith component of a vector. Reorganizing terms yields Q(n) W ˆβ(n) W−β⋆ W =−E[g(n) W(X(n), β⋆ W)g(n) W(X(n), β⋆ W)⊤]−1/2 g(n) W(X(n), β⋆ W) + G(n) W(X(n), β⋆ W)−E[G(n) W(X(n), β⋆ W)] (ˆβ(n) W−β⋆ W) +X 1≤i,j≤d1X 1≤k≤KnX m∈E(n) k,kZ1 0(1−t)σ′′(⟨β⋆ W+t(ˆβ(n) W−β⋆ W),∆ms(n) k,k(X(n) k,k)⟩)dt × ∆ms(n) k... | https://arxiv.org/abs/2503.13191v1 |
to normal of the esti- mator for any fixed n; if this is the focus of interest, then as in Remark 4.7, the mathematical setup of a sequence of random graphs is not needed. The next assumptions are used for asymptotical guarantees. Assumption 4.13. Assume that there exist ζW, ζB>0independent of nsuch that Υ(n) W≥ζWas we... | https://arxiv.org/abs/2503.13191v1 |
similar calculations as in the proof of Theorem 4.9, E[g(n) W(X(n), β⋆ W)g(n) W(X(n), β⋆ W)⊤]−1/2 ×X 1≤i,j≤d1X 1≤k≤KnX m∈E(n) k,kZ1 0(1−t)σ′′(⟨β⋆ W+t(ˆβ(n) W−β⋆ W),∆ms(n) k,k(X(n) k,k)⟩)dt × ∆ms(n) k,k(X(n) k,k) i ∆ms(n) k,k(X(n) k,k) j∆ms(n) k,k(X(n) k,k)ˆβ(n) W−β⋆ W iˆβ(n) W−β⋆ W j ≤√Kn√ζWd2 1 20Mn 2 L3 WM3... | https://arxiv.org/abs/2503.13191v1 |
define the vector V(g)∈Rnby letting the ith entry of V(g) be the variation of gatigiven by V(g)[i] = sup xj=yj j̸=i|g(x)−g(y)|. (23) Theorem A.1 ([9, Theorem 1]) .Letg:N → Rand let V(g)be as in (23). If∥D∥<∞then for anyα >0we have the inequality P(|g(X)−E[g(X)]| ≥α)≤exp −2α2 ∥D∥2∥V(g)∥2 . In Lemma A.3 we apply Theore... | https://arxiv.org/abs/2503.13191v1 |
This gives the bound from the statement of the lemma. We introduce the Orlicz norm for a real random variable Xby∥X∥ψ= inf C >0|E[ψ(|X|/C)]≤1 , where ψ: [0,∞)→[0,∞) is a convex function with ψ(0) = 0. It is straightforward to see that E[ψ(|X∥/C)]≤1 implies ∥X∥ψ≤C. We will only make use of the function ψ2(x) = exp( x2)... | https://arxiv.org/abs/2503.13191v1 |
we have for n≥4 dW21√nnX i=1Xi, Zd ≤ CPn i=1E[∥Xi∥4]1/2 n+ 2Pn i=1∥E[XiX⊤ i]∥2 F1/2 n +2 n1/21 nnX i=1X k>016k 2k(2k)!∥E[XiX⊤ i]∥2 F1/41 nnX i=1X k>016k 2k(2k)!∥E[XiX⊤ i]∥2 F +1 nnX i=18∥E[XiX⊤ i]∥2 FE[∥Xi∥2]2+ 4∥E[XiX⊤ i∥Xi∥2]∥2 F1/4 , where C= 8 +P k>04k kk!. 30 Lemma B.2. Suppose Assumption 4.3 holds. Let ... | https://arxiv.org/abs/2503.13191v1 |
et al. Stein’s method meets computational statistics: A review of some recent developments. Statistical Science , 38(1):120–139, 2023. 32 [2] A. Anastasiou and R. E. Gaunt. Wasserstein distance error bounds for the multivariate normal approximation of the maximum likelihood estimator. Electronic Journal of Statistics ,... | https://arxiv.org/abs/2503.13191v1 |
Degeneracy in sparse ERGMs with functions of degrees as sufficient statistics. Bernoulli , 26(2):1016 – 1043, 2020. [21] S. Mukherjee and Y. Xu. Statistics of the two star ERGM. Bernoulli , 29(1):24–51, 2023. [22] M. Raiˇ c. A multivariate Berry–Esseen theorem with explicit constants. Bernoulli , 25(4A):2824– 2853, 201... | https://arxiv.org/abs/2503.13191v1 |
arXiv:2503.13582v1 [cs.LG] 17 Mar 2025JOURNAL OF L ATEX CLASS FILES, VOL. 1, NO. 2, DECEMBER 2023 1 Spectrally-Corrected and Regularized QDA Classifier for Spiked Covariance Model Wenya Luo, Hua Li, Zhidong Bai, Zhijun Liu Abstract Quadratic discriminant analysis (QDA) is a widely used meth od for classification problems... | https://arxiv.org/abs/2503.13582v1 |
Renmin Stre et, Changchun, Jilin, China Zhijun Liu is with Northeast Normal University, 3-11 Wenhua Road, Heping District, Shenyang, Liaoning, China 0000–0000©2023 IEEE JOURNAL OF L ATEX CLASS FILES, VOL. 1, NO. 2, DECEMBER 2023 2 proposed by Friedman [7] and Remey et al. [8]. Other related w orks include Wu et al. [9]... | https://arxiv.org/abs/2503.13582v1 |
classification techniques, such as QDA, R-QDA, Im-QDA from Houssem et al. [17], support vector m achine (SVM) and k-nearest neighbor (KNN) through some simulation and empirical experiments. The remainder of this paper is organized as follows: Section II introduces a brief overview of QDA, R-QDA and the basic form of the... | https://arxiv.org/abs/2503.13582v1 |
that r1,iandr2,iare all perfect known. There are several e fficient methods for estimating them. Readers can refer to [12], [23] , [32], [33] for more details. For the empirical analysis later in this paper, we use the method given by Ke et a l. [33] to estimate r1,iandr2,i. And forσi, we give a new estimation method, an... | https://arxiv.org/abs/2503.13582v1 |
3. r1,iandr2,iare fixed and λ1,i>···> λ r1,i,i>√ J>0>−√ J> λ−r2,i,i>···> λ−1,i>−1, independently of pandn. JOURNAL OF L ATEX CLASS FILES, VOL. 1, NO. 2, DECEMBER 2023 5 The assumption 3 is the basis of our analysis and the basic pre mise of applying high-dimensional random matrix theory, which guarantees a one-to-one ma... | https://arxiv.org/abs/2503.13582v1 |
−r2,0,0,···,γ(2) −1,0,γ(1) 1,0,···,γ(1) r1,0,0,γ(2) −r2,1,1,···,γ(2) −1,1,γ(1) 1,1,···,γ(1) r1,1,1/parenrightigT, gi=˜i(1+λj,0aj,0)+iσ2 1 σ2 0ϕj,0+α0aj,0bj,0 j∈I0,−˜iα1aj,1bj,1+σ2 0 σ2 1ϕj,1−i(1+λj,1aj,1) j∈I1T bi=c1σ2 i σ2 1+c0σ2 i σ2 0+α˜iσ2 i σ2 ˜i... | https://arxiv.org/abs/2503.13582v1 |
CLT is a direct consequence of Theorem 5. Theorem 6. We assume the same conditions as in Theorem 5. Then we have p−(r1,i+r2,i) σ2 i√2Ji/parenleftig ˆσ2 ∗−σ2 i/parenrightigD→N (0,1). Theorem 7. Under Assumptions 1 to 4, we have /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 αi−1 ˆαi/vextendsin... | https://arxiv.org/abs/2503.13582v1 |
n=500,600. However, our method can outperform others when the sample si ze is small or moderate, due to the use of variance heterogeneity. JOURNAL OF L ATEX CLASS FILES, VOL. 1, NO. 2, DECEMBER 2023 10 TABLE I Comparisonforaccuracyrateof QDA, R-QDA, Im-QDAandSR-QDAwithdifferentvaluesof a. n 100 200 300 400 500 600 a=0.... | https://arxiv.org/abs/2503.13582v1 |
350 400 450 500 550 6000.40.50.60.70.80.91 SR-QDA QDA R-QDA Im-QDA (d)σ2 1=4 Fig. 1. Accuracy rate vs. training sample size nforp=150 andπ0=0.5. Comparison for QDA, RLDA, ILDA and SRLDA with different values of σ2 1. B. Empirical analysis For empirical analysis, we use five real datasets, repeating each experiment 500 ti... | https://arxiv.org/abs/2503.13582v1 |
4 0.9699 0.9726 0.9225 0.9782 0.9738 0.9728 0.9574 0.9738 0.9807 0.9839 0.9817 -QDA is the traditional QDA method; R-QDA is the regularized Q DA method; -Im-QDA is the improved QDA method proposed by [17]; SR-QDA is the method we give; -SVM (gauss) and SVM (linear) are the support vector machines with gaussian kernel a... | https://arxiv.org/abs/2503.13582v1 |
obtain that Yi(/tildewideH−1 0,/tildewideH−1 1)−pσ2 i1 σ2 1−1 σ2 0+κi+2yT iz−ξia.s.−→0, which is the conclusion of Theorem 1. /square Proof of Theorem 2. First, we recall the following results showing in [14] that w ill used throughout the proof: uT j,ivk,ivT k,iuj,i−aj,iδj,ka.s.−→0, uT j,ivk,˜i... | https://arxiv.org/abs/2503.13582v1 |
uT ℓ,0Σ1 2 iz/bracketrightbigg2/parenrightigg +1 σ4 1/summationdisplay j∈I2,1/parenleftig γ(2) j,1/parenrightig2Var/parenleftigg/bracketleftbigg uT j,1Σ1 2 iz/bracketrightbigg2/parenrightigg +2 σ4 1/summationdisplay j/nequalℓ∈I2,1γ(2) j,1γ(2) ℓ,1Cov/parenleftigg/bracketleftbigg uT j,1Σ1 2 iz/bracketrightbigg2 ,/b... | https://arxiv.org/abs/2503.13582v1 |
can derive: 1 n2 11T n1ZT 1Σ1 2 1/tildewideH−1 1Σi/tildewideH−1 1Σ1 2 1Z11n1−1 n1trΣ1/tildewideH−1 1Σi/tildewideH−1 1a.s.−→0, (A.27) 1 n2 01T n0ZT 0Σ1 2 0/tildewideH−1 0Σi/tildewideH−1 0Σ1 2 0Z01n0−1 n0trΣ0/tildewideH−1 0Σi/tildewideH−1 0a.s.−→0, (A.28) ReplacingΣiand/tildewideH−1 iby their expressions with the fact th... | https://arxiv.org/abs/2503.13582v1 |
of Statistics , vol. 43, no. 3, pp. 1243–1272, 2015. [4] B. Jiang, X. Wang, and C. Leng, “A direct approach for spar se quadratic discriminant analysis,” Journal of Machine Learning Research , vol. 19, no. 31, pp. 1–37, 2018. [5] T. T. Cai and L. Zhang, “A convex optimization approach to high-dimensional sparse quadrat... | https://arxiv.org/abs/2503.13582v1 |
vol. 65, no. 6, p. 066126, 2002. [23] S. Kritchman and B. Nadler, “Determining the number of c omponents in a factor model from limited noisy data,” Chemometrics and Intelligent Laboratory Systems , vol. 94, no. 1, pp. 19–32, 2008. [24] D. Passemier, Z. Li, and J. Yao, “On estimation of the noi se variance in high dime... | https://arxiv.org/abs/2503.13582v1 |
arXiv:2503.13986v1 [math.ST] 18 Mar 2025Stratified Permutational Berry–Esseen Bounds and Their Applications to Statistics Pengfei Tian1, Fan Yang2, and Peng Ding∗3 1Qiuzhen College, Tsinghua University, tpf24@mails.tsinghua.edu.cn 2Yau Mathematical Sciences Center, Tsinghua University, yangfan1987@tsinghua.edu.cn 3Depar... | https://arxiv.org/abs/2503.13986v1 |
[k]1/summationtext i∈I[k]ZiYidenote the k-th stratum sample mean. An unbiased estimator for γisˆγ=/summationtextK k=1w[k]ˆY[k]. We can rewrite ˆγasWAss,πwith Ass= diag w[k]n−1 [k]1 Y[k]11T n[k]10T n[k]0 ...... Y[k]n[k]1T n[k]10T n[k]0 k=1,...,K, where “ss” stands for “stratifi... | https://arxiv.org/abs/2503.13986v1 |
to the general matrix form defined in ( 1). Motoo(1956),H´ ajek(1961) andFraser(1956) further established Lindeberg–type cen- tral limit theorem (CLT) for the permutation statistic. von Bahr (1976) andHo and Chen (1978) provided its Kolmogorovdistance Berry–Esseen bounds (BEBs), achieving theorder O(n−1/2) under some bo... | https://arxiv.org/abs/2503.13986v1 |
conditions, it attains the n1/4order, and under additional yet plausible conditions, it attains the classic n1/2order. Section 4presents the sketch of the proof and derives some additional results as byproducts. Section 5revisits the motivating examples. Section 6applies our new results to analyze stratified permutation... | https://arxiv.org/abs/2503.13986v1 |
form matrix A= diag{GΩk}K k=1and re-index A= diag{A[k]}K k=1. Ifnh’s are uniformly bounded and ∞/summationdisplay h=1whn−2 h/parenleftigg/summationdisplay 1≤i,j≤nh|g0 h,ij|3/parenrightigg <∞, (2) 9 then asK→ ∞, K/summationdisplay k=1n[k]/summationdisplay i,j∈I[k]|as ij|3≍/summationtext∞ h=1whn−2 h/summationtext 1≤i,j... | https://arxiv.org/abs/2503.13986v1 |
we define f(n[k],R2 A[k]) =1 2−24n[k]R2 A[k] (n[k]−5)2−4/braceleftigg 1+(28n[k]−20)R2 A[k] (n[k]−5)2/bracerightigg1/2/braceleftigg n[k]R2 A[k] (n[k]−4)2/bracerightigg1/2 , ifn[k]≥6 fork= 1,...,K. Thef-function is increasing in n[k]and decreasing in R2 A[k]. It is upper bounded by 1 /2, and tends to 1 /2 whenn[k]→ ∞w... | https://arxiv.org/abs/2503.13986v1 |
Combininginequalities ( 5)and(6), wecanchoose C= max{C1+C2/f(˜n,1),2˜n3/(˜n−1)3/2} in Corollary 3. Corollary 4 Assume that the elements of A0are independent and identically distributed samples from YwithE|Y|6<∞, and the number of strata satisfies K=O(n1−ε)for some ε>0. Asmink=1,...,Kn[k]→ ∞,we have K/summationdisplay k=... | https://arxiv.org/abs/2503.13986v1 |
with each element bounded by 1 : M1 n={A∈Ms n:|aij| ≤1 for alli,j}. Consider a matrix A= [aij]∈Ms n. For any ε >0, when/summationtextK k=1βA[k]/n[k]≥ε, the following inequality holds trivially because the right-hand side is larger t han or equal to 2: sup t∈R|P(WA,π≤t)−Φ(t)| ≤2ε−1K/summationdisplay k=1βA[k]/n[k]. When/... | https://arxiv.org/abs/2503.13986v1 |
A,π−WA,π)|, A3=1 αE/bracketleftbigg |W∗ A,π−WA,π|/integraldisplay1 01[t,t+α]/braceleftbig WA,π+r(W∗ A,π−WA,π)/bracerightbig dr/bracketrightbigg .(13) The key is to bound A1,A2, andA3. ForA1, it is theL1norm difference between WA,πand its zero-bias transformation. We can prove Proposition S5in the supplementary material ... | https://arxiv.org/abs/2503.13986v1 |
Tuvaandorj (2024) obtained a CLT for WA,πwhenAis in the form of outer product of two vectors by proving dW(WAs,π,N(0,1))→0. Using Theorem 3, we can strengthen their results by relaxing their conditio ns. We will revisit this point in Corollary 10below. 22 5 Applications to the Motivating Examples We now apply Corollary... | https://arxiv.org/abs/2503.13986v1 |
apply Corollary 6and obtain a BEB for sup t∈R/vextendsingle/vextendsingle/vextendsingle/vextendsingleP/braceleftbiggˆγps−E(ˆγps|U1) σps(U1)≤t|U1/bracerightbigg −Φ(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle, whereσ2 ps(U1) = var(ˆγps|U1) is the conditional variance of ˆ γpsgivenU1. The post- stratificatio... | https://arxiv.org/abs/2503.13986v1 |
0 1−p>S2 τ, (24) the quantity inside of [·]δis of order 1/n1/2and the second term is O(1/n). Mimicking the proof of Corollary 8, we provide similar regularity conditions on the second and third moments of Y(1) andY(0). Considering both the treatment and con- trolgroupvariations, weimposetheconstraint on pasin(24). This... | https://arxiv.org/abs/2503.13986v1 |
in D’Haultfœuille and Tuvaandorj (2024), we only need σ2to tend to a constant and the third-moment term/summationtextK k=1n[k]M3 [k](Z)M3 [k](R)→0. With Corollary 10, the condition specified in D’Haultfœuille and Tuvaandorj (2024, Lemma 4) is redun- dant: we do not need to assume that the fourth-moment term in th e cond... | https://arxiv.org/abs/2503.13986v1 |
ffect in paired experiments’, Biometrika 105, 994–1000. Fraser, D. A. S. (1956), ‘A Vector form of the Wald–Wolfowitz–Hoe ffding theorem’, The AnnalsofMathematical Statistics 27, 540–543. Fuller, W. A. (1966), ‘Estimation employing post strata’, Journal oftheAmerican Statistical Association 61(316), 1172–1183. 32 Goldste... | https://arxiv.org/abs/2503.13986v1 |
20, 455–458. Panaretos, V. M. and Zemel, Y. (2019), ‘Statistical aspects of Wa sserstein distances’, AnnualReviewofStatistics anditsApplication 6, 405–431. Pashley, N. E. and Miratrix, L. W. (2021), ‘Insights on variance est imation for blocked and matched pairs designs’, JournalofEducational andBehavioral Statistics 4... | https://arxiv.org/abs/2503.13986v1 |
= 0H,var(Γ) =IH. S2 Now, we present BEB on the linear combination of multivariate stratifi ed linear permuta- tion statistics as follows. Corollary S1 For matrix G1,...,G Hsatisfying Condition S1and for any /bardblb/bardbl2= 1,b∈ RH, there exists a universal constant Csuch that sup t∈R/vextendsingle/vextendsingleP(bTΓ≤t... | https://arxiv.org/abs/2503.13986v1 |
S5 Proposition S4 GivenB†=k,I†=i,J†=j,P†=p,Q†=q,π†is a permutation that satisfies the following conditions: •π†(m) =π(m)for allm /∈ {i,j,π−1(p),π−1(q)}. •{π†(i),π†(j)}={p,q}. Furthermore, P(π†(m) =ξ† m,m /∈ {i,j},m∈ I[k]) =1 (n[k]−2)!, for all distinct ξ† m,m /∈ {i,j},m∈ I[k]withξ† m/∈ {p,q},ξ† m∈ I[k]. Proposition S4is... | https://arxiv.org/abs/2503.13986v1 |
ulation ofNitems, where w=Npare successes and pis the probability of success. Then Y= (n/W)1(W >0)satisfies −2 p(1−p)n/lessorequalslantE[Y]−1 p/lessorequalslant4 p21 n−1 p1 n+1+max/braceleftbigg/parenleftbiggn 2−4 p2n/parenrightbigg exp/parenleftbigg −p2 2n/parenrightbigg ,0/bracerightbigg . Lemma S9 Forpositive sequenc... | https://arxiv.org/abs/2503.13986v1 |
Ross(2011, Proposition 1.2, part 2))If the random variable Zhas Lebesgue density bounded by C1, then for any random variableW, we have dK(W,Z)≤/radicalbig 2C1dW(W,Z). (ii)(subadditivity and linearity, Panaretos and Zemel (2019))For indepen- dent{Xk}K k=1and independent {Yk}K k=1, we have that dW/parenleftiggK/summatio... | https://arxiv.org/abs/2503.13986v1 |
n[k]. C.6.2 Zero-Bias Transformation Recallht,αin Section 4and Eht−α,α(W(A′)s,π)−Φ(t)≤P(W(A′)s,π≤t)−Φ(t)≤Eht,α(W(A′)s,π)−Φ(t).(S12) ForN∼ N(0,1), because |Eht−α,α(N)−Φ(t)|=E[ht,0(N)−ht−α,α(N)]≤P(N∈[t−α,t])≤α√ 2π, S16 the left-hand side of ( S12) is bounded by |Eht−α,α(W(A′)s,π)−Φ(t)|=|Eht−α,α(W(A′)s,π)−Eht−α,α(N)+Eht−α... | https://arxiv.org/abs/2503.13986v1 |
3. Bound B0=/radicaligg var/parenleftbigg E/parenleftbigg1 2λ(W′ D,φ−WD,φ)21(|W′ D,φ−WD,φ| ≤δ)|WD,φ/parenrightbigg/parenrightbigg . 4. Give a lower bound and upper bound of EW2 D,φ. Step 1. Bound |L|/λWe have |L|=2 (n[b†]−l)(n−l−K)|/summationdisplay i∈R,j∈π(R)(a′)s ij| ≤2 (n[b†]−l)(n−l−K)/radicaligg l2/summationdispl... | https://arxiv.org/abs/2503.13986v1 |
in Chen and Fang (2015))We have E{/summationdisplay i∈I[b†]/R(a′)s iφ(i)}2=1 n[b†]−l−1/summationdisplay (i,j)∈D[b†]{(a′)s ij}2 +1 (n[b†]−l)(n[b†]−l−1)/summationdisplay (i,j)∈D[b†](a′)s ij{/summationdisplay k∈R,t∈π(R)(a′)s kt+/summationdisplay k∈π(R)(a′)s ik+/summationdisplay t∈R(a′)s tj}. (S35) By Lemma S15, we need to... | https://arxiv.org/abs/2503.13986v1 |
Chooseα=/summationtextK k=1(/summationtext i,j∈I[k]|(a′)s ij|3/n[k]), then sup t∈R|P(W(A′)s,π≤t)−Φ(t)| ≤(C′ 1+C′ 2 θ)K/summationdisplay k=1(/summationdisplay i,j∈I[k]|(a′)s ij|3/n[k]) = (C′ 1+C′ 2 θ)K/summationdisplay k=1(βA′[k]/n[k]). By Lemma S2and Lemma S1, forA∈Ms n, and/summationtextK k=1βA[k]/n[k]≤ε0, we have sup... | https://arxiv.org/abs/2503.13986v1 |
Simplify the right-hand side of (S48).By definition, we have WA,π−WA,π′′=ai,π(i)+aj,π(j)−ai,π(j)−aj,π(i)=b(i,j,π(i),π(j)). Corollary S2implies E(WA,π−WA,π′′)2=4σ2 A n−K. S33 Furthermore, combining the definition of Bandπ, we have (WA,π−WA,π′′)2 E(WA,π−WA,π′′)2·P(B=k,I=i,J=j,π(α) =ξα,α= 1,...,n) =b2(i,j,ξi,ξj) 4σ2 A/(n−K)... | https://arxiv.org/abs/2503.13986v1 |
C.12.3 Order of the second term in (21) Denoteg(x) = Φ(x−1/2σt). By the Taylor expansion, there exists x0betweenxandσ2 such that, Φ/parenleftbiggσt√x/parenrightbigg −Φ(t) =g(x)−g(σ2) =g′(σ)(x−σ2)+g′′(x0)(x−σ2)2. S37 Choosex=σ2(U1), thenx0=x0(U1) relies on U1. ByE[σ2 ps(U1)| D1] =σ2, we have that |E[Φ(σ σ(U1)| D1)−Φ(t)]... | https://arxiv.org/abs/2503.13986v1 |
10 Proof. We can choose A0= diag (R[k]1−¯R[k])(Z[k]1−¯Z[k])···(R[k]1−¯R[k])(Z[k]n[k]−¯Z[k]) ...... (R[k]n[k]−¯R[k])(Z[k]1−¯Z[k])···(R[k]n[k]−¯R[k])(Z[k]n[k]−¯Z[k]) k=1,...,K, ThenAs=A0/σ, and K/summationdisplay k=11 n[k]/summationdisplay i,j∈I[k]|as ij|3=K/summationdisplay k... | https://arxiv.org/abs/2503.13986v1 |
that there are positive constants αandβsuch that /bardblX/bardbl2≤αand/bardblD/bardbl2≤β. Then there exists a universal constant C, such that sup A∈C|P{Γ∈A}−P{ξH∈A}| ≤C/parenleftig H7/4αE/bardblD/bardbl2 2+H1/4β+H7/8α1/2B1/2 1+H3/8B2+H1/8B1/2 3/parenrightig where B2 1= Var/braceleftbig E/parenleftbig /bardblD/bardbl2... | https://arxiv.org/abs/2503.13986v1 |
A[k]/parenleftbigg 8+28 (n[k]−1)+4 (n[k]−1)2/parenrightbigg . Now, we can compute the unconditional expectation by incorp orating the conditional expectation: E|W∗ A,π−WA,π| ≤K/summationdisplay k=1σ2 A[k] σ2 A/summationtext i,j∈I[k]|aij|3 (n[k]−1)σ2 A[k]/parenleftbigg 8+28 (n[k]−1)+4 (n[k]−1)2/parenrightbigg =K/summati... | https://arxiv.org/abs/2503.13986v1 |
Chen et al. (2011, P179), we obtain |Φ(t−µA′ σA′)−Φ(t)| ≤2√ 2π|σA′−1|(1+|µA′|)+3 4|µA′|. From Lemma S1,|µA′| ≤c1/summationtextK k=1βA[k]/n[k]≤c1ε2and further 2√ 2π|σA′−1|(1+|µA′|)+3 4|µA′| ≤2c2√ 2πK/summationdisplay k=1βA[k]/n[k](1+c1ε2)+3c1 4K/summationdisplay k=1βA[k]/n[k] =(2c2√ 2π(1+c1ε2)+3c1 4)K/summationdisplay k... | https://arxiv.org/abs/2503.13986v1 |
THE COVARIANCE OF CAUSAL EFFECT ESTIMATORS FOR BINARY V -STRUCTURES The covariance of causal effect estimators for binary v-structures Jack Kuipers JACK .KUIPERS @BSSE .ETHZ .CH D-BSSE, ETH Zurich, Schanzenstrasse 44, 4056 Basel, Switzerland Giusi Moffa GIUSI .MOFFA @UNIBAS .CH Department of Mathematics and Computer Sc... | https://arxiv.org/abs/2503.14242v1 |
u1, u2, u3, u4, v1, v2, w1, w2}be all generating variables. Using this formulation E[M11R1] =1 NZ dw2Z dt4s2u4v2 t4w2∂3 ∂s2∂u4∂v2UN Gi=1 ∀i =Z dw2v2 w2∂ ∂v2(p3v1w1+p7v2w2+p2w1+p6w2)v2p7 (p6+p7v2)UN−1 Gi=1 ∀i =p6p7 (p6+p7)2(N−1)pZpX(1−pX)N−2F [1,1,2−N],[2,2],−pX 1−pX + (p4+p6)p7 Np2 X+E[M11]E[R1] (12) and likewise E[M... | https://arxiv.org/abs/2503.14242v1 |
5 KUIPERS AND MOFFA so that the asymptotics for the covariance becomes C[R, M ]·N=(q1+ (2pZ−1)C)(1−q1−(2pZ−1)C) + (q0+ (2pZ−1)C)(1−q0−(2pZ−1)C) +(1−pX)pZ(q1+C−(1−pZ)D)(1−q1−C+ (1−pZ)D) pX 1 +1 NpX +(1−pX)(1−pZ)(q1−C+pZD)(1−q1+C−pZD) pX 1 +1 NpX +pXpZ(q0+C+ (1−pZ)D)(1−q0−C−(1−pZ)D) 1−pX 1 +1 N(1−pX) +pX(1−pZ)(q0−C... | https://arxiv.org/abs/2503.14242v1 |
the compact form of SNfrom Equation (33) to easily obtain the result of N1(ρ5+ρ7), or E[R1] =v2 N1∂ ∂v2SN0,N1 v1=v2=1=v2(ρ5+ρ7)SN0,N1−1 v1=v2=1= (ρ5+ρ7) (37) Performing the same steps for R0we obtain E[R] = (ρ5+ρ7)−(ρ1+ρ3) (38) The Variance. To compute the variance V[R] =V[R1]−2C[R1, R0] +V[R0] (39) we first show that ... | https://arxiv.org/abs/2503.14242v1 |
[1,1,1−N0],[2,2],−pZ 1−pZ +N0N1ρ0ρ1(1 + ( N1−1)(1−pZ))(pZ)N0−1F [1,1,1−N0],[2,2],−pZ 1−pZ +Nρ7(1−ρ7) +Nρ5(1−ρ5) +Nρ3(1−ρ3) +Nρ1(1−ρ1) +ρ6ρ7 pZN0+ρ4ρ5 1−pZN0+ρ2ρ3 pZN1+ρ0ρ1 1−pZN1 + 2N(ρ7−ρ3)(ρ1−ρ5)−2Nρ3ρ7(1−pZ) pZ−2Nρ1ρ5pZ (1−pZ)(59) A.3 The covariance of RandM For this covariance we also need the two generating var... | https://arxiv.org/abs/2503.14242v1 |
ξ(1−ξ)[2C+ (2pX−1)D]2+O(N−3 2) (66) (C[R, M ]−V[R])·N=−pZ(1−pZ) ξ(1−ξ)[2C+ (2pX−1)D]2+O(N−3 2) (67) (C[R, M ]−V[M])·N=−q1(1−q1)(1−ξ) ξ2−q0(1−q0)ξ N(1−ξ)2+O(N−3 2) (68) which are the same as the results for random Xjust with a change of pXforξ. As for that case covered in the main text, also when Xis fixed we can see th... | https://arxiv.org/abs/2503.14242v1 |
arXiv:2503.14311v1 [math.ST] 18 Mar 2025Asymptotic properties of the MLE in distributional regression under random censoring Gitte Kremling∗, Gerhard Dikta Fachhochschule Aachen, Germany March 19, 2025 Abstract The aim of distributional regression is to find the best candi date in a given parametric family of conditiona... | https://arxiv.org/abs/2503.14311v1 |
censreg , see Henningsen (2020). However, it is based on a much simpler model, assuming that the censoring variables as well as the covariates are determini stic rather than random and that the regression function is linear. Moreover, the docum entation does not include any theoretical results. There are three related ... | https://arxiv.org/abs/2503.14311v1 |
the motivat ion ofLand do not need to be fulfilled in practical applications. Since all of the involved random variables, except for δ, are continuous, (1) cannot be used. Instead, we consider the approximate likelihood func tion Lh1,h2(ϑ;(x,z,d)) =Pϑ(x≤X≤x+h1,z≤Z≤z+h2,δ=d) forh1,h2≥0. Since argmax ϑ∈ΘLh1,h2(ϑ,(x,z,d)) ... | https://arxiv.org/abs/2503.14311v1 |
i.e. without any cen sor- ing,KG,Hcoincides with the extended Kullback-Leibler information without censoring KH(ϑ1,ϑ2) =/integraldisplay /integraldisplayc −∞log/parenleftbiggf(y|ϑ1,x) f(y|ϑ2,x)/parenrightbigg f(y|ϑ1,x)ν(dy)H(dx) defined in Dikta and Scheer (2021). Moreover, in case Yis independent of X, it matches the m... | https://arxiv.org/abs/2503.14311v1 |
covariance matrix Σ−1. Again, the conditions of the theorem are similar to the corre sponding result for para- metric GLMs without censoring as stated in Dikta and Scheer ( 2021, Theorem 5.55). Remark 6. IfF(C|ϑ0,X) = 1 , the value of the covariance matrix Σis undefined, so this case has to be handled separately. In the... | https://arxiv.org/abs/2503.14311v1 |
(MSE). The results are presented i n tables 1-4. In general, it can be said that the results look quite promisi ng. In all four setups, the MSE of all three parameter estimates decreases substantial ly for larger sample sizes. As it can be expected, the standard deviation of the estimates inc reases for a larger condit... | https://arxiv.org/abs/2503.14311v1 |
1 is given by k(t) = (t−1)+1 2∆(t−1)2 with∆between tand1. Now, let t1=f(y|ϑ1,x) f(y|ϑ2,x)andt2=1−F(c|ϑ1,x) 1−F(c|ϑ2,x). Then, KG,H(ϑ1,ϑ2) =/integraldisplay /integraldisplay /integraldisplayc −∞log/parenleftbiggf(y|ϑ1,x) f(y|ϑ2,x)/parenrightbiggf(y|ϑ1,x) f(y|ϑ2,x)f(y|ϑ2,x)ν(dy) +log/parenleftbigg1−F(c|ϑ1,x) 1−F(c|ϑ2,x)/... | https://arxiv.org/abs/2503.14311v1 |
the last inequality is a direct consequence of Lemma 3. Altogether, this establishes that for small enough ε limsup n→∞sup ϑ∈Vε(ϑ∗)1 n/parenleftBig ℓn(ϑ)−ℓn(ϑ0)/parenrightBig <0. SinceUis compact, there exist εiandϑ∗ isuch that U⊂ ∪m i=1Vεi(ϑ∗ i)which completes the proof. Proof of Theorem 5. For the first part of the pr... | https://arxiv.org/abs/2503.14311v1 |
E/bracketleftBig I{Y≤C}D(˜ℓ1(ϑ0))/parenleftbig D(˜ℓ1(ϑ0))/parenrightbigT/bracketrightBig =E/bracketleftBig E/bracketleftbig I{Y≤C}|X,Y/bracketrightbig D(˜ℓ1(ϑ0))/parenleftbig D(˜ℓ1(ϑ0))/parenrightbigT/bracketrightBig =E/bracketleftBig E/bracketleftbig I{Y≤C}|Y/bracketrightbig D(˜ℓ1(ϑ0))/parenleftbig D(˜ℓ1(ϑ0))/parenrig... | https://arxiv.org/abs/2503.14311v1 |
Confidence Intervals Using Turing’s Estimator: Simulations and Applications Jie Chang∗1, Michael Grabchak†1, and Jialin Zhang‡2 1Department of Mathematics and Statistics, UNC Charlotte 2Department of Mathematics and Statistics, Mississippi State University March 19, 2025 Abstract Turing’s estimator allows one to estima... | https://arxiv.org/abs/2503.14313v1 |
given dataset. The second is to develop a novel methodology for using these CIs for the problem of authorship attribution. The third is to theoretically verify when asymptotic normality and asymptotic Poissonity hold for Turing’s estimator when sampling from discrete uniform and geometric distributions. This is importa... | https://arxiv.org/abs/2503.14313v1 |
it can be estimated by ˆsr,n=q (r+ 1)2Nr+1,n+ (r+ 2)( r+ 1)Nr+2,n, see Chang & Grabchak (2023) for details. A number of asymptotic confidence intervals (CIs) for πr,nhave been proposed in the literature. These are based on various limit theorems. If sr,n→ ∞ , then, under mild conditions (see Appendix A), we have n ˆsr,... | https://arxiv.org/abs/2503.14313v1 |
better when sr,n→c∈(0,∞). In practice, of course, we do not know the asymptotics of sr,n. However, we can estimate sr,nby ˆsr,n, which suggests the following heuristic for choosing between the two types of CIs. First, select a threshold V >0. If ˆsr,n< V, use (8), otherwise use (5) or (6). We call this the heuristic CI... | https://arxiv.org/abs/2503.14313v1 |
Sample Sizer=0 , Fixed Uniform, K=1000 0.000.250.500.751.00 2074269100037151349050118 Sample Sizer=1 , Fixed Uniform, K=1000 0.000.250.500.751.00 2074269100037151349050118 Sample Sizer=2 , Fixed Uniform, K=1000 0.000.250.500.751.00 2074269100037151349050118 Sample Sizer=3 , Fixed Uniform, K=1000 CI Type Normal Poisson ... | https://arxiv.org/abs/2503.14313v1 |
CI. 10 What may be even more interesting is that the coverage of the Normal CI appears to converge to 0 .95 for large, but not too large, values of the sample size n. Such apparent con- vergence before the actual convergence is sometimes called pre-limit behavior, see Grabchak & Samorodnitsky (2010) and the references ... | https://arxiv.org/abs/2503.14313v1 |
are on the order of K1/γ, we should be close to normal and the normal CI should work well. However, for sample sizes nthat are much larger than K1/γ, we should no longer be close to normal. A similar result should hold for asymptotic Poissonity. Note that for r≥1, asymptotic Poissonity holds for larger values of γ(smal... | https://arxiv.org/abs/2503.14313v1 |
requires further study. We have asymptotic Poissonity when r= 2 and γ= 1.5. We can see that the Poisson CI works well in this case, while the Normal CI has very poor performance. In this case, the mean of the asymptotic Poisson distribution is 1 /6 and the normal approximation to the Poisson will not work well. We now ... | https://arxiv.org/abs/2503.14313v1 |
the situations look very much like what we would expect if asymptotic normality held. We leave the problem of verifying this theoretically as an important direction for future work. 15 Geometric Distribution The geometric distribution has a pmf given by pℓ=p(1−p)ℓ−1, ℓ= 1,2,3, . . . , (12) where p∈(0,1) is a parameter.... | https://arxiv.org/abs/2503.14313v1 |
the Normal CI works well when pis small. To define the dynamic geometric distribution, let {p(n)}be a sequence of real numbers with 0 < p(n)<1, let an=−1/log(1−p(n)), and note that p(n)= 1−e−1/an. We consider the sequence of geometric distributions with pmfs of the form p(n) ℓ=p(n)(1−p(n))ℓ−1= e1/an−1 e−ℓ/anℓ= 1,2, .... | https://arxiv.org/abs/2503.14313v1 |
al. (2018). A common approach to authorship attribution is to consider two writing samples and to estimate one or more diversity indices for each sample. One then compares these estimated indices to see if they are statistically different from each other. Of course, such approaches 19 0.000.250.500.751.00 2074269100037... | https://arxiv.org/abs/2503.14313v1 |
It follows that n2πr,n1= E[Ar|the corpus] and hence πr,n1= E[Dr|the corpus] ≈Dr. 21 Figure 5: The CIs are constructed from the Corpus and the detecting points are calculated from the testing sets. We now illustrate our methodology by applying it to real-world data. For simplicity and comparability, we calculate all CIs... | https://arxiv.org/abs/2503.14313v1 |
of the top X (then Twitter) users in 2017. The results are given in Figure 6. The plots do not include r= 0, as these would make it hard to zoom in on the main parts of the plot, and, as we have seen, they do not seem to be relevant in the context of this application. Overall, the method seems to work well. However, in... | https://arxiv.org/abs/2503.14313v1 |
larger this value, the longer the period during which the Normal CI is close to 0 .95 in its pre-limit apparent convergence phase. Furthermore, we observed that asymptotic normality and asymptotic Poissonity only hold in the dynamic case, where K→ ∞ , which is, again, as the tails get progressively heavier. Appendix A ... | https://arxiv.org/abs/2503.14313v1 |
x∈R. Note that s2 r,n=1 r!P∞ ℓ=1((r+ 1)hn,r+1(ℓ) +hn,r+2(ℓ)). Since g′ η(x) = xη−1e−x(η−x), gηis increasing on (0 , η), decreasing on ( η,∞), and max x≥0gη(x) = gη(η). Next, let x∗η n=−anlog ηn−1(e1/an−1)−1 and note that nfn(x∗η n) =η. Since fnis monotonically decreasing, it follows that hn,ηis increasing on ( −∞, x∗... | https://arxiv.org/abs/2503.14313v1 |
T., Chazdon, R., Colwell, R. & Gotelli, N. (2015), ‘Unveiling the species- rank abundance distribution by generalizing the good–turing sample coverage theory’, Ecology 96(5), 1189–1201. Chao, A., Lee, S.-M. & Chen, T.-C. (1988), ‘A generalized good’s nonparametric coverage estimator’, Chinese Journal of Mathematics 16(... | https://arxiv.org/abs/2503.14313v1 |
Data set. Harvard Dataverse. Tareaf, R. B., Berger, P., Hennig, P. & Meinel, C. (2018), Malicious behaviour identification in online social networks, inS. Bonomi & E. Rivi` ere, eds, ‘Distributed Applications and Interoperable Systems: DAIS 2018’, Vol. 10853 of Lecture Notes in Computer Science , Springer, Cham. 34 Zha... | https://arxiv.org/abs/2503.14313v1 |
arXiv:2503.14347v2 [math.PR] 9 May 2025A N EWPROOF OF SUB-GAUSSIAN NORM CONCENTRATION INEQUALITY Zishun Liu Georgia Institute of Technology Atlanta, GA 30332, US zliu910@gatech.eduSam Power University of Bristol Bristol, BS8 1QU, UK sam.power@bristol.ac.ukYongxin Chen Georgia Institute of Technology Atlanta, GA 30332, ... | https://arxiv.org/abs/2503.14347v2 |
sub-Gaussian norm concentration [ 1,3]. Following this technique, one constructs an ε-netN[1, Definition 1.17] for the Euclidean unit ball such that Nhas finite elements andmaxℓ∈Sn−1ℓ⊤X≤1 1−εmaxℓ∈Nℓ⊤X[3, Exercise 4.4.2]. The union bound can be applied since Nis a finite set. This method results in ( 3) with constants C1an... | https://arxiv.org/abs/2503.14347v2 |
(12) Sinceg(z)is monotonically increasing over z >0,G(z)is convex on z >0. Thus, by the definition of convexity, given any z0>0, G(z)≥g(z0)(z−z0)+G(z0). (13) Setz0=εn 1−ε2, then g(z0) =ε, G(z0) =nε2 1−ε2+n 2log(1−ε2). (14) Plugging ( 14) into ( 12)-(13) yields logφn(z)≥εz+n 2log(1−ε2). (15) The conclusion ( 8) and thus ... | https://arxiv.org/abs/2503.14347v2 |
the concentration inequality ( 24) does not depend on εdue to the additional optimization step ( 25). In practice, applying Theorem 2or Theorem 3can depend on the value of nandδ. Interestingly, Theorem 3coincides with a concentration bound established in [ 6, Remark 6] based on variational inequality. Theorem 3is only ... | https://arxiv.org/abs/2503.14347v2 |
of the Sub-Gaussian norm concentration based on the ε-net method [ 3, Chapter 4]. 4 Conclusion This paper presents an alternative proof of the sub-Gaussia n norm concentration inequality that applies to both ran- dom vectors and random matrices. The key to our proof is a modi fied MGF dubbed AMGF. The AMGF depends solel... | https://arxiv.org/abs/2503.14347v2 |
arXiv:2503.14381v1 [stat.ML] 18 Mar 2025Optimizing obliques splits Optimizing High-Dimensional Oblique Splits Chien-Ming Chi∗xbbchi@stat.sinica.edu.tw Institute of Statistical Science Academia Sinica Taipei, Taiwan Editor: Abstract Orthogonal-split trees perform well, but evidence suggest s oblique splits can enhance t... | https://arxiv.org/abs/2503.14381v1 |
a wide range of high-dimensional settings. Each split (# –w,c)defines a decision hyperplane that partitions the feature sp ace[0,1]pinto two subsets:{# –x∈[0,1]p:# –w⊤# –x > c}and{# –x∈[0,1]p:# –w⊤# –x≤c}. These splits are recursively applied to partition the feature space, formin g the foundation for subsequent steps i... | https://arxiv.org/abs/2503.14381v1 |
memory constraints, as the number of sp lits involved in optimization at each iteration is bounded by #Wl+1+#S(b)≤n×w0+2H−1. Our optimization approach is more efficient than existing met hods for optimizing oblique trees (see Section 1.1 for an overview and Sections 5–6 for a r untime comparison) because it does not requ... | https://arxiv.org/abs/2503.14381v1 |
for split opti- mization. To mitigate local minima, randomization is often incorporated. For instance, OC1, RR-RF (Blaser and Fryzlewicz, 2016), SPORF (Tomita et al., 20 20), and Forest-RC (Breiman, 2001) introduce random normal vectors into their oblique sp lit optimization to varying de- grees. Among these, SPORF and... | https://arxiv.org/abs/2503.14381v1 |
the sample version of the oblique splits s earching iteration. 2.2 Sample algorithm of oblique decision trees Let{Yi,Xi}n i=1be an observed sample of size nwhere(Y1,X1),...,(Yn,Xn),(Y,X)are i.i.d. Let/hatwideN1=···=/hatwideNH=∅, and/hatwideN0={(∅,(# –0,0),[0,1]p)}initially. For each h∈{1,...,H}, we perform the followin... | https://arxiv.org/abs/2503.14381v1 |
that it is not possible to uniformly sample inte- gers from{1,...,B}whenB=∞, as a uniform distribution over an infinite set is not well-defined. Second, the split set S(b)can be transferred and incorporated into standard tree-based models, such as random forests, to enhance predi ction accuracy. See Section 4 for the det... | https://arxiv.org/abs/2503.14381v1 |
t ree given sufficient compu- tational resources. Specifically, a Forest-RC model is an en semble of independently trained Forest-RC trees using bagging. At each node of a Forest-RC tr ee, the optimal split is se- lected from a random subset of size max_features from/hatwiderWp,s. Following reasoning similar to Propositio... | https://arxiv.org/abs/2503.14381v1 |
eachH≥1with√ 2n−1 2s/radicalbig H(1∨logs)logn≤1, and either (i)or(ii): (i) Condition 2(b) holds with #S×b≥n2HBlog(2HB), (ii) Condition 2(a) holds with #S×b≥D−1 min2H+s+1Blog(2HB)ands <logn 2, it holds that P(Bc)≤n−1+B−1. On the eventB, the progressive and one-shot trees are equivalent in the co ntext of Theorem 2. An i... | https://arxiv.org/abs/2503.14381v1 |
0}⊂{# –x∈[0,1]p:# –v(k)⊤ 1# –x− a(k) 1<0}. Furthermore, assume that no two hyperplanes intersect wit hin[0,1]p, that the total number of nonzero coordinates across all weight vecto rs satisfies #{j:|v(k) l,j|>0,l∈ {1,...,n k},k∈{1,...,K 1}}≤s0, for some integer s0>0, where# –v(k) l= (v(k) l,1,...,v(k) l,p)⊤, and that al... | https://arxiv.org/abs/2503.14381v1 |
trees. Define /hatwideR(X) =/summationtext (t′,(# –w,c),t)∈/hatwideN(b) H1X∈t×/hatwideβ(t), using the notation introduced in (3). In Theorem 4, c∈(0,1) represents a constant that can be arbitrarily close to 1. Thi s technical parameter should not be confused with the bias notation frequently used in this pa per. Theorem... | https://arxiv.org/abs/2503.14381v1 |
2024), where the estimation variance scales as n−1instead of n−1 2, at the cost of a slightly slower bias convergence rate in α0. However, our results address approximation error between a one-shot tree and the ideal tree, which is not considered by e xisting studies on tree con- vergence rates (Mazumder and Wang, 2024... | https://arxiv.org/abs/2503.14381v1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.