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from a list of candidates to study poverty using SQ(u), it should be compatible with the monotonic nature of the Sen index. While Sen index expressed in the general form (3.1) has q(F, x;z) = 2h 1−F(x) r(z)i 1−x z , where rzis the proportion below the poverty line, other prominent indices pro- posed in literature are... | https://arxiv.org/abs/2504.05713v1 |
=α+β γ 1−1−uδ δγ (4.1) where αis a location parameter and βis a scale parameter. Since the ran- dom variable Xhas to be non-negative, Q(0)≥0, giving α+β γ(1−δ−γ)≥0 along with β > 0. The advantage of (4.1) is that it contains as special cases, the Pareto distribution for δ= 1, Stoppa for γ < 0, δ > 0 and β=−γ(δ)γ, α... | https://arxiv.org/abs/2504.05713v1 |
should have λ+θ−ϕ≥0 and θ, ϕ > 0. An important property of the model is that it can satisfactory represent positively skewed data like the log-normal, gamma and beta laws. It contains the Pareto I and Pareto II distributions as special cases. The poverty measures are A1(u) =u−1 Qw(u)P5(u), L(u) =1 µP5(u), µQW(u) =1 uP5... | https://arxiv.org/abs/2504.05713v1 |
µ Q(u) u µ Q(u) u µ Q(u) 0.0345 19.7315 0.1207 1328.5588 0.2069 1911.2304 0.0577 304.2465 0.1379 1179.2204 0.2241 1876.7298 0.0690 755.8168 0.1552 1533.8510 0.2414 2206.1195 0.0862 1382.4415 0.1724 1484.0924 0.2586 2813.1330 0.1034 1495.4265 0.1897 1530.1792 0.2759 3604.7200 0.2931 4143.5181 0.5690 9230.4205 0.8276 195... | https://arxiv.org/abs/2504.05713v1 |
have a closed form dis- tribution function to calculate the conventional indices necessitating the need for quantile-based poverty measures for further analysis. The other poverty measures are the income gap ratio, IQ(u) =1 uQ(u)Zu 0pq(p)dp=µQ(u) Q(u) =α+βu αlogu+ 2βu+r and the poverty gap ratio, A1(u) =1 Q(u)Zu 0pq(p)... | https://arxiv.org/abs/2504.05713v1 |
IBM Journal of Research and Development , 38:251–258. Houghton, J. C. (1978). Birth of a parent: The wakeby distribution for modeling flood flows. Water Resources Research , 14:1105–1109. Kakwani, N. (1980). On a class of poverty measures. Econometrica: Journal of the Econometric Society , 48:437–446. Kakwani, N. C. an... | https://arxiv.org/abs/2504.05713v1 |
arXiv:2504.05819v1 [math.ST] 8 Apr 2025Nonparametric local polynomial regression for functional covariates Moritz Jirak Institut f¨ ur Statistik und Operations Research, Universi t¨ at Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria Email: moritz.jirak@univie.ac.at Alois Kneip Fachbereich Wirtschaftswissenschaften,... | https://arxiv.org/abs/2504.05819v1 |
to treat functional covariates. We mention approaches to nonp arametric density estimation for functional data, i.e. the wavelet-based method of [7] an d orthogonal series and ker- nel estimators by [9] and [10] for specific diffusion processes . While upper bounds on the convergence rates are provided under quite abstra... | https://arxiv.org/abs/2504.05819v1 |
formalized as follows: Write /⌊ard⌊lz/⌊ard⌊l2 N:=/summationtextn j=1z2 j·1N(Xj), Y:= (Y1,...,Y n)†and ˜PJ,K/parenleftbig /a\}⌊ra⌋ketle{tXj−x,ϕ1/a\}⌊ra⌋ketri}ht,...,/a\}⌊ra⌋ketle{tXj−x,ϕJ/a\}⌊ra⌋ketri}ht/parenrightbig =/summationdisplay k∈Kαk·ξj;k. Then the goal is to select α= (αk)ksuch that /⌊ard⌊lY−ξα/⌊ard⌊l2 Nwithξ:... | https://arxiv.org/abs/2504.05819v1 |
matrix M/nand its expected valueMnfor large n. Lemma 3.2. Forδ∈(0,1), it holds that Ee† 0(M+S)−1e0≤2−δ2 1−δ2·n−1·u0, withu0as in (3.3). We establish an upper bound on the term B1in (3.1) by the next lemma. Lemma 3.3. We have G†SG≤K−1/summationdisplay k=0Jk k!2/⌊ard⌊lg(k)(x;·)/⌊ard⌊l2, withGk(x)as in (2.6). Applying the... | https://arxiv.org/abs/2504.05819v1 |
J≍(logn)D0, (4.3) K=⌊D1(logn)/loglogn⌋, (4.4) with two constants D0,D1>0 which remain to be determined. Reconsidering Theorem 3.1 forβ=K, we deduce that exp/parenleftbig 8c1·(K−1)/parenrightbig ·J(8c1+2)·(K−1)·/summationdisplay j>J/a\}⌊ra⌋ketle{tϕj,ΓNϕj/a\}⌊ra⌋ketri}ht ≤CΓ,2·exp/braceleftbig 8c1·(K−1)+(8c1+2)·(K−1)·(lo... | https://arxiv.org/abs/2504.05819v1 |
y;X∗,1 1,[J]/parenrightbig dy ≥1 r(1−1/J)/integraldisplayr(1+1/J) r(1−1/J)/parenleftbig log|1−y/r|/parenrightbig ·pJ,δ/parenleftbig y;X∗,1 1,[J]/parenrightbig ydy+ log(1/J) =1 1−1/J/integraldisplay1+1/J 1−1/J/parenleftbig log|1−z|/parenrightbig ·pJ,δ/parenleftbig rz;X∗,1 1,[J]/parenrightbig rzdz−logJ ≥2c1J 1−1/J·1 J·/p... | https://arxiv.org/abs/2504.05819v1 |
≤n−1u0/(1−δ2). (5.7) 14 Combining (5.7) with the inequality e† 0(nMn+S)−1e0≤n−1·u0, completes the proof. /square Proof of Lemma 3.3 : We have G†SG=/summationdisplay kG2 k(x)//parenleftbigg|k| k1,...,kJ/parenrightbigg ≤K−1/summationdisplay k=0/⌊ard⌊lg(k)(x;·)/⌊ard⌊l21 k!2Jk/summationdisplay |k|=k/parenleftbiggk k1,...,k... | https://arxiv.org/abs/2504.05819v1 |
Deep Spatio-temporal Point Processes: Advances and New Directions Xiuyuan Cheng1, Zheng Dong2, and Yao Xie2 1Department of Mathematics, Duke University, Durham, U.S.A., 27708; email: xiuyuan.cheng@duke.edu 2H. Milton Stewart School of Industrial and Systems Engineering, Georgia Insitute of Technology, Atlanta, U.S.A., ... | https://arxiv.org/abs/2504.06364v1 |
likelihood of related crimes occurring in surrounding areas. Understanding such patterns is important for scientific inquiry and practical applications, such as forecasting future occurrences and clarifying event-to-event causal relationships. Spatio-temporal point processes (STPPs) provide a powerful statistical frame... | https://arxiv.org/abs/2504.06364v1 |
and then discuss the family of modern deep-learning kernel-based STPP frameworks, including the model architectures, survey key results, and their pros and cons. We also examine model inference techniques, including likelihood-based and likelihood-free methods, as well as recent developments in causal discovery and unc... | https://arxiv.org/abs/2504.06364v1 |
where the magnitude or shape of event-triggering effects changes over time or depends intricately on location. Traditional exponential-decay kernels are limited in their ability to capture complex phenomena such as sudden bursts, long-tail decays, or structural inhomo- geneities in space. To address these limitations, ... | https://arxiv.org/abs/2504.06364v1 |
1 provides a more complete summary of these approaches in terms of their model formulations and key features. In this article, we focus on the kernel-based modeling paradigm, as it offers an in- terpretable framework for capturing how past events influence future occurrences, thereby facilitating an understanding of co... | https://arxiv.org/abs/2504.06364v1 |
point processes. Then the conditional intensity is modeled by λ(x) = µ(x) +X j:tj<tk x, xj , where µ(x)>0 is a time- and location-dependent baseline term, and k(x, x′) :X × X → R is a kernel that captures the influence of a past event at x′= (t′, s′) on a future event atx= (t, s), with t′< t. This framework generaliz... | https://arxiv.org/abs/2504.06364v1 |
> t′. 5. www.annualreviews.org •Deep spatio-temporal point processes 7 Deep kernel: Main idea The main idea of the deep kernel point process is to represent the influence kernel using neural networks. Drawing ideas from Mercer’s decomposition, the basis functions are parameterized by neural architectures without the ne... | https://arxiv.org/abs/2504.06364v1 |
historical event embeddings or spatial displacements) are mapped into a hidden feature space through a shared multi-layer sub-network, commonly with Softplus activations. The resulting embeddings are passed to Rdistinct sub-networks, each producing one of the spatial basis functions {ϕr(x)}R r=1. Consequently, a kernel... | https://arxiv.org/abs/2504.06364v1 |
such cases, the mark vmay represent continuous or categorical characteristics of the event, such as location or type. A common approach to specifying kin the graph-based point process setting (Reinhart 2018) is given by k(t′, t, v′, v) =av,v′f(t−t′), where av,v′measures the influence of node v′onvthrough a graph-based ... | https://arxiv.org/abs/2504.06364v1 |
via the eigen-decomposition of L, paralleling the role of Fourier transforms in standard signal processing. From the algebraic viewpoint (Sandryhaila and Moura 2013, 2014), one treats the adjacency matrix Aas a shift operator and constructs graph filters as matrix polynomials in A. These two perspectives unify under th... | https://arxiv.org/abs/2504.06364v1 |
by intro- ducing a latent variable uifor each event. EM algorithm works numerically stable when the number of events is not too large; however, the EM algorithm is not scalable when the number of events is large since it needs to introduce a set of auxiliary variables for each event, scaling quadratically in n. Thus, f... | https://arxiv.org/abs/2504.06364v1 |
written as min θ−ℓ(θ) subject to −λ(t, s)≤0,∀t∈[0, T], s∈ S. To incorporate this constraint, we apply a log-barrier function (Boyd et al. 2004) that penalizes instances where λ(t, s) might approach zero. The log-barrier method preserves the linear form of λand promotes computational efficiency in evaluating the integra... | https://arxiv.org/abs/2504.06364v1 |
the j-th trajectory. The maximum likelihood estimation (MLE) for kernel recovery, while directly solving for the parameters of the kernel representation, can be interpreted as solving the following variational problem over kernels k: max k∈Kℓ[k]:=1 MMX j=1Z Xlogλj[k](x)dNj(x)−Z Xλj[k](x)dx , 10. where K ⊂C0(X ×X ) is... | https://arxiv.org/abs/2504.06364v1 |
population level L2loss in Equation 13. is equal to ERT 0R S λ(t, s)−λ∗(t, s)2ds dt, up to an additive constant indepen- dent from model intensity λ(t, s). By definition and the property of conditional expectation, LLS=EZT 0Z Sλ(t, s)2ds dt−EZT 0Z S2λ(t, s)E dN(t, s) Ht =EZT 0Z Sλ(t, s)2ds dt−ZT 0Z S2λ(t, s)dN(t, ... | https://arxiv.org/abs/2504.06364v1 |
data modeling and contagious dynamics modeling of crime incidents , as demon- strated in works such as Mohler et al. (2011), Mohler (2013), Zhu and Xie (2022), including studies on the impact of urban environments with street-network topology constraints and landmarks (Dong et al. 2024), as well as the effect of spatia... | https://arxiv.org/abs/2504.06364v1 |
these predictions with the observed final event in each sequence, thereby quantifying the accuracy of the model’s time and location forecasts. The results are shown in Table 2:DNSK provides more accurate predictions than other alternatives with higher event log-likelihood. 6.2. Atlanta police reports: Spatiotemporal da... | https://arxiv.org/abs/2504.06364v1 |
a 24-hour period. Event times are measured in hours, with the average sequence length being 15.6 events. For such a problem, we estimate the kernel to ˆk, using data; the meaning of the learning kernel is that it captures when the time gets close to the sepsis onset, how the different medical variables influence each o... | https://arxiv.org/abs/2504.06364v1 |
here—as well as most existing work—focuses on modeling the influence of past events as being additive. Ex- tensions to multiplicative influence effects (Duval et al. 2022) or other types of interactions (Perry and Wolfe 2013) represent promising directions for future research. Another important topic is uncertainty qua... | https://arxiv.org/abs/2504.06364v1 |
(2016). Recurrent marked temporal point processes: Embedding event history to vector. In Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining , pages 1555–1564. www.annualreviews.org •Deep spatio-temporal point processes 21 Duval, C., Lu¸ con, E., and Pouzat, C. (2022). Int... | https://arxiv.org/abs/2504.06364v1 |
H. (2024). Beyond point prediction: Score matching-based pseudolikelihood estimation of neural marked spatio-temporal point process. In Forty-first International Conference on Machine Learning . L¨ udke, D., Biloˇ s, M., Shchur, O., Lienen, M., and G¨ unnemann, S. (2023). Add and thin: Diffusion for temporal point proc... | https://arxiv.org/abs/2504.06364v1 |
process with monotonic nets. Perry, P. O. and Wolfe, P. J. (2013). Point process modelling for directed interaction networks. Journal of the Royal Statistical Society Series B: Statistical Methodology , 75(5):821–849. Reinhart, A. (2018). A review of self-exciting spatio-temporal point processes and their applications.... | https://arxiv.org/abs/2504.06364v1 |
derangements via continuous-time hawkes processes. In Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining , pages 2536–2546. Wu, W., Liu, H., Zhang, X., Liu, Y., and Zha, H. (2020a). Modeling event propagation via graph biased temporal point process. IEEE Transactions on Neural Networks... | https://arxiv.org/abs/2504.06364v1 |
Bounds in Wasserstein Distance for Locally Stationary Functional Time Series Jan Nino G. Tinio1,2, Mokhtar Z. Alaya1, and Salim Bouzebda1 1Universit ´e de Technologie de Compi `egne, Laboratoire de Math ´ematiques Appliqu ´ees de Compi `egne, CS 60 319 - 60 203 Compi `egne Cedex 2Department of Mathematics, Caraga State... | https://arxiv.org/abs/2504.06453v1 |
beneficial for modeling functional data. Many time series models, commonly observed in various physical phenomena and economic data, are non-stationary [48,65,16]. Conventional approaches become inappropriate when the (weak) stationarity assumption is violated. Specifically, global climate changes in meteorology affect... | https://arxiv.org/abs/2504.06453v1 |
transportation of multiple items along geodesic curves or straight paths. Among various metrics derived from OT, Wasserstein distance stands out for its robustness and versatility in comparing probability distributions [70, 50, 66]. The landscape of LSTS analysis is enriched by parametric and nonparametric approaches, ... | https://arxiv.org/abs/2504.06453v1 |
Xt,Tbelongs to a semi-metric space Hwith a semi-metric D(·,·). The semi-metric space Hcan be Banach or Hilbert spaces with norm ∥ · ∥. We consider the following regression estimation problem: Yt,T=m⋆ t T, Xt,T +εt,T,for all t= 1, . . . , T, (1) where {εt,T}t∈Zis a sequence of independent and identically distributed (i... | https://arxiv.org/abs/2504.06453v1 |
s∈I\N∥f(s)∥E, forp=∞. 2.2 Wasserstein distance LetPr(R)be the set of Borel probability measures in Rhaving finite r-th moment (r≥1), i.e., Pr(R) ={µ∈ P(R) :Z R|x|rµ(dx)<∞}. Given probability measures µ, ν∈ P r(R), we calculate the distance between them using the rth- Wasserstein distance, Wr(µ, ν), as follows Wr(µ, ν) ... | https://arxiv.org/abs/2504.06453v1 |
several forms of ϕ(r)that can also be found in [37, 17]. Fractional Brownian Motion. Considering the space C([0,1],R)with the supremum norm and its Cameron-Martin associated space F=C([0,1],R)CM. Using Theorems 3.1 and 4.6 in [ 49], for 0< η < 2, we have ∀x∈ F, C′ xer−2/η≤P[ζFBM∈B(x, r)]≤Cxer−2/η, 6 where ζFBMis the us... | https://arxiv.org/abs/2504.06453v1 |
a “just right” assumption in analyzing weakly dependent sequences [ 69]. There are different forms of β-mixing, such as exponentially β-mixing β(k) =O e−γk , forγ > 0, and arithmetically β-mixing β(k) =O k−γ [41]. Numerous common time series models, such as autoregressive moving average (ARMA) models [ 54], general... | https://arxiv.org/abs/2504.06453v1 |
bounded rate of change and is essential in obtaining upper bounds. First condition in (9) is a normalization, ensuring that the kernel can be interpreted as a probability density function. We assume that K2(·)is compactly supported in [0,1]; that is, it is a kernel of type II [ 41]. Second condition implies that K1(·)i... | https://arxiv.org/abs/2504.06453v1 |
in Wasserstein distance We establish the convergence rate of NW estimator ˆπt(·|x)wrt the Wasserstein distance. Theorem 1 below generalizes the convergence results in [66] to the functional or infinite-dimensional setting. Theorem 1. Suppose Assumptions 1 - 7 are satisfied and define Ih= [C1h,1−C1h]. Then sup x∈H,t T∈I... | https://arxiv.org/abs/2504.06453v1 |
the bias term can be obtained if we assume that F⋆ ·(·)is twice differentiable and satisfies the H ¨older condition. Proposition 2. Suppose Xt(u)is a fractal-type process and Assumptions 1 - 7 are satisfied. Let the bandwidth be chosen to be h=O(T−ξ), and the small ball probability take the form ϕ(h) =hτ0, where 0< ξ <... | https://arxiv.org/abs/2504.06453v1 |
aJ×Jmatrix with entries At/T(i, j)that are mutually independent zero-mean Gaussian random variables with variancet Ti6+ (1−t T)e−j−iand ∥A∥∞= sup∥x∥≤1∥Ax∥is a Schatten ∞-norm. Figure 1 shows the plot of Xt,T(τ)forT= 100 . This example was also used in [1]. (a)Xt,T(τ)for all tand some τ (b)Xt,T(τ)atτgiven some t (c)Xt,T... | https://arxiv.org/abs/2504.06453v1 |
from the true distribution. Remarkably, the largest sample size, T= 10000 , consistently achieves the minimum expected Wasserstein distance. This behavior is consistent across both processes under investigation. /uni00000013/uni00000011/uni00000017/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000017/uni0000001... | https://arxiv.org/abs/2504.06453v1 |
Yt,238(j))}t=1,...,238where j∈ {1, . . . , 60}. The behavior of the response variable is plotted in Figure 6c. We create copies of these datasets using the same method, Algorithm 2 used in [ 66], that relies on Gaussian smoothed procedure [ 56]. For a chosen jth continuous sample curve, we add Zt,T∼ N(0, σ2)to each dat... | https://arxiv.org/abs/2504.06453v1 |
a local weighting function with bandwidth g, analogous to h. Another possible extension is amending the NW estimator to handle missing data. While expanding our results to encompass functional ergodic data would be highly valuable, it requires substantial mathematical advancements and lies beyond the current scope of t... | https://arxiv.org/abs/2504.06453v1 |
Yt,T|Xt,T=xwith conditional CDF F⋆ t(y|x) =P[Yt,T≤y|Xt,T=x]. Observe that, by the definition of W1given in (4), E[W1(ˆπt(·|x), π⋆ t(·|x))] =EZ ˆFt(y|x)−F⋆ t(y|x) dy =Z E ˆFt(y|x)−F⋆ t(y|x) dy, using Fubini’s theorem. Observe that, using (6) and (7), 21 ˆFt(y|x)−F⋆ t(y|x) =PT a=1Kh,1 t T−a T Kh,2(D(x, X a,T))1Ya,T... | https://arxiv.org/abs/2504.06453v1 |
a a T =E K2 D(x, X a,T) h −K2 D x, X a a T h ≤L2 hE D(x, X a,T)−D x, X a a T ≤L2 hE D Xa,T, Xa a T ≤L2 hE 1 TUt,T a T ≤L2CU Th. (ii) We have E Kh,2(D(x, X a,T)) =E Kh,2(D(x, X a,T))−Kh,2 D x, X a a T +Kh,2 D x, X a a T =E Kh,2(D(x, X a,T))−Kh,2 D x, X a a T +E Kh,2 D x... | https://arxiv.org/abs/2504.06453v1 |
T−a T Kh,2 D x, X a a T 26 =1 Thϕ(h)TX a=1Kh,1 t T−a T E Kh,2 D x, X a a T . Using equation (4.3) in [41], we have E[eJt,T(t T, x)] =1 Thϕ(h)TX a=1Kh,1 t T−a T E 1(D(x,Xa(a T)))≤h =1 Thϕ(h)TX a=1Kh,1 t T−a T Ft/T(h;x) ≥1 ϕ(h)1 ThTX a=1Kh,1 t T−a T | {z } O(1)cdϕ(h)ψ(x)(using Assumption 3) ∼ψ(x)>0, w... | https://arxiv.org/abs/2504.06453v1 |
1, we have ΣΛ 1=1 (Thϕ(h))2vT−1X l=0l(rT+sT)+rTX a=l(rT+sT)+1K2 h,1 t T−a T E Z2 a,t,T =1 (Thϕ(h))2vT−1X l=0l(rT+sT)+rTX a=l(rT+sT)+1K2 h,1 t T−a T Eh K2 h,2(D(x, X a,T))(1Ya,T≤y−F⋆ t(y|x))2i . By Proposition 3. iii, we have Kh,1 t T−a T Eh K2 h,2(D(x, X a,T))(1Ya,T≤y−F⋆ t(y|x))2i ≤2C2Kh,1 t T−a T Eh Kh,2(D(x, X ... | https://arxiv.org/abs/2504.06453v1 |
Kh,2(D(x, X λ+n1,T))(1Yλ+n1,T≤y−F⋆ t(y|x))i ×Eh Kh,2(D(x, X λ+n2,T))(1Yλ+n2,T≤y−F⋆ t(y|x))i . By Proposition 3. iii, fori= 1,2,Kh,1 t T−λ+ni T E Kh,2(D(x, X λ+ni,T))(1Yλ+ni,T≤y−F⋆ t(y|x)) ≲ Kh,1 t T−λ+ni T 1 T+hϕ(h) , then ΣΛ 22≲1 (Thϕ(h))21 T+hϕ(h)2vT−1X l=0rTX n1=1 |n1−n2|>0rTX n2=1Kh,1 t T−λ+n1 T Kh,1 t T−λ+... | https://arxiv.org/abs/2504.06453v1 |
Control of the small blocks. Next, we deal with the small blocks. See that E Π2 t,T =EhvT−1X l=0Π2 l,t,T+vT−1X l=0 l̸=l′vT−1X l′=0Πl,t,TΠl′,t,Ti =Eh1 (Thϕ(h))2vT−1X l=0(l+1)(rT+sT)X a=l(rT+sT)+rT+1Kh,1 t T−a T Za,t,T2i +1 (Thϕ(h))2vT−1X l=0 l̸=l′vT−1X l′=0(l+1)(rT+sT)X a=l(rT+sT)+rT+1(l′+1)(rT+sT)X b=l′(rT+sT)+rT+... | https://arxiv.org/abs/2504.06453v1 |
T−λ+n2 T ≤C1 Thϕ2(h)1 T+hϕ(h)21 ThTX a=1Kh,1 t T−a T | {z } O(1) ≲1 Thϕ2(h)1 T+hϕ(h)2 ≲1 Thϕ2(h)1 T2+h2ϕ2(h) ≲1 T3hϕ2(h)+h T ≲1 Thϕ2(h). (33) Step 2.3. Control of ΣΠ 3.Now, let us deal with ΣΠ 3. ΣΠ 3=1 (Thϕ(h))2vT−1X l=0 l̸=l′vT−1X l′=0(l+1)(rT+sT)X a=l(rT+sT)+rT+1(l′+1)(rT+sT)X b=l′(rT+sT)+rT+1Kh,1 t T−a T K... | https://arxiv.org/abs/2504.06453v1 |
Za,t,TZb,t,T =1 (Thϕ(h))2TX a=vT(rT+sT)+1K2 h,1 t T−a T E Z2 a,t,T +1 (Thϕ(h))2TX a=vT(rT+sT)+1 a̸=bTX b=vT(rT+sT)+1Kh,1 t T−a T Kh,1 t T−b T Cov Za,t,T, Zb,t,T +1 (Thϕ(h))2TX a=vT(rT+sT)+1 a̸=bTX b=vT(rT+sT)+1Kh,1 t T−a T Kh,1 t T−b T E Za,t,T E Zb,t,T =:ΣΞ 1+ΣΞ 2+ΣΞ 3. Step 3.1. Control of ΣΞ 1.Consider... | https://arxiv.org/abs/2504.06453v1 |
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the local Gaussian correla- tion. Stat. Comput. , 28(2):303–321, 2018. (Cited on page: 10.) [58] M. Peligrad. Some remarks on coupling of dependent random variables. Statistics and Probability Letters , 60(2):201–209, 2002. (Cited on page: 8.) [59] G. Peyr ´e and M. Cuturi. Computational optimal transport. Foundations ... | https://arxiv.org/abs/2504.06453v1 |
Sparsified-Learning for Heavy-Tailed Locally Stationary Processes Yingjie Wang1*, Mokhtar Z. Alaya2, Salim Bouzebda2, and Xinsheng Liu3 1School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 2Universit´ e de technologie de Compi` egne, LMAC EA 2222, Compi` egne, France 3State Key Laboratory... | https://arxiv.org/abs/2504.06477v1 |
aims to develop a new method to solve the challenges posed by heavy-tailed and locally stationary behavior in time series data. Using sparsity techniques to deal with heavy-tailed behaviors, this paper aims to significantly improve the efficiency and accuracy of modeling these complex data structures, thereby advancing... | https://arxiv.org/abs/2504.06477v1 |
estimation of linear regression coefficients when covariates and noises are sampled from heavy-tailed distributions. Contributions. In this paper, we propose a novel approach for sparse learning specifically designed to handle heavy-tailed locally stationary process data. We incorporate suitable penalty functions to pr... | https://arxiv.org/abs/2504.06477v1 |
This concept is essential when dealing with time series data that exhibit non-stationary behavior. 2.1 Locally stationary processes We consider non-stationary processes with dynamics that change slowly over time and may thus behave as stationary at a local level. For example, consider a continuous function m: [0,1]→Ran... | https://arxiv.org/abs/2504.06477v1 |
heavier tail than any exponential distribution (Nair et al., 2022). In this section, we present two types of tail distribution functions. Let us start with the definition of the tail-capturing distribution. 5 Definition 3 (Tail-capturing distribution) .LetI:R→R+denote an increasing and continuous function with the prop... | https://arxiv.org/abs/2504.06477v1 |
ηyield distributions with heavier tails. 3 Sparse penalized estimation procedure We set Mto be the additive data-dependent hypothesis space defined by M=n mθ(u, x) =TX r=1dX j=1θr,jKh,1 u−r T Kh,2(xj−Xj r,T)o , where θ= (θ⊤ 1•, . . . , θ⊤ T•)⊤= (θ1,1, . . . , θ 1,d),(θ2,1, . . . , θ 2,d), . . . , (θT,1, . . . , θ T,... | https://arxiv.org/abs/2504.06477v1 |
( 4 ) can be written as follows ˆθ= arg min θ∈RTd RT(θ) +λΩ(θ) . (5) 8 We provide bounds for the generalization error R( ˆm, m⋆) =Eh1 TTX t=1 ˆm t T, Xt,T −m⋆ t T, Xt,T2i . Remark 2. For the weighted total variation, the estimator in ( 5 ) follows ˆθ= arg min θ∈RTd RT(θ) + Ω λ(θ) . (6) Block sparsity For all θ∈RT... | https://arxiv.org/abs/2504.06477v1 |
the (asymmetric) triangle and quadratic kernels (Silverman, 1986, Vapnik, 2000). 4 Concentration inequalities for heavy-tailed LSP We propose concentration inequalities for locally stationary β-mixing sub-Weibull random variables and regularly varying random variables. For the noise {εt,T}T t=1and the kernel function K... | https://arxiv.org/abs/2504.06477v1 |
4CK,L(2T+1))) γTh+ 2 exp −T7/2−4d1(γh)9/4−2d1 9(4CK,L(2T+ 1))9/4−2d1 , where d1∈(ϑ 3(1−ϑ)+1 4,1−2ϑ 2(1−ϑ)+1 8),CK=CK1CK2,φ >0and the constant CK,Ldepend on kernel bound and Lipschiz constant. 5 Oracle inequalities We provide the non-asymptotic oracle inequalities relating R(ˆm, m⋆) and R(mθ, m⋆). Here, R(mθ, m⋆) repr... | https://arxiv.org/abs/2504.06477v1 |
(Lasso penalization) .Let Assumptions 1-3 hold, {εt,T}T t=1follows the regularly varying heavy-tailed with index η2>3η1−1 η1−1and bounded slowly varying function L(·)and Ω(θ)is the Lasso penalization. Let 0< ϑ <(η1−1)(η2−1)−2η1 1+(2η1−1)η2, assume the sample size satisfies T > dc(2+1/η1−2d1) (clogd)(d1−1/η1)(η2−1)+1 ... | https://arxiv.org/abs/2504.06477v1 |
Lasso Weighted TV Bandwidth h=O(T−ξ) 0< ξ < 1/2 0< ξ < 1/2 0< ξ < ϑ <(η1−1)(η2−1)−2η1 1+(2η1−1)η20< ξ < ϑ <(η1−1)(η2−1)−2η1 1+(2η1−1)η2 Sample Size T≥c(logd)2 η−1T≥c(logd)2 η−1T > dc(2+1/η1−2d1) (clogd)(d1−1/η1)(η2−1)+1 1 (η2+3)d1−(η2+1)/η1−3T > dc(2+1/η1−2d1) (clogd)(d1−1/η1)(η2−1)+1 1 (η2+3)d1−(η2+1)/η1−3 Penalty... | https://arxiv.org/abs/2504.06477v1 |
guarantees reliability as dgrows. 5.2.2 Regularly varying heavy-tailed Theorem 7 (Lasso penalization) .Let Assumptions 1-3 and Assumption 4-(i) hold, κ(K, J(θ))> 0,{εt,T}T t=1follows the regularly varying heavy-tailed with index η2>3η1−1 η1−1and bounded slowly varying function L(·)andΩ(θ)is the Lasso penalization. Let ... | https://arxiv.org/abs/2504.06477v1 |
1 : oracle inequality for sub-Weibull distribution with Lasso By the minimizing property of θ, it follows that 1 T∥Y−Kˆθ∥2 2+λ∥ˆθ∥1≤1 T∥Y−Kθ∥2 2+λ∥θ∥1, which, using that Yt,T=m⋆ t T, Xt,T +εt,T,t= 1, . . . , T , yields 1 T∥M⋆+ε−Kˆθ∥2 2+λ∥ˆθ∥1≤1 T∥M⋆+ε−Kθ∥2 2+λ∥θ∥1, where M⋆= m⋆ 1 T, X1,T ,···, m⋆ 1, XT,T⊤∈RTandε=... | https://arxiv.org/abs/2504.06477v1 |
the minimizing property of θ, it follows that 1 T∥M⋆+ε−Kˆθ∥2 2+∥ˆθ∥TV,λ≤1 T∥M⋆+ε−Kθ∥2 2+∥θ∥TV,λ. Equivalently, we have 1 T∥M⋆−Kˆθ∥2 2+1 T∥ε∥2 2+2 T⟨M⋆−Kˆθ,ε⟩+∥ˆθ∥TV,λ ≤1 T∥M⋆−Kθ∥2 2+1 T∥ε∥2 2+2 T⟨M⋆−Kθ,ε⟩+∥θ∥TV,λ, we have 1 T∥M⋆−Kˆθ∥2 2≤1 T∥M⋆−Kθ∥2 2+2 T⟨K(ˆθ−θ),ε⟩+∥θ∥TV,λ− ∥ˆθ∥TV,λ. Define the block diagonal matrix D=... | https://arxiv.org/abs/2504.06477v1 |
1))2d1−1/η1−2 9T4d1−2/η1−3(λh)2d1−1/η1−2i ≃TdhL(TlogT) (TlogT)η2 +C1 T(d1−1/η1)(η2−1)λ1−(d1−1/η1)(η2−1)L((λT)d1−1/η1) +C2exp(−φλT) λ+ exp −C3T1+1/η1−2d1λ2+1/η1−2d1 +C4 T(d1−1/η1)(η2−1)(λh)1+(d1−1/η1)(η2−1)L((λTh)d1−1/η1) +C5exp(−φλTh ) λh+ exp −C6T1+1/η1−2d1(λh)2+1/η1−2d1i ≃TdhL(TlogT) (TlogT)η2+C7 T(d1−1/η1)(η2−1... | https://arxiv.org/abs/2504.06477v1 |
that R( ˆm, m⋆)≤inf θ∈RTd R(mθ, m⋆) + 2λ∥θ∥1 . A.5 Proof of Theorem 4: oracle inequality for regular varying heavy-tailed distribution with weighted total variation penalization Consider {εt,T}T t=1follows the regularly varying heavy-tailed with bounded slowly varying function L(·), we get the following result similar... | https://arxiv.org/abs/2504.06477v1 |
=λTX r=1⟨(−sign(θr•))Jr(θ),(ˆθr•−θr•)Jr(θ)⟩ −λTX r=1⟨(sign( θr•))J∁r(θ),(ˆθr•−θr•)J∁r(θ)⟩. Using a triangle inequality and the fact that ⟨sign(x), x⟩=∥x∥1, imply that −λ⟨g,ˆθ−θ⟩ ≤λTX r=1∥(ˆθr•−θr•)Jr(θ)∥1−λTX r=1∥(ˆθr•−θr•)J∁r(θ)∥1 (17) Note on Uλ Twith equation ( 16 ) and ( 17 ), we have λ 2TX r=1∥ˆθr•−θr•∥1+λTX r=1∥(... | https://arxiv.org/abs/2504.06477v1 |
we have 1 2TX r=1∥ˆθr•−θr•∥TV,λ+TX r=1∥(ˆθr•−θr•)Jr(θ)∥TV,λ−TX r=1∥(ˆθr•−θr•)J∁r(θ)∥TV,λ≥0, i.e., 3TX r=1∥(ˆθr•−θr•)Jr(θ)∥TV,λ≥TX r=1∥(ˆθr•−θr•)J∁r(θ)∥TV,λ, it also means that 3TX r=1dX j=1λj∥Dr(ˆθr•−θr•)Jr∥1≥TX r=1dX j=1λj∥Dr(ˆθr•−θr•)J∁r∥1. By Assumption 4-(ii) and Lemma 10, we have ˆθ−θ∈STV,JandD(ˆθ−θ)∈S1,J. 34 The ... | https://arxiv.org/abs/2504.06477v1 |
TκV,γ(J(θ))κ(K, J(θ))≤√ 288J⋆max r=1,...,T∥(λj)Jr(θ)∥∞ √ Tκ(K, J(θ)), (24) From the definition of κ(K, J(θ)) in Assumption 4-(ii), ∥∆J∥2≤1 κ(K, J(θ))∥K∆∥2√ T≤1 κ2(K, J(θ))κV,γ(J(θ)), 38 then ∥∆∥2=∥∆J∥2+∥∆J∁∥2≤q ∥∆J∁∥1∥∆J∁∥∞+∥∆J∥2. From ∆ ∈SJ,∥∆J∁∥1≤3∥∆J∥1. Since ∆ Jspans the largest coordinates of ∆ in absolute value, ... | https://arxiv.org/abs/2504.06477v1 |
>τT ≥CKbT ≤P 1 TCKTX t=1εt,T1 εt,T >τT ≥CKbT ≤P εt,T1 εt,T >τT ≥bT ≤P εt,T > τT 1(τT>bT)+P εt,T > τT 1(τT≤bT) ≤P εt,T > τT,for some 1 ≤t≤T ≤exp(− τT/Cεη2) ≤exp −(TlogT Cε)η2 .(27) 40 We now turn to the analysis of Φ 1 r T, Xj r,T . From Assumptiom 1, {Xt,T}T t=1is locally stationary sequence, whic... | https://arxiv.org/abs/2504.06477v1 |
CKp logT/T andT >4, we further get that P( Φt T, Xj t,T >2γ)≤exp −(TlogT Cε)η2 +Texp −(γT)η (2CKCε)ηC1 + exp −γ2T (2CKCε)2C2 +Texp −(γT2h)η (2CK,L(2T+ 1)Cε)ηC1 + exp −(wh)2T3 (2CK,L(2T+ 1)Cε)2C2 . Then, let h=O(T−ξ) with 0 < ξ <1 2, forγ >2CKp logT/T andT >4, we have P( Φr T, Xj r,T > γ)≤exp −(TlogT Cε)η... | https://arxiv.org/abs/2504.06477v1 |
ΣN∗ T I2j ≤BN, by the Hoeffding’s inequality in Lemma 4, we obtain the bound P sup l≤T X j≤[l/B]ΣN∗ T I2j ≥ϱ ≤exp −ϱ2 2µB2N2 . (32) Similarly, we also obtain P sup l≤T X j≤[l/B]ΣN∗ T I2j−1 ≥ϱ ≤exp(−ϱ2 2µB2N2). (33) 45 From (31)-(33), we have P sup l≤T Σl ≥6ϱ! ≤2T ϱcK η2−1N1−η2L(N) +2µBτ(B) ϱ+ 2 exp −ϱ2... | https://arxiv.org/abs/2504.06477v1 |
u∈[0,1] and CUis a constant, we can infer that Kh,1 t T−r T Kh,2(Xj t,T−Xj r,T)−Kh,1 u−r T Kh,2(Xj t(u)−Xj r,T) = Kh,1 t T−r T Kh,2(Xj t,T−Xj r,T)−Kh,1 t T−r T Kh,2(Xj t(u)−Xj r,T) +Kh,1 t T−r T Kh,2(Xj t(u)−Xj r,T)−Kh,1 u−r T Kh,2(Xj t(u)−Xj r,T) ≤Kh,1 t T−r T Kh,2(Xj t,T−Xj r,T)−Kh,2(Xj t(u)−Xj r,T) + Kh,1 t... | https://arxiv.org/abs/2504.06477v1 |
−φ(γT2h 4CK,L(2T+1))) γTh+ 2 exp −(4CK,L(2T+ 1))2d1−1/η1−2 9T4d1−2/η1−3(γh)2d1−1/η1−2 ≤(TlogT)−η2L(TlogT) +12T(d1−1/η1)(1−η2)γ(d1−1/η1)(1−η2)−1 21−η2(4CK)(d1−1/η1)(1−η2)−1L((γT/4CK)d1−1/η1 2) +24CKexp(−φγT/ 4CK) γ+ 2 exp −1 9T2d1−1/η1−1(γ/4CK)2d1−1/η1−2 +12T2(d1−1/η1)(1−η2)−1(γh)(d1−1/η1)(1−η2)−1 21−η2(4CK,L(2T+ 1)... | https://arxiv.org/abs/2504.06477v1 |
, Λ1(R)is the class of 1-Lipschitz functions from RtoR. Lemma 4 (Hoeffding’s inequality) .LetX1, X2, . . . , X nbe independent random variables such that ai≤Xi≤bialmost surely. Consider the sum of these variables by Sn=Pn i=1Xi and its expected value by E[Sn], then Hoeffding’s inequality states that for any t≥0, P(Sn−E... | https://arxiv.org/abs/2504.06477v1 |
inverse of matrix D, i.e., V D=I, where V=diag(V1, . . . , V T) is the Td×Tdmatrix with the ( d×d) lower triangular matrix Vr, and the entries Vr s,j= 0 if s < j and Vr s,j= 1 otherwise. To prove Theorem 6 and 8, we need the following results which give a compatibility property for the matrix V. For any concatenati... | https://arxiv.org/abs/2504.06477v1 |
100(2-3):533–553. (Cited on page: 2.) Bickel, P. J., Ritov, Y., and Tsybakov, A. B. (2009). Simultaneous analysis of lasso and Dantzig selector. Ann. Statist. , 37(4):1705–1732. (Cited on page: 15.) Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1989). Regular variation , volume 27 of Encyclopedia of Mathematics an... | https://arxiv.org/abs/2504.06477v1 |
Kuznetsov, V. and Mohri, M. (2018). Theory and algorithms for forecasting time series. (Cited on page: 5.) Lecu´ e, G. and Mendelson, S. (2012). General nonexact oracle inequalities for classes with a subexponential envelope. The Annals of Statistics , 40(2):832 – 860. (Cited on page: 12.) Li, Y., Mark, B., Raskutti, G... | https://arxiv.org/abs/2504.06477v1 |
and noninvertible distribution theory. (Cited on page: 1.) Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B , 58(1):267–288. (Cited on pages: 1, 2.) Vapnik, V. N. (2000). The nature of statistical learning theory . Statistics for Engineering and Information Science. ... | https://arxiv.org/abs/2504.06477v1 |
Estimation of rates in population-age-dependent processes by means of test functions Jie Yen Fan, Kais Hamza, Fima C. Klebaner and Ziwen Zhong Abstract This paper aims to develop practical applications of the model for the highly technical measure-valued populations developed by the authors in [2]. We consider the prob... | https://arxiv.org/abs/2504.06516v1 |
boundary between statistical learning and dynamical sys- tems, in which parameters are estimated from the observed trajectory of dynamics equations. This work is the first step in developing inference by using test functions, and has wide applicability in other areas. Another advantage of our approach is the ability to... | https://arxiv.org/abs/2504.06516v1 |
rate hand gives birth with rate b. These parameters are assumed to depend on the age of the individual xas well as on the population composition A, so that h=hA(x)andb=bA(x). Conditioned on the population composition, individuals act independently. Furthermore, we assume large carrying capacity K, so that all the quant... | https://arxiv.org/abs/2504.06516v1 |
Mf twith zero mean and quadratic variation ⟨Mf,Mf⟩t=Zt 0(f2(0)b¯As+h¯Asf2,¯As)ds. (6) Estimation of rates in population-age-dependent processes by means of test functions 5 3 Estimating Equations The idea is to use the limiting evolution equation with various test functions to ex- tract information about the rates. Rea... | https://arxiv.org/abs/2504.06516v1 |
of hwith different K.K 100 1000 10000 Sample Mean 0.39848 0.39848 0.39878 Sample Variance 0.00345 0.00040 0.00004 MSE 0.00342 0.00040 0.00004 Bias -0.00152 -0.00152 -0.00122 Table 2: Summary statistics of 100 estimates of bwith different K. 123456789100.150.200.250.300.350.400.45 K (in thousands)123456789100.150.200.25... | https://arxiv.org/abs/2504.06516v1 |
(hAft,A) =n ∑ i=1hi1Bift,A =n ∑ i=1hi(1Bift,A). Further, for ft(x) =xtm,(1Bift,A) =tm(x1Bi(x),A), giving ZT 0(hAsfs,As)ds=n ∑ i=1hiZT 0sm(x1Bi(x),As)ds. Thus we obtain a system of nlinear equations for hi’s. For m=0, Estimation of rates in population-age-dependent processes by means of test functions 9 n ∑ i=1hiZT 0(... | https://arxiv.org/abs/2504.06516v1 |
0h(i) ¯As(fs1Bi,¯As)ds = (fT,¯AT)−(f0,¯A0)−ZT 0(∂xfs+∂tfs,¯As)ds. Similar to the approach considered in Section 3.3, using test functions ft(x) =xtm andft(x) =tm, form=0,1,2,..., n−1, we can recover h(i) Aandb(i) A. For example, let hA(x) =α1(1J,A)1B1(x)+α2(1J,A)1B2(x), (13) and bA(x) =γ1(1J,A)1B1(x)+γ2(1J,A)1B2(x). (1... | https://arxiv.org/abs/2504.06516v1 |
Tby their estimated quantities: ˆVf T=ZT 0(f2 s(0)ˆbT+ˆhTf2 s,¯AK s)ds, (19) where ˆbTandˆhThere denote the estimates of bandh. 4.1 Constant parameters For the case of constant parameters, recall from Section 3.1 the estimator of hand b: ˆhT=(x,¯AK 0)−(x,¯AK T)+RT 0(1,¯AK s)ds RT 0(x,¯AKs)ds, ˆbT=(1,¯AK T)−(1,¯AK 0)+ˆh... | https://arxiv.org/abs/2504.06516v1 |
T)−(1,¯AK 0)+ˆλTRt 0(1J2,¯AK s)(1,¯AK s)ds RT 0(1,¯AKs)(1J1,¯AKs)ds. Taking f(x,t) =x, (Vx T)2=λZT 0(1J2,¯As)(x2,¯As)ds. Estimation of rates in population-age-dependent processes by means of test functions 17 Replacing ¯Awith ¯AKinVx T, from (16) a confidence interval of λis obtained by solving √ K λ−ˆλT ≤cα√ λqRT 0(1J... | https://arxiv.org/abs/2504.06516v1 |
Confidence regions of hhhin one sample with K=10000 using the direct approach (top left) and the approximate approach (top right). Point estimates of (h1,h2)in 100 samples for K=10000 (bottom). 4.4 Parameters depend on population and age The general case where parameters handbdepend on both Aandxcan be dealt with in a ... | https://arxiv.org/abs/2504.06516v1 |
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