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Then, for arbitrary but fixed x0∈R\SPX(ω)and for allε>0, the fact that PXhas a Lebesgue density being positive on Rimplies the existence of x1,x2∈SPX(ω)withx1< x0<x2. Moreover, from Lemma B.2, we know that there exists K∈N, s.t. |Φnjω k(x2)−Φnjω k(x1)|<ε 5 for everyk>K . By choosing K∈Nsufficiently large, we also have |ˆ... | https://arxiv.org/abs/2504.09564v1 |
expansion of Φn aroundx0the existence of νn(s)betweenx0andξn(s), s.t. bn (β+1)!(Φn(ξn(s))−Φn(x0))p(β) X(ξn(s))(ans)β+1 =δ−β n (β+1)!Φ(β) n(νn(s))(ξn(s)−x0)p(β) X(ξn(s))sβ+1 =1 (β+1)!Φ(β) 0(δnνn(s))(ξn(s)−x0)p(β) X(ξn(s))sβ+1. Combining the previous calculations, we have sup s∈[−S,S]/vextendsingle/vextendsingle/vextends... | https://arxiv.org/abs/2504.09564v1 |
⊂[−S,S], the marginals (Z1 n(s1),...,Z1 n(sk))converge weakly to (Z1(s1),...,Z1(sk)). We start with the weak convergence of the marginals, where we want to apply the Lindeberg- Feller central limit theorem. For this, let k∈Nbe arbitrary, let {s1,...,sk}⊂[−S,S]denote an arbitrary finite subset of [−S,S]and note that ... | https://arxiv.org/abs/2504.09564v1 |
as well as Z2 n(s)−→1 (β+1)!Φ(β) 0(0)pX(x0)sβ+1andZ3 n(s)−→PvpX(x0)sinℓ∞([−S,S]) forn−→∞ . The assertion now follows from the fact that Zn(s)=Z1 n(s)+Z2 n(s)−Z3 n(s) as well as that Z2 n(s)andZ3 n(s)converge to nonrandom functions. LEMMA 6.2. Under the same assumptions as in Lemma 6.1, the sequence of minimizers ˆsnofZ... | https://arxiv.org/abs/2504.09564v1 |
vanishing derivative in 0. Assume further that Xhas bounded moments up to order βand thatFXis invertible. Then the sequence of processes Wn(s)converges weakly in ℓ∞([0,1]) to the process W(s), as long asnδ2β n−→0forn−→∞ . PROOF . We will start with the convergence of W2 n(s). Note that by assumption on Φn, a Taylor exp... | https://arxiv.org/abs/2504.09564v1 |
choosing εn=√ δandMn=1, we have E/bracketleftig sup |s−t|<δ|W1 n(s)−W1 n(t)|/bracketrightig ≤J[]/parenleftbig δ1/2,Hn,δ,L2(PX,Yn)/parenrightbig/parenleftig 1+J[]/parenleftig√ δ,Hn,δ,L2(PX,Yn)/parenrightig δn1/2/parenrightig . By a standard bracketing argument, it follows that for some c onstantK >0, which may cha... | https://arxiv.org/abs/2504.09564v1 |
n(a)|≥xΛn(u)−au≤Λn(λ−1 n(a))−aλ−1 n(a)/bracerightig . THE WEAK-FEATURE-IMPACT EFFECT 39 Consequently, P/parenleftbig |˜Un(a)−λ−1 n(a)|>x/parenrightbig ≤P/parenleftig inf |u−λ−1 n(a)|≥xΛn(u)−au≤Λn(λ−1 n(a))−aλ−1 n(a)/parenrightig =P/parenleftig inf |u−λ−1 n(a)|≥xΛn(u)−Λn(λ−1 n(a))+aλ−1 n(a)−au≤0/parenrightig =P/par... | https://arxiv.org/abs/2504.09564v1 |
Dvoretzky, Kiefer and Wolfowitz (1956 ) P/parenleftbig |F−1 n(˜Un(a))−Φ−1 n(a)|≥x/parenrightbig =P/parenleftbig |F−1 n(˜Un(a))−F−1 X(λ−1 n(a))|≥x/parenrightbig ≤P/parenleftig sup u∈[0,1]/vextendsingle/vextendsingleF−1 n(u)−F−1 X(u)/vextendsingle/vextendsingle≥x/2/parenrightig +P/parenleftbig/vextendsingle/vextendsing... | https://arxiv.org/abs/2504.09564v1 |
=/integraldisplayΦn(T) Φn(−T)/integraldisplayT −T1{a≥Φn(t)}1{ˆΦn(t)≥a}dtda =/integraldisplayΦn(T) Φn(−T)/integraldisplayT F−1 n◦˜Un(a)1{Φ−1 n(a)≥t}1{ˆΦn(t)≥a}dtda =/integraldisplayΦn(T) Φn(−T)/integraldisplayΦ−1 n(a) F−1 n◦˜Un(a)1{ˆΦn(t)≥a}dtda =/integraldisplayΦn(T) Φn(−T)/integraldisplayΦ−1 n(a) F−1 n◦˜Un(a)1{F−1 n◦˜... | https://arxiv.org/abs/2504.09564v1 |
have /integraldisplayλn(1) λn(0)/vextendsingle/vextendsingle˜Un(a)−λ−1 n(a)−Bn(λ−1 n(a))√n/vextendsingle/vextendsingle1 pX(Φ−1n(a))da =/integraldisplayλn(0) λn(0)/vextendsingle/vextendsingle˜Un(aB n)−λ−1 n(a)/vextendsingle/vextendsingle1 pX(Φ−1n(a))da+oP(n−1/2). PROOF . Fori∈N0, letkn i..=λn(0)+i(nδ2 n)−1/3log(nδ2 n)an... | https://arxiv.org/abs/2504.09564v1 |
n(λ−1 n(a))/parenleftbig˜Un(aB n)−λ−1 n(a)/parenrightbig +1 2L′′ n(νn)/parenleftbig˜Un(aB n)−λ−1 n(a)/parenrightbig2. By definition of σ2 n(t), we have σ2 n(t)=E[(Yn−Φn(X))2|X=t]=Φn(t)(1−Φn(t)) and consequently, L′ n(λ−1 n(a))=Φn(Φ−1 n(a))(1−Φn(Φ−1 n(a)))≥Kmin{Φn(−T),1−Φn(T)}≥K for alla∈[λn(0),λn(1)]. Further, L′′ n(νn)... | https://arxiv.org/abs/2504.09564v1 |
10). This follows from the fact that Tn=n1 3(3q−5)δ−(9q−17) (9q−15) n=n1 3(3q−5)δ−1 3(3q−5) nδ−(3q−6) (3q−5) n=(n/δn)1 3(3q−5)δ−(3q−6) (3q−5) n and (n/δn)−1/3Tn=(n/δn)−(3q−6) 3(3q−5)δ−(3q−6) (3q−5)n=(nδ2 n)−(3q−6) 3(3q−5). Note further that (n/δn)1/3UL n(a)can differ from ˆUL n(a)only if(n/δn)1/3|UL n(a)|>Tn. But then,... | https://arxiv.org/abs/2504.09564v1 |
n+Kn−q/3δq/2 n ≤Kn1−q/3δ−q/6 n and E|X/bracketleftig sup |u|≤Tn|Rn(a,u)|q/bracketrightig =n2q/3 δq/6 nE|X/bracketleftig sup |u|≤Tn/vextendsingle/vextendsingle/integraldisplayL−1 n((n/δn)−1/3u+Ln(λ−1 n(a))) λ−1 n(a)Φn◦F−1 X(x)−Φn◦F−1 n(x) −Φ′ n(x)Bn(FX◦F−1 n(x))√npX◦F−1n(x)+Φ′ n(x)Bn(FX◦F−1 n(x))√npX◦F−1n(x) −Bn(λ−1 ... | https://arxiv.org/abs/2504.09564v1 |
we have by (Theorem 4, Durot ,2002 ) that there exists κ>0, s.t. P|X(Vn(t)/\e}atio\slash=˜Vn(t))≤P|X(Vn(t)>Sn)≤2exp(−κ2δ3 nS3 n/2)=2exp( −κ2log(nδ2 n)3/2) ≤(nδ2 n)−1/6/log(nδ2 n). Note further that under P|X, both ˜Vn(t) (L′n(t))4/3andVn(t) (L′n(t))4/3 have bounded moments of any order and that ηn(t)is bounded. So we h... | https://arxiv.org/abs/2504.09564v1 |
Mn, but still suffices for the consis- tency proof. LEMMA A.1. For everyn∈N, we haveMn(ˆΦn,Φ)≥0. PROOF . By concavity of the logarithm and the definition of ˆΦnas the maximizer of the log-likelihood, we have Mn(ˆΦn,Φ)=1 nn/summationdisplay i=1log/parenleftigpˆΦn(Xi,Yn i)+pΦ(Xi,Yn i) 2pΦ(Xi,Yn i)/parenrightig ≥1 nn/summ... | https://arxiv.org/abs/2504.09564v1 |
for GΦdefined as in Proposition A.3, we have sup Ψ∈F|Mn(Ψ,Φ)−M(Ψ,Φ)|= sup Ψ∈F/vextendsingle/vextendsingle/vextendsingle1 nn/summationdisplay i=1mΨ,Φ(Xi,Yi)−EΦ[mΨ,Φ(X,Yn)]/vextendsingle/vextendsingle/vextendsingle = sup g∈GΦ/vextendsingle/vextendsingle/vextendsingle1 nn/summationdisplay i=1g(Xi,Yi)−EΦ[g(X,Y)]/vextendsing... | https://arxiv.org/abs/2504.09564v1 |
−→0, forn−→∞ . APPENDIX B: PROOF OF L1(PX)- AND UNIFORM CONSISTENCY In this section, we give the necessary technical results for provingL1(PX)- and uniform convergence of the difference between the estimator ˆΦnand the true function Φn. We start with the following result, relating the L1-distance to the Hellinger dista... | https://arxiv.org/abs/2504.09564v1 |
b≤√a+2√ b√a+√ b≤2√a+√ b√a+√ b=2. By a combination of the first two statements, we obtain /vextendsingle/vextendsingle√a−√ b/vextendsingle/vextendsingle=2/vextendsingle/vextendsingle/radicalbigg a+b 2−√ b/vextendsingle/vextendsingle/parenleftig/radicalig a+b 2+√ b √a+√ b/parenrightig ≤4/vextendsingle/vextendsingle/rad... | https://arxiv.org/abs/2504.09564v1 |
DUROT , C. (2008). Monotone nonparametric regression with random design. Math. Methods Stat. 17327–341. https://doi.org/10.3103/S1066530708040042 DUROT , C., K ULIKOV , V. N. and L OPUHAÄ , H. P. (2012). The limit distribution of the L∞-error of Grenander- type estimators. Ann. Stat. 401578–1608. https://doi.org/10.121... | https://arxiv.org/abs/2504.09564v1 |
location of the maximum and asymmetric two-s ided Brownian motion with triangular drift. Stat. Probab. Lett. 29279–284. https://doi.org/10.1016/0167-7152(95)00183-2 TSYBAKOV , A. B. (2009). Introduction to nonparametric estimation .Springer Ser. Stat. New York, NY: Springer. VAN DE GEER, S. (1993). Hellinger-consistenc... | https://arxiv.org/abs/2504.09564v1 |
Parameters estimation of a Threshold Chan-Karolyi-Longstaff-Sanders process from continuous and discrete observations Sara Mazzonetto∗Benoît Nieto† April 15, 2025 Abstract. We consider a continuous time process that is self-exciting and ergodic, called threshold Chan-Karolyi-Longstaff-Sanders (CKLS) process. This proce... | https://arxiv.org/abs/2504.10022v1 |
or infinite horizon [23, 27, 28], low frequency ones [37, 15, 28]. Non- threshold processes represent a special class of TDs. Let us also mention the literature on parametric estimation for these processes. For CKLS, the drift maximum likelihood estimation has been considered in [30] in the continuous time setting, and... | https://arxiv.org/abs/2504.10022v1 |
according to the stationary distribution for all t≥0) and it is observed on a time-grid with Nobservations (not necessarily equally spaced) with maximal lag between two consecutive observations denoted by ∆N. Note that the law of the sample cannot be described to obtain a likelihood function because, except in very spe... | https://arxiv.org/abs/2504.10022v1 |
their asymptotic normality properties on a simulated dataset. Then, we apply these estimators to the ten-year US Treasury rate using the Federal Reserve Bank’s H15 dataset. In Section 5.2, we use a discretized version of the test introduced in [34] to assess the presence of one or more thresholds in the drift term of t... | https://arxiv.org/abs/2504.10022v1 |
time observations and discrete (not necessarily equally spaced) high frequency observations and infinite horizon. We denote θ⋆:= (a, b) = ( aj, bj)d j=0andσ⋆the parameters to be estimated. The parameters for which our results hold are further restricted according to the following requirements: the likelihood (resp. qua... | https://arxiv.org/abs/2504.10022v1 |
the last interval Idsatisfy (a0, b0)∈(0,+∞)×Rand (ad, bd)∈R×(0,+∞) when γ0̸= 0; and a0, ad∈Randb0, bd>0when γ0= 0. Moreover, with the latter choices (bd̸= 0), the stationary distribution µadmits non-negative moments of all order and, if γ0̸∈ {0,1/2}, negative moments of all order too. When γ0= 0one can only have finite... | https://arxiv.org/abs/2504.10022v1 |
T Qj,2−2γj T −Qj,1−2γj T Mj,1−2γj T Qj,−2γj TQj,2−2γj T −(Qj,1−2γj T )2,Mj,−2γj T Qj,1−2γj T −Qj,−2γj TMj,1−2γj T Qj,−2γj TQj,2−2γj T −(Qj,1−2γj T )2! .(3.4) The maximum of the quasi-likelihood q-LT(θ)is achieved at θ(q-L) T:= (aj,0 T, bj,0 T)d j=0, that is (aj,0 T, bj,0 T) = Mj,0 TQj,2 T−Qj,1 TMj,1 T Qj,0 TQj,2 T−(Qj,... | https://arxiv.org/abs/2504.10022v1 |
the T-CKLS process. We assume that the process is ergodic, µis the stationary distribution, and we introduce the following hypotheses: •HL:µadmits finite (−2γ0)-th and (2−2γd)-th moment, •Hq-L:µadmits finite (2 + 2 γd)-th moment. We have discussed these assumptions in Section 2.3, and we have provided a set of paramete... | https://arxiv.org/abs/2504.10022v1 |
to establish a lower bound for the asymptotic variance of estimators. For more details, we refer to the book [21]. Remark 3.8 (Estimator of the asymptotic variance) .We can obtain a consistent estimator for the asymptotic variance using the following construction. Define: Γ(L) j,T:= Γ(L,γj) j,T :=1 T Qj,−2γj T −Qj,1−... | https://arxiv.org/abs/2504.10022v1 |
{0}, then M0,m TN,N=f0,m+1(X0)−f0,m+1(XTN)−m 2σ2 0Q0,m+2γ0−1 TN,N +rm 1 M0,0 TN,N+f0 , Md,m TN,N=fd,m+1(XTN)−fd,m+1(X0)−m 2σ2 dQd,m+2γd−1 TN,N +rm d Md,0 TN,N−fd , where f0= min( XT, r1)−min(X0, r1),f0,m(x) =Rr1 xym−1dy1I0(x),fd,m(x) =Rx rdym−1dy1Id(x), and if j∈ {1, . . . , d −1}, then Mj,m TN,N=fj,m+1(XTN)−fj,m+1... | https://arxiv.org/abs/2504.10022v1 |
that the speed of convergence is larger than√TN. Theorem 4.3. Assume that (4.7)holds, that the T-CKLS Xis stationary and that Hypothesis Hq-Lholds. Then, the estimator σ2 TN,N= ((σj TN,N)2)d j=0in(4.6)is a consistent estimators of the diffusion coefficient vector σ2 ⋆= ((σj)2)d j=0, i.e. σ2 TN,NP− − − − → N→+∞σ2 ⋆. Und... | https://arxiv.org/abs/2504.10022v1 |
their rescaled difference vanishes asymptotically. Since all estimators depend on Mj,m TN,NandQj,k TN,N for suitable k, m, the proofs are based on the following result. Lemma 4.8. Letλ∈ {1,2}. Assume that lim N→+∞TN= +∞,lim N→+∞Tλ−1 N∆N= 0, and that the T-CKLS Xis stationary (see Section 2). Then lim N→∞T−1/λ NEh |Qj,k... | https://arxiv.org/abs/2504.10022v1 |
or a drifted version of the scheme in [35] when the diffusion coefficient is non-linear. More precisely we use the following scheme. Given X0∈(0,∞)and(Gk)k∈Na sequence of i.i.d. standard Gaussian random variables, next we set X(n) 0=X0and, we define for all k∈N X(n) (k+1)/n:= X(n) k/n+1 n a(X(n) k/n)−b(X(n) k/n)X(n) k... | https://arxiv.org/abs/2504.10022v1 |
a computationally efficient approach for calibrating the p-value. The test procedure, we describe below, simulta- neously estimates and tests the threshold. Note that we use the drift MLE and σ⋆estimators for CKLS process considered in this paper, but we do not have results on threshold estimation, 16 therefore we rely... | https://arxiv.org/abs/2504.10022v1 |
daily time intervals is dt= 0.046, where one unit of time represents one month. We assume that the data follow a T-CIR dynamics, i.e.γ≡1/2. We consider the ten year US Treasury rate for two different time window: Jan 2016 - Dec 2019, and Jan 2020 - Jan 2024 represented in Figure 2. 17 Figure 2: The figure shows the int... | https://arxiv.org/abs/2504.10022v1 |
could refer to [7, II.4] for a summary or find more details, e.g., in [32, Chapter VII, Section 3]. The scale function is continuous, unique up to a multiplicative constant, and its derivative satisfies S′(x) = exp −Rx r12(a(y)−b(y)y) σ(y)2y2γ(y)dy . The speed measure is given by m(x) dx=2 (σ(x))2|x|2γ(x)S′(x)dx. The... | https://arxiv.org/abs/2504.10022v1 |
we prove Theorem 4.3, Theorem 4.4, Lemma 4.8, Proposition 4.9 and Proposition 4.11. B.1 Likelihood discretization: M.,m TN,Nversus M.,m TN,N We derive M.,m TN,Nin (4.3) as a discretisation of an expression P-a.s. equal to Mj,m T, which would not involve any term Mj,kexcept if k= 0. The notation, in particular the funct... | https://arxiv.org/abs/2504.10022v1 |
TN,N2−σ2 j Qj,m+2γj−1 TN−Qj,m+2γj−1 TN,N . This, Item (c) in Lemma 4.8, and Theorem 4.3 ensure that T−1/λ N Qj,k TN,N−Qj,k TN P− − − − → N→+∞0and T−1/λ N Mj,m TN,N− Mj,m TN P− − − − → N→+∞0 with λ= 1to get the consistency of the MLE and λ= 2for the speed of convergence. In the caseℓ=q-L, the first term works analogous... | https://arxiv.org/abs/2504.10022v1 |
Itô-isometry, we obtain E |Mj,m TN−Mj,m TN,N| ≤ZTN 0Eh |Xm t−Xm ⌊t⌋∆N|(|aj|+bj|X⌊t⌋∆N|+bj|Xt−X⌊t⌋∆N|)i dt +ZTN 0E |X⌊t⌋∆N|m1Xt∈Ij,X⌊t⌋∆N/∈Ij max i=0,...,d|ai|+ max i=0,...,d|bi| |X⌊t⌋∆N|+|Xt−X⌊t⌋∆N| dt +ZTN 0E |X⌊t⌋∆N|m1Xt/∈Ij,X⌊t⌋∆N∈Ij max i=0,...,d|ai|+ max i=0,...,d|bi| |X⌊t⌋∆N|+|Xt−X⌊t⌋∆N| dt +√ 2 max... | https://arxiv.org/abs/2504.10022v1 |
by (B.8) if k= 1. In the other cases, i.e.k∈[−2,0)∪(0,1)∪(1,2], note that kappearing in Jk,jalways depends on j, i.e. k=kj. We show that Jkj,j≤C√∆Nfor some positive constant C. Note that if γ0= 0,µadmits no negative moments smaller or equal than −1. So in what follows about negative moments, unless γ≡0, assume γ0̸= 0. ... | https://arxiv.org/abs/2504.10022v1 |
(a) (QMLE θ(q-L) TN,N).The QMLE (4.5) involves the statistics Qj,kandMj,m, where m∈ {0,1},k∈ {0,1,2}, and j∈ {0, . . . , d }. The most restrictive assumptions on the moments of µis obtained for m= 1:µhas finite moment of order 2(1 + γd), that is Hq-L. Remark B.2.Ifσ⋆is known, one could replace Mj,1 TN,NbyMj,1 TN,Nin (4... | https://arxiv.org/abs/2504.10022v1 |
Burkholder-Davis-Gundy inequality ensures that there exists a constant C3∈(0,+∞)such that E |k| Zt sXk−1 uσ(Xu)Xγ(Xu) u dBu 2 ≤C3Zt sE X2(k−1+γ(Xu)) u du . 28 Note that Iifor all i∈ {1, . . . , d −1}is contained in [r1, rd]⊆(0,+∞), hence there exists a constant C∈(0,+∞)depending on k, a, b, σ such that E |Xk t−Xk ... | https://arxiv.org/abs/2504.10022v1 |
2(t−s)+bd(1−γd)ψ(Xs)2f(Xs), where K1andK2are two strictly positive constants, fis an explicit function which depends only on Xsand such that limx→+∞f(x)e−ψ(x)2= 0. Let us note that µ(x)1x≥rd=K32 σ2 dx1−2ad σ2 dexp −bd(1−γd)ψ(x)2 1x≥rd, with K3a strictly positive constant. Then, as s→t, andsince µhasafinite m-thmoment... | https://arxiv.org/abs/2504.10022v1 |
u)with ψ(x) =x1−γ0 σ0(1−γ0)(in particular Y0=X1−γ0s σ0(1−γ0)). So, dYu=a0 σ0((1−γ0)Yu)1−1 (1−γ0)−b0(1−γ0)Yu−γ0 2(1−γ0)Y−1 u du+ dBu. By the Comparison Theorem, it holds a.s. for all u∈[0, τ[ψ(Xs) 2,ψ(r1)])that Bν− u≤Yuand Y2 u≤ 1 2˜Bu−ψ(Xs)√n 2 2, where Bν−is drifted Brownian motion and ˜B:= (˜Bi)i≤na n-dimensional B... | https://arxiv.org/abs/2504.10022v1 |
, 4(2):123– 127, 2011. [15] Y. Hu and Y. Xi. Parameter estimation for threshold Ornstein–Uhlenbeck processes from discrete observations. Journal of Computational and Applied Mathematics , 411:114264, 2022. [16] N. Ikeda and S. Watanabe. Stochastic differential equations and diffusion processes , vol- ume 24 of North-Ho... | https://arxiv.org/abs/2504.10022v1 |
Spectral estimation for high-dimensional linear processes JAMSHID NAMDARI1,a, ALEXANDER AUE2,band DEBASHIS PAUL2,3,c 1Department of Biostatistics & Bioinformatics, Emory University ,ajamshid.namdari@emory.edu 2Department of Statistics, University of California, Davis ,baaue@ucdavis.edu 3Applied Statistics Unit, Indian ... | https://arxiv.org/abs/2504.10257v1 |
MANOVA, canonical correlations). Many inference problems in multivariate analysis consequently involve statistics that can be expressed as, or whose behavior is characterized by, the distribution of eigenvalues of appropriate symmetric random matrices, such as the sample covariance matrix or the “Fisher-matrix” involvi... | https://arxiv.org/abs/2504.10257v1 |
of “extreme” eigenvalues of the symmetrized sample autocovariance matrices𝑆𝑛,𝜏:=1 2𝑛Í𝑛−𝜏 𝑡=1 𝑌𝑡𝑌∗ 𝑡+𝜏+𝑌𝑡+𝜏𝑌∗ 𝑡, where𝑌𝑡=[𝑌1𝑡,...,𝑌𝑝𝑡]𝑇is the observed process such that𝑌𝑗𝑡=𝑏𝑗1(𝐿)𝑈1𝑡+···+𝑏𝑗𝑀(𝐿)𝑈𝑀𝑡+𝑋𝑗𝑡,𝑗=1,...,𝑝, the𝑈𝑘𝑡’s are underlying common factors, the𝑋𝑗𝑡’s are the i... | https://arxiv.org/abs/2504.10257v1 |
matrix of the innovation terms, by equating the Stieltjes transforms of the empirical and the limiting spectral dis- tributions. Secondly, by modeling the joint spectral distribution of the coefficient matrices as a discrete probability distribution over a pre-specified grid, we solve the estimating equations by a nume... | https://arxiv.org/abs/2504.10257v1 |
and nonnegative integers, respectively. A.0 The process(𝑍𝑡:𝑡∈Z)is represented as 𝑍𝑡=Σ1/2˜𝑍𝑡whereΣ1/2is a square-root of Σ, and the 𝑝-dimensional vectors ˜𝑍𝑡have iid entries ˜𝑍𝑗𝑡satisfying one of the following conditions: – (for complex-valued processes): ˜𝑍𝑗𝑡is complex valued, with zero mean, and E[ℜ( ˜... | https://arxiv.org/abs/2504.10257v1 |
To this end, denote the ESD of ˜S(𝑛) 𝑔by 𝐹(𝑛) 𝑔(𝑥)=1 𝑛𝑛∑︁ 𝑗=11{𝜉𝑔 𝑗≤𝑥}, where𝜉𝑔 1,...,𝜉𝑔 𝑛are the eigenvalues of ˜S(𝑛) 𝑔and1is the indicator function. We prove that the random distributions 𝐹(𝑛) 𝑔converge almost surely to a nonrandom limiting distribution under the following additional assumption... | https://arxiv.org/abs/2504.10257v1 |
and the role of the associated weighting matrix W𝑔are in order. The choice of an appro- priate window W𝑔in the frequency domain, followed by locally integrating the area under the curve of the resulting weighted sample periodogram, allows for a better discrimination of linear processes by isolating frequency bands th... | https://arxiv.org/abs/2504.10257v1 |
𝑖≤𝑥}, then𝐹𝐴,Σ=𝐹𝐴1···𝐹𝐴𝑞𝐹Σ andΛ0=Λ(1)×···×Λ(𝑞+1). Moreover, denoting 𝜔(𝑘)={𝜔(𝑘) 1,...,𝜔(𝑘) 𝑟𝑘}for𝑘=1,...,𝑞+1, the op- timization is over Δ={∪𝑞+1 𝑘=1𝝎(𝑘):Í𝑟𝑘 𝑖=1𝜔(𝑘) 𝑖=1, 𝑘=1,...,𝑞+1}. Note that the dimension of 𝝎∈Δ isÍ𝑞+1 𝑘=1𝑟𝑘, which is much smaller than 𝐽=Î𝑞+1 𝑘=1𝑟𝑘. 4. Esti... | https://arxiv.org/abs/2504.10257v1 |
across these estimates, with the goal of improving fidelity to the data. Another option is to try to impose an ordering among the elements{𝝀0 𝑗}by searching over all plausible permutations of the indices that lead to the greatest fidelity to the observed data. In the interest of keeping the discussions more focused, ... | https://arxiv.org/abs/2504.10257v1 |
the matrix G(𝚲0 𝐽)=1 2𝜋∫2𝜋 0h0(𝜃)(h0(𝜃))𝑇𝑑𝜃, (5.9) where h0(𝜃)=(ℎ(𝝀0 𝑗,𝜃))𝐽 𝑗=1. Note that G(𝚲0 𝐽)can be expressed as G(𝚲0 𝐽)=𝜸0 0(𝜸0 0)𝑇+2∞∑︁ ℓ=1𝜸0 ℓ(𝜸0 ℓ)𝑇, (5.10) 12 where 𝜸0 ℓ=(𝛾ℓ(𝝀0 1),...,𝛾ℓ(𝝀0 𝐽))𝑇, (5.11) and𝛾ℓ(𝝀0 𝑗)is the lag-ℓautocovariance function of the one-dimensional pr... | https://arxiv.org/abs/2504.10257v1 |
studying these factors, we made a few preliminary considerations as follow: •Note that when 𝑔(𝜃)=0, then𝑀(𝑧,𝜃)=0, which implies that the behavior of the integrand in S(𝑧)is the same as−1/𝑧. Thus, to avoid high fluctuations in neighborhoods of 𝑧=0, the function 𝑔was shifted by the constant value 0 .05. •Four fi... | https://arxiv.org/abs/2504.10257v1 |
daily closing prices of 486 companies included in the S&P 500, recorded from 5/08/2012 to 10/17/2016. The data was obtained from historical data available at yahoo.finance . Preprocessing involved adjustment for stock split such that before and after market capitalization of the companies remain the same. The goal of t... | https://arxiv.org/abs/2504.10257v1 |
estimating the variance covariance matrix of the data had the following ordering in 83 percent of the bootstrap samples 𝐴𝑅𝑀𝐴𝐼𝑛𝑑(1,1)≺𝐴𝑅(1)≺𝐼𝑛𝑑≺𝐴𝑅(2). lag 0 1 2 3 4 5 𝐴𝑅𝑀𝐴𝐼𝑛𝑑(1,1)≺𝐴𝑅(1)≺𝐼𝑛𝑑≺𝐴𝑅(2) 83 0 0 94 5 0 𝐴𝑅𝑀𝐴𝐼𝑛𝑑(1,1)≺𝐼𝑛𝑑≺𝐴𝑅(1)≺𝐴𝑅(2) 17 0 0 5.8 5 10.6 𝐼𝑛𝑑≺𝐴𝑅𝑀𝐴𝐼𝑛𝑑(... | https://arxiv.org/abs/2504.10257v1 |
we can write Xnin terms of the lag operator 𝐿=[0 :𝑒1:···:𝑒𝑛−1]asX𝑛=Z𝑛+A1Z𝑛𝐿, where Z𝑛=[𝑍1:···:𝑍𝑛]. The idea is to first approximate the lag operator 𝐿with a circulant matrix ˜𝐿, and then use the fact that circulant matrices are diagonalizable in the discrete Fourier basis. Formally, define˜˜X𝑛=Z𝑛+A1Z𝑛˜... | https://arxiv.org/abs/2504.10257v1 |
processes 19 S.1: DefineΩ0⊂ΩwithP(Ω0)=1 and show that the convergence statement in Theorem 2.1 holds for every 𝜔∈Ω0. S.2: Prove the existence and uniqueness of the solution 𝐾𝑔(𝝀,𝑧)to (2.5) on R𝑚× C+under the constraint that 𝐾𝑔(𝝀,𝑧)is a Stieltjes kernel. It will be shown that for every 𝜔∈Ω0, there exists a su... | https://arxiv.org/abs/2504.10257v1 |
first term on the right hand side of the above display is a positive definite function in 𝜼, (A.12) follows. Spectral estimation for high-dimensional linear processes 21 Proof of Proposition 5.1 In this case, 𝑋𝑡=A0𝑍𝑡, so thatΣ=A2 0=Var(𝑋𝑡). Then,𝐹𝐴,Σ≡𝐹Σ=Í𝐽 𝑗=1𝜔𝑗𝛿𝝀0 𝑗is the LSD of Σ, where 𝝀0 1,...,𝝀0... | https://arxiv.org/abs/2504.10257v1 |
some𝐶0>0, with𝑔(0)=1 and∫ 𝑔(𝜃)𝑑𝜃=1. Let G≡G𝛿={𝑔𝑘,𝛿}𝑁𝛿 𝑘=1. Let ℎ★(𝚲0 𝐽)=sup 𝜃∈[0,2𝜋]max 1≤𝑗≤𝐽ℎ(𝝀0 𝑗,𝜃). Then, for all 𝜃∈[0,2𝜋], sinceℑ(𝑧)≥𝑎andℑ(𝑀𝑔(𝑧,𝜃|𝝎0))≤ 0, 𝑎≤ℑ(𝑧) ≤ ℑ(𝑐𝑀𝑔(𝑧,𝜃|𝝎0)−𝑧) ≤ 𝑐𝑀𝑔𝑘,𝛿(𝑧,𝜃|𝝎0)−𝑧 (A.24) ≤ 𝑐𝐾∑︁ 𝑗=1𝜔0 𝑗ℎ(𝝀0 𝑗,𝜃) 1+𝐾𝑔𝑘,𝛿(𝝀0 𝑗,𝑧|𝝎0) ... | https://arxiv.org/abs/2504.10257v1 |
and Lippi, M. (1999). Aggregation of linear dynamic microeconomic models. Journal of Mathematical Economics ,31, 131–158. Hachem, W., Loubaton, P., and Najim, J. (2006). The empirical distribution of the eigenvalues of a Gram matrix with a given variance profile. Annales de l’IHP Probabilités et statistiques ,42, 649–6... | https://arxiv.org/abs/2504.10257v1 |
0.5 1 (1,2) (0.5,0.5)mean median sd0.0435 0.0365 0.02840.0329 0.0318 0.01850.0423 0.0308 0.03320.0295 0.0225 0.02190.0312 0.0262 0.0234 0.5 1 (1,2)(0.75, 0.25)mean median sd0.0310 0.0231 0.02580.0295 0.0235 0.02000.0240 0.0185 0.01410.0222 0.0206 0.00850.0212 0.0209 0.0094 (-0.5, 0.8)(0.5, 0.5)1 1mean median sd0.0344 0... | https://arxiv.org/abs/2504.10257v1 |
1 1.5 2 2.5 3−0.200.20.40.60.811.2cdf 0.5 1 1.5 2 2.5 3−0.200.20.40.60.811.2cdf (𝑝,𝑛)=(200,800) ( 𝑝,𝑛)=(200,400) 0.5 1 1.5 2 2.5 3−0.200.20.40.60.811.2cdf 0.5 1 1.5 2 2.5 3−0.200.20.40.60.811.2cdf Figure B.6 . Plot of Median and 90% confidence band for spectral cdf of 𝚺corresponding to the case 2.2 .Dash- Dot Red ... | https://arxiv.org/abs/2504.10257v1 |
BAYESIAN ANALYSIS OF REGRESSION DISCONTINUITY DESIGNS WITH HETEROGENEOUS TREATMENT EFFECTS LAST UPDATE: Wed 16thApr, 2025 AT 00:07 KEVIN TAO, Y. SAMUEL WANG, DAVID RUPPERT Abstract. Regression Discontinuity Design (RDD) is a popular framework for esti- mating a causal effect in settings where treatment is assigned if a... | https://arxiv.org/abs/2504.10652v1 |
posterior consistency for our estimation procedure; and most notably, the proposed procedure shows very strong empirical results when compared to previously proposed procedures. We provide extensive simulations which show that the point estimator has smaller mean squared error than existing methods. In addition, the cr... | https://arxiv.org/abs/2504.10652v1 |
(1) are typically estimated by applying local linear regression (LLR) (Fan and Gijbels, 1994) on both sides of the cut-off. However, the performance of LLR depends heavily on the bandwidth selection. Imbens and Kalya- naraman (2012) propose a data-driven procedure for selecting the mean squared error optimal bandwidth.... | https://arxiv.org/abs/2504.10652v1 |
j. As before, Tijis the treatment indicator, Zijis the running variable, and Yijis the observed outcome. Additionally, Yij(0), Yij(1) denote the potential outcomes so that Yij=TijYij(1) + (1 − Tij)Yij(0). Most notably, we allow the conditional expectations E(Yij(1)|Zij) and E(Yij(0)|Zij) to differ across sub-population... | https://arxiv.org/abs/2504.10652v1 |
effects ( δ1, . . . , δ J) are drawn from a Gaussian distribution with covariance Kδ. In most settings where the analyst does not have strong prior knowledge about which groups may have similar treatment effects, Kδshould be set so that the δjare independent. However, in settings where the similarity between groups may... | https://arxiv.org/abs/2504.10652v1 |
evaluating the covariance kernel KgonZ. Similarly, let f−andf+denote the vector formed by evaluating the appropriate fjonZ−andZ+respectively, and let f= [f−,f+]. BAYESIAN RDD WITH HETEROGENEOUS TREATMENT EFFECTS 9 •H∈ {0,1}N+×J, where the Hi,jis 1 if the ith element of Z+is from the jth sub-population and Hi,j= 0 other... | https://arxiv.org/abs/2504.10652v1 |
through the joint density [ Y, θ]. Following the practice in Bayesian statistics literature, we will BAYESIAN RDD WITH HETEROGENEOUS TREATMENT EFFECTS 11 use the following notations for densities of random variables: [X] = density of X [X, Y] = joint density of X, Y [X|Z] = conditional density of X|Z. We will use mj(·|... | https://arxiv.org/abs/2504.10652v1 |
adjusting for multiple testing. We also consider a localized version of HGPR, termed HGPR-CUT , which only considers observations within a window around the threshold z= 0 instead of all observations. BAYESIAN RDD WITH HETEROGENEOUS TREATMENT EFFECTS 13 Branson et al. (2019) also apply local Gaussian process regression... | https://arxiv.org/abs/2504.10652v1 |
square exponential kernel. To encourage the functions fjto be highly varied, we set σa= 10. We also consider 2 different distribu- tions for the treatment effects, and 2 different distributions for the random errors: (I)δ∼N(0, S), with Sij=ρ|i−j|forρ=.8, BAYESIAN RDD WITH HETEROGENEOUS TREATMENT EFFECTS 15 (II)P(δj=τ1)... | https://arxiv.org/abs/2504.10652v1 |
0.306 DGP3, (B-I), nj= 200 0.132 0.179 0.634 0.508 0.447 0.289 DGP3, (B-II), nj= 200 0.154 0.208 0.632 0.517 0.457 0.307 methods. However, although HGPR-CUT maintains nominal coverage, the empirical cov- erage of HGPR dips below the nominal level when nj= 200. This may be because the increased sample size makes the mod... | https://arxiv.org/abs/2504.10652v1 |
HGPR-CUT LLR-IK LLR-RBC GPR HRDD DGP1, nj= 100 0.08 0.08 0.37 0.44 0.27 2.72 DGP1, nj= 200 0.06 0.06 0.23 0.28 0.18 2.34 DGP2, J= 25 1.56 1.16 2.25 5.53 2.78 2.25 DGP2, J= 50 1.43 2.24 5.67 2.78 2.15 DGP3, (A-I), nj= 100 0.69 0.90 2.38 5.69 2.63 2.70 DGP3, (A-II), nj= 100 0.81 1.05 2.37 5.59 2.64 2.61 DGP3, (B-I), nj= ... | https://arxiv.org/abs/2504.10652v1 |
(1.12,8.39) 5.02 (3.56,6.32) 3.30 (-1.47,8.07) 1974-1993 2.36 (-1.20,6.33) 6.30 (4.97,7.87) 11.85 (6.05,17.65) 1994-2010 4.90 (0.79,9.27) 5.57 (2.73,8.59) 11.48 (-1.76,24.72) 6.Discussion In this paper, we propose a procedure for estimating heterogeneous treatment effects using a RDD with known sub-populations. Specifi... | https://arxiv.org/abs/2504.10652v1 |
nearly universal insurance coverage on health care utilization and health: Evidence from medicare. Working Paper 10365, National Bureau of Economic Research. Cattaneo, M. D., Frandsen, B. R., and Titiunik, R. (2015). Randomization inference in the regression discontinuity design: An application to party advantages in t... | https://arxiv.org/abs/2504.10652v1 |
11702, National Bureau of Economic Research. Reguly, A. (2021). Heterogeneous treatment effects in regression discontinuity designs. arXiv preprint arXiv:2106.11640 . Rubin, D. B. (1974). Estimating causal effects of treatments in randomized and non- randomized studies. Journal of Educational Psychology , 66:688–701. S... | https://arxiv.org/abs/2504.10652v1 |
− − → 0, then Π(UC ϵ|Y,Z)a.s. P θ0− − − − → 0. Therefore, it suffices to show posterior consistency for a single group junder a Gauss- ian process setup for RDD. Lemma 2 and 3 below show that under Assumptions 2 and 3, the Gaussian process setup satisfies Assumption A and B in Theorem 2. Thus we have that Eq. (12) hold... | https://arxiv.org/abs/2504.10652v1 |
be shown for when Zi<0. Thus, letting γ(ϵ) = min {1/2, σ2 +0ϵ}, we have B(γ(ϵ))⊆ {θ:Ki(θ0, θ)< ϵ∀i}. This implies that for any ϵ >0 (16)Π (B(1/2)∩ {θ:Ki(θ0, θ)< ϵ,∀i})≥Π (B(1/2)∩B(γ(ϵ))) = Π ( B(γ(ϵ)))>0, which proves (A2) BAYESIAN RDD WITH HETEROGENEOUS TREATMENT EFFECTS 29 To show (A1), we use the law of total varian... | https://arxiv.org/abs/2504.10652v1 |
Ψ3n[fl, δl′, γ] = 1(X i:zi<0Yj−fli σ−02 < n−(1−γ2)) Ψ4n[fl, δl′, γ] = 1(X i:zi≥0bYj−f0i−δ0 σ+0 >2cn√n+) Ψ5n[fl, δl′, γ] = 1(X i:zi≥0Yj−f0i−δ0 σ+02 > n +(1 +γ)) Ψ6n[fl, δl′, γ] = 1(X i:zi≥0Yj−f0i−δl′ σ+02 < n +(1−γ2)) .(20) BAYESIAN RDD WITH HETEROGENEOUS TREATMENT EFFECTS 31 We will define the test: (21) Ψ n[fl... | https://arxiv.org/abs/2504.10652v1 |
We consider the null hypothesis: (24) H0:f=f0, σ−=σ−0, σ+=σ+0, δ=δ0. There is a total of 15 different possible alternatives, yet whenever dPZ(f, f0)> ϵor |σ−/σ−0−1|> ϵoccurs in the alternative, we can use Theorem 2 from (Choi and Schervish, 2007) to obtain exponential bounds on the Type II error by operating on EP(1−Ψ1... | https://arxiv.org/abs/2504.10652v1 |
across subgroups, as well as mean absolute error (MAE), calculated as:PJ j=1|ˆδj−δj|/J. To measure inferential performance, we include the multivariate coverage (Multi-Cover), which is calculated using Bonferroni correction for competing methods, and using the posterior ellipsoid method for HGPR and HGPR- CUT. Finally,... | https://arxiv.org/abs/2504.10652v1 |
RDD WITH HETEROGENEOUS TREATMENT EFFECTS 39 Table 8. Simulation result for GPR on DGP3 Settings Length Cover |Bias|RMSE MAE Multi-Cover V1/J (A-I), nj= 100 2.628 0.957 0.025 0.651 0.509 0.946 3.698 (A-II), nj= 100 2.637 0.961 0.018 0.644 0.504 0.959 3.711 (B-I), nj= 100 2.618 0.957 0.028 0.664 0.511 0.939 3.671 (B-II),... | https://arxiv.org/abs/2504.10652v1 |
1934−1953 treatment 0 1000 2000 3000 4000024681012 0 5 100.000.050.100.150.20 Parameter estimateDensityDensity − 1934−1953 treatment 0 5 100.000.050.100.150.20 0 1000 2000 3000 4000−20246810 IterationValueTrace − 1954−1973 treatment 0 1000 2000 3000 4000−20246810 0 5 100.000.050.100.150.20 Parameter estimateDensityDens... | https://arxiv.org/abs/2504.10652v1 |
arXiv:2504.10866v1 [math.ST] 15 Apr 2025Gaussian Approximation for High-Dimensional U-statistics with Size-Dependent Kernels Shunsuke Imai∗1and Yuta Koike†2 1Graduate School of Economics, Kyoto University 2Graduate School of Mathematical Sciences, University of To kyo April 16, 2025 Abstract Motivated by small bandwidt... | https://arxiv.org/abs/2504.10866v1 |
require some restrictions on the tuni ng parameters and data generating processes so that the influential functions have asymptotically linea r forms and the quadratic terms are ignored. Recently, Cattaneo et al. (2024 ) have established Edgeworth expansions for the DWAD estima tor (standardization and studentization ar... | https://arxiv.org/abs/2504.10866v1 |
which address U-statistics and their generalizations ( Chen ,2018 ;Chen and Kato ,2019 ,2020 ;Song et al. ,2019 ,2023 ;Cheng et al. ,2022 ;Chiang et al. ,2023 ;Koike ,2023 ). Nonetheless, existing CCK-type results for U-statistics are almost essentially concerned with the non- degenerate case. The exceptions are Chen a... | https://arxiv.org/abs/2504.10866v1 |
provid e several additional results alongside a general result (Theorem 1). Specifically, (i) Theorem 2presents a result for r= 2with a simple bound that is sufficiently sharp for practical purposes; and (ii) Co rollary 2serves a bound expressed in terms of moments of kernels rather than those of Hoeffding projecti ons. Th... | https://arxiv.org/abs/2504.10866v1 |
respectively. TheHoeffding decomposition ofJr(ψ)is given by Jr(ψ) =r/summationdisplay s=0/parenleftbiggn−s r−s/parenrightbigg Js(πsψ) =E[Jr(ψ)]+r/summationdisplay s=1/parenleftbiggn−s r−s/parenrightbigg Js(πsψ). (2) Whenψ∈L2(Pr), the variance of Jr(ψ)is decomposed as (cf. Eqs.(2.8) and (2.10) in D¨obler and Peccati (201... | https://arxiv.org/abs/2504.10866v1 |
small bandwidth asymptotics, we found th at the bound of Theorem 1, and hence Corollary 1, is not sharp enough to recover the weakest possible conditi on on the lower bound of bandwidths (see Remark 5). This is caused by the second term on the right-hand side of ( 7) whose derivation relies on a somewhat crude argument... | https://arxiv.org/abs/2504.10866v1 |
asymptotical ly linear, and the lower bound condition on the bandwidth coincides with, up to a logarithmic factor, the weakest condition to ensure that the variance of the estimator converges to 0asn→ ∞ . This indicates that our high-dimensional Gaussian approximation holds under nearly the same conditi ons as small ba... | https://arxiv.org/abs/2504.10866v1 |
x,y∈Rp. We begin by evaluating the order of the variances σ2 j:= Var[ˆθj],j∈[p]. Recall that by ( 4), σ2 j=n(n−1)2Var[Pψj(X1)]+n(n−1) 2Var[ψj(X1,X2)]. Let us evaluate the first term. Observe that for any integer m≥1, P(ψm j)(x) =1 hm−1 n/parenleftbign 2/parenrightbigm/integraldisplay RK(u)mfj(xj+uhn)du. (12) 12 In parti... | https://arxiv.org/abs/2504.10866v1 |
2.3, we assume that Pdoes not depend on n, and so does d. Assume that Phas density f. We aim to test whetherfis equal to a prespecified density function f0or not, based on the data X1,...,X n. Namely, we consider the following hypothesis testing problem: H0:f=f0 vsH1:f/ne}ationslash=f0. LetK:Rd→Rbe a bounded positive de... | https://arxiv.org/abs/2504.10866v1 |
nlogn→0asn→ ∞ for anyδ>0. We can take¯hn=e−√lognfor example. We have defined Hnso that the smallest bandwidth hn:= minHn satisfies the following condition. log5n=O(n2hd n)asn→ ∞. (24) Note that unlike Li and Yuan (2024 ), we take the maximum over a finite set of bandwidths. Apart fr om mathematical tractability, this is c... | https://arxiv.org/abs/2504.10866v1 |
Distinguishable separation r atesρn(α)are smaller; Li and Yuan (2024 ) need to replace√loglogninρad n(α)byloglogn. Note that the remaining√loglognfactor is not an artifact but is essential; see Theorem 1 in Ingster (2000 ). We also mention that in the context of two-sample testing, Schrab et al. (2023 ) have addressed ... | https://arxiv.org/abs/2504.10866v1 |
po ssible to examine cases with high- dimensional censored outcomes, such as top-coded incomes g rouped by occupations in large labor markets and high-dimensional corner solutions in markets w ith numerous goods, without relying on the normality and homoscedasticity of the error term assume d in the standard method ( T... | https://arxiv.org/abs/2504.10866v1 |
asymptotics ( Cattaneo et al. ,2018a ,b), due to the involvement of the inverse of a product of projection matric es in dominant terms in such settings. Addressing such situations is an important direction for fu ture research. 20 Appendix A Proofs for Section 2 Throughout the discussions, we will frequently use the fo... | https://arxiv.org/abs/2504.10866v1 |
right-hand side of ( 27). In our application, we regard X= (Xi)n i=1as a random element taking values in (Sn,S⊗n)and construct an exchangeable pair (X,X′)in a standard way. Then we apply Theorem 5to(Y,Y′) = (X,X′)with Gj=/summationtextr s=0s−1/parenleftbign−s r−s/parenrightbig {Js,X′(πsψj)−Js,X(πsψj)}; see Step 1 of th... | https://arxiv.org/abs/2504.10866v1 |
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