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(2) continuous treatment with functional co-variates and outcomes which are temporally misaligned. 5.1 Binary Treatment with Synthetic Data The first experiment simulates functional outcomes Y(t)generated as time-dependent curves influ- enced by a binary treatment X∈ {0,1}and baseline covariates V. We consider two scen... | https://arxiv.org/abs/2503.05024v3 |
brackets) of causal effect estimators for binary treatment and monotonic path outcomes Y. Method n= 50 n= 100 n= 250 IPW ATE 0.88 (0.77) 0.89 (0.77) 0.89 (0.79) Doubly Robust ATE 0.67 (0.67) 0.68 (0.67) 0.66 (0.70) Kernel ATE 0.87 (0.78) 0.89 (0.78) 0.89 (0.79) Operator Kernel ATE 0.65 (0.62) 0.62 (0.60) 0.62 (0.60) SR... | https://arxiv.org/abs/2503.05024v3 |
X∈Rby a simple change of kernel kXdescribing the feature maps ψX. We simulate functional outcomes as: Y(t) =µV(t) +βX(t)X+ϵ(t), (37) where µV(t)represents a curve effect (arc parameterized with expected peak location and hight) dependent on the covariates, βX(t)is the time-varying treatment effect, and ϵ(t)is independe... | https://arxiv.org/abs/2503.05024v3 |
heuristic with regularization terms set using hyperparameter grid search optimizing the out-of-sample performance holding 20 %of the data for testing. Figure 5 present a box plot of the DS estimation error across simulations, while Table 3 summarizes the mean absolute error and standard deviation across the time grid. ... | https://arxiv.org/abs/2503.05024v3 |
two weeks. For our analysis, we estimate causal effects based on predicted symptom trajectories over this 2-week period. For PD patients with tremor, we selected data from the most affected arm (determined by MDS-UPDRS Part III, items 3.15 & 3.17). For those without tremor, we selected the corresponding side matched fo... | https://arxiv.org/abs/2503.05024v3 |
12 hours, and the final outcome curves are constructed by aggregating across available days per time bin. We model the diagnostic category (binary PD vs. non-PD control) as the intervention variable Dwhile controlling for the confounding effects of hours awake andgender. Figure 8 presents both the average treatment eff... | https://arxiv.org/abs/2503.05024v3 |
in PD; (Right) displays the minimal assumptions made for factors affecting tremor probability estimated from a wrist-worn device. 6 Discussion This work introduces a novel framework for causal inference in settings involving functional data, extending traditional methods to accommodate dynamic and non-linear domains. T... | https://arxiv.org/abs/2503.05024v3 |
application, we demonstrate how the dynamic ATE, φdATE, re- veals different ways in which therapy affects symptoms, highlighting its previously underexplored potential (Section 5.3). In summary, this work bridges a gap in the literature by introducing a non- parametric methodology for causal inference in functional dat... | https://arxiv.org/abs/2503.05024v3 |
Fréchet means. arXiv preprint arXiv:2408.07534 , 2024. Alexandre Belloni, Victor Chernozhukov, Ivan Fernandez-Val, and Christian Hansen. Program evaluation and causal inference with high-dimensional data. Econometrica , 85(1):233–298, 2017. Bastiaan R. Bloem, William J. Marks, Ana Luisa Silva de Lima, Marjolein L. Kuij... | https://arxiv.org/abs/2503.05024v3 |
SivaramanBalakrishnan, andL.A.Wasserman. Semiparametriccounterfactual density estimation. Biometrika , 110(4):875–896, 2023. Daisuke Kurisu, Yidong Zhou, Taisuke Otsu, and Hans-Georg Müller. Geodesic causal inference. arXiv preprint arXiv:2406.19604 , 2024. Darrick Lee and Harald Oberhauser. The signature kernel. arXiv... | https://arxiv.org/abs/2503.05024v3 |
and Arthur Gretton. Kernel methods for causal functions: dose, hetero- geneous and incremental response curves. Biometrika , 111(2):497–516, 2024. Anuj Srivastava and Eric P Klassen. Functional and shape data analysis , volume 1. Springer, 2016. Anuj Srivastava, Wei Wu, Sebastian Kurtek, Eric Klassen, and James Stephen... | https://arxiv.org/abs/2503.05024v3 |
q∗ 2= (q2◦γ∗)√˙γ∗be the optimally-warped SRSF of f2, and f∗ 2=f2◦γ∗be the corresponding optimally-warped f2which is best aligned to f1. Estimation of γ∗can be sensitive to function noise, as taking derivatives to compute the SRSF will exacerbate the noise. As a remedy, 35 one can add an additional penalty term to the l... | https://arxiv.org/abs/2503.05024v3 |
curved manifold in L2, and the quotient cannot inherit a linear or Hilbert space structure. Consider the restricted action group of phase shifts: Γc={γc(t) =t+cmod 1 : c∈[0,1)}. Here, the group acts by isometric translations on the circle. The associated quotient space F/Γc now satisfies some improved properties: •The ... | https://arxiv.org/abs/2503.05024v3 |
an integer T≥2and let D= Y= (Y(1), . . . , Y (T))⊤∈RT:Y(1)<···< Y(T) ,T=D ∩[ 0, T−1−δT−1]T, for some 0< δ < 1. Assume ni.i.d. observations Y1, . . . ,Yn∈ TwithYi= (Yi(1), . . . , Y i(T))⊤. Each Yiis obtained by evaluating an underlying positive function fi∈ Gat the design points {1/T, . . . , T/T }, where G=B(T)is the... | https://arxiv.org/abs/2503.05024v3 |
i=1wi√ai=T−1X i=1wip bi. Repeating the same argument recursively (using that wT−1> w T−2>···) we conclude that ai=bi for all i= 1, . . . , T. Thus, Fis injective on the domain D. Step 2: almost-sure uniqueness. Because ϕEFRis not convex on RT, classical M–estimator theory cannot guarantee a single minimiser. We therefo... | https://arxiv.org/abs/2503.05024v3 |
D) where the uniqueness is ensured. Corollary 14. Let λ(g) :=EPx ϕEFR(Y,g) ,g∈RT, 41 and assume the Jacobian∂λ ∂g|g=¯f=:Λis non-singular. Then √n fn−fd− → N 0,Λ−1CΛ−⊤ , where C= Cov Px ϕEFR(Y,¯f) . Proof.Since fnis a Huber–type ρ-estimator (Theorem 4) and ϕEFRis continuously differentiable onT, Theorem 6.6 of H... | https://arxiv.org/abs/2503.05024v3 |
convergence in distribution to the norm of a mean-zero Gaussian process, i.e. ∥Z∥ L2forZin a suitable function space; see (Testa et al., 2025, Theorem 3.9) for a more detailed study of the asymptotic normality of the residuals of ˆ∆in the functional setting. For the infinite-dimensional scenario, the primary additional... | https://arxiv.org/abs/2503.05024v3 |
of the feature vectors, using the data from the unscripted activities of all 24 PD patients (both with and without annotated tremor episodes) and all 24 non-PD controls. These features are then used to train a logistic classifier (i.e., applying l1-regularization) to predict the annotated gait and tremor episodes in th... | https://arxiv.org/abs/2503.05024v3 |
Optimal and fast online change point estimation in linear regression Annika H¨ uselitz1, Housen Li1,2, and Axel Munk1,2 1Institute for Mathematical Stochastics, University of G¨ ottingen 2Cluster of Excellence “Multiscale Bioimaging: from Molecular Machines to Networks of Excitable Cells” (MBExC), University of G¨ otti... | https://arxiv.org/abs/2503.05270v1 |
modern large-scale data analysis, they remain largely unexplored, with only a few exceptions (Kov´ acs et al., 2024; Romano et al., 2023, 2024; Ward et al., 2024). A more detailed discussion of the literature from the perspective of this paper, with particular focus on minimax optimality and computational efficiency, w... | https://arxiv.org/abs/2503.05270v1 |
simultaneously on the statistical optimality and the computation and memory efficiency. Our contributions are two folds: i. We introduce the Fast Limited-memory Optimal Change (FLOC) detector, which operates in constant time per observation and requires only constant memory, particularly independent of the sample size.... | https://arxiv.org/abs/2503.05270v1 |
Computational efficiency Despite their statistical optimality, the aforementioned methods will be con- fronted with computational challenges in large-scale data sets, as both their runtime and memory usage scale linearly with the number of cumulatively observed data samples for each incoming sample. Recently, in the su... | https://arxiv.org/abs/2503.05270v1 |
a change if the test statistic exceeds the threshold, with ρJ= 1.5 and ρK= 0.3, displayed by the dashed lines in the plot. For a jump this happens at observation 521 and for a kink at observation 521 as well. For visibility, the values of threshold and test statistic are multiplied by 5 for the kink case (lower panel).... | https://arxiv.org/abs/2503.05270v1 |
nand can be chosen as r∗ J= 9·106·4−1δ−4 0+ 1. ii. Let ˆτK,nbe the CUSUM kink detector in (3c)and(3e)with bin size NK= (300 /c2)1/3n2/3log1/3(n)and threshold ρK= 4c/5n. Then it holds for sufficiently large n sup θ=(τ,α−,α+,β−,β+)∈ΘK δ0Eτ"n1/3 log1/3n(ˆτK,n−τ)2# ≤r∗ K, where r∗ Kis independent of nand can be chosen as... | https://arxiv.org/abs/2503.05270v1 |
weighted sums W1, W2, W3 Input: latest data point Xtat time t, jump bin size NJ, kink bin size NK, jump threshold ρJ, kink threshold ρK, estimate of pre-change signal ˆf− 1:compute rJ←tmod NJ 2:compute rK←tmod NK 3:compute scaling factor d←(2NK+rK+1)(2 NK+rK+2)(4 NK+2rK+3) 6 4:ifrJ= 0then 5: update jump sums SJ,1←SJ,2,... | https://arxiv.org/abs/2503.05270v1 |
In contrast, our FLOC detector, designed for a more general model, attains the same detection delay rate in this specific submodel, and requires a computational and memory costs independent of time t, see also Section 1.2. 4 Numerical experiments In this section, we first introduce an empirical method for the selection... | https://arxiv.org/abs/2503.05270v1 |
20 133 9 20 52 15 1000 500 0.651 0.03 0.54 11 22 192 8 18 48 9 255075 0.25 0.50 0.75 false alarm probabilityexpected detection delay80100120 0.25 0.50 0.75 false alarm probabilityexpected detection delayFigure 4: Comparison of the type I error of a false alarm probability for τ= 1000 and the type II error of the expect... | https://arxiv.org/abs/2503.05270v1 |
NJ=NK= 5) performs better for detecting larger changes (jump sizes greater than 1 and slope changes greater than 0 .05). Notably, this difference occurs in the regime where the expected detection delay is between 20 and 30. In such cases, the larger window size (10) needs to encompass nearly the entire change point to ... | https://arxiv.org/abs/2503.05270v1 |
4 for visibility. The horizontal dashed line marks the detection threshold, and the vertical red line indicates the detected change in slope. than abrupt changes in values (see again Figures 5 and 6). Therefore, we apply only the kink component of the FLOC algorithm by setting the jump threshold as infinity. We choose ... | https://arxiv.org/abs/2503.05270v1 |
will denote by f−(·):=β−(· −τ)+α−andf+(·):=β+(· −τ)+α+ the linear functions, which correspond to the two segments of fθ0. For a jump Now we assume that a jump at the change point occurs, namely, |α+−α−| ≥δ0, for some δ0>0. We take (3d) as our detector. For this, we set NJ:=blogn, with some bindependent of n, which we w... | https://arxiv.org/abs/2503.05270v1 |
≥y ≤nexp −y2 2 . (8) When we set y=√10 log n >1, we have for nlarge enough Pτ0 max k<m≤n|Zm| ≥p 10 log n ≤nexp{−5 logn}=n−4. (9) We can use Korostelev and Korosteleva (2011, Theorem 7.5) to show that ˆα∼ N α−−β−τ0,4cn+ 2 c2n2−cn andˆβ∼ N β−,12n c3n2−c . Define Zα= (α−−β−τ0−ˆα)r c2n2−cn 4cn+ 2∼ N (0,1) and Zβ=... | https://arxiv.org/abs/2503.05270v1 |
parts (12a) and (13b), it holds under the event Awith b= 300/c21/3that 1 dMMX i=1iεm−M+i ≤max k<m≤n1√dM|Zm|<√8 logn√dM=r 48 log n 2M3+ 3M2+M≤s 24 log n (2N)3 =s 3 logn b3n2logn=c 10n. And for the parts (12b) and (13c), it holds under the event Athat 1 dMMX i=1i f−m−M+i n −ˆf−m−M+i n <1 dMMX i=1ic(2M+ 1) 30n =6 ... | https://arxiv.org/abs/2503.05270v1 |
fj(i):=ftj(i/n) and fM(i):= ftM(i/n). Note that fj(i) = 0 = fM(i) for i≤tj< tM. Then dPtM dPtj=Qn i=1expn −1 2(Xi−fM(i))2o Qn i=1expn −1 2(Xi−fj(i))2o = exp nX i=⌊tjn⌋+11 2 (Xi−fj(i))2−(Xi−fM(i))2 , and 1 2 (Xi−fj(i))2−(Xi−fM(i))2 =1 2 2Xi(fM(i)−fj(i)) +f2 j(i)−f2 M(i) =εi(fM(i)−fj(i))−1 2(fM(i)−fj(i))2, ... | https://arxiv.org/abs/2503.05270v1 |
reveal a phase transition between the jump and kink scenarios, which echo the understanding in the offline setup (Goldenshluger et al., 2006, Frick et al., 2014b, Chen, 2021; see also Table 1). The FLOC detector is specifically designed to achieve asymptotically minimax optimal rates. While the constants involved have ... | https://arxiv.org/abs/2503.05270v1 |
changepoint detection. J. R. Stat. Soc. Ser. B. Stat. Methodol. , 84(1):234–266. Chen, Y., Wang, T., and Samworth, R. J. (2024). Inference in high-dimensional online changepoint detection. J. Amer. Statist. Assoc. , 119(546):1461–1472. Cho, H. and Kirch, C. (2024). Data segmentation algorithms: univariate mean change a... | https://arxiv.org/abs/2503.05270v1 |
Ann. Statist. , 38(6):3445–3457. Roberts, S. W. (1966). A comparison of some control chart procedures. Technometrics , 8:411–430. Romano, G., Eckley, I. A., and Fearnhead, P. (2024). A log-linear nonparametric online changepoint detection algorithm based on functional pruning. IEEE Trans. Signal Process. , 72:594–606. ... | https://arxiv.org/abs/2503.05270v1 |
arXiv:2503.05323v1 [stat.ML] 7 Mar 20251–16 Graph Alignment via Birkhoff Relaxation Sushil Mahavir Varma SUSHIL -MAHAVIR .VARMA @INRIA .FR INRIA, DI/ENS, PSL Research University, Paris, France Ir`ene Waldspurger WALDSPURGER @CEREMADE .DAUPHINE .FR CNRS, Universit ´e Paris Dauphine, INRIA, Paris, France Laurent Massouli... | https://arxiv.org/abs/2503.05323v1 |
,2024 ;Mao et al. ,2023a ,2021 , 2023b ;Araya and Tyagi ,2024 ). A popular class of algorithm in practice with good empirical and computational performance is the convex relaxations ( Fan et al. ,2023a ,b;Araya and Tyagi ,2024 ) of ( 1). We are interested in the Birkhoff relaxation, which is a tight convex relaxation, ... | https://arxiv.org/abs/2503.05323v1 |
the true permutation. 1.3. Related Work Regarding practical algorithms for graph alignment, there is a rich literature on optimization-based methods ( Kezurer et al. ,2015 ;Dym et al. ,2017 ;Ling ,2024 ;Fan et al. ,2023a ;Araya and Tyagi , 2024 ) and simple spectral-based methods ( Umeyama ,1988 ;Feizi et al. ,2019 ;Ga... | https://arxiv.org/abs/2503.05323v1 |
following optimization problem fo r˜Π⋆=I: min X∈Bn(2+σ2)(n+1)/bardblX/bardbl2 F−2Tr(X)2−2/an}bracketle{tX,XT/an}bracketri}ht, whose optimal value is given by ¯X⋆=ǫI+1−ǫ nJ, whereǫ=2 2+σ2(n+1). The above can be verified using the KKT conditions. Now, using the above characterization, we get /bardblI−¯X⋆/bardblF= (1−ǫ)/ve... | https://arxiv.org/abs/2503.05323v1 |
is true because /bardblZ/bardbl2 F=2 n/bardbl˜z/bardbl2 2, where˜z∼N(0,In(n+1)/2). Thus, by (Fan et al. ,2023a , Lemma 15), with probability at least 1−e−√n, we have /bardblZ/bardbl2 F≥n−˜c√n≥ n/2, for a sufficiently large nsince˜c >0is a constant independent of n. Next, we upper bound 5 VARMA WALDSPURGER MASSOULI ´E ma... | https://arxiv.org/abs/2503.05323v1 |
have /summationdisplay j/negationslash=iXij≤4n1.75+3ǫ/4/bardblAX−XA/bardblF+2n1−ǫ/8w.p.1−on(1). Now to show that the bound obtained in the above lemma is small enough, we upper bound /bardblAX− XA/bardblFfor the optimal solution X=X⋆in the following lemma. Lemma 4 Under the setting of Theorem 1, there exists a constant... | https://arxiv.org/abs/2503.05323v1 |
handle on Cusing the following claim, proved at the end of this section. Claim 5 There exists a constant c >0such that, for large enough n >0, we have, with a probability of at least 1−e−cn1−ǫ/4,#/braceleftbig |/an}bracketle{tui,1/an}bracketri}ht| ≤n−ǫ/4/bracerightbig ≤3n1−ǫ/4. 9 VARMA WALDSPURGER MASSOULI ´E To prove ... | https://arxiv.org/abs/2503.05323v1 |
is equal in distribution to√nz /bardblz/bardbl2, forz∼N(0,In). In particular, P/parenleftBig #/braceleftBig |/an}bracketle{tui,1/an}bracketri}ht| ≤n−ǫ/4/bracerightBig >3n1−ǫ/4/parenrightBig =P/parenleftBig #/braceleftBig |zi| ≤n−1/2−ǫ/4/bardblz/bardbl2/bracerightBig >3n1−ǫ/4/parenrightBig ≤P/parenleftBig #/braceleftBig... | https://arxiv.org/abs/2503.05323v1 |
result suggests t hat Birkhoff relaxation combined with post-processing could succeed beyond σ∼n−0.5. In the right plot of Figure 1, we test the performance of Birkhoff relaxation as a functio n of n∈ {100,200,300,400,500}. Although the fraction of correctly matched vertices as a fu nction of σworsens as nincreases, th... | https://arxiv.org/abs/2503.05323v1 |
of Computational Mathe- matics , 23(5):1511–1565, 2023a. Zhou Fan, Cheng Mao, Yihong Wu, and Jiaming Xu. Spectral grap h matching and regularized quadratic relaxations II: Erd˝ os-r´ enyi graphs and univer sality. Foundations of Computational Mathematics , 23(5):1567–1617, 2023b. Soheil Feizi, Gerald Quon, Mariana Reca... | https://arxiv.org/abs/2503.05323v1 |
´E Appendix A. Proof of Claim 6 Fix˜ǫ >0and note that for any i/ne}ationslash=j∈[n], we have E/bracketleftBigg 1 (|λj−λi|+n−1.5−˜ǫ)2/bracketrightBigg =/integraldisplay∞ 0P/parenleftBigg 1 (|λj−λi|+n−1.5−˜ǫ)2> x/parenrightBigg dx =/integraldisplay∞ 0P/parenleftbigg |λj−λi|<1√x−n−1.5−˜ǫ/parenrightbigg dx =/integraldispla... | https://arxiv.org/abs/2503.05323v1 |
Stochastic dominance of sums of risks under dependence conditions Jorge Navarro1,∗and Jose Miguel Zapata1 1Facultad de Matemáticas, Universidad de Murcia, Murcia, Spain. March 10, 2025 Abstract We provide conditions for the stochastic dominance comparisons of a risk X and an associated risk X+Z, where Zrepresents the u... | https://arxiv.org/abs/2503.05348v1 |
future research. 2 Preliminary results LetL1be the set of integrable random variables over an atomless probability space (Ω,F,Pr). Let X∈ L 1be a random variable representing a random payoff or wealth andlet F(t) = Pr( X≤t)beitscumulativedistributionfunction(CDF).Itssurvival(or reliability)functionis ¯F(t) = 1−F(t) = P... | https://arxiv.org/abs/2503.05348v1 |
order is already defined with a property of this type where Z is assumed to be independent of W. So it implies the ST order. Analogously, from these properties, it is clear that the ST order implies both the ICV and ICX orders (a well known property that can also be deduced from the definitions). Now we state the impro... | https://arxiv.org/abs/2503.05348v1 |
(Z|X)property is sometimes easy to check than that property. For example, it holds if (X, Z)is PQD. We have similar results for the ICV and CX orders. 5 Proposition 3.3. ForX, Z∈ L 1,X≥ICVX+Zholds if E(Z)≤0and(3.1)holds. TheproofisanalogoustothatofProposition3.2byusingnow (3.2)andProposition 2.4. Proposition 3.4. ForX,... | https://arxiv.org/abs/2503.05348v1 |
holds). Now we want to obtain similar results by using weaker conditions than (3.6) for the survival copula bC. For the first result we need the following symmetry property for Z. Definition 3.5. A random variable Zwith CDF Gis symmetric around zero if G(−z) = 1−G(z)for all zsuch that G(z−) =G(z). A random variable Zis... | https://arxiv.org/abs/2503.05348v1 |
the copula Cof(X, Z)satisfies C(u, v) +C(u,1−v)≥ufor all u, v∈[0,1], (3.9) then X≥ICVX+Z. Proof.The conditional distribution function of (Z|X > x )is Pr(Z≤z|X≤x) =Pr(X≤x, Z≤z) Pr(X≤x)=C(F(x), G(z)) F(x) 9 for all xsuch that F(x)>0. Hence, from (2.1), E(Z|X≤x) =Z∞ 0(1−Pr(Z≤z|X≤x))dz−Z0 −∞Pr(Z≤z|X≤x)dz =Z∞ 0 1−C(F(x), G... | https://arxiv.org/abs/2503.05348v1 |
by using this new dependence measure. Proposition 3.11. If the generalized Gini measure of association of a random vector with copula Cis defined as δC= 6Z1 0Z1 0[C(u, v) +C(u,1−v)−u]dvdu, then δC=ρC. Proof.First we note that if (U, V)is a random vector with CDF C, then ρC= 12E(UV)−3(see [9], p. 167). We also note that... | https://arxiv.org/abs/2503.05348v1 |
Z)has an absolutely continuous distri- bution, the PDF g(z|x)of(Z|X > x )is g(z|x) =−∂ ∂z¯G(z|x) =∂2bC(¯F(x),¯G(z)) ¯F(x)g(z), where g=−¯G′is the PDF of Zand∂2bCrepresents the partial derivative of bCwith respect to its second variable. Hence E(Z|X > x ) =Z∞ −∞zg(z|x)dz =Z∞ −∞∂2bC(¯F(x),¯G(z)) ¯F(x)zg(z)dz =Z∞ 0∂2bC(¯F... | https://arxiv.org/abs/2503.05348v1 |
note that ϕ(u, v) =ϕ(u,1−v)and so (3.10) also holds for u, v∈[0,1]. The relationships between the dependence notions used in this paper are sum- marized in Table 1. This table shows that the new positive dependence properties sPQD (Z|X)andwPQD (Z|X)(which only depends on copula properties) imply a positive Spearman rho... | https://arxiv.org/abs/2503.05348v1 |
with PDF fu(v) =kuv(0.5−u)2(1−v)(0.5−u)2forv∈[0,1] ifu∈(0,1/2]and the following PDF f∗ u(v) = 2−fu(v) = 2−kuv(0.5−u)2(1−v)(0.5−u)2forv∈[0,1] ifu∈(1/2,1), where ku= 1/β(1 + (0 .5−u)2,1 + (0 .5−u)2)is the normalizing constant and β(·,·)is the beta function. It is easy to see that f∗ uis a proper PDF for allu∈(1/2,1)since... | https://arxiv.org/abs/2503.05348v1 |
3.13). We can also confirm that it is neither wPQD (U|V)norwNQD (U|V). For example, C(u, v) +C(1−u, v)−v= 2uv > 0 foru∈(0,1/2)andv∈(0,1/4)and C(u, v) +C(1−u, v)−v=−2u(1−v)<0 20 foru∈(0,1/2)andv∈(3/4,1). Hence, this example proves that this concept is not symmetric with respect to the two different conditioning. Let us ... | https://arxiv.org/abs/2503.05348v1 |
to stronger stochastic dominance concepts as the hazard rate or the likelihood ratio orders. We should also study if these dependence notions hold for copulas that do not have the PQD property. Finally, these properties should be applied to real data with different dependence structures in several fields (actuarial sci... | https://arxiv.org/abs/2503.05348v1 |
Comparing regularisation paths of (conjugate) gradient estimators in ridge regression Laura Hucker1, Markus Reiß1and Thomas Stark2 1Institute of Mathematics, Humboldt-Universität zu Berlin, Germany, e-mail: huckerla@math.hu-berlin.de ;mreiss@math.hu-berlin.de 2Department of Statistics and Operations Research, Universit... | https://arxiv.org/abs/2503.05542v2 |
feature vectors xiandR-valued responses yisatisfying yi=x⊤ iβ0+εi, i= 1,...,n. Here,β0∈Rpis the unknown true coefficient vector. The error variables εisatisfy E[εi|Xi] = 0and Var(εi|Xi) =σ2for some noise level σ>0. Using the notation y:= (y1,...,yn)⊤,X:= (x1,...,xn)⊤ andε:= (ε1,...,εn)⊤for the response vector, the feat... | https://arxiv.org/abs/2503.05542v2 |
paths of CG, GD and RR indeed closely resemble each other, see Section 4. L. Hucker, M. Reiß, T. Stark/Comparing regularisation paths 3 2. Ridge regression, gradient flow and conjugate gradients Let us first fix some standard notation. The Euclidean norm and scalar product are denoted by ∥·∥and ⟨·,·⟩, respectively. For... | https://arxiv.org/abs/2503.05542v2 |
˜pwith value one at zero and vanishing at these eigenvalues so that ∥P˜p(/hatwideΣλ)yλ∥= 0. Thus, the CG algorithm stops at k=˜pat the latest and ˆβCG λ,˜p=ˆβRR λ. To avoid indeterminacies before (compare the stopping criterion in Algorithm 1), we assume throughout the paper /summationdisplay j=1,...,p:sj=si⟨X⊤y,vj⟩2>0... | https://arxiv.org/abs/2503.05542v2 |
the definitions and the fact that the cross term has conditional mean zero whenRis deterministic. For the conditional variance term note E[ελε⊤ λ|X] =σ2 n/hatwideΣ−1 λ/hatwideΣ. We can immediately compare the regularisation path of gradient flow in time t>0with that of ridge regression with penalties larger than λ. In ... | https://arxiv.org/abs/2503.05542v2 |
analysing the CG residual polynomial globally. The left symbolic plot shows a CG residual polynomial RCG tand the corresponding residual filters RGF ρt,RRR λ,λ+1/ρtfor gradient flow and ridge regression, respectively, with ρt=|(RCG t)′(0)|.RCG t(x)is up to its first zero x1,tupper bounded by RGF ρt(x), but afterwards i... | https://arxiv.org/abs/2503.05542v2 |
other hand, we have 2t>maxi(˜si+λ)−1=∥/hatwideΣ−1 λ∥for this case, implying (2t∧/hatwideΣ−1 λ) =/hatwideΣ−1 λ. This showsRin λ,γ(ˆβCG λ,τt) =σ2 ntrace/parenleftbig (2t∧/hatwideΣ−1 λ)/hatwideΣ/parenrightbig and yields the asserted bound also in this case. Based on the previous result, it remains to compare the CG and GF... | https://arxiv.org/abs/2503.05542v2 |
λ. This argument shows in view of /hatwideΣ−1 λ/hatwideΣ=h(/hatwideΣλ) and(1−e−1/2)(2tx∧1)⩽1−RGF t(x) ∀t⩾t0: (1 +C0,λ) trace/parenleftbig (Ip−RGF t(/hatwideΣλ))2/hatwideΣ−1 λ/hatwideΣ/parenrightbig ⩾(1−e−1/2) trace/parenleftbig (2t∧/hatwideΣ−1 λ)/hatwideΣ/parenrightbig . This implies the asserted bound for Rin λ,γ(ˆβCG... | https://arxiv.org/abs/2503.05542v2 |
it is not clear whether the CG risk decays monotonically as well because the residual polynomial RCG tis data-dependent, Theorem 3.7 then yields a monotone upper bound. Proposition 3.11. Lets1⩾...⩾spdenote the eigenvalues of /hatwideΣandv1,...,vpcorresponding normalized eigenvectors. Then for the target vector γ=β0and ... | https://arxiv.org/abs/2503.05542v2 |
is a constant C=C(L)>0such that for all z⩾1with probability at least 1−e−z ∥Σ−1/2 λ(Σ−/hatwideΣ)Σ−1/2 λ∥=∥Σ′−/hatwideΣ′∥⩽C/parenleftig/radicalig N(λ) n∨N(λ) n∨/radicalig z n∨z n/parenrightig . (3.24) This gives the first claim. Jensen’s inequality then implies directly the second statement. L. Hucker, M. Reiß, T. S... | https://arxiv.org/abs/2503.05542v2 |
1000Monte Carlo runs. The penalisation parameter is chosen as λ= 3. As a proxy for the gradient flow estimator ˆβGF λ,twe consider the gradient descent iterates ˆβGD λ,η,kwith learning rate η= 2/(2λ+∥/hatwideΣ∥). In Figure 2(left) we plot the Monte Carlo prediction risks ˆβCG λ,k,ˆβGD λ,η,kandˆβRR λ+1/(ηk)as a function... | https://arxiv.org/abs/2503.05542v2 |
Mathematical Statistics in the Information Age – Statistical Efficiency and Computational Tractability is gratefully acknowledged. L. Hucker, M. Reiß, T. Stark/Comparing regularisation paths 16 References [1]Ali, A., Dobriban, E., Tibshirani, R.: The implicit regularization of stochastic gradient flow for least squares... | https://arxiv.org/abs/2503.05542v2 |
Filtering of partially observed polynomial processes in discrete and continuous time Jan Kallsen Ivo Richert Christian-Albrechts-Universität zu Kiel∗ Abstract This paper is devoted to filtering, smoothing, and prediction of polynomial processes that are partially observed. These problems are known to allow for an expli... | https://arxiv.org/abs/2503.05588v1 |
of an optimal prediction problem. If 𝑠=𝑡, the problem is termed a filtering problem, while the case 𝑠 >𝑡 is commonly referred to as an optimal smoothing problem. As noted above, the general filtering, prediction and smoothing problems allow for an ex- plicit solution in discrete-time Gaussian state space models and... | https://arxiv.org/abs/2503.05588v1 |
suitable dimension. We write I𝑑for the identity matrix in ℝ𝑑×𝑑. We use the notation 𝑘∶𝓁for the range 𝑘,𝑘+1,…,𝑘+𝓁, for example 𝑋1∶𝑑∶=(𝑋1,…,𝑋𝑑). 2 Polynomial processes and state space models We first recall the properties of (time-inhomogeneous) continuous-time polynomial processes. 2.1 Polynomial processes... | https://arxiv.org/abs/2503.05588v1 |
𝑚)ofℕ𝑑 𝑚⧵{0}has been chosen, the coefficients 𝑎𝑐(𝑡) ∈ℝ̃ 𝑚and𝐴𝑐(𝑡) ∈ℝ̃ 𝑚×̃ 𝑚in Remark 2.3 for the process (𝑋𝜆)𝜆∈ℕ𝑑𝑚⧵{0}can be written in the form 𝑎𝑐 𝑗(𝑡)=𝑏𝑐 𝜆𝑗,0(𝑡), 𝐴𝑐 𝑗,𝑘(𝑡)=𝑏𝑐 𝜆𝑗,𝜆𝑘(𝑡). 4 The components of the processes (1,𝑋)⊗𝑚are of the form 𝑋𝜆for some𝜆satisfying |𝜆|≤𝑚. S... | https://arxiv.org/abs/2503.05588v1 |
third differential characteristic of (𝑋,𝑍), then we can apply the proof of Theorem 5.3 of Agoitia Hurtado and Schmidt [1], which yields that the extended gener- ator of the process (𝑋,𝑍)at𝑡maps𝑥𝜆to a polynomial in 𝑥of degree at most |𝜆|with lo- cally bounded coefficients. (2.2) however holds by assumption beca... | https://arxiv.org/abs/2503.05588v1 |
F0-measurable ℝ𝑘-valued random variable. Finally, let the ℝ𝑘-valued process 𝑍be given by 𝑍(𝑡)=𝑌(𝑡)𝑍(𝑡−1)+𝑐(𝑡)+𝐶(𝑡)𝑋(𝑡) (2.8) for𝑡∈ℕ∗, or, alternatively, given by 𝑍(𝑡)=𝑌(𝑡)𝑍(𝑡−1)+𝑐(𝑡)+𝐶(𝑡)𝑋(𝑡−1) (2.9) for𝑡∈ℕ∗. Then the following two statements hold true: 1. The process (𝑋,𝑍)is anℝ𝑑+𝑘-val... | https://arxiv.org/abs/2503.05588v1 |
Let𝐹(𝑡)∈ℝ𝑑×𝑑denote the unique solution to d𝐹(𝑡)=𝐴𝑐(𝑡)𝐹(𝑡)d𝑡 with𝐹(0)=I . Set𝑁(𝑡𝑘)∶=𝐹(𝑡𝑘)∫𝑡𝑘 𝑡𝑘−1𝐹(𝑠)−1d𝑀(𝑠). Then 𝔼(𝑁(𝑡𝑘)|F𝑡𝑘−1)=0 and (𝑋𝜆(𝑡𝑘))𝜆∈ℕ𝑑𝑛⧵{0}=𝑎(𝑘)+𝐴(𝑘)(𝑋𝜆(𝑡𝑘−1))𝜆∈ℕ𝑑𝑛⧵{0}+𝑁(𝑡𝑘), (2.11) where𝐴(𝑘) ∶=𝐹(𝑡𝑘)𝐹(𝑡𝑘−1)−1and𝑎(𝑘) ∶=𝐹(𝑡𝑘)∫𝑡𝑘 𝑡𝑘−1𝐹(�... | https://arxiv.org/abs/2503.05588v1 |
polynomial model. Proposition 3.3. Let𝑋be a polynomial state space model of order 1 as in (2.3) with finite second moments. Moreover, consider a linear Gaussian state space model 𝑌as in Definition 3.1 with𝔼(𝑌(0))= 𝔼(𝑋(0)),Cov(𝑌(0))=Cov( 𝑋(0)), and coefficients 𝑎(𝑡),𝐴(𝑡),𝐵(𝑡)satisfying 𝐵(𝑡)𝐵(𝑡)⊤=Cov(𝑁... | https://arxiv.org/abs/2503.05588v1 |
global version of the Picard–Lindelöf theorem. For the proof of the final assertion, we only need to observe that 𝔼(𝑋(𝑡)𝑋(𝑠)⊤)=𝔼(𝑋(𝑠)𝑋(𝑠)⊤)+∫𝑡 𝑠(𝑎𝑐(𝑢)+𝐴𝑐(𝑢)𝔼(𝑋(𝑢)𝑋(𝑠)⊤))d𝑢 for𝑡≥𝑠. Finally, we are able to use Lemma 3.6 to construct the Gaussian Ornstein–Uhlenbeck pro- cess sharing the first two... | https://arxiv.org/abs/2503.05588v1 |
It ¯o isometry for locally square-integrable martingales, we have ‖‖‖∫𝑡 0𝛾(𝑛)(𝑠)d𝑋(𝑠)−∫𝑡 0𝛾(𝑠)d𝑋(𝑠)‖‖‖2→0. Since∫𝑡 0𝛾(𝑛)(𝑠)d𝑋(𝑠)∈(𝑋,𝑡,𝑑′), this proves “ ⊇”. Let̃𝐿2(𝑋,𝑡,𝑑′)denote the right-hand side of equation (3.10). For “ ⊆” it suffices to show that (𝑋,𝑡,𝑑′)⊆̃𝐿2(𝑋,𝑡,𝑑′)and that ̃𝐿2(�... | https://arxiv.org/abs/2503.05588v1 |
0,𝐶(𝑡)and 0 in our setup. From Anderson and Moore [2, Theorem 5.2.1] it follows that the inverses above can be replaced by pseudoinverses. Let us briefly consider the problems of smoothing and prediction which extend the above filtering problem slightly. For prediction the goal is to determine the optimal predictor ̂... | https://arxiv.org/abs/2503.05588v1 |
, as in Proposition 4.2. Then ̂𝑋(𝑡,𝑠) is an optimal linear predictor for 𝑋(𝑡)and𝔼[(𝑋(𝑡)−̂𝑋(𝑡,𝑠))(𝑋(𝑡)−̂𝑋(𝑡,𝑠))⊤]=̂Σ(𝑡,𝑠)for 𝑠 <𝑡 . Proof. This follows from Proposition 4.2 together with Proposition 3.3. Corollary 4.6 (Smoothing) .Suppose that 𝑋is a polynomial state space model with finite second mo... | https://arxiv.org/abs/2503.05588v1 |
𝜅= 1,𝑚= 0.42,𝜎= 0.3and𝜌= −0.5. Moreover, it displays the corresponding optimal linear filter ̂ 𝑣(𝑡,𝑡)(1)based on observations of both Δ𝑌and(Δ𝑌)2as well as the optimal linear filter ̂ 𝑣(𝑡,𝑡)(2)based on observations of only Δ𝑌. The initial distribution used for both filters is the stationary distribution of ... | https://arxiv.org/abs/2503.05588v1 |
a different spacing in time in the sense that one considers 𝑋(𝑘)=(𝑣,𝑌,𝑌2,̃𝑌,̃𝑌2,(Δ̃𝑌)2)(𝑘Δ𝑡),𝑘∈ℕfor some positive increment Δ𝑡 >0. 4.2 Continuous-time filtering of polynomial processes The ideas of the previous section allow for a natural continuous-time counterpart. Accord- ingly, let now 𝐼=ℝ+. Analogous ... | https://arxiv.org/abs/2503.05588v1 |
𝔼(𝑋(𝑡)̃𝑋(𝑟)⊤)=̂Σ(𝑡)Γ(𝑡)−1Γ(𝑟)because 𝔼(𝑋(𝑡)̃𝑋(𝑡)⊤)=𝔼(̃𝑋(𝑡)̃𝑋(𝑡)⊤)=̂Σ(𝑡). All in all we get d d𝑟𝔼[𝑋(𝑡)̃𝑊(𝑟)⊤]=̂Σ(𝑡)Γ(𝑡)−1Γ(𝑟)𝐴𝑐 o,∶(𝑟)⊤𝐶o(𝑟)−1 2. Inserting this into (4.9) proves (4.7). Note that ̃𝑋(𝑡,𝑠)∶=𝑋(𝑡,𝑠) −̂𝑋(𝑡,𝑠)is of the form ̃𝑋(𝑡,𝑠)=̃𝑋(𝑡)+∫𝑠 𝑡̂Σ(𝑡)Γ(𝑡)−1Γ(𝑟)�... | https://arxiv.org/abs/2503.05588v1 |
. 2nd Edition. Springer, Berlin. [13] G. Kallianpur (2013). Stochastic Filtering Theory . Vol. 13. Springer, New York, NY. [14] J. Kallsen and I. Richert (2025). Parameter estimation for partially observed affine and polynomial processes . Preprint. [15] R. S. Liptser and A. N. Shiryaev (2013). Statistics of Random Pro... | https://arxiv.org/abs/2503.05588v1 |
Parameter Estimation for Partially Observed Affine and Polynomial Processes Jan Kallsen Ivo Richert Christian-Albrechts-Universit ¨at zu Kiel∗ Abstract This paper is devoted to parameter estimation for partially observed polynomial state space models. This class includes discretely observed affine or more generally pol... | https://arxiv.org/abs/2503.05590v1 |
. . 60 5.4 Proofs for Section 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 A Tools 64 A.1 Ergodic theorems for Markov chains . . . . . . . . . . . . . . . . . . . . . . . 64 A.2 Results from matrix analysis and linear systems theory . . . . . . . . . . . . . 66 A.3 Results from probability the... | https://arxiv.org/abs/2503.05590v1 |
of Generalised Ornstein–Uhlenbeck processes, which have been applied in financial mathematics e.g. in the context of the COGARCH model of Kl ¨uppelberg et al. [76]. Other ex- amples of the vast applications of polynomial processes in finance include interest rate theory (Delbaen and Shirakawa [35], Filipovi ´c and Will... | https://arxiv.org/abs/2503.05590v1 |
components of the underlying process are observable for estimation, which renders even these procedures infeasible. This happens for example in stochastic volatility models or in latent-factor interest rate models. In this case, parameter estimation effectively involves a stochastic filtering problem. In this study we ... | https://arxiv.org/abs/2503.05590v1 |
some technical tools that are needed in different parts of the paper. Frequently used notation throughout the paper We finish this section by introducing and recalling the basic notation and some theoretical preliminaries. Throughout, we let ℕ∶={0,1,2,…}denote the set of natural numbers includ- ing 0 and ℕ∗∶=ℕ∗⧵{0}. Fo... | https://arxiv.org/abs/2503.05590v1 |
a shorthand for 𝜕𝛼1𝑥1…𝜕𝛼𝑘𝑥𝑘. We let∇2 𝑥𝑓stand for the (𝑘×𝑘)Hessian matrix of 𝑓∈C2(𝐸,ℝ). If𝑓∈C1(𝐸,ℝ𝑑), we write∇𝑥𝑓∶𝐸→ℝ𝑑×𝑘to denote the Jacobian matrix of 𝑓. Let𝐸be a subset of ℝ𝑑. We call𝐸a proper state space if 𝐸is a closed and connected smooth manifold, contains 0 ∈ℝ𝑑, and is not contained ... | https://arxiv.org/abs/2503.05590v1 |
same first and second moments. Consistency and asymptotic normality of the so-obtained QML estimator is stated in Section 3 under suitable ergodicity assumptions on the polynomial process. 2.1 Parametric polynomial state space models We fix some filtered space (Ω,F,(F𝑡)𝑡∈ℕ)along with some 𝐸-valued adapted process 𝑋... | https://arxiv.org/abs/2503.05590v1 |
∑ 𝑖=1𝐴𝑖𝑋(𝑡−𝑖)=𝐵0𝜀(𝑡)−𝑞 ∑ 𝑖=1𝐵𝑖𝜀(𝑡−𝑖) for some matrices 𝐴𝑖,𝐵𝑖∈ℝ𝑑×𝑑and where (𝜀(𝑡))𝑡∈ℕis an uncorrelated sequence of mean- zero variables. As noted in the introduction, consistency and asymptotic normality for QML estimators of VARMA models (however with homoskedastic noise sequence) have been sh... | https://arxiv.org/abs/2503.05590v1 |
in particular for the computation of the asymptotic covariance matrix. But just as importantly, first and second moments of the original process are typically not enough to estimate high-dimensional parameter vectors of a polynomial process. Instead, we will later consider the Gaussian equivalent of the higher-dimensio... | https://arxiv.org/abs/2503.05590v1 |
the Gaussian quasi log-likelihood is of the form 𝐿𝜃(𝑡)∶=log𝑞𝜃 𝑡(𝑋𝜗 o(1),…,𝑋𝜗 o(𝑡))=𝑡 ∑ 𝑠=1𝐿𝜃(𝑠,𝑠−1)∶=𝑡 ∑ 𝑠=1log𝑞𝜃 𝑡|𝑡−1(𝑋o(𝑡)|𝑋o(1),…,𝑋o(𝑡−1)) (2.5) for𝑡∈ℕ∗with 𝐿𝜃(𝑡,𝑡−1)=−1 2[log||det̂Σ𝜃 o(𝑡,𝑡−1)||+𝜀𝜃(𝑡)⊤̂Σ𝜃 o(𝑡,𝑡−1)−1𝜀𝜃(𝑡)]. (2.6) We naturally refer to a maximiser ̂𝜃(𝑡)o... | https://arxiv.org/abs/2503.05590v1 |
affine process (𝑋(𝑡))𝑡∈ℝ+and fix an even 𝑝∈ℕ∗. If𝑋(𝑡)∈𝐿𝑝(ℙ𝜃)and𝜌(𝐴𝜃)<1for any𝑡∈ℕand𝜃∈Θ, then 𝑋is bounded in 𝐿𝑝(ℙ𝜃). Hence, if𝑝≥6and𝜌(𝐴𝜃)<1, condition 2 of Assumption B is fulfilled. We can now prove an ergodic theorem for the process 𝑋which moreover gives a statement about the speed of convergenc... | https://arxiv.org/abs/2503.05590v1 |
Most of the time when Assumption C is fulfilled, it is however actually the case that it is fulfilled already with 𝑢=0, i.e. the parameter 𝜃is identifiable from the observed part of the stationary mean and covariance matrix. We could therefore replace Assumption C by the following stronger assumption, which is easier... | https://arxiv.org/abs/2503.05590v1 |
Gaussian quasi-log-likelihood once more, we obtain the following rep- resentation for the ℝ𝑘×𝑘-valued process ∇𝜃𝑍𝜃(𝑡)=∑𝑡 𝑠=1∇𝜃𝑍𝜃(𝑠,𝑠−1). Proposition 3.6. For𝑖,𝑗∈{1,…,𝑘}, the𝑖𝑗-th component of ∇𝜃𝑍𝜃(𝑡,𝑡−1) is given by ∇𝜃𝑍𝜃(𝑡,𝑡−1)𝑖𝑗=−1 2[𝜇𝜃 𝑖𝑗(𝑡)+2𝜈𝜃 𝑖𝑗(𝑡)𝜀𝜃(𝑡)+𝜀𝜃(𝑡)⊤𝜓𝜃 𝑖𝑗... | https://arxiv.org/abs/2503.05590v1 |
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