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2−1/hatwidestDµhn(t) 3 Heinrichs Note that for V=R, we obtain the usual local linear estimator, and for V=C([0,1]), the estimator defined in (2) equals the pointwise defined local linear estimation, used in the literature (Bastian and Dette, 2025). The performance of the estimators ˜µnand/tildewidestDµndepend on the pr... | https://arxiv.org/abs/2503.15039v1 |
Brownian motions and (η(2) i)i∈Nan i.i.d. sequence of Brownian bridges. We generated the following 5 Heinrichs 0.0 0.2 0.4 0.6 0.8 1.0Space 0.00.20.40.60.81.0 Time1.0 0.5 0.00.51.01.52.0 Function Value 0.0 0.2 0.4 0.6 0.8 1.0Space 0.00.20.40.60.81.0 Time1.71.81.92.02.12.22.3 Figure 1: Different choices of the mean oper... | https://arxiv.org/abs/2503.15039v1 |
of the MSE. The processing time of all estimators seems to grow approximately linearly with n. As expected, there are no substantial differences between the variants of µandε. The processing time of the Jackknife estimator is about twice as long as that of the local linear estimator, which is clear from the definition ... | https://arxiv.org/abs/2503.15039v1 |
screen. Only recordings without technical errors have been considered.The EEG part of the dataset consists of 4 EEG signals recorded at different positions on the scalp with a sampling rate of 256 Hz. We first transformed the four dimensional time series (Yi)n i=1to a (discretized) multivariate functional time series b... | https://arxiv.org/abs/2503.15039v1 |
Nadaraya-Watson estimators to smooth the video, with a bandwidth of 50frames, which corresponds to 2s. Subsequently, we calculated the 10 Local Linear Regression for Time Series in Banach Spaces 0 500 1000 1500 2000Frame Number 468101214161820L2-normNW hn n Figure 4: Sup norm of the residuals for various estimators. 0 ... | https://arxiv.org/abs/2503.15039v1 |
nhnn/summationdisplay i=1Xi(i−nt nhn/parenrightbigℓK/parenleftbigi−nt nhn/parenrightbig,(4) forℓ∈{0,1,2}andt∈[0,1]. By convexity of fand Proposition 4, any b∈V2with df(b;ψ) = 0for anyψ∈V2is a global minimum of f. Note thatSℓ(t)∈RandRℓ(t)∈V, fort∈[0,1]. Further note, that df(b;ψ) = 0for anyψ∈V2, if Rℓ(t)−b0Sℓ(t)−b1hnSℓ+... | https://arxiv.org/abs/2503.15039v1 |
series. Annals of the Institute 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., Dette, H., andHeinrichs, F.(2023). Apo... | https://arxiv.org/abs/2503.15039v1 |
0.57 0.26 1.66 0.81 0.37 3.49 1.74 0.75 (BM) 0.87 0.43 0.21 1.12 0.56 0.26 1.69 0.82 0.36 3.53 1.74 0.76 (FAR-BB) 0.87 0.43 0.20 1.15 0.57 0.26 1.65 0.81 0.36 3.60 1.71 0.73 (FAR-BM) 0.87 0.43 0.20 1.14 0.55 0.25 1.65 0.80 0.35 3.38 1.68 0.73 (tvBM) 0.88 0.43 0.21 1.15 0.58 0.27 1.64 0.80 0.36 3.47 1.83 0.76 (tvFAR1) 0... | https://arxiv.org/abs/2503.15039v1 |
Asymptotic Normality in LAD Polynomial Regression and Hilbert Matrices Sa¨ ıd Maanan∗1, Azzouz Dermoune2, and Ahmed El Ghini1 1Mohammed V University of Rabat, LEAM, Rabat, Morocco 2Universit´ e de Lille, Laboratoire Paul Painlev´ e, Lille, France Abstract This paper investigates the asymptotic properties of least absol... | https://arxiv.org/abs/2503.15041v2 |
asymptotic variance of the sample median. Pollard (1991) further refined these results by showing that, under certain regularity conditions, 2f(0)"TX t=1xtx′ t#1/2 (ˆβ−β)→N(0, Ip+1), where Ip+1is the identity matrix of dimension ( p+ 1)×(p+ 1). These results firmly establish the asymptotic normality of LAD estimators i... | https://arxiv.org/abs/2503.15041v2 |
the following conditions (using the Euclidean norm ∥ · ∥): max t≤T∥[TX t=1xtx′ t]−1/2xt∥ →0 as T→ ∞ , 3 and the scaled design matrix converges: diag((TX t=1x2 t0)−1/2, . . . , (TX t=1x2 tp)−1/2)[TX t=1xtx′ t]diag((TX t=1x2 t1)−1/2, . . . , (TX t=1x2 tp)−1/2)→Q Proposition 1. The LAD estimator satisfies: diag((TX t=1x2 ... | https://arxiv.org/abs/2503.15041v2 |
T−(p+1/2))˜Hdiag( T−1/2, . . . , T−(p+1/2))→H+O(1/T). From that we derive that Pollard’s condition: max t=1,...,T∥x′ t˜H−1xt∥ →0 as T→ ∞ , is satisfied. We conclude that: diag( T1/2, . . . , Tp+1/2)(ˆβ−β)→N(0, H−1/4f2(0)). 3 Multiscale Central Limit Theorem and the Speed of Convergence in Probability and Almost Surely ... | https://arxiv.org/abs/2503.15041v2 |
the convergence properties of the estimators and their robustness to dif- ferent noise characteristics. To ensure variability, independent noise trajectories are generated without using a fixed seed, guaranteeing randomness across iterations and sample sizes. The first model under consideration is yt=β0+et, where the r... | https://arxiv.org/abs/2503.15041v2 |
noise variability on the behavior of the LAD estimator. Supremum Analysis. The maximum values of T1/2|ˆβ0(T,seed)−β0|were calculated for 50 independent trajectories per noise type. The results of the experiment are displayed in Figures 1 and 2. In Figure 1, we present the results for the original noise variance, where ... | https://arxiv.org/abs/2503.15041v2 |
the robustness of the LAD estimator under heavy-tailed noise. 3.2.2 Impact of Noise Variance on Pathwise Supremum Simulations were conducted to analyze the behavior of supT T1/2|ˆβ0(T,seed)−β0| for the model yt=β0+etunder Laplace, Gaussian, and Cauchy noise. The objective was to assess the impact of noise variability... | https://arxiv.org/abs/2503.15041v2 |
that the empirical probability of the estimator ˆβ0falling within a symmetric interval around β0, scaled by the theoretical standard deviation (1 .96σ0), converges to the theoretical asymptotic probability of 0.95. To investigate the finite-sample behavior of this result, we analyze the gap between the empirical probab... | https://arxiv.org/abs/2503.15041v2 |
0.95, while the black dashed line denotes the average empirical probability across all Tvalues. The empirical probability analysis highlights the alignment of finite-sample probabilities with theoretical asymptotic predictions for different values of αand noise distributions. While the results confirm that the LAD esti... | https://arxiv.org/abs/2503.15041v2 |
condition for these noise types. However, these distribu- tions require larger sample sizes ( T > 50,000) to align more closely with the 95% theoretical probability. These results validate the robustness of the estimator ˆβ0under different noise 16 distributions and emphasize the differences in convergence rates and be... | https://arxiv.org/abs/2503.15041v2 |
within the interval across 1000 simulations for sample sizes T∈ {100,500,1000 ,5000,10000 ,50000 ,100000 ,500000 }. 18 Results. The empirical coverage probabilities are summarized in Figure 8. Key observations include the following: For the standard CLT interval ( α= 0.5), the coverage probabilities converge to the nom... | https://arxiv.org/abs/2503.15041v2 |
the confidence inter- vals for the peak emission level ( a), the turning point in GDP ( l), and the dispersion parameter (s2). The gradient expressions are given by: ∇a= a,−aβ1 2β2,aβ2 1 4β2 2 , ∇l= 0,−1 2β2,β1 2β2 2 , ∇s2= 0,0,1 β2 2 . The confidence intervals for each parameter under different noise assumptions... | https://arxiv.org/abs/2503.15041v2 |
improved coverage probabilities. The analysis demonstrated that the LAD estimator is robust to different noise charac- teristics, including heavy-tailed distributions like Cauchy noise. However, the convergence of standard confidence intervals, scaled by T1/2, to the nominal level of 95% was shown to be slow, particula... | https://arxiv.org/abs/2503.15041v2 |
Semiparametric plug-in estimation, sup-norm risk bounds, marginal optimization, and inference in BTL model Vladimir Spokoiny∗ Weierstrass Institute Berlin, HSE and IITP RAS, Mohrenstr. 39, 10117 Berlin, Germany spokoiny@wias-berlin.de April 23, 2025 Abstract The recent paper Gao et al. (2023) on estimation and inferenc... | https://arxiv.org/abs/2503.15045v2 |
sup-norm estimation 15 4.1 Conditional and marginal optimization . . . . . . . . . . . . . . . . . . . 15 4.1.1 Partial optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.1.2 Conditional optimization under (semi)orthogonality . . . . . . . . 17 4.1.3 Semiparametric bias . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2503.15045v2 |
υL(υ) = argmax (θ,η)L(θ,η). The profile MLE eθis the θ-component of eυ. This estimator possesses a number of nice theoretical properties like asymptotic efficiency and normality; see e.g. Bickel et al. (1993), Robinson (1988), Andrews (1994), Van der Vaart (2000), Chernozhukov et al. (2018). Spokoiny (2025b) explained ... | https://arxiv.org/abs/2503.15045v2 |
high-dimensional situation. Sup-norm estimation is studied as a special setup in the semiparametric plug-in approach: one component of the vector υis treated as the target while the rest is the nuisance. For a special case with a stochastically dominant information matrix F= −∇2EL(υ∗) , we prove a coordinate-wise expan... | https://arxiv.org/abs/2503.15045v2 |
θfor each fixed value of the nuisance parameter s: eθ(s)def= argmax θL(θ,s),θ∗(s)def= argmax θEL(θ,s). We also assume that each partial problem can be easily solved. Further, let a pilot estimate bsof the nuisance parameter sbe given. This leads to the plug-in estimator bθ=eθ(bs).The profile MLE is a special case of th... | https://arxiv.org/abs/2503.15045v2 |
(2.2) It is important that the radius rDcorresponds to the dimension of the target parameter θ. Now we apply the general results from Section A to the partial pMLEs eθ(s) . Proposition 2.1. Assume (ζ), and let the θ-component ∇θζof∇ζsatisfy (∇ζ). Let also (2.1) hold with ω≤1/3. For any s∈H0, let fs(θ) = EL(θ,s)satisfy ... | https://arxiv.org/abs/2503.15045v2 |
One of them corresponds to the already mentioned additive case f(θ,s) =f1(θ) +f2(s) . If f(θ,s) is quadratic, then orthogonality can be achieved by a linear transform of the nuisance parameter s. For a general function f, such a linear transform helps to only ensure the one-point orthogonality condition ∇θ∇sf(υ∗) = 0 .... | https://arxiv.org/abs/2503.15045v2 |
(T∗ 3|s),(T∗ 3,S)from Section 4.1. The first one requires that f(θ,s) is smooth in θforsfixed, while the second one describes the smoothness prop- erties of f(θ∗,s) w.r.t. s. These conditions are weaker than the full dimensional smooth- ness condition (T∗ 3)and only involve partial derivatives in θand cross-derivatives... | https://arxiv.org/abs/2503.15045v2 |
∥D F−1∇θζ∥. The corresponding relation can be formulated as n−1/2(𝕡D+𝕡H) =o(√𝕡D). It is important to stress that the value 𝕡H= E{∥H(bs−s∗)∥2 ◦1 IΩ(x)}only involves the ∥ · ∥◦-norm of the error H(bs−s∗) of the pilot bs. For the case of sup-norm estimation, this gradually helps to relax the critical dimension conditi... | https://arxiv.org/abs/2503.15045v2 |
max j≤p|Dj(eυj−υ∗ j)| ≤r∞. Furthermore, D−1 F(eυ−υ∗)− ∇ζ ∞≤τ∞∥D−1∇ζ∥2 ∞, D(eυ−υ∗−F−1∇ζ) ∞≤τ∞ 1−ρ1∥D−1∇ζ∥2 ∞, D(eυ−υ∗)−D−1∇ζ) ∞≤τ∞ 1−ρ1∥D−1∇ζ∥2 ∞+ρ1 1−ρ1∥D−1∇ζ∥∞. As a corollary, we obtain a component-wise expansion: on Ω(x) Dj(eυj−υ∗ j)−D−1 j∇jζ) ≤τ∞ 1−ρ1∥D−1∇ζ∥2 ∞+ρ1 1−ρ1∥D−1∇ζ∥∞ ≤τ∞ 2r2 ∞+ρ1√ 2r∞. 4 Marginal optimiz... | https://arxiv.org/abs/2503.15045v2 |
the condition Fxs= 0 yields xs≡x∗and the bias vanishes. If f(υ) is not quadratic, the orthogonality condition ∇s∇xf(x,s)≡0 for all ( x,s)∈ W still ensures a vanishing bias. Lemma 4.3. Letf(x,s)be continuously differentiable and ∇s∇xf(x,s)≡0. Then the point xs= argmaxxf(x,s)does not depend on s. Proof. The condition ∇s∇... | https://arxiv.org/abs/2503.15045v2 |
3,S)with𝔻2≤𝔽. Fix s∈ S, set ωdef=τ21∥ℍ(s−s∗)∥◦, and assume (T∗ 3|s)with𝔻s≡𝔻,rs≡r, and τ3, such that ω <1,r≥3ρ2 2(1−ω)∥ℍ(s−s∗)∥◦,ρ2τ3 1−ω∥ℍ(s−s∗)∥◦≤4 9, forρ2from (4.13) . Then the partial solution xsobeys ∥𝔻(xs−x∗)∥ ≤3ρ2 2(1−ω)∥ℍ(s−s∗)∥◦. (4.14) Moreover, with ρ∗from (4.12) , it holds ∥Q{xs−x∗+𝔽−1Fxs(s−s∗)}∥ ≤ ∥ ... | https://arxiv.org/abs/2503.15045v2 |
ρ2 1≤max j=1,...,psup ∥z∥∞≤1X m̸=j1 𝔻2 j𝔻2mF2 jmz2 m≤max j=1,...,p1 𝔻2 jX m̸=jF2 jm 𝔻2m as claimed. It is mandatory for the proposed approach that ρ1<1 . The next result provides an upper bound on ∥υ◦−υ∗∥∞. Proposition 4.8. Letf(υ)be concave function and g(υ)be a linear perturbation of f(υ)with a vector A. For a di... | https://arxiv.org/abs/2503.15045v2 |
nitems of interest. The items jandmare compared if and only if (jm)∈ E. One observes independent paired comparisons Y(ℓ) jm,ℓ= 1, . . . , N jm, and Y(ℓ) jm= 1−Y(ℓ) mj. For modeling and risk analysis, Bradley-Terry-Luce (BTL) model is frequently used; see Bradley and Terry (1952), Luce (1959). The chance of each item wi... | https://arxiv.org/abs/2503.15045v2 |
one condition per connected component of the graph G. Alternatively, one can use penalization with a quadratic penalty ∥Gυ∥2/2 . A “non- informative” choice is G2=g2In; cf. Chen et al. (2019). Another option is to replace the constraintP jυj= 0 by the penalty ∥Gυ∥2=g2⟨υ,e⟩2with G2=g2ee⊤. It is obvious to see that this ... | https://arxiv.org/abs/2503.15045v2 |
5.3: Comparison of the leading term and the remainder for different n. References Andresen, A. and Spokoiny, V. (2016). Convergence of an alternating maximization procedure. Journal of Machine Learning Research , 17(63):1–53. Andrews, D. W. K. (1994). Asymptotics for Semiparametric Econometric Models via Stochastic Equ... | https://arxiv.org/abs/2503.15045v2 |
Syst. Sci. , 26(5):98–101. 30Semiparametric plug-in estimation, sup-norm risk, and marginal optimization Ostrovskii, D. M. and Bach, F. (2021). Finite-sample analysis of M-estimators using self-concordance. Electronic Journal of Statistics , 15(1):326 – 391. Pankevich, S. and Spokoiny, V. (2025). Estimation and inferen... | https://arxiv.org/abs/2503.15045v2 |
function f(υ) = EL(υ) within a local vicinity of the point υ∗. The notion of locality is given in terms of a metric tensor D∈Mp. In most of the results later on, one can use D= F1/2. In general, we only assume D2≤κ2Ffor some κ>0 . Let f(υ) be three or sometimes even four times Gateaux differentiable in υ∈Υ. Introduce t... | https://arxiv.org/abs/2503.15045v2 |
specific, we only consider the 3S-results of Theorem A.1. Also, assume κ≡1 . The important constant τ3is identified by(S∗ 3):τ3=c3/√n, where the scaling factor nmeans the sample size. It can be defined as the smallest eigenvalue of the Fisher operator F. First, we discuss the case Q=D= F1/2. It appears that in this ful... | https://arxiv.org/abs/2503.15045v2 |
t∈[0,1] ⟨∇3 xxsf(x∗,s∗+t(s−s∗)),(𝔻−1z)⊗2⊗(s−s∗)⟩ ≤τ21∥z∥2∥ℍ(s−s∗)∥◦. This yields (C.2). The next result describes some corollaries of (C.2). Lemma C.2. Assume 𝔻2≤κ2𝔽and let some other matrix 𝔽1∈Mpsatisfy ∥𝔻−1(𝔽1−𝔽)𝔻−1∥ ≤κ−2ω (C.3) with ω <1. Then ∥𝔽−1/2(𝔽1−𝔽)𝔽−1/2∥ ≤ ω , (C.4) ∥𝔽1/2(𝔽−1 1−𝔽−1)𝔽1/2∥ ≤ω 1... | https://arxiv.org/abs/2503.15045v2 |
1) ≤ρ1∥ℍ1(s1−s∗ 1)∥∞. (C.13) For any s1, define υ◦ 1(s1)def= argmax υ1g(υ1,s1). 40Semiparametric plug-in estimation, sup-norm risk, and marginal optimization Now we apply Proposition 4.6 with Q=𝔻1,𝔽=F11,ℍ=ℍ1= diag( 𝔻2, . . . ,𝔻p) , and∥ℍ1(s1−s∗ 1)∥∞≤r∞. As𝔻1F−1 11≤𝔻−1 1, bound (4.18) yields 𝔻1{υ◦ 1(s1)−υ∗ 1−F−1 ... | https://arxiv.org/abs/2503.15045v2 |
A Bivariate Poisson-Gamma Distribution: Statistical Properties and Practical Applications Indranil Ghosh1, Mina Norouzirad2∗, Filipe J. Marques2,3 1Department of Mathematics and Statistics, University of North Carolina, Wilmington, USA 2Center for Mathematics and Applications (NOVA Math), NOVA School of Science and Tec... | https://arxiv.org/abs/2503.15062v1 |
between light and severe accidents. Nelson (1985) built upon the research of Arbous & Kerrich (1951) by including a Dirichlet distribution into the analysis of cross-sectional data and permitting individual rates to change at the start of the second period in the study of two-period lon- gitudinal data. Lemaire (1979) ... | https://arxiv.org/abs/2503.15062v1 |
discussed in this paper for the bivariate distribution with joint density in (3), which comprises both continuous and discrete random variables, enabling the modeling of several phenomena. Such forms of bivariate distributions are rare in the statistical literature. For example, identifying bivariate distributions in w... | https://arxiv.org/abs/2503.15062v1 |
joint and marginal distributions of BPGC for various choices of the model parameters. 6 2.2 Exponential family Theorem 2.1. The distribution BPGC( m)is a member of the exponential family with five parameters. Proof. The BPGC distribution in (3) is represented in the form f(x, y|m) = exp exp 5X j=1δj(m)Uj(x, y)−∆ (m)!! ... | https://arxiv.org/abs/2503.15062v1 |
test exists for any one-sided hypothesis when the other parameters are fixed. Assuming independent random variables XandYwith cumulative distribution func- tions (c.d.f.’s) FXandFY, respectively, we have 1)X≥stYindicates that Xis greater than Yin the stochastic order if, for all x, FX(x|m)≤FY(x|m). 2)X≥hrYdenotes that ... | https://arxiv.org/abs/2503.15062v1 |
right-hand-side of (7) becomes non-positive for all y∈(0,∞). This implies that the distribution satisfies the reverse rule of order 2. i.e., m12 y−m11≤0∀y∈(0,∞). Thus, this condition holds true if and only if m11> m 12. The next theorem establishes the non-linearity of regression when ( X, Y) follows the BPGC distribut... | https://arxiv.org/abs/2503.15062v1 |
is particularly evident due to the complexity of the log-likelihood function in (8). 12 4 Simulation Study This section provides details of the simulation study based on the methodology outlined in section 3. The Gibbs sampling technique (Gelfand, 2000) is employed to generate random samples from the BPGC distribution,... | https://arxiv.org/abs/2503.15062v1 |
steps by handeling param- eter estimation, simulating the second sample, and performing the FF test, thus providing a comprehensive evaluation of how goodness-of-fit of the BPGC model to the observed data. The test statistics and p-values are presented in Table 2, showing that the fitted dis- tribution using the MLEs i... | https://arxiv.org/abs/2503.15062v1 |
it can be confidently asserted that the BBCD model provides an excellent fit to the hospital data. 6 Concluding Remarks There are several well-established bivariate mixtures distributions in the rich literature of distribution theory, either both continuous or discrete. Interestingly, relatively few distribu- tions ari... | https://arxiv.org/abs/2503.15062v1 |
disease or of repeated accidents. Journal of the Royal Statistical Society, 83 (2), 255–279. Jones, M. C. (2002). Multivariate t and beta distributions associated with the multivariate F distribution. Metrika, 54 , 215–231. Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions (Vol. 2... | https://arxiv.org/abs/2503.15062v1 |
Nonlinear Bayesian Update via Ensemble Kernel Regression with Clustering and Subsampling Yoonsang Lee Abstract Nonlinear Bayesian update for a prior ensemble is proposed to extend traditional ensemble Kalman filtering to settings characterized by non-Gaussian priors and nonlinear measurement operators. In this framewor... | https://arxiv.org/abs/2503.15160v1 |
Gaussian mixture–based inference schemes. The remainder of the paper is organized as follows. Section 2 outlines the state and measure- ment models and reviews the standard Kalman update, which is interpreted as a linear regression between observed and unobserved components. Building on this framework, Section 3 introd... | https://arxiv.org/abs/2503.15160v1 |
= ˜u+˜Cuv˜C−1 vv(v−˜v), (12) 3 and Cu|v=˜Cuu−˜Cuv˜C−1 vv˜Cvu, (13) respectively. Thus, the relation between the posterior means of uandvcan be expressed as ˆu=µu|v(ˆv), (14) demonstrating that the update of uis obtained through the linear regression of uonvunder the Gaussian prior. Note that ˜Cuv˜C−1 vvrepresents the l... | https://arxiv.org/abs/2503.15160v1 |
the nonlinear regression uses only the subsampled ensemble µu|v(v) =PM m=1Φv(v; ˜vkm)˜ukmPM m=1Φv(v; ˜vkm). (20) The subsampled ensemble size may become very small - or even empty - so that the available prior samples provide little to no information for estimating the unobserved component at ˆ v. In such cases, due to... | https://arxiv.org/abs/2503.15160v1 |
at v= ˆvand hierarchical clustering is applied; the mean of the most populated cluster is then used as the refined estimate of u. If clustering is not enabled, the nonlinear regression (via the Nadaraya–Watson estimator) is used directly to compute the conditional mean. Several tuning parameters must be carefully consi... | https://arxiv.org/abs/2503.15160v1 |
the 40-dimensional Lorenz 96 model. As a linear update method to be compared with the nonlinear method, we choose the Ensemble adjustment Kalman Filter [2] along with multiplicative inflation and localization if necessary. For KDE, we choose the kernels Φ uand Φ vto be Gaussian with covariance matrix estimated from sam... | https://arxiv.org/abs/2503.15160v1 |
with and without subsam- pling (SS) and clustering (Cl). Regarding the linear update using EAKF, we use the best results after testing various multiplicative inflation levels ranging from 1 to 1.5 with an increment of 0.05. The nonlinear method shows at least a 8% error decrease in the worst-case scenario (NlBU with cl... | https://arxiv.org/abs/2503.15160v1 |
F= 6 F= 8 Prior error Post error Prior error Post error EAKF 1 .09×10−16.58×10−22.86×10−11.09×10−1 NlBU w/ SS 1 .35×10−18.16×10−21.94×10−17.98×10−2 NlBU w/ SS Cl 1 .33×10−17.88×10−21.90×10−17.88×10−2 4.3 A high-dimensional case: 40-dimensional Lorenz 96 The last experiment is a high-dimensional test using the Lorenz 96... | https://arxiv.org/abs/2503.15160v1 |
mitigated by employing local subsampling and unsupervised clustering techniques. The proposed strategy was evaluated using a variety of test problems, including Lorenz systems and a PDE-constrained optimization problem related to subsurface Darcy flow. Overall, numerical experiments demonstrate that the nonlinear Bayes... | https://arxiv.org/abs/2503.15160v1 |
William F Campbell, and Patrick A Rei- necke. Quadratic polynomial regression using serial observation processing: Implementation within dart. Monthly Weather Review , 145(11):4467–4479, 2017. [13] Peter L Houtekamer and Fuqing Zhang. Review of the ensemble kalman filter for atmospheric data assimilation. Monthly Weath... | https://arxiv.org/abs/2503.15160v1 |
OPTIMAL DATA SPLITTING FOR HOLDOUT CROSS -VALIDATION INLARGE COVARIANCE MATRIX ESTIMATION Lamia Lamrani1, Christian Bongiorno1, Marc Potters2 1Université Paris-Saclay, CentraleSupélec, Laboratoire de Mathématiques et Informatique pour la Complexité et les Systèmes, 91192 Gif-sur-Yvette, France 2Capital Fund Management,... | https://arxiv.org/abs/2503.15186v1 |
fall within such limit should be regarded as noise and should be shrunk to a single average value [ 5,15]. However, the limits of the MP distribution depend only on the measured aspect ratioq, i.e., the size of the matrix ndivided by the number of data points t. Notably, the measured aspect ratio is not always equal to... | https://arxiv.org/abs/2503.15186v1 |
helps to assess how well the model generalizes to unseen data and is particularly useful to detect overfitting [ 27]. When overfitting occurs, the model may perform well on the training set but will show a noticeable drop in performance on the testing set. The most popular CV methods are the k-fold CV , the Leave-One-O... | https://arxiv.org/abs/2503.15186v1 |
population covariance is an inverse Wishart, both the Bayesian estimator and linear shrinkage coincide with the oracle [ 36,37]. We believe that studying the holdout method is a necessary step to understand the k-fold CV better. Furthermore, while the holdout method may be less efficient than k-fold CV on stationary da... | https://arxiv.org/abs/2503.15186v1 |
proportional to the identity matrix. To ensure that the expected value of Σis the identity matrix, we choose the scale parameter appropriately. Lemma 2.1. LetΣ∼ W−1 n(t∗,(t∗−n−1)1)witht∗> n+ 1. Then E[Σ] =1. (6) Proof. The expected value of an inverse Wishart-distributed matrix is given by E[Σ] =Ψ t∗−n−1. (7) By settin... | https://arxiv.org/abs/2503.15186v1 |
distribution of the population covariance [16]. Property 2.3. ForEa sample covariance matrix from multivariate normal data X∈Rn×twith population covariance Σ, in the high-dimensional limit, the expected Frobenius error between EandΣis equal to E τ((E−Σ)2) =q. (17) In order to define the oracle estimator of the covari... | https://arxiv.org/abs/2503.15186v1 |
the holdout method and the k-fold CV . Additionally, we introduce two variations of these methods, one with and one without rotational invariance. To explain these methods, we need first to define the index partitions. Definition 2.11. Lettbe the total number of observations, and let toutbe an integer between 1andt−1. ... | https://arxiv.org/abs/2503.15186v1 |
observations Xout,l={xi|i∈ I out,l}(test set). Let Vin,landVbe eigenvectors of Ein,landErespectively ranked according to their eigenvalues. Then the k-fold CV eigenvalues are ΛCV=1 kkX l=1ΛH l=1 kkX l=1Diag(VT in,lEout,lVin,l). (30) The rotational invariant k-fold CV estimator is then ΞRCV=VΛCVVT. (31) (rotational inva... | https://arxiv.org/abs/2503.15186v1 |
function of the log of the train-test ratio factor k, integers which divides tthe number of data points. The averages are computed with 100simulations of inverse Wishart population matrices with n= 200 andp= 1.5, with Gaussian data with q= 0.5 Proof. Let us now prove the proposition by deriving the expression of the Fr... | https://arxiv.org/abs/2503.15186v1 |
(40) can be written as E τ((ΞH−Σ)2) =V λtrue −V λO + 2E[λtrue]2+V λO tout. (44) 3.2 White inverse Wishart case If the population matrix is drawn from a white inverse Wishart, then equation (40) has a closed form in the high dimensional limit. Proposition 3.2. Let us consider Σ∼ W−1 npa white inverse Wishart of ... | https://arxiv.org/abs/2503.15186v1 |
bias is still observable, the position of the minimum seems not affected. This motivates us to use equation (45) to derive an analytical expression for the optimal k. 3.2.1 Optimal split One interesting consequence of the expression of the holdout error is the computation of the optimal number kthat minimizes it. Corol... | https://arxiv.org/abs/2503.15186v1 |
2025 It is worth remarking that the minimum in the right panel of figure 2, around k= 1.5, is a prediction confirmed by our formula written in equation (54). Such a minimum is counterintuitive since for k= 1.5, the prediction of the formula implies a test set double the train set, which is against every recommendation ... | https://arxiv.org/abs/2503.15186v1 |
83(7):1467, 1999. [6]O Ledoit, M Wolf, and I Honey. shrunk the covariance matrix-problems in mean-variance optimization. The Journal of Portfolio Management , 30(4):110–119, 2004. [7]Ester Pantaleo, Michele Tumminello, Fabrizio Lillo, and Rosario N Mantegna. When do improved covariance matrix estimators enhance portfol... | https://arxiv.org/abs/2503.15186v1 |
detecting and preventing overfitting. School of Computer Science Carneigie Mellon University , 133, 2001. [28] Bruce G Marcot and Anca M Hanea. What is an optimal value of k in k-fold cross-validation in discrete bayesian network analysis? Computational Statistics , 36(3):2009–2031, 2021. [29] Tzu-Tsung Wong and Po-Yan... | https://arxiv.org/abs/2503.15186v1 |
Systemic Risk Management via Maximum Independent Set in Extremal Dependence Networks Qian Hui∗aand Tiandong Wang†a,b aShanghai Center for Mathematical Sciences, Fudan University bShanghai Academy of Artificial Intelligence for Science Abstract The failure of key financial institutions may accelerate risk contagion due ... | https://arxiv.org/abs/2503.15534v1 |
extremal losses occur simultaneously across institutions, while extremograms track how extreme events propagate over time. A detailed comparison of these methods can be found in [24]. ∗qhui24@m.fudan.edu.cn. †Corresponding author, td wang@fudan.edu.cn. 1arXiv:2503.15534v1 [q-fin.PM] 3 Mar 2025 In this study, we analyze... | https://arxiv.org/abs/2503.15534v1 |
risk measurement indicators such as ES for each maximum independent set, and construct a portfolio optimization model by minimizing the overall risk. Output: Optimal portfolio with minimum expected shortfalls. 2 Extremal dependence measure The extremal dependence measure (EDM) (cf. [24]) quantifies the tendency for lar... | https://arxiv.org/abs/2503.15534v1 |
[24], the authors highlight that EDM can be interpreted as the limit of the cross moment between normalized Z1andZ2when R=∥Z∥is large, i.e. EDM( Z1, Z2) = lim x→∞EZ1 RZ2 R R > x . (6) Based on this relationship, they proposed an estimator for EDM( Z1, Z2), which is defined as \EDM( Z1, Z2) =1 NnnX i=1Zi1 RiZi2 Ri1[Ri... | https://arxiv.org/abs/2503.15534v1 |
between vertices iandj. 3.1.3 Clustering coefficient The clustering coefficient measures the degree of clustering or cohesion among vertices in a network. It is defined as the probability that any two neighbors of a given vertex are connected. This is calculated as the ratio of the actual number of connections between ... | https://arxiv.org/abs/2503.15534v1 |
1 2 5 10 20 Degree1−CDF 1e−02 1e−01 1e+00 (d) Threshold=0.24 Figure 2: Log-log plots of the complementary cumulative distribution function (1-CDF) for the degrees at different thresholds in the U.S. S&P 500 market. 5 We start by examining the empirical degree distributions. In the Chinese A-shares market (Figure 1), ne... | https://arxiv.org/abs/2503.15534v1 |
we proceed with θ= 0.22 for both China and the U.S.. Then, we visualize the graphs for both the Chinese A-shares and the U.S. S&P 500 markets. As shown in Figure 3(a) and Figure 4(a), both networks exhibit clear community structures. In the sequel, we refer to such graphs as extremal dependence networks for stocks, and... | https://arxiv.org/abs/2503.15534v1 |
closure indicates strong extremal dependence among the three banks, positioning them as key transmission vertices if any extreme events occur in the state-owned banks. Such dependence may be due to their shared regional focus in Beijing, the capital of China, and similar exposure to local government projects. The yello... | https://arxiv.org/abs/2503.15534v1 |
the community structure, betweenness centrality is another important measure for the constructed extremal dependence network since it identifies key vertices controlling the spread of systemic risks. Vertices with high betweenness are potentially vulnerable points, where failure may trigger widespread financial contagi... | https://arxiv.org/abs/2503.15534v1 |
independent set in graph G. IfV∗is not contained in any other independent set, it is called a maximal independent set. If the size of V∗is the largest among all maximal independent sets, it is referred to as the maximum independent set. The maximum independent set problem (MISP) is a classic combinatorial optimization ... | https://arxiv.org/abs/2503.15534v1 |
stock jto stock iby ∆CoVaRi|j q= CoVaRi|Xj=VaRj q −CoVaRi|Xj=VaRj 50q . (14) In particular, by (14), if asymptotic independence holds between XiandXj, then both ∆CoVaRi|j qand ∆CoVaRj|i q are equal to 0. To proceed, we set q= 0.99 and calculate the ∆CoVaR for each pair of stocks. Then we generate heatmaps based on the ... | https://arxiv.org/abs/2503.15534v1 |
that institutions with high betweenness centrality scores do not reveal concerning exposure to systemic risk, showing that the U.S. regulations on financial institutions are now effective. 11 4 Empirical study and results In this section, we propose a portfolio strategy to minimize the risk of extremal loss, where one ... | https://arxiv.org/abs/2503.15534v1 |
Chinese and U.S. portfolios further confirm the effectiveness of our method in avoiding investments in higher-risk institutions. Moreover, the entire Chinese portfolio exhibits a lower ES of 2.17% with a less uniform weight distribution compared to the U.S. portfolio, which achieves a portfolio ES of 3.14%, and stocks ... | https://arxiv.org/abs/2503.15534v1 |
March 7 and March 20, 2024. Hence, the MIS portfolio gives a competitive return profile that mirrors the performance of the five state-owned banks. This alignment may reflect the potential influence of government intervention and the hierarchical structure within the Chinese financial system. In terms of risk, the SSE ... | https://arxiv.org/abs/2503.15534v1 |
shares and 37 U.S. S&P 500 stocks to compare the systemic risk structures of the two markets. By investigating key network characteristics, we identify unique properties in the Chinese and U.S. financial systems, offering insights into how systemic risk propagates and can be mitigated. We also construct portfolios base... | https://arxiv.org/abs/2503.15534v1 |
and Sidney Resnick. Living on the multidimensional edge: seeking hidden risks using regular variation. Advances in Applied Probability , 45(1):139–163, 2013. [14] Richard A Davis and Thomas Mikosch. The extremogram: A correlogram for extreme events. Bernoulli: Official Journal of the Bernoulli Society for Mathematical ... | https://arxiv.org/abs/2503.15534v1 |
arXiv:2503.15705v1 [math.ST] 19 Mar 2025On the Functoriality of Belief Propagation Algorithms on Finite Partially Ordered Sets Gr´ egoire Sergeant-Perthuis∗1,2, Toby St Clere Smithe3, and L´ eo Boitel1 1CQSB, Sorbonne Universit´ e, Paris, France 2Ouragan, Inria Paris, France 3VERSES Research, Topos Institute March 21, ... | https://arxiv.org/abs/2503.15705v1 |
costly with respect t o the number of variables, |I\J|. It requires computing the marginal likelihood PXJ, which involves summing over all possible value configurations of XJ, consisting of N|J|terms, assuming that all Xitake values in a set of cardinality N. One resorts instead to approximate inference, which is infe r... | https://arxiv.org/abs/2503.15705v1 |
poset. Methods such as homotopy continuation can be appl ied to Belief Propagation in order to modify the fixed points of the Belief P ropagation algorithm into those of a simpler algorithm in a continuous f ashion [10]. The homotopy continuation is performed in the space of all al gorithms: one follows a straight line ... | https://arxiv.org/abs/2503.15705v1 |
their fixed po ints. Finally, we provide a proof of the main theorems of this article in Appe ndix E. 4 2 Preliminaries 2.1 Graphical Models, Hypergraphs, Factor Graphs LetG= (V,E) beanundirectedgraph, where Visafiniteset ofvertices and Eis a collection of undirected edges, i.e., subsets of Vof cardinality 2. For v∈V, le... | https://arxiv.org/abs/2503.15705v1 |
Let Pa∈P>0(Ea) denote the marginal distribution of Xa; it is defined as Pa(za) =/summationtext ya∈EaP(za,ya), whereza∈Ea, anda=V\a is the complement of a. Letd(v) denote the degree of node v∈V, i.e., the cardinality of ∂v. Then, for any x∈EV, P(x) =/producttext e∈EPe(xe) /producttext v∈VPd(v)−1 v. (2.1) Recall thattheen... | https://arxiv.org/abs/2503.15705v1 |
of a po set A, denotedζ, is the operator from/circleplustext a∈AR→/circleplustext a∈AR, where/circleplustext a∈ARis identified with the set of functions from Ataking values in R, defined as, for 7 anyλ∈/circleplustext a∈ARand anya∈A,ζ(λ)(a) =/summationtext b≤aλb. It is a central result in combinatorics that the zeta-oper... | https://arxiv.org/abs/2503.15705v1 |
PFa b:P(Ea)→P(Eb), which sends P∈P(Ea) to the pushforward measure defined as follows: for any x∈Eb, PFa b(P)(x) =/summationtext y∈Ea1[Fa b(y) =x]P(y). The optimization problem for variational inference graphi cal presheaves F, as introduced in [32, 20], involves minimizing the Bethe Fr ee Energy FBethe(Qa;a∈A) under the... | https://arxiv.org/abs/2503.15705v1 |
by /tildewideF∗. When a scalar product ∝a\}bracketle{t·,·∝a\}bracketri}htais specified on each vector space /tildewideFa, there is an isomorphism f/a\}bracketle{t·,·/a\}bracketri}hta:F∗ a→Fawhich allows us to identify /tildewideFa,∗ bwith its adjoint /tildewideFa,† b, defined as /tildewideFa,† b=f/a\}bracketle{t·,·/a\}br... | https://arxiv.org/abs/2503.15705v1 |
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