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z1, x2): that is, these subscripts are intended to disambiguate the role of the two arguments in the product space. The form of the influence function (22) arises from the fact that the data consist of two i.i.d. parts, LnandUN, which are independent of each other. By Definition 4.2, if ˆθn,Nis asymptotically linear, t... | https://arxiv.org/abs/2502.17741v2 |
of θ(·)atθ∗relative to P. Ifθ(·)is well-specified at P∗ Y|Xrelative to PXin the sense of Definition 3.1, then the efficient influence function of θ(·)atθ∗relative to Punder the OSS is φ∗ η∗(z1). Thus, an efficient supervised estimator of a well-specified parameter can never be improved via the use of unlabeled data. 4.... | https://arxiv.org/abs/2502.17741v2 |
theorem establishes the asymptotic properties of ˆθeff. n,N. Theorem 4.4. Suppose that the supervised estimator ˆθn=ˆθ(Ln)satisfies Assumptions 3.1 and 3.2, and˜ηnis an estimator of η∗as in Assumption 3.1 (c). Further, suppose that {gk(x)}∞ k=1is a set of basis functions of Sα Msuch that GKn(x) = g1(x)⊤, . . . , g Kn(... | https://arxiv.org/abs/2502.17741v2 |
to improve upon existing PPI estimators. Since f1, . . . , f Kare independent of Ln∪UN, existing PPI estimators fall into the category of OSS estimators, and can be shown to be regular and asymptotically linear in the sense of Definitions 4.1 and 4.2. Therefore, Theorem 4.1 suggests that their asymptotic variances are ... | https://arxiv.org/abs/2502.17741v2 |
3.1, Assumption 5.1 requires that the class of functions {gη(x) :η∈ O} is a Donsker class; when η(·) is finite-dimensional, the next proposition provides sufficient conditions for Assumption 5.1. Proposition 5.1. When η(·)is a finite-dimensional functional, the following condition implies As- sumption 5.1 (b): under th... | https://arxiv.org/abs/2502.17741v2 |
and only if Wi= 1, and P∗ Wis a Bernoulli distribution with known probability 1 −γwhere γ= lim n→∞N n+N∈(0,1). Assume that P∗∈ P, where Pis defined in (1) as in previous sections. The underlying model of ( Z, W ) is thus Q={P×P∗ W:P∈ P} . (46) The next proposition derives the efficiency bound relative to (46). Proposit... | https://arxiv.org/abs/2502.17741v2 |
(47) as the initial supervised estimator of θ∗. Under regularity conditions such as those stated in Theorems 5.7 and 5.21 of Van der Vaart [2000], the M-estimator (47) is a regular and asymptotically linear estimator of θ∗with influence function φη∗(z) =−V−1 θ∗∇mθ∗(z), (48) where Vθ(P) =∂2EP[mθ(Z)] ∂θ∂θ⊤andVθ∗=Vθ(P∗)(P... | https://arxiv.org/abs/2502.17741v2 |
that takes values in Y∞×R, where Y∞is the space of uniformly bounded functions h:Y →Requipped with the uniform metric ∥h1−h2∥∞= supY|h1(y)−h2(y)|. Denoting η1= (h1, θ1) and η2= (h2, θ2), it follows that (52) is R-Lipschitz with respect to the metric ρ(η1, η2) =∥h1−h2∥∞+∥θ1−θ2∥. We have already established that U-statis... | https://arxiv.org/abs/2502.17741v2 |
used ( z1, u2) = (( u1, a1, y1), u2) as arguments to the influence function. We note that if Uhas no confounding effect, i.e. µ(u,P∗) =θ∗, then ϕ∗ η∗(u) = 0 and the efficiency bound (55) is the same as in the supervised setting. Example 7.6 (Additional data on confounders and treatment) .Suppose that additional observa... | https://arxiv.org/abs/2502.17741v2 |
prediction model. Each of these estimators is constructed by modifying an efficient supervised estimator, whose perfor- mance we also consider. Additionally, we include the PPI++ estimator proposed by Angelopoulos et al. [2023b], using the same two prediction models as for ˆθPPI n,N. In each simulation setting, we also... | https://arxiv.org/abs/2502.17741v2 |
n= 1,000. Left: When the conditional influence function is linear (Setting 1), ˆθsafe n,N with g(x) =xachieves the efficiency lower bound in the OSS setting. Center: When the conditional influence function is non-linear (Setting 2), ˆθsafe n,Nwith g(x) =xis no longer efficient, whereas ˆθeff. n,Nis efficient with a suf... | https://arxiv.org/abs/2502.17741v2 |
confidence intervals for each method. All methods achieve the nominal coverage. Figure 2 displays the standard error of the first parameter for each method, averaged over 1,000 simulated datasets. Results for the standard error of the second parameter are similar, and are displayed in Figure 3 in Appendix F.1. We first... | https://arxiv.org/abs/2502.17741v2 |
efficient semi-supervised estimators in the presence of high- or infinite-dimensional nuisance parameters. Second, a general theoretical framework for efficient semi-supervised estimation in the presence of covariate shift also remains a relatively open problem, despite some promising preliminary work [Ryan and Culp, 2... | https://arxiv.org/abs/2502.17741v2 |
Abhishek Chakrabortty and Guorong Dai. A general framework for treatment effect estimation in semi-supervised and high dimensional settings. arXiv preprint arXiv:2201.00468 , 2022. 23 Hanan Ahmed, John HJ Einmahl, and Chen Zhou. Extreme value statistics in semi-supervised models. Journal of the American Statistical Ass... | https://arxiv.org/abs/2502.17741v2 |
Conference on Artificial Intelligence and Statistics , pages pp. 7433–7449, 2022. Aad van der Vaart and Jon Wellner. Weak convergence and empirical processes: with applications to statistics . Springer Science & Business Media, 2013. Bruce Hansen. Econometrics . Princeton University Press, 2022. Mark J van der Laan. Ef... | https://arxiv.org/abs/2502.17741v2 |
=f′ a(x)+g′ a(z) such that f′ a(x)∈ TPX(P∗ X) and g′ a(z)∈ TPY|X(P∗ Y|X). However, ga(z)/∈ TPY|X(P∗ Y|X), and hence ga(z)̸=g′ a(z). By definition, TPX(P∗ X)⊆ L2 1,0(P∗ X), and TPY|X(P∗ Y|X)⊆ L2 1,0(P∗ Y|X). Therefore, a⊤φ∗ η∗(z) =f′ a(x) +g′ a(z) is another decomposition of f′ a(x) such that f′ a(x)∈ L2 1,0(P∗ X), and ... | https://arxiv.org/abs/2502.17741v2 |
gradient DP∗(z) ofθ(·) atP∗relative to P. Since this holds true for any s(x)∈ TPX(P∗ X), we see that any gradient DP∗(z) satisfies DP∗(z)⊥ TPX(P∗ X). Consider the the efficient influence function φ∗ η∗(z) ofθ(·) relative to PatP∗, which is a gradient of θ(·) relative to PatP∗by definition. Further, by the definition of... | https://arxiv.org/abs/2502.17741v2 |
η∗) g0 L2(P∗). Since∥L(z)∥L2(P∗)<∞andρ(˜ηn, η∗) =op(1) by Assumption 3.1, and g0 L2(P∗)<∞, we have that ∥L(z)∥L2(P∗)ρ(˜ηn, η∗) g0 L2(P∗)=op(1). Further, by Assumption 3.1, P∗{˜ηn∈ O} → 1, and hence P∗h φ˜ηn(g0)⊤i −P∗h φη∗(g0)⊤i 2=op(1). (63) Combining (61) and (63), we have ˆBg n−Bg 2=op(1) as (60). Since√nPn g0 =Op(... | https://arxiv.org/abs/2502.17741v2 |
Pnh G0 Kn(G0 Kn)⊤i −IKn =Kn+op(1) =Op(Kn), where we used continuous mapping for the tr( ·) function. Since L(z) is square integrable, we then have Pn(L2)Pnh (G0 Kn)⊤G0 Kni ρ2(˜ηn, η∗) =Op(ρ2(˜ηn, η∗)Kn), which implies that I =Opp Knρ(˜ηn, η∗) =op(1). For II, by the bias-variance decomposition, ∥ϕη∗−BKnGKn(x)∥2 L2 1,... | https://arxiv.org/abs/2502.17741v2 |
x2),is a gradient at Q(P∗) relative to P, dθ(t∗) dt=⟨lt∗(z1, x2), DQ(P∗)(z1, x2)⟩H =⟨lt∗(z1, x2), DQ(P∗)(z1, x2)−φ∗ η∗(z1, x2) +φ∗ η∗(z1, x2)⟩H =⟨lt∗(z1, x2), φ∗ η∗(z1, x2)⟩H+⟨lt∗(z1, x2), DQ(P∗)(z1, x2)−φ∗ η∗(z1, x2)⟩H =⟨lt∗(z1, x2), φ∗ η∗(z1, x2)⟩H, which shows that φ∗ η∗(z1, x2) is a gradient. Since further a⊤φ∗ η∗(... | https://arxiv.org/abs/2502.17741v2 |
Lemmas E.1 and E.2, the asymptotic variance of ˆθsafe n,Ncan be represented as Var φη∗(Z)−γBgg0(X) + Varhp γ(1−γ)Bgg0(X)i = Var φη∗(Z)−γBgg0(X) +γ(1−γ)Var Bgg0(X) = Var [ φη∗(Z)−ϕη∗(X)] + Var ϕη∗(X)−Bgg0(X) + (1 −γ)Bgg0(X) +γ(1−γ)Var Bgg0(X) = Var [ φη∗(Z)−ϕη∗(X)] + Var ϕη∗(X)−Bgg0(X) + (1−γ)2Var Bgg0(X) ... | https://arxiv.org/abs/2502.17741v2 |
} III+Pnh rKn(ˆG0 Kn)⊤i ˆΣ−1 | {z } IV +Pnh ϵ(ˆG0 Kn)⊤i ˆΣ−1 | {z } V+Pnh (φ˜ηn(z)−φη∗(z))(ˆG0 Kn)⊤i ˆΣ−1 | {z } VI. We first consider ˆΣ: Pn ˆG0 Kn ˆG0 Kn⊤ =Pnh G0 Kn G0 Kn⊤i +Pn G0 Kn Pn+Nh G0 Kn⊤i +Pn+N G0 Kn Pnh G0 Kn⊤i +Pn+N G0 Kn Pn+Nh G0 Kn⊤i , where, by Lemma E.8, we have that Pn G0 Kn Pn+Nh... | https://arxiv.org/abs/2502.17741v2 |
ˆg0 ˜ηn(Xi) =g˜ηn(Xi)−Pn+N(g˜ηn) =g˜ηn(Xi)−P∗(g˜ηn)−Pn+N[g˜ηn−P∗(g˜ηn)] =g0 ˜ηn(Xi)−Pn+N(g0 ˜ηn). 37 For the estimator ˆθsafe n,Ndefined in (41) and ˆBg n,Ndefined in (40), √n ˆθsafe n,N−θ∗ =√n ˆθn−θ∗ −ˆBg n,N√nPn ˆg0 ˜ηn =√nPn(φη∗)−ˆBg n,N√nPn g0 ˜ηn +ˆBg n,N√nPn+N g0 ˜ηn +op(1) =√nPn(φη∗)−γˆBg n,N√nPn g0 ˜... | https://arxiv.org/abs/2502.17741v2 |
Op(1). We showed that III= Op(1) in the proof of Theorem 4.3. For IV, by Lemma E.5, both Pn+N(g0 ˜ηn) and Pn+N(g0 η∗) areop(1). Consequently, Pn+Nh ˆg0 ˜ηn(ˆg0 ˜ηn)⊤−ˆg0 η∗(ˆg0 η∗)⊤i 2 = Pn+Nh g0 ˜ηn(g0 ˜ηn)⊤−g0 η∗(g0 η∗)⊤i −Pn+N(g0 ˜ηn)Pn+N(g0 ˜ηn)⊤+Pn+N(g0 η∗)Pn+N(g0 η∗)⊤ 2 ⩽ Pn+Nh g0 ˜ηn(g0 ˜ηn)⊤−g0 η∗(g0 η∗)⊤i 2+ P... | https://arxiv.org/abs/2502.17741v2 |
+wsY|X(z) is the score function at P∗×P∗ Wof {P×P∗ W:P∈ PT}, where PT= pt,X(x)pt,Y|X(z) :t∈T⊂R , which proves sX(x)+wsY|X(z)∈ TQ(P∗×P∗ W) and hence M ⊆ T Q(P∗×P∗ W). Pathwise differentiability of θ(·) atP∗relative to P∗implies, for any one-dimensional parametric sub-model PT⊂ P such that pt∗(z) corresponds to the dens... | https://arxiv.org/abs/2502.17741v2 |
semi-supervised estimators of θ∗that are regular and asymptotically linear in the sense of Definitions 4.1 and E.2, with influence function φf A(z1, x2) =−V−1 θ∗h ∇mθ∗(z1)−A∇mf θ∗(x1) +A∇mf θ∗(x2)i , (76) where A∈Rp×p. The above class includes a number of existing PPI estimators, such as the proposal of Angelopoulos et... | https://arxiv.org/abs/2502.17741v2 |
(78), it then follows that D φf A(z1, x2)−φf A∗(z1, x2), φf A∗(z1, x2)⊤E H= 0, which then finishes the proof. Proposition C.1 shows that (41) provides optimal efficiency among estimators with influence func- tion of the form (76). We note that, for fair comparison, we consider the case where there is only one machine l... | https://arxiv.org/abs/2502.17741v2 |
Lipshitz condition of Proposition 3.4. Denote η(1)(P) = EP b(2)(X⊤θ(P))XX⊤−1∈Rp×pandη(2)(P) =θ(P). Consider the metric ρ(η1, η2) = η(1) 1−η(1) 2 2+ η(2) 1−η(2) 2 . Assuming that both Xand Θ are compact, which is a mild assumption, we have: ∥φη1(z)−φη2(z)∥ = η(1) 1−η(1) 2 yx−η(1) 1b(1) x⊤η(2) 1 +η(1) 2b(1) x⊤η(2... | https://arxiv.org/abs/2502.17741v2 |
with O= h(y)−θ:R2→R,∥h∥∗ V⩽M, θ∈Θ , and the estimator ˜ηn= 1 nnX i=1I{(Ui−u)(Vi−v)>0},ˆθn! . 46 Proof. We first validate part (b) of Assumption 3.1. Clearly, h∗(y)−θ∗∈ O, and we have shown that the influence function of U-statistics of kernel RisR-Lipshitz in Section 7.2. Therefore, it remains to prove that {φη:η∈ O} ... | https://arxiv.org/abs/2502.17741v2 |
A)denote the projection operator that project aonto a linear space A. Proof. See Theorem 3.3 of Tsiatis [2006]. Lemma E.3. For a random variable Z∼P∗, and let f, gbe any functions such that f∈ L2 p(P∗)and g∈ L2 q(P∗). Then, P∗ fg⊤ 2⩽∥f∥L2(P∗)∥g∥L2(P∗). Proof. Consider any non-random vector a∈Rqsuch that ∥a∥2= 1. By t... | https://arxiv.org/abs/2502.17741v2 |
such that ∥a∥=O(√Kn). Then •1 nPn i=1a⊤Xi=Opq Kn n • 1 nPn i=1Xi =Opq Kn n Proof. We have: E 1 nnX i=1a⊤Xi!2 =∥a∥2 n=OKn n . E 1 nnX i=1Xi!⊤ 1 nnX i=1Xi! =Kn n. 50 Therefore, by Markov’s inequality, for any ϵ >0, P( 1 nnX i=1a⊤Xi > ϵ) =P 1 nnX i=1a⊤Xi!2 > ϵ2 ⩽Eh 1 nPn i=1a⊤Xi2i ϵ2 =∥a∥2 nϵ2, ... | https://arxiv.org/abs/2502.17741v2 |
a sigma-algebra on Z. Let (X,FX,P∗ X) be the corresponding probability space of ( Z,F,P∗) over X. Consider the product probability space (Z × X ,F ⊗ F X,Q(P∗)), where F ⊗ F Xrepresents the product sigma-algebra of FandFX, and Q(P∗) =P∗×P∗ Xrepresents the product measure of P∗andP∗ X. Note that the probability measure h... | https://arxiv.org/abs/2502.17741v2 |
Consider an arbitrary model PofP∗. We now extend the notion of a tangent space to the OSS setting. Similar to the i.i.d. setting, the tangent set relative to PatQ(P∗) is defined as the set of all score functions at Q(P∗) of one-dimensional regular parametric sub-models of P. The tangent space atQ(P∗) relative to P, den... | https://arxiv.org/abs/2502.17741v2 |
The next lemma establishes the connection between pathwise differentiability under the OSS setting and pathwise differentiability in the usual sense. Lemma E.14. Ifθ:P → Θ⊆Rpis a pathwise differentiable relative to PatP∗, then it is pathwise differentiable relative to PatQ(P∗). Moreover, if DP∗is a gradient of θ(·)rela... | https://arxiv.org/abs/2502.17741v2 |
0.25 0.50 0.75 Proportion of Unlabeled DataAverage Standard ErrorSetting 2: well−specified model PPI++ (Noisy Predictor) PPI++ (Random Forest) θ^ n, NPPI , (Noisy Predictor)θ^ n, NPPI , (Random Forest) θ^ n, Nsafe θ^ n, Neff , Kn=4θ^ n, Neff , Kn=9 θ^ n, Neff , Kn=16 θ^ nOSS Lower Bound ISS Lower BoundFigure 3: Estimat... | https://arxiv.org/abs/2502.17741v2 |
Forest) 0.957 0.963 0.962 0.946 0.941 ˆθPPI n,N(Noisy Predictor) 0.957 0.952 0.956 0.937 0.949 PPI++ (Random Forest) 0.939 0.959 0.960 0.939 0.946 PPI++ (Noisy Predictor) 0.940 0.948 0.948 0.960 0.934 Table 1: 95% coverage of confidence intervals for methods of mean estimation in three settings, averaged over 1,000 sim... | https://arxiv.org/abs/2502.17741v2 |
influence function is ϕη∗(x) = 2P∗ (U′−u)(V′−V)>0|X=x −2θ∗, where Y′= (U′, V′) is an independent copy of Y. We generate the response Y= (U, V) asU=E[U| X] +ϵandV=E[V|X] +ϵ′, where ( ϵ, ϵ′)⊤∼N(0,I2). We set E[U|X=x] =E[V|X=x] :=h(x), and consider two settings for h(x): 1.Setting 1 (non-linear model): h(x) =−1.70x1x2−6.... | https://arxiv.org/abs/2502.17741v2 |
For details of numerical experiments, see Section 8.2. Table 5 reports the coverage of 95% confidence intervals for each method. All methods achieve the nominal coverage. The results are similar to those of Section 8. In Setting 1 (non-linear model), ˆθeff. n,Nwith Kn= 16 has the lowest standard error. ˆθsafe n,Nwith g... | https://arxiv.org/abs/2502.17741v2 |
Conformal Prediction Under Generalized Covariate Shift with Posterior Drift Baozhen Wang Xingye Qiao Binghamton University Binghamton University Abstract In many real applications of statistical learn- ing, collecting sufficiently many training data is often expensive, time-consuming, or even unrealistic. In this case,... | https://arxiv.org/abs/2502.17744v1 |
target data, but the conditional distributions of the label given the features (that is, the posterior class probabilities) remain the same between the source and the target data. PD(Cai and Wei, 2021), on the other hand, occurs when the conditional distributions of the label given the features differ between the sourc... | https://arxiv.org/abs/2502.17744v1 |
while our framework generalizes CSPDbyrelaxing monotonicity conditions. Liu et al. (2024) pro- pose a multi-source conformal inference framework that reweights data from multiple biased sources, whereas our work focuses on a single-target setting under a more general shift model. Despite these advancements, to our best... | https://arxiv.org/abs/2502.17744v1 |
and some dataset D. A high value of S((x, y), D)indicates that the point (x, y)“conforms” to D. Then we evaluate the score function on the second part to obtain the conformity scores V(x,y) i =S(Zi, ZS1), for all i∈ S 2. In binary classification where y∈ {0,1}, Lei (2014) proposed split-conformal classification with a ... | https://arxiv.org/abs/2502.17744v1 |
assumes ηP,1(x) =ϕ(ηQ,1(x)) for some strictly increasing function ϕ, for all x. Com- pared to CS,CSPDrelaxes the requirement that ηP,1=ηQ,1; compared to PD,CSPDdropped the requirement that PX=QX. BothCSandPDare special cases of CSPD. Cai and Wei (2021) considered a special case of PDusing a specific ϕjfunctions. Note t... | https://arxiv.org/abs/2502.17744v1 |
both the target sample Z(m+1):( m+n)and the new test target data point ZT.Following the work of Lei (2014), a natural choice of the prediction set is ˆC(x) = j: ˆηQ,j(x)>ˆt∗ j,α , that is, based on comparing the estimated conditional class probabilities ˆηQ,j(x)with a threshold ˆt∗ j,α. One challenge is that in the ab... | https://arxiv.org/abs/2502.17744v1 |
=dQX|Y=j(x)/dPX|Y=j(x). To compute the weight for each data point in S2∪ T, we first define a series of initial weight functions , one for each data point: wij(x) = 1for data point i∈ S 2∩ Ij; and wij(x) =wj(x)for data point i∈ T ∩ Ij. We then assign a weight of ˜wijto data point iin class j, defined as, ˜wij=P σ:σ(T)=... | https://arxiv.org/abs/2502.17744v1 |
σ(T) =i, many of them lead to the same prod- uct, and hence, the unique number of product terms Conformal Prediction Under Generalized Covariate Shift with Posterior Drift that one needs to compute is reduced by a factor of NS j!·NT j!times. In other words, for each iin class j, we “only” need to consider the “ (Nj−1)c... | https://arxiv.org/abs/2502.17744v1 |
unchanged as we switch the thresholding inequality, we need ϕj(t1)< ϕj(tj,α)< ϕj(t2), so that no data instances that satisfied ηQ,j(x)≥t∗ j,αwould turn out to be ηP,j(x)< tj,αafter the switch, and vice versa. Theorem 1 still remains true with the replacement of the CSPDassumption by g-CSPD at α. Note that if g-CSPD is ... | https://arxiv.org/abs/2502.17744v1 |
each trial, we first sample 3000 points from the following Conformal Prediction Under Generalized Covariate Shift with Posterior Drift 0.000.250.500.751.00 0.000.250.500.751.00Class 1 0.000.250.500.751.00 0.000.250.500.751.00Class 2 0.000.250.500.751.00 0.000.250.500.751.00Class 3(a) Empirical assessment of g−CSPD assu... | https://arxiv.org/abs/2502.17744v1 |
refers to the expected size of the prediction set. Results. In all scenarios where the CSPDassump- tion holds, WCC-CSPD consistently achieved the de- sired 1−αcoverage, regardless of whether the oracle weight was available or not. Additionally, even when the marginal coverage of two baselines falls short, the Baozhen W... | https://arxiv.org/abs/2502.17744v1 |
Hongji Wei. Transfer learning for non- parametric classification: Minimax rate and adaptive classifier. The Annals of Statistics , 49(1):100–128, 2021. Maxime Cauchois, Suyash Gupta, Alnur Ali, and John C Duchi. Robust validation: Confident predic- tions even when distributions shift. Journal of the American Statistica... | https://arxiv.org/abs/2502.17744v1 |
Volodya Vovk, and Alex Gammerman. Inductive confidence ma- chines for regression. In European Conference on Machine Learning , pages 345–356. Springer, 2002.Peter Prettenhofer and Benno Stein. Cross-language text classification using structural correspondence learning. In Proceedings of the 48th annual meeting of the a... | https://arxiv.org/abs/2502.17744v1 |
joint distribution can be factorized as f(v1, . . . , v n) =nY i=1wi(vi)·g(v1, . . . , v n), where gdoes not depend on the ordering of its inputs, i.e.,g(vσ(1), . . . , v σ(n)) =g(v1, . . . , v n)for any permu- tation σof1, . . . , n. According to Lemma 2 in (Tibshirani et al., 2019), un- derCSandCSPD, allZiare weighte... | https://arxiv.org/abs/2502.17744v1 |
data set used to estimate ˆηP,j, and nj=|Rj|as the size of calibration set for class j. Underg-CSPD assumption atαand assumptions (A),(B), ifQX|Y=jis absolutely continuous with respect to PX|Y=jandwjare bounded, then for each r >0, there exists a positive constant c such that for mandnjlarge enough, with probability at... | https://arxiv.org/abs/2502.17744v1 |
where ηQ,j(x) = ϕ−1(ηP,j(x)). For class 3, the posterior probability ηQ,3(x) = 1 −ηQ,2(x)−ηQ,1(x). We can verify that there exist an increasing function ϕ3for class 3 (see the right panel of Figure 4) that satisfies ηP,3(x) = ϕ3(ηQ,3(x)).Then we sample the label based on ηQ,j(x) Remark 1. One nice property of binary cl... | https://arxiv.org/abs/2502.17744v1 |
w(x5), w(x20), . . . , w (x1)then forms one term in the numerator of (6) (see Figure 5 for an illustration). C MULTI-SOURCE In this section, we discuss an extension of our method when multiple source samples are available. First of all, we may no longer set the initial weight for those instances from the source samples... | https://arxiv.org/abs/2502.17744v1 |
arXiv:2502.17830v1 [econ.EM] 25 Feb 2025Certified Decisions Isaiah Andrews and Jiafeng Chen Abstract. Hypothesis tests and confidence intervals are ubiquitous in empirica l research, yet their connection to subsequent decision-making is o ften unclear. We develop a theory of certified decisions that pairs recommended deci... | https://arxiv.org/abs/2502.17830v1 |
then constructing confidence sets for t he resulting loss or welfare. Kitagawa and Tetenov (2018) select a policy by maximizing an empirical welfare function, and in their supplementary materials discuss how o ne may construct a confidence set for the true welfare via simultaneous inference an d projection (see, also,Pon... | https://arxiv.org/abs/2502.17830v1 |
by some as-if decision (δ,R) corresponding to some confidence set, in the sense that R(·)≤˜R(·) almost surely and the loss estimate R(·) is thus weakly tighter than the loss estimate ˜R(·). This establishes that all reasonable P-certified decisions can be cast as as-if optimization ( 1), but it does not distinguish certa... | https://arxiv.org/abs/2502.17830v1 |
incompleteness . The decision-maker has access to a default action a0/ne}ationslash∈ A, which yields a loss C which is known to the decision-maker and does not depend on θ.To assess whether to adopt the analyst’s recommendation instead, the decision-make r also asks for an assessment of the loss L(δ(Y),θ). In this sect... | https://arxiv.org/abs/2502.17830v1 |
average between a Bayes expected utility and the worst-case expected ut ility. One may inter- pret this weighted average as a Lagrange-multiplier form of ( 6). P-certified decisions let the decision-maker ensure that the worst -case risk con- straint is respected, while still sometimes adopting the recommenda tion. Supp... | https://arxiv.org/abs/2502.17830v1 |
Accepting any recommendation with R(Y)>Cstrictly increases the worst-case risk over a rich M(P,α) DM. On the other hand, accepting any R(Y) =r≤Cwith probability less than uworsens the objective in ( 6) without relaxing the constraint. Thus, the optimal acceptance decision is u /BD(r≤C). In particular, if the decision-m... | https://arxiv.org/abs/2502.17830v1 |
low-income households withchildren tomove tohigh-opportunityneighborhood s. Inthiscontext, eacha∈ Acould index a potential set of “recommended” neighborhoods, while the default action a0corresponds to the status quo of not making any recommendation . The decision-maker’s goal is to maximize some bounded outcome, for in... | https://arxiv.org/abs/2502.17830v1 |
rejects the null that it increases loss relative to t he status quo. 3.Optimal P-certificates In settings where P-certificates are useful, it seems reasonable t hat both analysts and decision-makers will prefer tighter bounds on the loss. That is, given the choice between level-1 −αP-certificates R(Y) and˜R(Y) such that R... | https://arxiv.org/abs/2502.17830v1 |
rather than jointly over ( X,ϑ). We also consider evaluations of the certificate R(·) beyond its expectation, using a weaker preference ordering (Definition 3.1 ). 3.1.Example: inference on winners, continued. Theorem 3.2 establishes a strong sense in which as-if optimization is reasonable. Dominance ord ering is de- man... | https://arxiv.org/abs/2502.17830v1 |
compact, L(a,θ)is weakly decreasing in θ, and R(θ) = inf a∈AL(a,θ)is achieved by some a∈ Afor everyθ∈Θ. Letˆθ(Y)be a1−α uniformly most accurate confidence lower bound, assumed to exist. Let (δ,R)be a certified decision that as-if optimizes against [ˆθ(Y),∞)∩Θ: R(Y)≡L(δ(Y),ˆθ(Y)) = inf a∈AL(a,ˆθ(Y)). Then for any other 1−... | https://arxiv.org/abs/2502.17830v1 |
evidence against H0—in the sense that it is an unlikely event under H0—much like a small p-value is evidence against H0. 15 As with confidence sets, we can use e-variables to certify the qualit y of decisions. Throughout this section, we assume L(a,θ)>0, but no longer assume L(a,θ)≤1. Definition 4.1. For fixedγ >0, a pair... | https://arxiv.org/abs/2502.17830v1 |
andthusthatinferential anddecision goalsare, att hevery least, partially aligned. Appendix A.Proofs Proposition 2.2. Foru∈[0,1], any adoption decision q(a,r)≤u /BD(r≤C)has maximum risk sup (θ,P)∈M(P,α) DMEP[L(δQ(Y),θ)]≤C+uα(1−C). Proof.Fix (θ,P)∈ M(P,α) DM, and letAbe the event that L(δ(Y),θ)≤R(Y). Then EP[L(δQ(Y),θ)−C... | https://arxiv.org/abs/2502.17830v1 |
By construction, R(Y)≤sup θ∈ˆΘ(Y)L(˜δ(Y),θ)≤˜R(Y). SinceR(Y) and˜R(Y) are ordered almost surely, they are also ordered in the sense of stochastic dominance. This concludes the proof. /square Theorem 3.4. Suppose Θ⊂Ris compact, L(a,θ)is weakly decreasing in θ, and R(θ) = inf a∈AL(a,θ)is achieved by some a∈ Afor everyθ∈Θ... | https://arxiv.org/abs/2502.17830v1 |
only if q(r)(1−C)≤α au(1−C)−1−a asup rq(r)(r−C) for all (a,r). If suprq(r)(r−C)/(1−C)>uαthis constraint necessarily fails, while if suprq(r)(r−C)/(1−C)≤uαthe right hand side is minimized at a=α,so the 23 constraint is equivalent to sup rq(r)≤u−1−α αsup rq(r)(r−C) 1−C. (10) By the law of iterated expectations, Eπ,q[QL+(... | https://arxiv.org/abs/2502.17830v1 |
C. (2024). Creating moves to opportunity: Experimental evidence o n barriers to neighborhood choice. American Economic Review ,114(5), 1281–1337. 10 Chernozhukov, V. ,Lee, S.,Rosen, A. M. andSun, L. (2025). Policy learning with confidence. 2,4,13 Gr¨unwald, P. D. (2023). The e-posterior. Philosophical Transactions of th... | https://arxiv.org/abs/2502.17830v1 |
Generating Correlation Matrices with Graph Structures Using Convex Optimization Ali Fahkar∗, K´evin Polisano∗, Ir`ene Gannaz‡, Sophie Achard∗ ∗Univ. Grenoble Alpes, CNRS, Grenoble INP, Inria, LJK, F-38000 Grenoble, France ‡Univ. Grenoble Alpes, CNRS, Grenoble INP, G-SCOP, 38000 Grenoble, France Abstract —This work deal... | https://arxiv.org/abs/2502.17981v1 |
with correlation ma- trices and precision matrices is similar. Therefore, we fo- cus on correlation matrices. Generating a correlation matrix involves constructing a symmetric PSD matrix that satisfies condition (1) [19, Problem 7.1.]. A graph Gis a mathematical structure used to represent pair- wise relations between ... | https://arxiv.org/abs/2502.17981v1 |
example in [22] with a characterization of the uniform distribution over the space of correlation matrices. Common approaches include the vines and onion [25]. For a broader review of available techniques, we refer to the bibliographic surveys in [11], [29]. Most existing methods, however, cannot be extended to generat... | https://arxiv.org/abs/2502.17981v1 |
of rows {qj.s.t.(i, j)/∈ E andj < i}. In [10], the authors suggest first triangulating the graph Gto obtain a chordal graph, and then applying the Cholesky-based procedure from [9], as described above. The resulting matrix is the initialization of the partial orthogonalization algorithm. In contrast, our proposed metho... | https://arxiv.org/abs/2502.17981v1 |
(5) yields a matrix with a minimum eigenvalue close to zero while negative, which indicates that the matrix is not strictly PSD. To address this, we apply a shift and normalization strategy. Specifically, we add a small positive constant ϵto the diagonal of the solution ˜C, i.e., ˜Cϵ=˜C+ϵI. Subsequently, we normalize t... | https://arxiv.org/abs/2502.17981v1 |
is 30 minutes with a repetition time of 0.5 seconds, and 3600 time points are thus available at the end of the experiment. After the preprocessing explained in [6], time series of 51 brain regions for each rat were extracted. We then calculate the wavelet transform with the Daubechies wavelet of order 8 of the 51 signa... | https://arxiv.org/abs/2502.17981v1 |
by our algorithm for different random graph models and the Chordal graph, given a graph edge density of d= 0.5forC∈R51×51. Results are obtained over 50 runs, with a threshold constraint of b= 0.2. varying graph densities and models. Overall, the execution time decreases as the graph density increases. 0.0707 0.1 0.5 0.... | https://arxiv.org/abs/2502.17981v1 |
117–124. Springer, 2018. [10] Irene C ´ordoba, Gherardo Varando, Concha Bielza, and Pedro Larra ˜naga. On generating random Gaussian graphical models. International Journal of Approximate Reasoning , 125:240–250, 2020. [11] Irene C ´ordoba. Unifying methodologies for graphical models with Gaussian parameterization . Ph... | https://arxiv.org/abs/2502.17981v1 |
Sequential Outlier Detection in Non-Stationary Time Series Florian Heinrichs f.heinrichs@fh-aachen.de FH Aachen Heinrich-Mußmann-Straße 1 52428 Jülich, Germany Patrick Bastian patrick.bastian@rub.de Ruhr-Universität Bochum Universitätsstraße 150 44801 Bochum, Germany Holger Dette holger.dette@rub.de Ruhr-Universität Bo... | https://arxiv.org/abs/2502.18038v1 |
whereµdenotes the (unknown) mean function and (εi)i∈Zis a sequence of centered errors. Now, if we have a suitable estimator ˆµofµ, we can estimate the residuals ˆεi=Yi−ˆµi. If the residual is larger than a critical value c, for some time point i, we reject the null hypothesis H(i) 0 in (2) and the corresponding observa... | https://arxiv.org/abs/2502.18038v1 |
for the sequential detection of change points. For a recent review on outlier detection in time series, see Blázquez-García et al. (2021). In the field of machine learning, it was first proposed to estimate µthrough LSTMs (Malhotra et al., 2015). Subsequently, other neural networks were used as well, such as CNNs (Muni... | https://arxiv.org/abs/2502.18038v1 |
we assume in the following thatcitakes on sufficiently large values under the alternative, as specified in Corollary 8 (iii). When sequentially testing for various hypotheses, we run into the multiple testing problem and the probabilities of falsely rejecting some null hypothesis accumulate. If we do not adapt the leve... | https://arxiv.org/abs/2502.18038v1 |
the sequences satisfyan ar=/parenleftbign r/parenrightbigγandbn=br+ ar(n r)γ−1 γ, forr=o(n). Remark 4 Assumptions 1 and 2 are standard in the time series literature (see, e.g., Bücher et al., 2021; Heinrichs and Dette, 2021) Assumption 3 is relatively mild as it is satisfied by many distributions of practical interest.... | https://arxiv.org/abs/2502.18038v1 |
straightforward calculations yieldq1−α(ˆθn)−bn an=q1−α(θρ) + oP(1). Assumption 6 is rather mild. For estimators based on maximum likelihood estima- tion or the probability weighted moment method, stronger results like asymptotic normality or almost sure convergence were shown under various conditions (see, e.g., Dombry... | https://arxiv.org/abs/2502.18038v1 |
i, that are not detected as outliers. We will refer to this adapted version of the decision rule (5) as the “partial” version. The rationale behind the partial version is that outliers potentially distort the estimator ˜µn, yielding larger deviations of |Xi−˜µn(i n)|, so that the null hypothesis tends to be rejected mo... | https://arxiv.org/abs/2502.18038v1 |
without outliers. In the former case, outliers were randomly added to the original time series at 5% of the observations. For each distribution, a minimal outlier height was defined in accordance with the assumption from Corollary 8 (iii). This minimal height was multiplied with a random factor, sampled uniformly from ... | https://arxiv.org/abs/2502.18038v1 |
between our “full” and 11 F. Heinrichs, P. Bastian, H. Dette “partial” methods seems negligible, which contrasts the aforementioned differences in specificity. The sensitivity of Ho2018 is almost 100%across all cases, suggesting that the low specificity is a result of rejecting the null hypothesis frequently. For n= 50... | https://arxiv.org/abs/2502.18038v1 |
http://www.bom.gov.au/climate/data/index.shtml . 12 Sequential Outlier detection in non-stationary time series Table 1: Empirical specificity of various methods for IID errors, under different mean functions and error distributions. n 50 100 200 µ N UExp Par 1Par 2N UExp Par 1Par 2N UExp Par 1Par 2 Panel A: Specificity... | https://arxiv.org/abs/2502.18038v1 |
99.7 99.7 99.9 99.8 99.8 99.8 99.8 99.9 99.9 99.9 99.9 99.9 Panel B: Specificity for µ2 Ours (full) 87.2 95.7 87.8 92.5 90.2 87.5 96.8 88.5 94.9 92.9 87.3 97.4 88.1 95.9 94.3 Ours (partial) 95.5 96.3 95.6 94.9 94.7 99.0 98.2 99.2 98.9 98.3 99.7 98.9 99.5 99.5 99.2 Ca2018 99.9 100.0 99.9 99.7 99.7 100.0 100.0 99.9 99.8 ... | https://arxiv.org/abs/2502.18038v1 |
93.6 99.9 99.9 99.8 100.0 100.0 We2024 2.1 0.5 3.1 2.4 3.1 1.1 0.2 1.6 1.3 1.7 0.6 0.1 0.9 0.7 0.9 Panel H: Sensitivity for µ0 Ours (full) 90.8 48.7 87.7 63.2 63.9 96.1 55.7 91.5 61.0 59.5 97.5 61.5 94.6 61.2 59.7 Ours (partial) 92.9 46.9 89.3 64.7 65.6 97.6 54.3 93.2 62.1 60.6 99.1 60.3 96.3 62.0 60.5 Ca2018 84.6 46.4... | https://arxiv.org/abs/2502.18038v1 |
318 59 047 No. outliers 956 182 2 256 233 2 2 468 916 148 measurement errors. We selected the first consecutive 365 days without outlier as the initial period and discarded the data before. We used generally the same options for the different methods as in Section 3.1. To select an adequate bandwidth for the Jackknife ... | https://arxiv.org/abs/2502.18038v1 |
minimize gn(b0,b1) =1 nhn/summationdisplay i∈N/parenleftig Xi−b0−b1/parenleftbigi n−t/parenrightbig/parenrightig2 K/parenleftbigi−nt nhn/parenrightbig. By convexity of gn,(b0,b1)is a global minimum if the gradient of gnvanishes. Straight- forward calculations yield ∂gn ∂bℓ=−2/parenleftbighℓ nRℓ(t)−b0hℓ nSℓ(t)−b1hℓ+1 ... | https://arxiv.org/abs/2502.18038v1 |
of Statistical Mathematics , 72:1055–1094. Bücher, A., Dette, H., and Heinrichs, F. (2021). Are deviations in a gradually varying mean relevant? a testing approach based on sup-norm estimators. The Annals of Statistics , 49(6):3583–3617. Bücher, A. and Segers, J. (2017). On the maximum likelihood estimator for the gene... | https://arxiv.org/abs/2502.18038v1 |
memory networks for anomaly detection in time series. In Esann, volume 2015, page 89. Montiel, J., Halford, M., Mastelini, S. M., Bolmier, G., Sourty, R., Vaysse, R., Zouitine, A., Gomes, H. M., Read, J., Abdessalem, T., et al. (2021). River: machine learning for streaming data in python. Journal of Machine Learning Re... | https://arxiv.org/abs/2502.18038v1 |
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 Mu2018 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 Panel C: Specificity for µ3 Ours (full) 97.5 97.6 98.1 97.8 97.8 98.9 98.8 99.4 99.3 99.3 99.5 99.5 99.7 99.8 99.7 Ours (partial) 97.0 96.7 97.9 97.4 9... | https://arxiv.org/abs/2502.18038v1 |
99.9 99.9 99.9 99.9 99.9 Ma2015 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 Mu2018 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 Panel C: Specificity for µ3 Ours (full) 97.6 97.7 98.2 97.9 97.7 98.9 98.9 99.2 99.2 99.1 99.5 99.5 9... | https://arxiv.org/abs/2502.18038v1 |
99.7 99.7 99.7 99.6 99.6 99.8 99.8 99.8 99.8 99.8 99.9 99.9 99.9 99.9 99.9 Ma2015 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 Mu2018 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 Panel C: Specificity for µ3 Ours (full) 97.5 97.5 9... | https://arxiv.org/abs/2502.18038v1 |
68.8 70.8 67.9 41.0 58.9 71.8 73.7 72.3 62.0 68.2 We2024 99.7 99.7 99.7 99.7 99.7 99.8 99.8 99.8 99.8 99.8 99.9 99.9 99.9 99.9 99.9 Ma2015 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 Mu2018 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 ... | https://arxiv.org/abs/2502.18038v1 |
0.0 0.0 0.0 0.0 0.0 0.0 Mu2018 0.0 0.0 0.0 0.0 0.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Panel G: Sensitivity for µ3 Ours (full) 90.7 49.0 86.5 62.9 64.5 96.5 56.3 91.7 60.9 61.7 97.5 60.6 95.5 58.4 60.4 Ours (partial) 91.6 47.1 88.1 64.1 66.5 97.6 54.6 93.5 61.5 62.9 99.1 59.2 97.2 59.2 61.2 Ca2018 77.3 33.2 81.3 74... | https://arxiv.org/abs/2502.18038v1 |
99.9 99.9 Ma2015 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 Mu2018 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 Panel C: Specificity for µ3 Ours (full) 86.5 92.2 86.7 89.9 88.7 86.9 93.9 86.6 92.2 91.2 84.7 94.4 85.7 92.8 91.5 O... | https://arxiv.org/abs/2502.18038v1 |
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