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19.61 0.406 0.526 0.394 1.054 9.38 7.884 25000 45.026 28.7 28.614 28.7 28.614 35.684 0.0 0.014 0.008 0.948 12.724 10.0 37 Table 8: Estimated L2risks for densities 1-8(where applicable). Density n Wand AIC BIC BR Knuth SC RIH RMG-B RMG-R TS L2CV KLCV 1 50 0.165 0.194 0.195 0.19 0.191 0.189 0.202 0.32 0.217 0.234 0.209 0... | https://arxiv.org/abs/2505.22034v1 |
0.072 0.077 25000 0.043 0.041 0.053 0.041 0.046 0.042 0.05 0.05 0.05 0.031 0.043 0.045 10 50 0.145 0.15 0.136 0.137 0.138 0.142 0.136 0.176 0.138 0.205 0.142 0.162 200 0.131 0.116 0.128 0.125 0.119 0.111 0.129 0.137 0.13 0.165 0.132 0.134 1000 0.123 0.071 0.083 0.071 0.071 0.069 0.086 0.109 0.09 0.082 0.083 0.085 5000 ... | https://arxiv.org/abs/2505.22034v1 |
rate: A practical and powerful approach to multiple testing. Journal of the Royal Statistical Society: Series B (Methodological) , 57:289–300, 1995. doi:10.1111/j.2517-6161.1995.tb02031.x. L. Birgé and Y. Rozenholc. How many bins should be put in a regular histogram. ESAIM: Probability and Statistics, 10:24–45, 2006. d... | https://arxiv.org/abs/2505.22034v1 |
2020. doi:10.1093/biomet/asz081. J.S.MarronandM.P.Wand. Exactmeanintegratedsquarederror. TheAnnalsofStatistics ,20:712–736,1992. doi:10.1214/aos/1176348653. P.Massart. ConcentrationInequalitiesandModelSelection: Ecoled’EtédeProbabilitésdeSaint-FlourXXXIII-2003 . Springer Berlin, Heidelberg, 2007. doi:10.1007/978-3-540-... | https://arxiv.org/abs/2505.22034v1 |
arXiv:2505.22206v1 [math.ST] 28 May 2025Directional ρ-coefficients Enrique de Amoa, David Garc´ ıa-Fern´ andezb,∗, Manuel ´Ubeda-Floresa aDepartment of Mathematics, University of Almer´ ıa, 04120 Almer´ ıa, Spain edeamo@ual.es, mubeda@ual.es bResearch Group of Theory of Copulas and Applications, University of Almer´ ıa... | https://arxiv.org/abs/2505.22206v1 |
theoretical implications, alongside providing a more general response to the conjecture presented in [14] and correct- ing [7, Equation (10)]. In Section 4, we present nonparametric rank-based estimators with the objective of estimating the values of the directional ρ-coefficients from the observed data in a sample. Po... | https://arxiv.org/abs/2505.22206v1 |
it by ρ(C), and its expression is given by ρ(C) = 12Z I2C(u, v)dudv−3. Consequently, when analyzing their properties, we can assume that the random variables XandYare uniformly distributed on I. For a detailed exploration of their characteristics, refer to [13] and the sources cited therein. In this paper, we will focu... | https://arxiv.org/abs/2505.22206v1 |
, n .Xis said to be smaller than Yin the positive dependence order according to the direction α, denoted by X≤PD(α)Y, if, for every x∈Rn, we have P"n\ i=1(αiXi> xi)# ≤P"n\ i=1(αiYi> xi)# . 3 Multivariate directional ρ-coefficients The directional ρ-coefficients were first introduced in [14] order to measure directional... | https://arxiv.org/abs/2505.22206v1 |
families of n- copulas. Example 1 (Directional ρ-coefficients for Π n).IfC= Π nthen we have ρα n(Πn) = 0 since, in this case, P[Tn i=1(αiXi> xi)] =Qn i=1P[αiXi> xi]. Example 2 (Directional ρ-coefficients for Mn).For the minimum copula C=Mn, we have X1= X2=···=Xnin distribution. Since the marginals of the copula are uni... | https://arxiv.org/abs/2505.22206v1 |
karises from the structure of the joint probability involving minj∈Juj−max i∈Iui, which effectively reverses the roles of the minimum and maximum compared to the even kcase where we had mini∈Iui−max j∈Juj. However, the integral of max(0 , a−b)is the same as the integral of max(0 , b−a)over the appropriate domains, lead... | https://arxiv.org/abs/2505.22206v1 |
(10)], as well. 8 Theorem 4. LetXbe an n-dimensional random vector whose marginals are uniform on [0,1], letCbe its associated n-copula and α∈Rnsuch that αi∈ {− 1,1}for all i= 1, . . . , n . Let I⊆ {1,2, . . . , n }such thatαi=−1ifi∈Iandαi= 1ifi∈J={1,2, . . . , n } \I. Then ρα n(C) =2n(n+ 1) 2n−(n+ 1)X S⊆J(−1)|S|2|I|+|... | https://arxiv.org/abs/2505.22206v1 |
already warn of the lack of good properties of the estimators resulting from this method; for example, these may fall outside the parametric space ([16, Example 1]). That is why we have decided that the estimators, bρα n, proposed in this paper are based on ranks. Inspired by [7], where a nonparametric estimator for tr... | https://arxiv.org/abs/2505.22206v1 |
l∈I∪S(1−Ul)# . This identity also applies to empirical averages for a sample U1j, . . . , U djforj= 1, . . . , n : 1 nnX j=1 Y i∈IUij! Y i∈J(1−Uij)! =X S⊆J(−1)|S| 1 nnX j=1Y l∈I∪S(1−Ulj) . Proof. Consider the product termQ i∈IUiQ i∈J(1−Ui). We can rewrite each Uifori∈Ias 1−(1−Ui): Y i∈IUi=Y i∈I(1−(1−Ui)). Applying ... | https://arxiv.org/abs/2505.22206v1 |
1 nnX j=1Y l∈I∪S(1−Ulj)−1 2|I|+|S| . This identity allows us to decompose Kemp αinto a sum of terms related to products of (1 −Ulj). From the definition of bρ− K, we can isolate the parenthesized term: 1 nnX j=1Y i∈K(1−Uij)−1 2|K| =2|K|−(|K|+ 1) 2|K|(|K|+ 1)bρ− K. Substituting this, and by setting K=I∪S, we have... | https://arxiv.org/abs/2505.22206v1 |
dimension dand, therefore, guarantees the weak convergence of our process. Theorem 7. LetHβ,αbe a|β|-dimensional distribution function, given by (5) with continuous marginal distributions Fi,i∈βand with Cβ,αgiven by (6), where β⊆ {1, ..., d}andα∈Rd,αi={−1,1}. Under Condition 1 on the function Cβ,α, when n−→ ∞ √n{Cβ,α,n... | https://arxiv.org/abs/2505.22206v1 |
with θ∈[0,∞), is given by Cθ(u1, . . . , u d) = dX i=1u−θ i−d+ 1!−1 θ ,∀u∈[0,1]d (see [13]). We consider two dimensions, d={3,4}, and four sample sizes n={20,50,100,500}. We have generated 1000 Monte Carlo replicates of size nfor each parameter value and for each direction. In each of these cases, we have calculated th... | https://arxiv.org/abs/2505.22206v1 |
Applications In this final section, we explore potential applications of the proposed directional dependence estimators in various fields, with a particular focus on health, climate, and rainfall studies. By leveraging these estimators, researchers can gain deeper insights into the directional dependencies that exist b... | https://arxiv.org/abs/2505.22206v1 |
values of all three variables tend to occur simultaneously, and the same with “small” values. 6 Conclusion In this paper, the authors have explored the use of copulas and directional ρ-coefficients as effective tools for measuring directional association between random variables. We have presented the generalization of... | https://arxiv.org/abs/2505.22206v1 |
with copulas . Chapman & Hall, New York. [11] M¨ uller, A., Scarsini, M. (2006). Archimedean copulae and positive dependence. J. Multivariate Anal. 93, 434–445. [12] Nelsen, R.B. (1996). Nonparametric measures of multivariate association. In: Distributions with Fixed Marginals and Related Topics , Vol. 28 (L. R¨ uschen... | https://arxiv.org/abs/2505.22206v1 |
arXiv:2505.22371v1 [stat.OT] 28 May 2025Adaptive tail index estimation: minimal assumptions and non-asymptotic guarantees Johannes Lederer∗Anne Sabourin†Mahsa Taheri‡ Abstract A notoriously difficult challenge in extreme value theory is the choice of the number k≪n, where nis the total sample size, of extreme data poin... | https://arxiv.org/abs/2505.22371v1 |
(2007a, 2008). Estimating the tail index in this context has been the subject of a wealth of works, one main reason being that estimating γalso allows to estimate high quantiles with the help of the Weissman estimator (Weissman, 1978). A particularly popular estimator for γif the Hill estimator proposed in Hill (1975) ... | https://arxiv.org/abs/2505.22371v1 |
guarantees under the minimal Condition (1.1). In addition, it achieves strong (nearly minimax, up to a power oflog log n) guarantees under the more restrictive von Mises Condition 3. A key feature of our approach is the consideration of a grid Kforkof size |K| ≪ n, which aligns with common practice among practitioners ... | https://arxiv.org/abs/2505.22371v1 |
any particular condition on the bias term except that it can be made smaller than the variance term for certain values of the tuning parameter (here, for small k). To date, these general tools have not been considered within the field of EVT. Under the minimal regular variation assumption (1.1), we propose an ‘Extreme ... | https://arxiv.org/abs/2505.22371v1 |
behind the factor γin the required error decomposition (2.1) is that the standard deviation of the Hill estimator is γ/√ kin the ideal case of a Pareto distribution. The function Bshould be seen as a bias term: the larger k, the less extreme the considered data are, the larger B. We emphasize that explicit knowledge of... | https://arxiv.org/abs/2505.22371v1 |
of k∗(δ, n), without a bias term. The following result derives immediately from the very definition of k∗, from the upper bound (2.1) and from the monotonicity requirements on the functions Vand B. Proposition 1 (Explicit error bound for the oracle) Letˆγsatisfy Condition 1. For any δ∈ (0,1),n≥1andKsatisfying Condition... | https://arxiv.org/abs/2505.22371v1 |
approaches involving chaining techniques are discussed in Remark 6. Second, we need to consider only extreme sample sizes ksuch that the error due to variance is less than 1/2. This is because our rule involves division by 1−2V(k, δK). We thus restrict the search to k≥k0(δ), where k0(δ)satisfies k0(δ)≥inf{k≥1 :V(k, δK)... | https://arxiv.org/abs/2505.22371v1 |
K. Thus, on the latter eventE, it holds that for all k∈ K ∩ { 1, . . . , k∗(δK, n)},|ˆγ(k)−γ| ≤γV(k, δK) +B(k, n, δ K). This proves the first inequality in (2.8). The second inequality in (2.8) derives immediately from the definition of the oracle k∗in (2.2). Also on E, it holds that for all k∈ Ksuch that k≤k∗(δK, n), ... | https://arxiv.org/abs/2505.22371v1 |
merely a factor ofp log|K|, meaning that for grids of logarithmic size, this difference is O(√log log n). Thus, the adaptive extreme sample size ˆkEAVis almost optimal in the sense that its associated error is bounded by a multiple of the error bound associated with the (unknown) oracle k∗up to a quasi-constant factor.... | https://arxiv.org/abs/2505.22371v1 |
if and only if there exists two functions b:R+→Randa:R+→R+such that limt→∞b(t) = 0and limt→∞a(t) =A≥0, such that for some t0≥0and all t≥t0, L(t) = a(t) expZt t0b(u) udu . (3.1) Summarizing, for ¯Fsatisfying (1.1) and Qthe quantile function of Fas above, we may write Q(t) =L(t)tγ, where Lis as in (3.1). Define ¯b(t) = s... | https://arxiv.org/abs/2505.22371v1 |
Appendix, Section A.1. Lemma 3 (Concentration of U(k+1))With probability at least at least 1−δ, 1−U(k+1)≤k+ 1 n 1 +R(k+ 1, δ) , (3.10) withR(k, δ) =p 3 log(1 /δ)/k+ 3 log(1 /δ)/k. To treat the second term of (3.9) as a bias term, we need to bound Zkfrom above with high proba- bility. This can be done using a quantile... | https://arxiv.org/abs/2505.22371v1 |
analysis also facilitates direct comparisons with existing results in the literature under comparable conditions. Our main result demonstrates that the oracle indeed attains the optimal rate, and that for ‘well chosen’ grids (in a sense that shall be made precise shortly) of size O(logn)spanning the full interval [0, n... | https://arxiv.org/abs/2505.22371v1 |
sequence kopt(n)such that kopt(n)∼λn−2ρ/(1−2ρ)is asymptotically optimal in terms of asymptotic mean squared error of the Hill estimator, and the associated asymptotic rate of convergence is nρ/(1−2ρ), meaning that n−ρ/(1−2ρ)(ˆγ(kopt(n))−γ)converges in distribution to a non degenerate limit. A lower bound derived in Hal... | https://arxiv.org/abs/2505.22371v1 |
in two steps: In Section 4.2 we establish a lower bound on k∗under Condition 3, thereby obtaining an explicit expression for the minimum sample size n0(δ)introduced in (2.5). In Section 4.3 we leverage this lower bound together with previously proved upper bounds on the oracle error involving k∗, to establish an explic... | https://arxiv.org/abs/2505.22371v1 |
sample size n0(δ)defined in (2.5)with V(k, δ)as in Theo- rem 2, is no greater than the first integer nsuch that 36 log(4 |K|/δ)≤β−1C2(ρ)γ2/(1−2ρ) log(4|K|/δ)n−2ρ/(1−2ρ)−1 . 14 4.3 Oracle and adaptive error bounds The following error bound is a consequence of the previously established lower bound of k∗(Propo- sition ... | https://arxiv.org/abs/2505.22371v1 |
follows: inspection of the proof of Theorem 3 reveals that the only 15 necessary condition on the grid is that km∗+1/km∗is not too large, where m∗is the grid index such thatk∗=km∗. Therefore, the structure of the grid matters only in the neighborhood of k∗. For large enough n,k∗is sufficiently large so that even with a... | https://arxiv.org/abs/2505.22371v1 |
tail behavior. The adaptive rules of Boucheron & Thomas (2015) and Drees & Kaufmann (1998) are imple- mentedwiththeexactsameparametersandcalibratedconstantsastheonesproposedbyBoucheron & Thomas (2015, Page 30; Equations (5.1) and (5.2)). For ˆkBTandˆkDK, we set the lower limit of the admissible range for ktoln= 30, fol... | https://arxiv.org/abs/2505.22371v1 |
with c= (eα/β )βdefined on the domain [x0,∞)where x0= exp( β/α). This distribution is referred to as the ‘Perturb’ distribution in Resnick (2007a), p. 87. In our simulation we fix α= 2andβ= 1and we denote this distribution as L2,1. This distribution has standardized Karamata representation (4.1) but it does not satisfy... | https://arxiv.org/abs/2505.22371v1 |
γ−12i ,where ˆγ(k)is the Hill es- timator for kextreme order statistics. As an illustration, Figure 1 displays the standardized root mean squared error RMSE (k) =p MSE(k)for datasets of sample size n= 10 000 generated from one representant of the 6 distribution families described above, as a function of k. The expecta... | https://arxiv.org/abs/2505.22371v1 |
100, for ˆk∈ {ˆkEAV,ˆkBT,ˆkDK}, over N= 500 experiments. n= 1 000 d.f. γ [MSE(ˆkEAV); (stderr) ×100[MSE(ˆkBT); (stderr) ×100[MSE(ˆkDK); (stderr) ×100 C2,2/30.5 01.63 ; (0.04) 04 .50; (0 .22) 07 .15; (0 .18) C2,1/20.5 02.77 ; (0.07) 07.60;(0.26) 09.15; (0 .19) S1.7 1/1.7 52.48; (0 .71) 04.62;(0.15) 06.73; (0 .21) S1.5 1... | https://arxiv.org/abs/2505.22371v1 |
+ t/σ2) . Inverting the above inequality yields that with probability greater than 1−δ, 1−U(k)≤k n+r 3σ2log(1/δ) n+3 log(1 /δ) n =k n 1 +r 3 log(1 /δ) k+3 log(1 /δ) k . □ A.2 Proof of Theorem 2 In the setting of Lemma 2, combining Lemmas 2, 3, 4 we obtain the following upper bound on the error: with probability at l... | https://arxiv.org/abs/2505.22371v1 |
/δ)>4, so that D≥(4/7)1−ρ. Combining the two latter bounds we obtain ˜C2(δ, ρ)1−2ρ 2≥c1 (21/4)1−ρ√ 2C log(4/δ)−(1−2ρ) 2, hence, ˜C2(δ, ρ)≥c2 1 2C21/(1−2ρ) ×1 (21/4)2×1 log(4/δ). Theresultfollowsfromtheabovedisplaycombinedwith(A.4)andtheassumptioninthestatement thatkm∗/km∗+1≥β−1. 23 A.5 Proof of Theorem 3 From Propo... | https://arxiv.org/abs/2505.22371v1 |
a Potter bound. The latter quantity converges to zero much slower than sρfor any ρ <0. We conclude that for the perturb distribution, we cannot have ¯b(t)< Ctρfor some ρ <0, thus Condition 3 is not satisfied. A.7 Counter-example distribution The general idea behind the proposed counter-example distribution is to start ... | https://arxiv.org/abs/2505.22371v1 |
by km=⌊mn/|K|⌋ for1≤m≤ |K|, where |K|= log n/log(β)andβ= 1.1. The results, shown in Table 4 and Table 5, indicate that our EAV estimator is largely robust to changes in the grid. d.f. γ ˆkEAV;¯kEAVˆkBT;¯kBTˆkDK;¯kDK C2,2/30.5 [0717,1051];0922 [31,0127];0066 [001,0011];0007 C2,1/20.5 [0368,1399];0595 [31,0109];0043 [001... | https://arxiv.org/abs/2505.22371v1 |
&Sabourin, A. (2025). Weak signals and heavy tails: Machine-learning meets extreme value theory. arXiv:2504.06984 . Comte, F. &Lacour, C. (2013). Anisotropic adaptive kernel deconvolution. Annales de l’IHP Probabilités et statistiques 49, 569–609. Csorgo, S. ,Deheuvels, P. &Mason, D. (1985). Kernel estimates of the tai... | https://arxiv.org/abs/2505.22371v1 |
on Artificial Intelligence and Statistics . PMLR. Lederer, J. (2022).Fundamentals of High-Dimensional Statistics: with exercises and R labs . Springer Texts in Statistics. Lepski, O. V. (1990). A problem of adaptive estimation in gaussian white noise. Teoriya Veroy- atnostei i ee Primeneniya 35-3, 459–470. Lepski, O. V... | https://arxiv.org/abs/2505.22371v1 |
arXiv:2505.22417v1 [math.ST] 28 May 2025High-Dimensional Binary Variates: Maximum Likelihood Estimation with Nonstationary Covariates and Factors Xinbing Kong1, Bin Wu∗2, and Wuyi Ye2 1Southeast University, Nanjing 211189, China 2University of Science and Technology of China, Hefei 230026 , China Abstract This paper in... | https://arxiv.org/abs/2505.22417v1 |
probit). Both xitandf0tare integrated of order one, i.e., I(1) processes. It is certainly that yitcan be extended to other types of variate like counts. We consider two cases for the single index z0it, one is nonstationary I(1) index and one is cointegrated I(0) index. In the nonstationary univariate regression sett in... | https://arxiv.org/abs/2505.22417v1 |
index. The modeling framework of this paper has a wide range of appli cations; we apply it specif- ically to jump arrival events in financial markets. We find tha t the model captures well the potential jump arrival factor, which is nonstationary. Add itionally, we find that the jump ar- rival factor effectively explains t... | https://arxiv.org/abs/2505.22417v1 |
The vector {εit= (εe′ it,εv′ t)′}are i.i.d. with mean zero and satisfies E/bardblεit/bardblι<∞for someι >8. The distribution of (εit)is absolutely continuous with respect to the Lebesgue measure and has a characteristic function ϕi(t)such that ϕi(t) =o(/bardblt/bardbl−κ)as/bardblt/bardbl → ∞ for some κ >0. (ii) Define Ui... | https://arxiv.org/abs/2505.22417v1 |
whereD∗is a diagonal matrix. We then sort the diagonal elements of Λ∗(F∗′F∗/T2)1/2Q∗in descending order to obtain ˆΛ. Similarly, we sort F∗(F∗′F∗/T2)−1/2Q∗according to the same order to obtain ˆF. Remark 1. If the covariate component β0ixitis absent, the binary probability simplifies to Ψ(λ′ 0if0t), reducing the model t... | https://arxiv.org/abs/2505.22417v1 |
=∂ ∂AvlogL(A,F),JNT,11(A,F) =∂2 ∂Av∂A′vlogL(A,F),SNT,2(A,F) =∂ ∂Fv logL(A,F), andJNT,22(A,F) =∂2 ∂Fv∂F′vlogL(A,F). The score function with respect to Ais expressed as SNT,1(A,F) =/parenleftbigg/parenleftig S(1)′ NT,1(α1,F),...,S(N)′ NT,1(αN,F)/parenrightig′/parenrightbigg N(q+r)×1. The corresponding Hessian is JNT,11... | https://arxiv.org/abs/2505.22417v1 |
Specifically, we define the following quantities: θ0i:=Q′ iα0i= (θ(1) 0i,θ(2)′ 0i)′whereθ(1) 0i=/bardblα0i/bardbl,θ(2) 0i=Q(2)′ iα0i= 0, h0it:=Q′ ig0it= (h(1) 0it,h(2)′ 0it)′whereh(1) 0it=α′ 0ig0it//bardblα0i/bardbl=z0it//bardblα0i/bardbl,h(2) 0it=Q(2)′ ig0it. In the general case, we define θi:=Q′ iαiandhit:=Q′ igit. With... | https://arxiv.org/abs/2505.22417v1 |
RK(/bardblα0i/bardblm)dm andΩf,t= limN→∞tδ N/summationtextN i=1E[K(z0it)]λ0iλ′ 0i, whereL1i(s,0) =LH1i(s,0)σH1iwithLH1i(s,0) being the local time of H1iandσH1iits variance. The asymptotic behavior of the estimator ˆftvaries with t, influenced by the Hessian matrix. Recall the notation of DT, two distinct limiting di... | https://arxiv.org/abs/2505.22417v1 |
results hold: −√ T/bracketleftig J(i) NT,11(ˆαi,ˆF)/bracketrightig−1 ,−√ T/bracketleftig J(i) NT,11(ˆαi,ˆF)/bracketrightig−1 →P¯ωθ,i 11α0iα′ 0i /bardblα0i/bardbl2, −Nt−δ/bracketleftig J(t) NT,22(ˆA,ˆft)/bracketrightig−1 ,−Nt−δ/bracketleftig J(t) NT,22(ˆA,ˆft)/bracketrightig−1 →PΩ−1 f,t. Based on Corollary 3.4an... | https://arxiv.org/abs/2505.22417v1 |
generalized here, e.g. Trapani (2018) andYu et al. (2024). 16 3.3 Cointegrated Single Index Inthis subsection, weexamine thecase wherealinear cointe gration happensamongcomponents ofg0itfor alli= 1,...,N. In other words α′ 0ig0it∼I(0) for every i. Withinthiscontext, wesolvetheoptimization probleminEq uation(4)underthea... | https://arxiv.org/abs/2505.22417v1 |
Assumptions 2,5, and6, asN,T→ ∞, DT(ˆθi−θ0i)→DΩ∗−1 θ,iξi,and√ N(ˆft−f0t)→DN(0,Ω∗−1 f,t), whereξi= (ξ1i,ξ′ 2i)′with ξ1i=/radicalbigg E/bracketleftig M(/bardblα0i/bardblh(1) 0i1)h(1) 0i1/bracketrightig2/integraldisplay1 0dUi(s)andξ2i=/radicalbigg E/bracketleftig M(/bardblα0i/bardblh(1) 0i1)/bracketrightig2/integraldi... | https://arxiv.org/abs/2505.22417v1 |
unobservable. For each generated dataset, we first determine the number of f actors. Then, we evaluate the parameter estimates by measuring the error according to the following criteria, where M denotes the number of iterations. MAE 1 =1 NTMN/summationdisplay i=1T/summationdisplay t=1M/summationdisplay j=1|ˆz(j) it−z0it... | https://arxiv.org/abs/2505.22417v1 |
0ixit+λ′ 0if0t)+uit, i= 1,...,N;t= 1,...,T, 21 where Jumpitindicates whether or not asset iundergoes jumps on day t, with 1 representing presence of jumps and 0 representing absence of jumps. 5.1 Data We collected intraday observations of S&P 500 index constit uents from January 2004 to De- cember 2016.2Using these hig... | https://arxiv.org/abs/2505.22417v1 |
the factors. 2006 2008 2010 2012 2014 20160501001501st factor 2006 2008 2010 2012 2014 2016-150-100-500501002nd factor 2006 2008 2010 2012 2014 2016-150-100-500501001503rd factor 2006 2008 2010 2012 2014 2016-100-50050100First-order difference for 1st factor 2006 2008 2010 2012 2014 2016-100-50050100First-order differe... | https://arxiv.org/abs/2505.22417v1 |
captured by the Fa ma–French–Carhart five factors. Motivated by the Fama–French–Carhart five factors model (e. g.,Fama and French 2015 ), 24 Figure 4: Canonical correlations and asset pricing results. Notes. The left panel displays the canonical correlation coefficients between the four jump arrival facto rs and the Fama-F... | https://arxiv.org/abs/2505.22417v1 |
properties have been established. First, the convergence rates differ between the two cases, with an elevated rate when the single index is cointegrated. Second, while the con vergence rate for factor estima- tors depends on time tin the nonstationary case—necessitating a larger sample si zeN—but is independent of tin t... | https://arxiv.org/abs/2505.22417v1 |
Econometrics 202 (1), 18–44. Fama, E. F. and K. R. French (2015). A five-factor asset pricin g model. Journal of Financial Economics 116 (1), 1–22. Fama, E. F. and J. D. MacBeth (1973). Risk, return, and equili brium: Empirical tests. Journal of Political Economy 81 (3), 607–636. Fan, J., Y. Liao, and M. Mincheva (2013)... | https://arxiv.org/abs/2505.22417v1 |
arXiv:2505.22423v1 [math.ST] 28 May 2025Max-laws of large numbers for weakly dependent high dimensional arrays with applications Jonathan B. Hill∗ Dept. of Economics, University of North Carolina, Chapel Hill, NC This draft: May 29, 2025 Abstract We derive so-called weak and strong max-laws of large numbers for max 1≤i... | https://arxiv.org/abs/2505.22423v1 |
first attempt to derive and compare possible laws and their resulting bounds on knunder various serial or cross-coordinate dependence and heterogeneity settings. A very few examples where max-WLLN’s appear include HD model inference under independence (Dezeure, B¨ uhlmann and Zhang, 2017; Hill, 2025b) or weak dependenc... | https://arxiv.org/abs/2505.22423v1 |
2004, 2005). The latter construction along with other recent mixing concepts, like mixingale and related moment-based constructions (Gordin, 1969; McLeish, 1975), were proposed to handle stochastic processes that are not, e.g., uniform σ-field based α-,β-, or ϕ-mixing. This includes possibly infinite order functions of... | https://arxiv.org/abs/2505.22423v1 |
different than near-martingale (EFi−1,txi,n,t = EFi−1,txi−1,n,t),weak-martingale , orlocal-martingale (cf. Kallenberg, 2021). 4 wart, 2017), and physical dependent processes (Wu, 2005).3In most cases the random variables are assumed bounded or sub-exponential, and in many cases only 1-Lipschitz functions are treated. W... | https://arxiv.org/abs/2505.22423v1 |
asymptotics for√n(ˆθn,ˆln−θ0,i∗). We instead study max 1≤i≤kn|√nˆθn,i|to test H0:θ0,i= 0∀i⇔H0:θ0,i∗= 0, under weak dependence, allowing for non-stationarity, and high dimensionality kn>> n , where kn→ ∞ andkn/n→ ∞ are allowed. We do not explore, nor do we need, an endogenously selected optimal covariate index ˆlnunder ... | https://arxiv.org/abs/2505.22423v1 |
(cf. Dedecker and Prieur, 2004), and is implied by α-mixing and nests Lp- mixingales (Hill, 2024). Let Λ 1(Rr) denote the class of 1-Lipschitz functions f:Rr→ R, letAbe a σ-subfield of F, and define for Rr-valued random variable Xas in Dedecker and Prieur (2004): τ(1)(A, X) :=||supf∈Λ1(Rr)|EAf(X)−Ef(X)| ||1. If we writ... | https://arxiv.org/abs/2505.22423v1 |
C.2). We next have a corresponding max-SLLN that exploits a maximal concentration in- equality for max 1≤l≤n|1/nPl t=1xi,n,t|. The proof is similar to the one for Theorem 2.6.b under physical dependence and therefore presented in Hill (2024, Appendix G). Theorem 2.3 (max-SLLN: τ(p)-mixing) .Let{xi,n,t: 1≤i≤kn}n t=1sati... | https://arxiv.org/abs/2505.22423v1 |
p∈(1,2), or a moment bound due to Dedecker and Doukhan (2003), cf. Rio (2017, Chapt. 2.5). See also Jirak and K¨ ostenberger (2024, Lemma 21). We can evidently also set p∈(0,1] by using related general Doob-type bounds (e.g. K¨ uhn and Schilling, 2023, Theorem 4.4). Remark 2.4. max i∈Nγi(α)<∞nests a well known polynomi... | https://arxiv.org/abs/2505.22423v1 |
E max 1≤i≤kn,1≤l≤n 1 nlX t=1xi,n,t p)1/p ≤( Emax 1≤l≤nknX i=1 1 nlX t=1xi,n,t p)1/p ≤ B pkn np(1−1/p′)max 1≤i≤kn,1≤t≤nn Θ(p) i,n,top1/p forp >1. Ifp≥2 then np(1−1/p′)=np/2and the upper bounds are virtually identical since cosmetically Θ(p) i,n,t≤2||xi,n,t||p p. The major differences are Mies and Steland (2023) ( i) o... | https://arxiv.org/abs/2505.22423v1 |
now impose restrictions across coordinates ito improve bounds on kn. 2.2.2 Martingale Coordinates Write Si,n:=1 nnX t=1xi,n,t, and let the filtrations {Fi,n}i∈Nbe such that σ {xi,n,t}n t=1 ⊆Fi,nandEFi,n xi+1,n,t = xi,n,t∀i, n, t. Then EFi,n Si+1,n =Si,n, hence the collection {Si,n,Fi,n}i≥1forms a martingale. Doob... | https://arxiv.org/abs/2505.22423v1 |
with |ρn,j|<1 and |ρn,j| → 1 as n→ ∞ at a sufficiently slow rate. For example ρn,j= (1−ζ/ln(kn))jhence δj=jζ(see Example 6 below). By (2.7) it follows for√n/bkn→ ∞ andakn/√n→0, thus ln( kn) =o(n), P(Mn> u)≤P max 1≤i≤kn ˜Zi,n −akn bkn>√n bknu max 1≤i≤knVi,n−akn√n →0∀u≥0. 15 Compare this to ln( kn) =O(√n) for sub-G... | https://arxiv.org/abs/2505.22423v1 |
reducing to Example 7. This is a mild improvement over Theorem 2.5.b where ln( kn) =O(n1/2)for any 1< α≤2. EXAMPLE 9 (Stable Domain ).Suppose max 1≤i≤knP(Pn t=1xi,n,t/[n1/φh(n)]> u)→ Sφ(u)∀u∈R, some φ∈(1,2), slowly varying h(n) that may be different in different places, and some zero mean non-degenerate distribution Sφ... | https://arxiv.org/abs/2505.22423v1 |
definite, and bHnisa.s.positive definite ∀n≥n0, and some n0∈N. d.lim inf n→∞infλ′λ=1E[(PL h=1λhZn(h))2]>0for each L ∈N. Remark 3.1. (a)-(c) allow us to use Theorem 2.2 for key summands by exploiting the fact that geometric α-mixing implies geometric τ(p)-mixing. ( c) is standard for least squares identification. (d) is... | https://arxiv.org/abs/2505.22423v1 |
Test statistic Consider a scalar outcome yand set of covariates x= [xi]kn i=1, with variances v(y), v(xi) ∈(0,∞). We want to test the hypothesis that no covariate is linearly related to y, H0:cov(y, xi) = 0∀i= 1, ..., k nand each n (4.1) H1:cov(y, xi)̸= 0 for some ∀i= 1, ..., k nasn→ ∞ , where the number of covariates ... | https://arxiv.org/abs/2505.22423v1 |
where max i∈Nψi,m= O(m−λ−ι) for some size λ≥1,ψi,0= 1, and logically d(p) i,t≤K||ˇz||n,p. Then λ≥1 yields max 1≤i≤kn,1≤t≤n{Θ(p) i,t∨˜Θ(p) t} ≤K||ˇz||n,pP∞ m=1m−λ−ι=K||ˇz||n,p. Define Hi:=E˜xi,t˜x′ i,t. Assumption 2. a.(xi,t, yt)are covariance stationary, governed by non-degenerate distributions uniformly over (i, t),Lp... | https://arxiv.org/abs/2505.22423v1 |
max-test statistic max 1≤i≤kn|√nˆθi,n|. Theorem 4.3. Let Assumption 2 and H0hold. Assume {xi,t, yt}areLp-physical de- pendent, p≥4, with size λ > 2. Then max 1≤i≤kn|√nˆθi,n|d→max i∈N|Zi|for any {kn} satisfying ln(kn) =o(ns(b,λ))where by case ifb∈(0,1/6]then s(b, λ) =( 1 4ifλ≥28 5 λ 8+2λ1 (7/6)∨(1+b)ifλ <28 5(4.2) ifb∈(... | https://arxiv.org/abs/2505.22423v1 |
max-statistic sidesteps HAC estimation and therefore inversion of a large dimension matrix, both of which may lead to poor inference. See Hill and Motegi (2020), Hill, Ghysels and Motegi (2020) and Hill (2025b) for demonstrations of asymptotic max-test superiority in models with (potentially very) many parameters. The ... | https://arxiv.org/abs/2505.22423v1 |
allow for nonstationarity, (5.1) need not be the true model, and within-group dependence can be arbitrary as discussed above. Nonstationarity allows for heteroscedasticity and other forms of heterogeneity, and a max-test allows us to by-pass covariance matrix estimation entirely (it is ipso facto het- eroscedasticity r... | https://arxiv.org/abs/2505.22423v1 |
Prieur (2004, Lemma 5) and Merlev` ede et al. (2011, p. 460), directly imply (A.1) holds under τ(p). Indeed max 1≤i≤knτ(1) i,n(m)≤ {max 1≤i≤knτ(p) i,n(m)}1/p ≤a1/pe−(b/p)mγ1by Lyapunov’s inequality and (2.1). Hence Merlev` ede et al. (2011, proof of Theorem 1) arguments go through with ( a, b) replaced with ( a1/p, b/p... | https://arxiv.org/abs/2505.22423v1 |
constant Cp:= 18 p3/2/(p−1)1/2, andC′ p:=pCp/(p−1). Case 1 ( p∈(1,2)). Apply Lemma 2.2 in Li (2003) to ||Pn l=1y(r) i,n,l||p, cf. Wu and Shao 30 (2007, Lemma 1), to yield nX t=1y(r) i,n,t p≤ C p nX t=1 y(r) i,n,t p p!1/p ≤ C pn1/pmax 1≤t≤n y(r) i,n,t p. Hence ||max 1≤t≤n|Zi,t|||p≤ C′ pn1/p−1/2max 1≤t≤nP∞ r=0||y(r) i,n,... | https://arxiv.org/abs/2505.22423v1 |
1≤i≤kn 1 nnX t=1xi,n,t ≤1 λln kn e+1 Knα/2/λα−1 =ln(kn) ln(kn) + ln ln( n)+1 ln(kn) + ln ln( n)ln e+1 K √n ln(kn)+ln ln( n)α −1 =o(1). Finally, set λ=ξ√nfor any ξ∈(0,K1/α) to yield Emax 1≤i≤kn 1 nnX t=1xi,n,t ≤ln(kn) ξ√n+1 ξ√nln e+1 K(√n/[ξ√n])α−1 =ln(kn) ξ√n+1 ξ√nln e+1 K/ξα−1 =ln(kn) ξ√n+O1√n , hence ... | https://arxiv.org/abs/2505.22423v1 |
that max i∈N˚γi(α)<∞, max 1≤i≤knP max 1≤l≤n 1 nlX t=1xi,t > u! ≤2 exp −Cnα(1−b)uα . (A.12) Step 2 proves for some C>0, any ξ∈(0,C), and any positive λ <(C −ξ)nα(1−b), max 1≤i≤knE" exp( λmax 1≤l≤n 1 nlX t=1xi,t )# ≤e+2λ ξnα(1−b). (A.13) We then prove the claim in Step 3. Step 1 (A.12). By arguments in the proofs of ( a... | https://arxiv.org/abs/2505.22423v1 |
supposition. Apply Doob’s inequality to yield max 1≤i≤kn,1≤l≤n lX t=1xi,t tb p≤p p−1 max 1≤l≤n lX t=1xkn,t tb p=o Kp¯d(p) n for some p >1. Thus max 1≤i≤kn,1≤l≤n|Pl t=1xi,t/tb|=op(¯d(p) n). This implies max 1≤i≤kn|Pnr t=1xi,t/tb|/¯d(p) nra.s.→0 asr→ ∞ for some sequence of positive integers {nr}r∈N. Now use (A.10) and ... | https://arxiv.org/abs/2505.22423v1 |
2023. Testing the martingale difference hypothesis in high dimension. J. Econometrics 235, 972–1000. Chazottes, J.R., Gouezel, S., 2012. Optimal concentration inequalities for dynamical sys- tems. Comm. Math. Phys. 316, 843–889. Chernick, M.R., 1981. A limit theorem for the maximum of autoregessive processes with unifo... | https://arxiv.org/abs/2505.22423v1 |
31, 1527–1551. Hill, J.B., Motegi, K., 2020. A max-correlation white noise test for weakly dependent time series. Econometric Theory 36, 907–960. 41 Hsing, T., H¨ usler, Reiss, R.D., 1996. The extremes of a triangular array of normal random variables. Ann. Appl. Probab. 6, 671–686. Ibragimov, I.A., 1962. Some limit the... | https://arxiv.org/abs/2505.22423v1 |
22, 1679–1706. Pollard, D., 1984. Convergence of Stochastic Processes. Springer Verlag, New York. P¨ otscher, B.M., Prucha, I.R., 1989. A uniform law of large numbers for dependent and heterogeneous data processes. Econometrica 5, 675–683. Rio, E., 1995. The functional law of the iterated logarithm for stationary stron... | https://arxiv.org/abs/2505.22423v1 |
arXiv:2505.22646v1 [math.ST] 28 May 2025Path-Dependent SDEs: Solutions and Parameter Estimation Pardis Semnani, Vincent Guan, Elina Robeva, and Darrick Lee May 29, 2025 Abstract . We develop a consistent method for estimating the parameters of a rich class of path- dependent SDEs, called signature SDEs , which can mode... | https://arxiv.org/abs/2505.22646v1 |
may depend on the entire history of the path, dYt=a(Y[0,t])dt+b(Y[0,t])dW t. A simple construction of a path-dependent SDE is given in [ 29]. First, consider the three- dimensional linear SDE, dY(1) t=dW(1) t dY(2) t=Y(3) tdt+dW(2) t (1.1) dY(3) t=Y(1) tdt. Although this three-dimensional SDE is Markovian, we note that... | https://arxiv.org/abs/2505.22646v1 |
While the integrals in (1.2) are well-defined if the path Yis sufficiently smooth, such integrals may not exist if Yis highly irregular. Instead, arough path is a path Y, together with postulated signature terms Yk, for k≤p, where pis determined by the regularity of the path. Equipped with this additional data, integra... | https://arxiv.org/abs/2505.22646v1 |
for N ≥N0and r≥r0, the polynomial system (1.5) has a solution θr,N∈Bε(θ0). In fact, we show in Proposition 5.12 that Pis locally Lipschitz, and thus differentiable al- most everywhere. Our assumptions and proof of consistency are distinct from those of [ 35]. In particular, [ 35] assumes a priori that a unique solution... | https://arxiv.org/abs/2505.22646v1 |
respectively defined as the direct sum and product of all such tensor powers T(V):=∞M k=0V⊗kand T( (V) ):=∞ ∏ k=0V⊗k. Note that the individual Hilbert space structure of V⊗kdoes not induce a Hilbert space structure onT( (V) ), but we may restrict to finite norm elements to obtain a Hilbert space, H( (V) ):=( s= (sk)∞ k... | https://arxiv.org/abs/2505.22646v1 |
used to measure the regularity of paths. Definition 2.4. Acontrol is a continuous non-negative function ω:∆T→R≥0such that ω(t,t) =0 and ω(s,t) +ω(t,u)≤ω(s,u)for all s≤t≤u. PATH-DEPENDENT SDES: SOLUTIONS AND PARAMETER ESTIMATION 7 Now, we turn to the definition of a rough path. Definition 2.5. Letp≥1. A p-rough path is ... | https://arxiv.org/abs/2505.22646v1 |
to do so, we consider the notion of Lip (γ)functions in the sense of [ 42, Section VI]. We state the definition for such functions on Banach spaces U; for the more general definition for closed sets F⊂U, see [ 42, Section VI.2.3] and [ 27, Definition 1.21]. Definition 2.11. [15, Definition 10.2] Let γ>0, and U,Vbe Bana... | https://arxiv.org/abs/2505.22646v1 |
(X,ζ)toY=πV(Z)is continuous in the p-variation topology. (3)LetZ(r)be the sequence of Picard iterations defined in (2.5) , and define Y(r):=πV(Z(r)). The solution is given as the limit Y=lim r→∞Y(r). (4)Letωbe a control for the p-variation of X. For all ρ>1, there exists some T ρ∈(0,T]such that ∥Y(r)k s,t−Y(r+1)k s,t∥ ... | https://arxiv.org/abs/2505.22646v1 |
understanding the path-dependent SDEs in (1.3), by considering a (deterministic) path-dependent rough differential equation. Throughout this article, assume Uand Vare Banach spaces whose tensor powers are endowed with norms which satisfy the usual requirements of symmetry and consistency [ 27, Definition 1.25]. Moreove... | https://arxiv.org/abs/2505.22646v1 |
such that for all Lip(α)functions f :K→U, we have ∥f∥Lip(α)≤C∥f∥Lip(α). We will apply this result by first factoring the vector field F:eV→L(U,eV)as defined in (3.3) into two components. We define the map tens : eV→L(V,eV)to be the truncated tensor product ineV; in particular, for s= (s0, . . . , sq)∈eVand v∈V, we have... | https://arxiv.org/abs/2505.22646v1 |
geometric p-rough path in GΩp(V). Before we continue, we will briefly discuss some notational conventions. Notation 3.7. We denote signatures of signatures or rough paths valued in T(eV)using calli- graphic symbols Y. Integer superscripts will continue to denote the outer level of the signature, for instance Yk∈eV⊗k. G... | https://arxiv.org/abs/2505.22646v1 |
IF,M(X) =lim r→∞Y(r)in the p-variation rough path topology. Then, restricting this to the level 1 component of YandY(r), we obtain the desired result. Finally, part (4) also follows directly from the analogous result in part (4) of Theorem 2.15 and considering the level 1 component. □ Remark 3.10. An important point in... | https://arxiv.org/abs/2505.22646v1 |
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