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resulting in the prior f⋆1≃f⋆2. This prototypical example of transfer learning is explored in the following paragraphs. Such inexact constraints can be enforced by adding a penalty λ∥Cθ∥2 2in the empirical risk (2), where λ >0is an hyperparameter. (Equivalently, this 12 FORECASTING TIME -SERIES WITH CONSTRAINTS … … … …... | https://arxiv.org/abs/2502.10485v1 |
. , 1)⊤∈R27, and defining 1(j)∈R27by1(j) i= 1ifPj−1 k=1zk≤i≤Pj k=1zk 0 otherwise, we have that S= (I 27|1(1)| ··· | 1(7)|1)⊤. The goal of hierarchical forecasting is to take advantage of the summation constraints defined by Sto improve the predictions of the vector Y representing all hierarchical nodes. This context c... | https://arxiv.org/abs/2502.10485v1 |
parameterized by a mapping ϕℓ(e.g., a Fourier map or an additive model) and a coefficient vector θℓ, such that fθℓ(Xℓ,t) =⟨ϕℓ(Xℓ,t), θℓ⟩. Therefore, the model for the lower nodes Yb,tcan be expressed as Φtθ, where θ= (θ1, . . . , θ ℓ2)⊤is the vector of all coefficients, and Φtis the fea- 14 FORECASTING TIME -SERIES WIT... | https://arxiv.org/abs/2502.10485v1 |
matrix S to estimate the higher levels, WeaKL-G relies directly on global information, which is subsequently penalized by S. In the next paragraph, we complement the WeaKL-BU estimator by adding transfer learning constraints. Estimator 3. Hierarchy-informed transfer learning: WeaKL-T. In many hierarchical forecast- ing... | https://arxiv.org/abs/2502.10485v1 |
then computed asˆYBU t=SˆYBU ℓ,t, where Sis the summation matrix. The Independent (Indep) model involves running separate linear regressions for each target time series using its own lags. This results in 415 linear regressions of the form ˆYIndep ℓ,t=P24 j=1aℓ,jYℓ,t−jfor1≤ℓ≤ℓ1. Rec-OLS is the estimator resulting from ... | https://arxiv.org/abs/2502.10485v1 |
can be formulated as a linearly constrained quadratic program. While this generally increases the complexity of the optimization, it can also lead to efficient algorithms for certain con- straints. In particular, when d= 1, imposing a non-decreasing constraint on fθreduces the problem to isotonic regression, which has ... | https://arxiv.org/abs/2502.10485v1 |
Makonin, W. Su, and H. Zareipour. Day-ahead electricity demand forecasting competition: Post-COVID paradigm. IEEE Open Access Journal of Power and Energy , 9:185–191, 2022. M. Fasiolo, S. N. Wood, M. Zaffran, R. Nedellec, and Y . Goude. Fast calibrated additive quantile regression. Journal of the American Statistical A... | https://arxiv.org/abs/2502.10485v1 |
equations. The Annals of Statistics , 52:1825–1844, 2024. D. Obst, J. de Vilmarest, and Y . Goude. Adaptive methods for short-term electricity load forecasting during COVID-19 lockdown in France. IEEE Transactions on Power Systems , 36:4754–4763, 2021. C. L. Pennings and J. van Dalen. Integrated hierarchical forecastin... | https://arxiv.org/abs/2502.10485v1 |
is to provide detailed proofs of the theoretical results presented in the main article. Appendix A.2 elaborates on the formula that characterizes the unique minimizer of the WeaKL empirical risks, while Appendix A.3 discusses the integration of linear constraints into the empirical risk framework. A.1 A useful lemma Le... | https://arxiv.org/abs/2502.10485v1 |
Let ΛandMbe injective matrices, and let λ≥0be a hyperparameter. Let ˆθbe the WeaKL given by (3)and let ˆθCbe the WeaKL obtained by replacing Mwith(√ λC⊤|M⊤)⊤in(3). Then, almost surely, 1 nnX j=1∥fθ⋆(Xtj)−fˆθC(Xtj)∥2 2+∥M(θ⋆−ˆθC)∥2 2≤1 nnX j=1∥fθ⋆(Xtj)−fˆθ(Xtj)∥2 2+∥M(θ⋆−ˆθ)∥2 2. Proof Recall from (3) that ˆθ=P−1nX j=1Φ... | https://arxiv.org/abs/2502.10485v1 |
=pX ℓ=1(p−1+hθℓ(t))ˆYℓ t, 24 FORECASTING TIME -SERIES WITH CONSTRAINTS where hθℓ(t) =⟨ϕ(t), θℓ⟩,ϕis the Fourier map ϕ(t) = (exp( ikt/2))⊤ −m≤k≤m, and θℓ∈C2m+1. Thep−1term introduces a bias, ensuring that hθℓ= 0 corresponds to a uniform weighting of the forecasts ˆYℓ. The function f⋆is thus estimated by minimizing the l... | https://arxiv.org/abs/2502.10485v1 |
variables (X1, X2, Y1, Y2). The goal is to construct three estimators ˆY1,ˆY2, and ˆY3ofY1, Y2, and Y3:=Y1+Y2. Benchmark. We compare four techniques. The bottom-up (BU) approach involves running two separate ordinary least squares (OLS) regressions that independently estimate Y1andY2without using information about Y1+Y... | https://arxiv.org/abs/2502.10485v1 |
j=1|Y1,n+j+Y2,n+j−ˆY3,n+j|2onY1+Y2. The hierarchical error is defined as the sum of these three MSEs, which are visualized in Figure 4. Results. Figure 4 clearly shows that all hierarchical models (Rec, MinT, and WeaKL) outper- form the naive bottom-up model for all four MSE metrics. Among them, our WeaKL consistently ... | https://arxiv.org/abs/2502.10485v1 |
nonlinear, or categorical) of the effects are specified. Thus, as detailed in Section 3, (i)If the effect ⟨ϕ1,ℓ(Xℓ,t), θ1,j⟩is assumed to be linear, then ϕ1,j(Xℓ,t) =Xℓ,t, (ii)If the effect ⟨ϕ1,ℓ(Xℓ,t), θ1,ℓ⟩is assumed to be nonlinear, then ϕ1,ℓis a Fourier map with 2mℓ+ 1Fourier modes, (iii)If the effect ⟨ϕ1,ℓ(Xℓ,t), ... | https://arxiv.org/abs/2502.10485v1 |
(λ,m)opt, which can be a grid. Then, we compute the minimizer ˆθ(λ,m)of the loss (8) on the training period, and the resulting estimation of (λ,m)optis the set of hyperparameters (λ,m)such that the MSE of fˆθ(λ,m) on the validation period is minimal. The performance of the online WeaKL is thus measured by the performan... | https://arxiv.org/abs/2502.10485v1 |
(see, e.g. Lahiri, 2013, Corollary 31 DOUM `ECHE , BACH, BEDEK , BIAU, BOYER , GOUDE 2.1). In addition, block resampling introduces a bias, as Ztnbelongs to only a single block and is therefore less likely to be resampled than Zt⌊n/2⌋. This explains why E(¯Z∗ n|Zt1, . . . , Z tn)̸=¯Zn. To address both problems, Politis... | https://arxiv.org/abs/2502.10485v1 |
Precisions on the Use case 1 on the IEEE DataPort Competition on Day-Ahead Electricity Load Forecasting In this appendix, we provide additional details on the two WeaKLs used in the benchmark for Use case 1 of the IEEE DataPort Competition on Day-Ahead Electricity Load Forecasting. The 32 FORECASTING TIME -SERIES WITH ... | https://arxiv.org/abs/2502.10485v1 |
(2022), ensuring a fair comparison between the two approaches. Then, we run an online WeaKL, where the effects ˆgℓ,1≤ℓ≤7, are inherited directly from the previously trained additive WeaKL. The weights of this online WeaKL are determined using the hyperparameter selection technique described in Appendix D.1. The trainin... | https://arxiv.org/abs/2502.10485v1 |
those in the GAM model, aiming to identify an optimal configuration. Specifically, we consider the additive WeaKL with X= (FcloudCover corr1 ,Load1D ,Load1W ,DayType ,FTemperature corr1 , FWindDirection ,FTemps95 corr1 ,Toy,t), where 34 FORECASTING TIME -SERIES WITH CONSTRAINTS (i)the effects of FclouCover corr1 ,Load1... | https://arxiv.org/abs/2502.10485v1 |
process. 35 DOUM `ECHE , BACH, BEDEK , BIAU, BOYER , GOUDE Online WeaKL. Next, we train an online WeaKL to update the effects of the additive WeaKL. To achieve this, we apply the hyperparameter selection technique detailed in Appendix D.1. The train- ing period spans from 1February 2018 to1April 2020 , while the valida... | https://arxiv.org/abs/2502.10485v1 |
Implicit vs. explicit regularization for high-dimensional gradient descent Thomas Stark and Lukas Steinberger University of Vienna Abstract In this paper we investigate the generalization error of gradient descent (GD) applied to an ℓ2-regularized OLS objective function in the linear model. Based on our analysis we dev... | https://arxiv.org/abs/2502.10578v1 |
the generalization performance of the natural benchmark. Another reason why early stopping of GD can be understood as a kind of implicit regularization is the fact that for an appropriate choice of iteration number its risk is very close to that of the optimal benchmark, which in our setting is explicitly ℓ2regularized... | https://arxiv.org/abs/2502.10578v1 |
In particular, our contributions are the following. •We provide a precise finite sample analysis of the (out-of-sample) generalization error of (fixed step size) regularized gradient descent (cf. Section 2.4). •In particular, we find that the generalization error of RGD is monotonically decreasing in the iteration numb... | https://arxiv.org/abs/2502.10578v1 |
t>0and initialized at ˆβ0(λ,t) =θ∈Rpto (1.1) the iterations take the following form: ˆβm(λ,t) =ˆβm−1(λ,t)−t∇L(ˆβm−1(λ,t)), where∇L(β) =1 n/parenleftig −X⊤(y−Xβ) +λnβ/parenrightig .(2.1) As we can see from (2.1), calculating one RGD iteration takes O(np)flops. Thus, as long asm≤min{n,p}we are computationally better of... | https://arxiv.org/abs/2502.10578v1 |
training data (X,y)in terms of the (out-of-sample) generalization error Riskout(ˆβ) =E((xT 0ˆβ−y0)2|X) (2.3) =E(E((ˆβ−β)Tx0xT 0(ˆβ−β)|X,y)|X) +σ2 =E((ˆβ−β)TΣ(ˆβ−β)|X) +σ2. Since the irreducible error term σ2does not depend on ˆβ, we analyse only the quantity RΣ(ˆβ):=E((ˆβ−β)TΣ(ˆβ−β)|X). 2.3 The random effects assumptio... | https://arxiv.org/abs/2502.10578v1 |
sample analysis. Since these are technical but conceptually straight forward alternative views on the random effects assumption, we do not include the details here. 2.4 Generalization error of RGD In this section we present our first main result on the generalization properties of regularized gradient descent (2.1). Am... | https://arxiv.org/abs/2502.10578v1 |
computational budget. However, by Theorem 2.5(b) we have RΣ(ˆβm(λ∗,t))→RΣ(ˆβR(λ∗)), asm→∞fort∈(0,2/(s1+λ)), which is no surprise given that ˆβm(λ∗,t)→ˆβR(λ∗)asm→∞fort∈(0,2/(s1+λ))(cf. Proposition 2.1). Hence, only optimally tuned RGD can achieve the benchmark to arbitrary precision, provided the algorithm runs long eno... | https://arxiv.org/abs/2502.10578v1 |
their result is that minimization of CVnhas to be done on a pre-specified compact interval that is known to contain the optimal tuning parameter λ∗, that is, they require λ1≤λ∗≤λ2. Another problem with cross-validation in general – and even in this simple setup where there is the short cut formula – is the computationa... | https://arxiv.org/abs/2502.10578v1 |
we assume that uis independent of the Zi,jandβfor alln. (f) For all ˜β∈Sp−1={v∈Rp:∥v∥2= 1}andk∈{1,2}, we assume ˜β⊤Σk n˜β−1 ptr(Σk n)→0 asn→∞. (g)Letβ∈Rp, whereτn=∥β∥2is uniformly bounded in n(i.e., supn∈N∥β∥2<∞). For eachn∈N,uis an-dimensional random vector with independent entries satisfying E(ui) = 0,E(u2 i) =σ2 n, ... | https://arxiv.org/abs/2502.10578v1 |
every point x∈Rat which ¯Fis continuous. The corresponding Stieltjes transform v(z) =m¯F(z)withz∈C+is the unique solution to v(z) =m¯F(z) =−/parenleftbigg z−γ/integraldisplay∞ 0tdH(t) 1 +tm¯F(z)/parenrightbigg−1 . (3.6) 13 The limit distribution F=Fγ,His written with dependence of γandH, since the limit only depends on... | https://arxiv.org/abs/2502.10578v1 |
i.i.d.data forZwith finite 12-th moments and Σnto be positive-definite for all n, but without assuming that the largest eigenvalue of Σnis uniformly bounded from above. The second result of independent interest is the following Lemma, where the first statement is similar to Ledoit and Péché [13][Lemma 1] with the diffe... | https://arxiv.org/abs/2502.10578v1 |
Statistics 50(2), 949–986 (2022). https://doi.org/10.1214/21-AOS2133 [11]Hastie, T., Tibshirani, R., Friedman, J.H.: The elements of statistical learning: data mining, inference, and prediction. New York, Springer (2009) [12]Hucker, L., Reiß, M.: Early stopping for conjugate gradients in statistical inverse problems. P... | https://arxiv.org/abs/2502.10578v1 |
that we can write (Ip−tnCm) =Ip−(ˆΣ +λIp)−1(Ip−Am)ˆΣ =Ip−(ˆΣ +λIp)−1(ˆΣ +λIp−λIp−AmˆΣ) = (ˆΣ +λIp)−1(λIp+AmˆΣ) and(Ip−tnCm)is symmetric, since all matrices involved are simultaneously diagonalizable and thus commute. Hence we obtain (I) =τ2 ptr((Ip−tnCm)⊤Σ(Ip−tnCm)) =τ2 ptr(Σ(ˆΣ +λIp)−2(λIp+AmˆΣ)2)and 20 (II) =σ2t2 ntr... | https://arxiv.org/abs/2502.10578v1 |
since am i is monotonically decreasing in mfort∈(0,1/(λ+sp)]andλ∈[λ∗,∞), the result in (a) follows. Comparing (2.5)and(2.6)we see that limm→∞ei=fi, as long as, t∈ (0,2/(s1+λ))and therefore the second statement follows. For the third statement we need to check if tr(ΣE) =RΣ(ˆβm(λ,t))<RΣ(ˆβR(λ)) =tr(ΣF)form∈N,λ∈[0,λ∗)and... | https://arxiv.org/abs/2502.10578v1 |
p−1tr(A)a.s.−→0. By (4.11) we can write 1 ptr(A) =1 pnn/summationdisplay i=1/parenleftbiggtr/parenleftbig(ˆΣn−zIp)−1Σn/parenrightbig 1 +x⊤ i(nˆΣn,−i−nzIp)−1xi−x⊤ i(ˆΣ−zIp)−1xi/parenrightbigg =1 nn/summationdisplay i=11 p/parenleftbiggtr/parenleftbigΣn(ˆΣn−zIp)−1/parenrightbig−x⊤ i(ˆΣn,−i−zIp)−1xi 1 +x⊤ i(nˆΣn,−i−nzIp)−... | https://arxiv.org/abs/2502.10578v1 |
Since∥(λvn(−λ)Σ +λIp)−1∥2 2≤λ−1and∥Σn∥2 2≤C, we can conclude by the same arguments as in Theorem 3.7 that p−1tr(A)a.s.−→0. To complete the first statement, we are going to show 1 ptr/parenleftbigΣn(λvn(−λ)Σn+λIp)−1/parenrightbiga.s.−→1−λmF(−λ) 1−γ(1−λmF(−λ)). Forλ∈R+we writexn=xn(λ) =λvn(−λ)and by Theorem 3.3 the empir... | https://arxiv.org/abs/2502.10578v1 |
=1 n2E(β⊤(X⊤X)2β+ 2u⊤XX⊤Xβ+u⊤XX⊤u|X,β) =1 n2β⊤(X⊤X)2β+σ2 n ntr/parenleftbiggXTX n/parenrightbigg =β⊤ˆΣ2 nβ+σ2 nγnˆm1. So we obtain E(ˆσ2|X,β) =1 ˜m2E/parenleftbigg∥y∥2 2 nˆm2−ˆm1∥XTy∥2 2 n2/vextendsingle/vextendsingle/vextendsingle/vextendsingleX,β/parenrightbigg 32 =1 ˜m2/parenleftbiggˆm2 nβ⊤X⊤Xβ−ˆm1 n2β⊤(X⊤X)2β+σ2( ˆ... | https://arxiv.org/abs/2502.10578v1 |
in (4.18) we get R(ˆβR(λ∗))≤R(ˆβR(λ)), for allλ>0and therefore proving the optimality of λ∗. Lemma 4.2. Letm∈Nandx≥0. Then,|1−(1−x)m|≤max(1,|(1−x)m−1|)|x|m. Proof of Lemma 4.2. |1−(1−x)m|=|m(1−ζ)m−1x|<max (1,|(1−x)|m−1)|x|m. (4.19) The first equality follows from the mean value theorem for some zeta in the open interva... | https://arxiv.org/abs/2502.10578v1 |
i=1Σ1/2 nziz⊤ iΣ1/2 n)2) =1 n2n/summationdisplay i=1tr((Σ1/2 nziz⊤ iΣ1/2 n)2) +1 n2n/summationdisplay i=1n/summationdisplay i1=1 i1̸=itr((Σ1/2 nziz⊤ iΣ1/2 n)(Σ1/2 nzi1z⊤ i1Σ1/2 n)) =1 n2n/summationdisplay i=1(z⊤ iΣnzi)2+1 n2n/summationdisplay i=1n/summationdisplay i1=1 i1̸=i(z⊤ iΣnzi1)2=1 n2n/summationdisplay i=1n/summ... | https://arxiv.org/abs/2502.10578v1 |
k=1σ4 jk≤pC4, because tr(Σ4 n) =p/summationdisplay j=1p/summationdisplay k=1(p/summationdisplay l=1σjlσlk)2= (p/summationdisplay j=1σ2 1j)2+ (p/summationdisplay j=1σ1jσj2)2 +···+ (p/summationdisplay j=1σ1jσjp)2+···+ (p/summationdisplay j=1σ2 pj)2+···+ (p/summationdisplay j=1σ1jσpj)2 ≥p/summationdisplay j=1p/summationdi... | https://arxiv.org/abs/2502.10578v1 |
m=1 m̸=lajjσkkallσmmE/parenleftbig(z2 i,jz2 i,k−1)(z2 i,lz2 i,m−1)/parenrightbig ≤ν2 4 pn2p/summationdisplay j=1p/summationdisplay k=1 k̸=ja2 jjσ2 kk+ν2 4 pn2p/summationdisplay j=1p/summationdisplay k=1 k̸=jajjσjjakkσkk +ν4 pn2p/summationdisplay j=1p/summationdisplay k=1 k̸=jp/summationdisplay m=1 m̸=j m̸=ka2 jjσkkσmm+... | https://arxiv.org/abs/2502.10578v1 |
I=|ˆσ2 n−E(ˆσ2 n|X)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 ˜m2/parenleftbigg ˆm2||y||2 2 n−ˆm1∥|X⊤y||2 2 n2/parenrightbigg −1 ˜m2/parenleftbigg ˆm2β⊤ˆΣnβ−ˆm1β⊤ˆΣ2 nβ/parenrightbigg −σ2 n/vextendsingle/vextendsingle/vextendsingle/vextendsingle (4.28) Substituting∥y∥2 2/n=β⊤ˆΣnβ+2u⊤Xβ/n +u⊤u/nand∥X⊤y∥... | https://arxiv.org/abs/2502.10578v1 |
assumption (a), 1 nn/summationdisplay i=1p/summationdisplay j=1P(|zi,j|≥√n)≤1 nn/summationdisplay i=1p/summationdisplay j=1E(|zi,j|4) n2=ν4p n2=o(1). Since the singular values of a symmetric p×pmatrixAare the positive square root of the eigenvalues of A⊤Aand by the triangle inequality for the spectral norm we observe t... | https://arxiv.org/abs/2502.10578v1 |
A Power Transform Jonathan T. Barron barron@google.com Power transforms, such as the Box-Cox transform [5] and Tukey’s ladder of powers [3], are a fundamental tool in mathe- matics and statistics. These transforms are primarily used for normalizing and standardizing datasets, effectively by raising values to a power. I... | https://arxiv.org/abs/2502.10647v1 |
λ= 1 λ· 1−λq 1 +1−λ λx−1 if 0< λ < +∞ ∧λ̸= 1 x ifλ= 0 −λ λ+1· 1−1 λxλ+1−1 if− ∞ < λ < 0∧λ̸=−1 log(1 + x) if λ=−1 1−exp(−x) if λ=−∞(2) 1arXiv:2502.10647v1 [cs.LG] 15 Feb 2025 There are other non-singular values for λthat yield familiar forms of f(x, λ): f(x,−3) =−3 2·1 (1 +x/3)2−1 , f (x,3) = 3 · 1p 1−2x/3−1! , ... | https://arxiv.org/abs/2502.10647v1 |
1−1 Charbonnier Loss (10) ρ(x,−2, c) =2x2 4c2+x2Geman-McClure Loss (11) Here is a visualization of this family of robust losses: 12(x/c)2 /radicalbig(x/c)2+ 1−1 log(1 +1 2(x/c)2)exp(12(x/c)2)−1 −4c−3.5c−3c−2.5c−2c−1.5c−c−0.5c00.5cc1.5c2c2.5c3c3.5c4c x0.00.51.01.52.02.53.0ρ(x,λ,c ) 2x2 4c2+x2 1−exp(−1 2(x/c)2)−log/paren... | https://arxiv.org/abs/2502.10647v1 |
c )), Z (λ) =Z∞ −∞exp(−ρ(x, λ,1))dx (18) This family includes several existing probability distributions: P(x,∞, c) =3 4c√ 2·max 0,1−1 2(x/c)2 Epanechnikov Distribution [9] (19) P(x,0, c) =1 c√ 2π·exp −1 2(x/c)2 Normal Distribution (20) P(x,−1/2, c) =1 2cK1(1)·exp −p 1 + (x/c)2 “Smooth Laplace” Distribution (21) ... | https://arxiv.org/abs/2502.10647v1 |
λ+=−∞λ−=∞ λ+=−2λ+=−1log(1 +x) λ+=−1/2λ+= 0λ+=1/2λ+= 1λ+= 2λ+=∞The upper left quadrant is sigmoid-shaped, the lower right quadrant is logit-shaped, the lower left quadrant is “log-shaped”, and the upper right quadrant is “exp-shaped.” exp(x)−1,log(1 + x), and the ELU activation [8] are members of this family, as shown. ... | https://arxiv.org/abs/2502.10647v1 |
in the Box-Cox transform. This shift also makes it easier to reason about the inverse f−1(x, λ) =f(x,−λ), and induces an aesthetically pleasant symmetry to the math and to the visualizations. Here are plots of the Box-Cox transform h(x, λ)(left), the normalized variant ˆh(x, λ)(center), and f(x, λ)(right). 0 1 2 3 4 x0... | https://arxiv.org/abs/2502.10647v1 |
geometric mean of the absolute error for x∈[0.01,1]) of the naive and stable implementations of f(x, λ)for different values of λ. On the right are plots of the stable implementation (dotted lines) and the naive implemen- tation (solid lines) for values of λclose to ±1. The numerical instability of the naive implementat... | https://arxiv.org/abs/2502.10647v1 |
8 [3] H. Beyer. Exploratory Data Analysis . Addison-Wesley, 1977. 1 [4] Michael J Black and Paul Anandan. The robust estimation of multiple motions: Parametric and piecewise-smooth flow fields. Computer vision and image understanding , 1996. 3 [5] George EP Box and David R Cox. An analysis of transformations. Journal o... | https://arxiv.org/abs/2502.10647v1 |
arXiv:2502.10772v1 [math.ST] 15 Feb 2025Convergence Analysis of a Greedy Algorithm for Conditionin g Gaussian Random Variables Daniel Winkle, Ingo Steinwart, and Bernard Haasdonk University of Stuttgart Faculty 8: Mathematics and Physics Stuttgart, Germany daniel.winkle@mathematik.uni-stuttgart.de ingo.steinwart@mathem... | https://arxiv.org/abs/2502.10772v1 |
the kernel setting, the so-called P-greedy algorithm (Santin and Haasdonk, 2017). In simple te rms, we select the measurement that provides the ‘most’ additional informati on, taking into account all the prior measurements. This approach is the same as in Bayesian optimization, when o ne maximes the covariance function... | https://arxiv.org/abs/2502.10772v1 |
set L2(A) :=L2(Ω,A,µ)as the Hilbert space of µ-equivalence classes of square integrable functions. Note that here we emphasize the dependence on the σ-algebraAby taking it as argument of the space. The reason for this is that conditioning changes the σ-algebra, while the measure µusually remains unchanged in this work.... | https://arxiv.org/abs/2502.10772v1 |
(X)e′:=E(e′(X)X)for alle′∈E′. Givene′∈E′the one-dimensional conditional variance is defined by cov(e′(X)|Y) :=E/parenleftbig (e′(X)−E(e′(X)|Y))2|Y/parenrightbig , as in (Spanos, 2019, Chapter 4). We want to generalize this to the infinite dimensional case. To this end recall that given e1,e2∈Ethe elementary tensor e1⊗e2:... | https://arxiv.org/abs/2502.10772v1 |
W)⊆WZand thus we can consider the new operator MW:WY→WZgiven by MWwy:=L∗ Wwy=ˆVZˆV∗ Ywy. (8) Note that MWis a bounded and linear operator with /bardblMW/bardbl ≤1. In addition MWis surjective since ˆVZ: L2(A)→WZis surjective and ker (ˆVZ)⊥=GZ⊆GY=Im(ˆV∗ Y), see Lemmata 15 and 28. Finally, MWis in general notan isometry,... | https://arxiv.org/abs/2502.10772v1 |
true, where the convergence of the series is almost eve rywhere and in Lp(A,E)for allp∈[1,∞). Additionally the following statements hold true: i) The orthogonal projection ΠG:G(X,Y)→G(X,Y)ontoG(Z,Y)=GYis given by ΠGg=E(g|Y). 6 ii) The orthogonal projection ΠW:W(X,Y)→W(X,Y)ontoW(Z,Y)is given by ΠW(wx,wy) = (L∗ wwy,wy). ... | https://arxiv.org/abs/2502.10772v1 |
exists an extension ˆM:F→˜EofMWwhich is an isometric isomorphism and satisfies ˆMY=Z. We note that one could instead of changing the norm on Echange the norm on Fto obtain a continuous extension of MWinto the space E. Having shown criteria for extending the operator MWto an operator M:F→E, we make the following assumpti... | https://arxiv.org/abs/2502.10772v1 |
LWw=w|[1/2,1], see Lemma 17. To this end, we first note that for u∈WXandv∈WYwe have 2·u(1/2)v(1/2)+/integraldisplay1 1/2u′(t)v′(t)dt=/a\}bracketle{tLWu,v/a\}bracketri}htWY=/a\}bracketle{tu,L∗ Wv/a\}bracketri}htWX=/integraldisplay1 0u′(t)(L∗ Wv)′(t)dt. In particular, for u∈WXwith supp(u)⊂[1/2,1]we can conclude that /inte... | https://arxiv.org/abs/2502.10772v1 |
that for ˜E:=Fwe obtain Mw=was a continuous linear mapping, see Example 41. 5 Proofs Lemma 15. Let Assumption A be satisfied and VX:GX→WXbe the map given by VXg:=/integraldisplay ΩgXdµ. Then the following statements hold true: i)VXis an isometric isomorphism and its adjoint V∗ XofVXis given by V∗ X=V−1 X. ii)WXis a Hilb... | https://arxiv.org/abs/2502.10772v1 |
be found in (Veraar and Weis, 2016, Theor em 3.3.2). Theorem 20. Let(An)n∈Nbe a filtration in the probability space (Ω,A,µ)and we define A∞:= σ(An:n≥1). Then for all p∈[1,∞). and allX∈Lp(A,E), we have lim n→∞E(X|An) =E(X|A∞), where the convergence is almost everywhere and in the norm of Lp(A,E). 14 Lemma 21. Let Assumpti... | https://arxiv.org/abs/2502.10772v1 |
2∈E′we havee′ 1(ΠGX)∈ G ⊆Gande′ 2(X)∈GX⊆G, and thus e′ 1(ΠGX)+e′ 2(X) is as an element of GGaussian. We conclude ΠGXis jointly Gaussian with X. Moreover, since each e′(Zn)is anR-valued Gaussian random variable, so is the limit e′(Z∞), see e.g. (Steinwart, 2024, Theorem C.4). To verify the last assertion we first note th... | https://arxiv.org/abs/2502.10772v1 |
21, Identity (4), and Lemma 25 leads to e′(E(X|Y)) =E(e′(X)|Y) =E(e′(X)|GY) = ΠGYe′(X) =e′(ΠGYX), whereΠGYdenotes the orthogonal projection ΠGY:G(X,Y)→G(X,Y)ontoGY. Using Lemma 24 and Lemma 25, we obtain E(X|Y) = ΠGYX=/summationdisplay j∈Jrj/integraldisplay ΩrjXdµ, (23) where the series converges almost everywhere and ... | https://arxiv.org/abs/2502.10772v1 |
allB∈ E, then we have Z∈E(X|B). Proof. We consider the Banach space valued mappings Q,P:B →Egiven by Q(B) :=/integraldisplay BXdµ P(B) :=/integraldisplay BZdµ. Moreover we define D:={B∈ B|Q(B) =P(B)}. Note that Ω∈ E by assumption, and (24) implies E ⊆ D . Let us now show that Dis a Dynkin system in the sense of (Cohn, 2... | https://arxiv.org/abs/2502.10772v1 |
B∩. With the help of Lemma 32 and Lemma 31, we conclude that (27) h olds for all B∈ˆB. This finishes the proof of Z∈E(X|ˆB). Next, we note that for ˆZ∈E(X|ˆB)we haveµ(Z/\e}atio\slash=ˆZ) = 0 by the almost sure uniqueness of the conditional expectation of XgivenˆB. Our next goal is to show that Z∈E(X|C). SinceB ⊆ C , it ... | https://arxiv.org/abs/2502.10772v1 |
have P(E) = 0 if/summationtext∞ j=1P(Ej)<∞. Using a well known tail bound for one dimensional standard normal ra ndom variables we find P(|rj|> t)≤2e−t2/2 for allt >0. We thus obtain ∞/summationdisplay j=1P(Ej) =∞/summationdisplay j=1P/parenleftbigg |rj|>√ 2·/radicalBig ln(jln2(j+1))/parenrightbigg ≤∞/summationdisplay j... | https://arxiv.org/abs/2502.10772v1 |
a closed subspace of L2(A)we then find the desired inclusion. To prove the other inclusion GY⊆GE, we fix an a′∈ℓ2(N). Analogously to (29) we have /parenleftbigg αja′(ej) j/parenrightbigg ∈ℓ2(N). By (32) we conclude that a′(Y)∈GE, and since GEis a closed space we then find GY⊆GE. Proof of Theorem 4. We setGE:=span{e′(X)|e′... | https://arxiv.org/abs/2502.10772v1 |
7. We first note that by Lemma 39, we have that MWis an isometry. We now define a norm onWZby /bardblw/bardbl˜E0:=/bardblM−1 Ww/bardblF for allw∈WZ. SinceMWis invertible as an isometric isomorphism, this norm is well -defined. We define ˜E to be the completion of the space (WZ,/bardbl · /bardbl˜E0). Our first goal is to sho... | https://arxiv.org/abs/2502.10772v1 |
sup f∈BHYsup f′∈BF′|(f−ΠF′nf)(f′)|2 = sup f′∈BF′/bardblkY(·,f′)−ΠF′nkY(·,f′)/bardbl2 HY = sup f′∈F′/bardblcov(Y)−cov(Yn)/bardblF′→F. Again using Theorem 4 and (Santin and Haasdonk, 2017, Lemma 2 .3) we obtain dn(F)2≤sup f∈BHY/bardblf−ΠF′nf/bardbl2 C(BF′) =/bardblcov(Y)−cov(Y⋆ n)/bardblF′→F≤Cn−α. Using (DeV ore et al., ... | https://arxiv.org/abs/2502.10772v1 |
adjoint given u∈WXandv∈WYwe obtain /a\}bracketle{tLWu,v/a\}bracketri}htWY=/a\}bracketle{tL† WLWu,L† Wv/a\}bracketri}htWX=/a\}bracketle{tu,(L† WLW)∗L† Wv/a\}bracketri}htWX=/a\}bracketle{tu,L† Wv/a\}bracketri}htWX, where in the last step we used one of the defining properties of the Moore-Penrose inverse. In other words w... | https://arxiv.org/abs/2502.10772v1 |
(wj)j∈J. In other words, we have αj=λ−1/2 jwjfor allj∈J. In view of (39) we conclude that the sought minimizer satisfiesαi= 0for alli∈I\Jand therefore we find MWw=/summationdisplay j∈Jαj/radicalbig λjej=/summationdisplay j∈Jwjej with convergence in WX. Since this shows /bardblMWw/bardblH≤ /bardblw/bardblℓ2(J)for allw∈ℓ2(... | https://arxiv.org/abs/2502.10772v1 |
Tarieladze, V ., and Chobanyan, S. (2012). Probability distributions on Banach Spaces . Springer. Veraar, T. H. J. V . N. M. and Weis, L. (2016). Analysis in Banach Spaces: Volume I: Martingales and Little wood- Paley Theory . Springer. Wenzel, T., Santin, G., and Haasdonk, B. (2022). Analysis of target data-dependent ... | https://arxiv.org/abs/2502.10772v1 |
Submitted to Electronic Journal of Statistics ISSN: 1935-7524 Spectral analysis of spatial-sign covariance matrices for heavy-tailed data with dependence Hantao Chen and Cheng Wang School of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong University, e-mail: htchen2000@sjtu.edu.cn ;chengwang@sjtu.edu.cn Abstract: Th... | https://arxiv.org/abs/2502.10943v1 |
Bai and Silverstein (2010) for example. From a statistical perspective, the moment condition EZ2 11<∞is neces- sary for the existence of the population covariance matrix, cov (X), and for the consistent estimation of the sample covariance matrix. For heavy-tailed distri- butions where EZ2 11=∞, the sample covariance ma... | https://arxiv.org/abs/2502.10943v1 |
studied by Gao et al. (2017). For general Σ, it is referred to Heiny (2022) for LSD and Zheng et al. (2019) for CLT. See also Dörnemann and Heiny (2022)andYin,ZhengandZou(2023).Forheavy-tailedpopulationswith Σ=I, Heiny and Yao (2022) found a new curious LSD under infinite second moment 4 and Heiny and Parolya (2024) pr... | https://arxiv.org/abs/2502.10943v1 |
the moments of self-normalized variables Y1,...,Yp, we consider the moment EYk1 1···Ykr r, for general integers k1,···,kr>0andr≤p. 6 By the identity of Gamma function 1 xβ=1 Γ(β)/integraldisplay∞ 0e−txtβ−1dt, and Fubini’s theorem, we have EYk1 1···Ykr r=E/parenleftigg Zk1 1···Zkr r1 Γ(k 2)/integraldisplay∞ 0e−s(Z2 1+·... | https://arxiv.org/abs/2502.10943v1 |
•α≥2: E/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay i̸=jaijYiYj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglek =/braceleftigg O(p−1)∥A∥k, k = 2, o(p−1)∥A∥k, k≥3. •α>4: E/vextendsingle/vextendsingle/vextendsingle/vextendsingl... | https://arxiv.org/abs/2502.10943v1 |
X⊤ 1X1−1 ptrA/parenrightbigg/parenleftbiggX⊤ 1BX 1 X⊤ 1X1−1 ptrB/parenrightbigg . (4) Both two steps are quite similar to that in proving LSD, but we need to find more precise results here under a stronger condition. For (4), we derive the results in Proposition 2.7. Next, we consider T. By Proposition 2.7, pEY2 i Y⊤ΣY... | https://arxiv.org/abs/2502.10943v1 |
norm, i.e., ∥FB− Fy,H∥∞. In Figure 2, we set p∈{200,800,2000,5000}withy= 0.5, and the degrees of freedom α∈{0.5,1.0,1.5,2.0,2.5,3.0}. On the one hand, if α < 2, the error terms converge to a positive level whenpandntending to infinity, and it increases dramatically as αdecreases to zero. On the other hand, if α≥2, the ... | https://arxiv.org/abs/2502.10943v1 |
we have ∗/summationdisplay i1,···,iraij1ij2···aij2k−1ij2k=/braceleftigg O(p)∥A∥k, r = 2,3, O(pr+⌊r 4⌋ 2)∥A∥k, r≥4,(7) wherej1,···,j2k∈{1,···,r}andj2l−1̸=j2lforl= 1,···,k. Proof.For simplicity, we denote A⊗mas the Hadamard product of mmatrices A. Whenr= 2, (7) can only be ∗/summationdisplay i1,i2ak i1i2= 1⊤ pA⊗k1p−tr(A... | https://arxiv.org/abs/2502.10943v1 |
proof is complete. Lemma A.7. Suppose (Zi)∞ i=1are i.i.d. and regularly varying with index α>4, (ai)∞ i=1is a positive sequence with infai>0. We denote J1={1≤i≤r:ki= 1}, J 2={1≤i≤r: 2≤ki≤4}, J3={1≤i≤r:ki≥5}, 20 then for any 4<β < min{α,5}, EZk1 1···Zkrr/parenleftbig a1Z2 1+···+apZ2p/parenrightbigk 2=O(p−3 2|J1|−/summat... | https://arxiv.org/abs/2502.10943v1 |
B.2. Proof of Proposition 2.2 Proof.By Lemma A.4 Giné, Götze and Mason (1997), the conclusion holds for the case where at least one of k1,···,krequals to one. We extend the result to general case where at least one of k1,···,kris odd by induction. Without loss of generality, we assume k1is odd and set k1= 2m+1. Ifm= 0,... | https://arxiv.org/abs/2502.10943v1 |
i=1aiiY2 i/parenrightigg + 2var /summationdisplay i̸=jaijYiYj =o(1)∥A∥2, and E/vextendsingle/vextendsingle/vextendsingle/vextendsingleY⊤AY−trA p/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 =var/parenleftbig Y⊤AY/parenrightbig +/vextendsingle/vextendsingle/vextendsingle/vextendsingleEY⊤AY−trA p/vextend... | https://arxiv.org/abs/2502.10943v1 |
p3tr(Σ◦Σ) +2 p3tr(Σ2) +1−τ p2/bracketrightbigg +o(p−1)∥A∥∥Σ∥2, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleEY⊤AY/parenleftbig Y⊤ΣY−1/parenrightbig3 Y⊤ΣY/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤∥A∥ λmin(Σ)E/vextendsingle/vextendsingleY⊤ΣY−1/vextendsingle/vextendsin... | https://arxiv.org/abs/2502.10943v1 |
+1 p/summationdisplay i̸=j/parenleftbigg p·EYiYj Y⊤ΣY/parenrightbigg2 . It remains to show p·EY2 i Y⊤ΣY−1→0,uniformly in 1≤i≤p, (15) p3 2·/vextendsingle/vextendsingle/vextendsingle/vextendsingleEYiYj Y⊤ΣY/vextendsingle/vextendsingle/vextendsingle/vextendsingle→0,uniformly in 1≤i,j≤p. (16) •Diagonal elements: As for (15... | https://arxiv.org/abs/2502.10943v1 |
concluded that EY2 i(1−Y⊤ΣY)2=2tr(Σ2) p3+tr(Σ◦Σ)(τ−3) p3−τ−1 p2+o(p−2).(20) Then, we consider EY2 i(1−Y⊤ΣY)4 Y⊤ΣY≲EY2 i(1−Y⊤ΣY)4 ≲EY2 i /summationdisplay j(σjj−1)Y2 j 4 +EY2 i ∗/summationdisplay j,kσjkYjYk 4 . (21) For the first term, we expand the fourth power and it consists of the summation like ∗/summationd... | https://arxiv.org/abs/2502.10943v1 |
kZlZs (/summationtextp k=1σkkZ2 k)3 +∗/summationdisplay k,l,s,tσklσstEZiZjZkZlZsZt (/summationtextp k=1σkkZ2 k)3 =4σ2 ijEZ3 iZ3 j (/summationtextp k=1σkkZ2 k)3+ 8/summationdisplay k̸=i,jσ2 ikEZ3 iZjZ2 k (/summationtextp k=1σkkZ2 k)3 + 2∗/summationdisplay k,l̸=i,jσ2 klEZiZjZ2 kZ2 l (/summationtextp k=1σkkZ2 k)3+ 16σij/s... | https://arxiv.org/abs/2502.10943v1 |
sign covariance matrix with unknown location. Journal of Multivariate Analysis 130107– 117. Dürre, A. ,Vogel, D. andFried, R. (2015). Spatial sign correlation. Journal of Multivariate Analysis 13589–105. spatial-sign covariance matrices for heavy-tailed 47 Feng, L. ,Zou, C. andWang, Z. (2016). Multivariate-sign-based h... | https://arxiv.org/abs/2502.10943v1 |
A robust and p-hacking-proof significance test under variance uncertainty Xifeng Li,∗Shuzhen Yang,†Jianfeng Yao‡ Abstract P-hacking poses challenges to traditional hypothesis testing. In this paper, we propose a robust method for the one-sample significance test that can protect against p-hacking from sample manipulati... | https://arxiv.org/abs/2502.11038v1 |
historical information in order to artificially inflate the significance of the test (see details in Section 4). Because of lack of information on the variance of the data which can be in particular time-varying, it is actually difficult to construct an accurate one-sample test that respects a preassigned significance ... | https://arxiv.org/abs/2502.11038v1 |
machine learning, see, for example, Epstein and Ji (2013), Lin et al. (2016), Peng and Zhou (2020), Jin and Peng (2021), Peng et al. (2023), Ji et al. (2023), among others. We refer to Peng’s plenary talk at ICM 2010 (Peng, 2010) and his monograph (Peng, 2019) for a com- plete account of the theory of nonlinear/subline... | https://arxiv.org/abs/2502.11038v1 |
function φ, lim n→∞sup {σi}∈Σ(σ,σ)E[φ(√n(¯Xn−µ))] = u(1,0;φ), (3.4) where {u(t, x;φ) : (t, x)∈[0,∞)×R}is the unique viscosity solution to the Cauchy problem, ut=1 2 ¯σ2(uxx)+−σ2(uxx)− , u(0, x) =φ(x). (3.5) In the above expression, ut=∂u/∂t ,uxx=∂2u/∂x2, and a+anda−denote the positive and negative parts of a, respect... | https://arxiv.org/abs/2502.11038v1 |
{σ2,σ2}of the variance interval [ σ2,σ2] are unknown, we need to estimate them from the data. Clearly, further information on the data generating process is needed for otherwise, it is hard to find good estimates for the two variance bounds governing the variability of the data sequence. In this paper, we assume that t... | https://arxiv.org/abs/2502.11038v1 |
. . . , Z k} is the smallest unbiased estimator for the lower mean µ. Applying this result to the asymptotic maximum distribution M[σ2,σ2] establishes the conclusions. In practice, the ksub-sample sizes may not be equal, and we do not know the exact change points of the sub-samples. We can still construct 8 optimal est... | https://arxiv.org/abs/2502.11038v1 |
level α. In Section 4.1, we present an optimal manipulation strategy that the experimenter will adopt to maximize the probability of {√n(¯Zn−µ0)> c} when µ=µ0, where cis a given constant. Then in Section 4.2, we compare the defense capability of the traditional test procedure with our robust pro- cedure. Although the n... | https://arxiv.org/abs/2502.11038v1 |
the actual type I error rate. Proposition 4.1 (Peng and Zhou (2020)) .For a given nominal significance level α(0< α < 0.5), if the experimenter chooses data Z1, . . . , Z nas in Theorem 4.1 with c= ¯σΦ−1(1−α), then the actual type I error rate of the rejection region given by the traditional test procedure is at least ... | https://arxiv.org/abs/2502.11038v1 |
given by the sublinear expectation in (3.8) can ensure that the actual asymptotic type I error rate does not exceed α. Obviously, for sufficiently large n (c1,∞)⊂(SnΦ−1(1−α),∞), a.e.. Thus, if the experimenter chooses data Z1, . . . , Z naccording to the optimal strategy and constructs the rejection region ( SnΦ−1(1−α)... | https://arxiv.org/abs/2502.11038v1 |
than 0.05 and does not approach 0.05 as nincreases: indeed the error rate inflation is constantly between 35% and 55%! Note also that the empirical type I error 14 Figure 1: Empirical type I error rate ( µ=µ0) plot over 5,000 repetitions for Simulation 5.1 Figure 2: Empirical power plots over 5,000 repetitions for Simu... | https://arxiv.org/abs/2502.11038v1 |
data sequence using an optimal strategy to artificially inflate the rejec- tion probability. Particularly in this situation, the traditional procedure will report an inflated and wrong test significance while our new method will resist to such wrong inflation without loosing much test power. Several interesting directi... | https://arxiv.org/abs/2502.11038v1 |
identically distributed as X1and independent of (X1, . . . , X i). Definition A.7. (Maximal distribution) A random variable ηon a sublinear expectation space (Ω,H,bE)is called maximal distributed if bE[φ(η)] = sup µ≤y≤µφ(y),∀φ∈CLip(R), where µ=bE[η]andµ=−bE[−η].This distribution is denoted by ηd=M[µ,µ]. Theorem A.3. (L... | https://arxiv.org/abs/2502.11038v1 |
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