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0.24 0.27 0.12 0.15 0.18 0.21 0.24 Orig 0.219 0.202 0.189 0.180 0.173 0.174 0.157 0.144 0.135 0.128 (0.933) (0.930) (0.926) (0.909) (0.880) (0.940) (0.938) (0.933) (0.918) (0.881) Setting: b“1.2h TB 0.232 0.209 0.192 0.178 0.167 0.181 0.160 0.145 0.134 0.124 (0.863) (0.859) (0.860) (0.868) (0.867) (0.858) (0.858) (0.85... | https://arxiv.org/abs/2504.01535v1 |
arXiv:2504.01562v1 [math.ST] 2 Apr 2025Asymptotic analysis of the finite predictor for the fractional Gaussian noise P. Chigansky and M. Kleptsyna ABSTRACT . The goal of this paper is to propose a new approach to asymp- totic analysis of the finite predictor for stationary sequen ces. It produces the exact asymptotics of... | https://arxiv.org/abs/2504.01562v1 |
are orthogonal standard random variables. The real numbe rs(cj)and (aj), are called, respectively, the MA( ∞) and AR( ∞) coefficients of X. A positive measurable function ℓ(·), defined on some neighborhood of infinity, is called slowly varying if lim x→∞ℓ(tx)/ℓ(x) =1 for t>0. In the next theorem, ASYMPTOTIC ANALYSIS OF TH... | https://arxiv.org/abs/2504.01562v1 |
MA( ∞) and AR( ∞) coefficients (cn)and(an)of the process X. Then, in view of (1.8) and ( 1.9), the quantities associated with the finite predictor P[1,n−1]can also be expressed in terms of these coefficients. Thus it is po ssible to derive a useful representation for partial correlation coefficient s in terms of (cn)and(an... | https://arxiv.org/abs/2504.01562v1 |
1.4. Let X be the fGn ARIMA type process (1.10)where f 0(λ)is spectral density (1.16)of the fGn with d ∈(−1 2,1 2)\{0}and polynomials φ(·)and θ(·)as in Theorem 1.3. Then the partial correlation satisfies (1.7)and the relative prediction error follows the asymptotics δ(n)∼σ2 0d2 n,n→∞, (1.17) where σ2 0is given by (1.1)w... | https://arxiv.org/abs/2504.01562v1 |
be viewed as a general roadmap to the proof. Each theorem is pr oved in one of the separate sections that follow. Section 3summarizes the relevant properties of the sectionally holomorphic extension Q(·)of the fGn spectral density. Some calculations and auxiliary results are moved to the appendi ces. 2. Proof of Theore... | https://arxiv.org/abs/2504.01562v1 |
OF THE FINITE PREDICTOR FOR FGN 9 Then equation ( 2.1) can be rewritten as gL n(j)+gR n(j)+∞ ∑ k=−∞gn(k)γ(j−k)=γ(j),j∈Z. (2.5) It follows from ( 2.2)–(2.3) that the prediction error and the partial correlation are related to the sequences in ( 2.4) through the formulas: σ2(n)=gL n(0)and α(n)=gR n(n) gLn(0)n≥2. (2.6) 2.... | https://arxiv.org/abs/2504.01562v1 |
where arg (·)takes values in (−π,π], and let X(z):=/braceleftigg z−1X0(z),d∈(0,1 2), X0(z), d∈(−1 2,0).(2.18) This function is sectionally holomorphic in C\[0,1]and satisfies the homogeneous boundary condition X+(t) X−(t)=Q+(t) Q−(t),t∈(0,1). (2.19) Define the function h(s):=1 2i1 sin(πd)φ(es) φ(e−s)/parenleftbiggX(es) ... | https://arxiv.org/abs/2504.01562v1 |
the sequence which drives the ARIMA process. For the fGn, it has a rather complicated fo rm (2.7), which involves special functions, namely, polylogarithms. The n ext theorem derives its main properties relevant to our purposes. THEOREM 3.1. Let d∈(−1 2,1 2)\{0}. (i)The function Q (·)has the symmetries Q(z)=Q(z−1), Q(z... | https://arxiv.org/abs/2504.01562v1 |
CHIGANSKY AND M. KLEPTSYNA at any λ/\e}atio\slash=0. The function G0(z)is continuous on the punctured unit circle: lim z→eiλ,|z|<1G0(z)=2π/parenleftbig 1−2π/hatwidegn(λ)/parenrightbig f(λ)−2π/hatwidegR n(λ)= 2π/hatwidegL(λ)= lim z→eiλ,|z|>1G0(z),λ∈(−π,π]\{0}. The first equality here holds by definition ( 2.9), limit ( 4.... | https://arxiv.org/abs/2504.01562v1 |
S(z):=Φ0(z)+Φ1(z) 2X(z), D(z):=Φ0(z)−Φ1(z) 2X(z),(5.1) where Φ0(z)andΦ1(z)solve the Hilbert problem ( H1)-(H4) from Theorem 2.2. All functions on the right-hand side of ( 5.1) are sectionally holomorphic in C\ 18 P. CHIGANSKY AND M. KLEPTSYNA [0,1], and since X(z)is non-vanishing, it follows that S(z)andD(z)are section... | https://arxiv.org/abs/2504.01562v1 |
of X(z)aszapproaches infinity. This, in turn, governs the asymptotics of S(z)andD(z)in (5.2). Any k≤0 ensures that {S(z),D(z)}=O((z−1)−d)aszap- proaches 1, so that the functions on the right-hand side of ( 5.3) are integrable, and thus the Sokhotski-Plemelj theorem applies. The function i n (2.18) corresponds to the sim... | https://arxiv.org/abs/2504.01562v1 |
5.7) that the functions S(et)andD(et)are their particular solutions. The next lemma shows that these solutions belong to the funct ion class LN=/intersectiondisplay n≥N/braceleftbig f:/bardblAnf/bardbl<∞/bracerightbig for some N. LEMMA 5.4. The functions S (et)and D(et), defined in (5.1), belong to LN with N=n0+q+3, whe... | https://arxiv.org/abs/2504.01562v1 |
asz→∞from ( 3.10), this translates to Φ0(z)∼/parenleftbig aq+1,n+bq+1,n/parenrightbig zq, Φ1(z)∼/parenleftbig aq+1,n−bq+1,n/parenrightbig zq,z→∞, and, in turn, due to definitions ( 2.12), to G0(z)∼aq+1,n+bq+1,n, G1(z)∼aq+1,n−bq+1,n,z→∞, where we used the normalization φ(0)=1. The formulas ( 2.25) now follow from (2.11).... | https://arxiv.org/abs/2504.01562v1 |
( 6.11) is therefore β 2e⊤V−1e=−1 2σ2 0s0q ∏ j=1(−1/zj)1 ∏q+1 j=1(−ζj)=1 2σ2 0∏ j:|zj|<11 z2 j, where we used definition ( 6.5). The expression 1⊤V−1e, being the sum over the last column of V−1, equals the sum of coefficients of Pq+2(x), that is, 1⊤V−1e=Pq+2(1)=q+1 ∏ k=11−ζk 0−ζk=q+1 ∏ k=1ζk−1 ζk. (6.13) Similarly, e⊤V−1... | https://arxiv.org/abs/2504.01562v1 |
30 P. CHIGANSKY AND M. KLEPTSYNA As the radius of the contour tends to infinity and its base appr oaches the real line, the integral in ( A.9) converges: /contintegraldisplay C+(logζ)−2d(ζ+z) (ζ−z)3dζ→/integraldisplay0 −∞(log(−t)+πi)−2d(t+z) (t−z)3dt+ e−2πdi/integraldisplay1 0(logt−1)−2d(t+z) (t−z)3dt+/integraldisplay∞ ... | https://arxiv.org/abs/2504.01562v1 |
( A.7) satisfies B(t)=−2d Γ(1−2d)(logt)−2d−1 t/parenleftig 1+c1(logt)−1+O((logt)−2)/parenrightig , t→∞, for some constant c1∈C. Plugging this and ( A.6) into ( A.5) gives Q+(t)=1 2πd Γ(1−2d)(1−t)2(logt)−2d−1 t/parenleftig 1+c2 logt+O((logt)−2)/parenrightig ,t→∞, for some constant c2∈C. In view of ( 3.3), it follows ... | https://arxiv.org/abs/2504.01562v1 |
THE FINITE PREDICTOR FOR FGN 35 /integraldisplay1 2+iε z0Q′(ζ) Q(ζ)dζ=/integraldisplay1 2+iε z0U′(ζ)+iV′(ζ) U(ζ)+iV(ζ)dζ= /integraldisplay1 2+iε z0U′(ζ)U(ζ)+V′(ζ)V(ζ) U(ζ)2+V(ζ)2dζ+i/integraldisplay1 2+iε z0U(ζ)V′(ζ)−U′(ζ)V(ζ) U(ζ)2+V(ζ)2dζ= /integraldisplay1 2+iε z0d dζlog|Q(ζ)|dζ+i/integraldisplay1 2+iε z0d dζarccotU... | https://arxiv.org/abs/2504.01562v1 |
(3.9) and ( 3.11) and property ( B.2) we obtain ( 3.13). It remains to argue that s0is the only zero of Q(z)inside the unit disk and it is simple. The function Q(z), being holomorphic in the compact set Ω, may have at most finitely many zeros in it, say k, in addition to s0. In this case, let us redefine the region Ωby e... | https://arxiv.org/abs/2504.01562v1 |
( 2.21) are derived by the same argument. /square LEMMA C.2. There exists a constant C such that /vextendsingle/vextendsingleuj,n(t)−(ejt−1)−qn(t)/vextendsingle/vextendsingle≤Cn−1qn(t), (C.5) for all sufficiently large n. PROOF . The first series of equations in ( 2.21) can be rewritten as uj,n(t)−(ejt−1)=sin(πd) π/integ... | https://arxiv.org/abs/2504.01562v1 |
Dj,n(z)similarly to ( C.3) and estimate each of the obtained terms as in subsection C.1.2 using ( C.11 ). This yields the second estimate in ( 6.1). Appendix D. Proof of Theorem 6.2 Computation of the constants in ( 6.2) is based on a somewhat hidden connec- tion between equations ( 6.3) and the integral equations on t... | https://arxiv.org/abs/2504.01562v1 |
0e−zxe−t|x−y|sign(x−y)dx/parenrightbigg dydt= 1 Γ(α)/integraldisplay∞ 0tα−1 t−z/integraldisplay1 0u(y)/parenleftig e−ty−e−zy/parenrightig dydt+ 1 Γ(α)/integraldisplay∞ 0tα−1 t+z/integraldisplay1 0u(y)/parenleftig e−yz−e−z−t(1−y)/parenrightig dydt= 1 Γ(α)/integraldisplay∞ 0tα−1 t−z/parenleftig U(t)−U(z)/parenright... | https://arxiv.org/abs/2504.01562v1 |
It follows from ( D.4) that sin(πα/2) π/integraldisplay∞ 0e−τq(τ)dτ=lim t→∞t(q(t)−1), (D.12) and it remains to compute the value of the latter limit. By Lem maD.4 t(q(t)−1)=t/parenleftbigg1 bt−α/2Ψ(−t)−1/parenrightbigg ,t∈R+. Thus we need to establish the precise asymptotics of Ψ(−t)ast→∞. To this end, it follows from ... | https://arxiv.org/abs/2504.01562v1 |
z1,...,zr−1are distinct zeros of θ(z)with multiplicities µ1,...,µr−1re- spectively and let zr:=s0andµr=1, where s0is the only zero of Q(z)inside the unit disk, see Lemma 3.3. In order to keep the previous notations as much as possible, we will assume, without loss of generality, that ∑r j=1µj=q+1. With ζj’s as in ( 6.5... | https://arxiv.org/abs/2504.01562v1 |
(E.8) By the above construction none of Ej’s affects the last line of Vε. Hence E−1 j’s are also elementary matrices which do not affect the last lin e, i.e., E−1 1...E−1 ke=e. ASYMPTOTIC ANALYSIS OF THE FINITE PREDICTOR FOR FGN 53 Consequently, in view of ( 6.12), e⊤V−1 εE−1 1...E−1 ke=e⊤V−1 εe=µ1−1 ∏ k=01 −ζ(kε) 1...... | https://arxiv.org/abs/2504.01562v1 |
+O(n−2),n→∞. Plugging the obtained asymptotic expressions for a1,nandb1,ninto ( 2.25) we get σ2(n)−σ2 0=σ2 0d2+1 n+O(n−2),n→∞. and α(n)=d−1 n+O(n−2),n→∞. Similarly, for odd n, b1,n=1 2σ2 0/parenleftig 1−1 nd(1−d)/parenrightig +O(n−2),n→∞. The second equation in ( F.2) becomes /parenleftig 2S′ 0,n(−1)+(n+2+2c)S0,n(−1... | https://arxiv.org/abs/2504.01562v1 |
ients. Izv. Vysˇ s. Uˇ cebn. Zaved. Matematika , 1959(2(9)):158–166, 1959. [17] B. V . Pal′cev. Asymptotic behavior of the spectrum and eigenfunction s of convolution operators on a finite interval with the kernel having a homogeneous Four ier transform. Dokl. Akad. Nauk SSSR , 218:28–31, 1974. [18] B. V . Pal′tsev. Asy... | https://arxiv.org/abs/2504.01562v1 |
arXiv:2504.01781v2 [math.ST] 13 May 2025Proper scoring rules for estimation and forecast evaluation Kartik Waghmare and Johanna Ziegel∗ Abstract Proper scoring rules have been a subject of growing interest in rec ent years, not only as tools for evaluation of probabilistic forecasts but also as methods f or estimating ... | https://arxiv.org/abs/2504.01781v2 |
ignorance score in meteorology and is a strictly proper scoring rule ( Good ,1952), see Example 5for details. Example 3 (Continuous ranked probability score (CRPS)) .For a real-valued outcome y∈R, the continuous ranked probability score (CRPS) is defined as CRPS(P,y) =/integraldisplay (FP(x)−1{y≤x})2dx, whereFPis the cu... | https://arxiv.org/abs/2504.01781v2 |
saying that the expectation S(·,Q) =EY∼Q[S(·,Y)] is minimized pre- cisely when the model and data distributions are equal. It is reasonable to expect that given an iid sample {Yj}n j=1fromQwith empirical distribution ˆQn, the minimizer of the empirical mean S(·,ˆQn) = (1/n)/summationtextn j=1S(·,Yj) would give reasonab... | https://arxiv.org/abs/2504.01781v2 |
the lower score. Justification of this approach, even in the absence of stationarity of forecast-observation pai rs, can be found in Modeste et al. (2023). Inference methods for differences in predictive performance are discussed by Diebold and Mariano (2002);Giacomini and White (2006);Lai et al. (2011);Henzi and Ziegel ... | https://arxiv.org/abs/2504.01781v2 |
andstrongly equivalent if, in addition, α= 1. Definition 4 (Proper Scoring Rules) .A scoring rule S:P×Y →Ris called proper if S(Q,Q)≤S(P,Q) (4) for every P,Q∈Pandstrictly proper if Equation 4holds with equality if and only if P=Q. We have defined (strict) propriety of a scoring rule with rega rd to all measures in P. Sin... | https://arxiv.org/abs/2504.01781v2 |
and Raftery , 2007), sphere ( Takasu et al. ,2018), general Riemannian manifolds ( Mardia et al. ,2016), functional data or stochastic processes ( Hayati et al. ,2024), point processes ( Brehmer et al. ,2024) and dis- crete or continuous trajectories ( Bonnier and Oberhauser ,2024). With the goal of forecast evaluation... | https://arxiv.org/abs/2504.01781v2 |
should be noted that with the sole exception of the Kullback-Leibler divergence, these do not correspond to proper scoring rules ( Csisz´ ar ,1991), although they have been shown to be intimately related ( Mohamed and Lakshminarayanan ,2017;Gao et al. ,2020). Proper scoring rules have been studied from functional anal ... | https://arxiv.org/abs/2504.01781v2 |
divergence can be said to impose 8 a manifold structure on the space of probability measures wi th the Hessian ∇2 PHof the entropy acting as the metric tensor by relating the “distance” d(P+ ∆P,P) between two infinitesimally close points P,Q∈Pto the difference ∆ P=Q−Pbetween their “coordinates” PandP+ ∆Pas d(P,P+ ∆P)≈1 4... | https://arxiv.org/abs/2504.01781v2 |
are intimately connected to energy sta tistics (Sz´ ekely and Rizzo ,2013). The relationship between energy statistics and MMDs is ca refully stud- ied in Sejdinovic et al. (2013), see also Lyons (2013). For any conditionally negative definite kernel h, the entropy at Equation 11is concave, and hence the induced scoring... | https://arxiv.org/abs/2504.01781v2 |
forces in physics (Sz´ ekely and Rizzo ,2013). Unlike kernel scores with Gaussian or Laplacian kernels, energy scores arehomogeneous orscale-free , that is, replacing x,x′andybycx,cx′andcyrespectively for some c >0 only scales the score by cβ. The most popular kernel score is the CRPS, which is a special case of an ene... | https://arxiv.org/abs/2504.01781v2 |
negative kernel is a positive definite fu nction, then µis a finite measure. It follows that different translation invariant kernel score s place different weight on the different frequencies of the density. For example, the Gaussian and La placian kernel scores for λ >0, and the energy score for β∈(0,2) inRd, correspond to... | https://arxiv.org/abs/2504.01781v2 |
outcomes, it elegantly connects to the conceptually simple r point forecasting setting. Indeed, it reduces to the absolute error loss when point forecasts are i ssued. Moreover, homogeneity implies that scaling the observations scales the predictive distri bution accordingly even when the CRPS is away from the minimum.... | https://arxiv.org/abs/2504.01781v2 |
arbitrarily small neighborhood around x. LetY ⊆Rdbe open and let Pk acdenote the set of probability measures on Ywith Lebesgue densities, which are k-times differentiable. Definition 22. Alocal scoring rule of order kis a proper scoring rule S:Pk ac×Y →Rof the form S(P,y) =s(y,p(y),∇yp(y),...,∇k yp(y)) wheres:Y ×R×···×Rd... | https://arxiv.org/abs/2504.01781v2 |
Ghosh et al. (2025) establish minimax rate optimal- ity (up to log factors) of denoising. A fairly general characterization of local scoring rules of arbitrary order has been obtained by Parry et al. (2012) using a calculus of variations approach. In particular, th ere are no local scoring rules of odd orders, and all ... | https://arxiv.org/abs/2504.01781v2 |
Good ,1971) given by SSpherical (P,y) =−p(y) /bracketleftbig/integraltext p(x)2dµ(x)/bracketrightbig1/2andSpSpherical (P,y) =−p(y)α−1 /bracketleftbig/integraltext p(y)αdµ(x)/bracketrightbig1−1/α, respectively are instances of gf-scores with f(u) =uαandg(u) =−u1/αforα= 2 andα > 1, respectively. In this case, strict conc... | https://arxiv.org/abs/2504.01781v2 |
decomposed into components qua ntifying miscalibration, discrimination ability, and uncertainty. The familiar bias-variance decomposition of the squared er ror loss can be generalized to Bregman divergences ( Pfau,2013). Since proper scoring rules are extensions of Bregman loss functions, we can derive a similar decomp... | https://arxiv.org/abs/2504.01781v2 |
Estimating θvia maximum likelihood for such complicated models would re quire evaluating the normalization constant cθ=/integraltext expηθ(x) dx, which is prohibitively expensive even for moderately high dimensions. The Hyv¨ arinnen score allows for a much mor e tractable estimation of such models by circumventing the ... | https://arxiv.org/abs/2504.01781v2 |
merical weather prediction models ( Chen et al. ,2024), Bayesian generalized likelihood-free infer- ence ( Pacchiardi et al. ,2024b ), extrapolating neural networks ( Shen and Meinshausen ,2024), un- certainty quantification of dynamical systems ( B¨ ulte et al. ,2024), training neural SDEs ( Issa et al. , 2023), repres... | https://arxiv.org/abs/2504.01781v2 |
a different approach to mod- ifying scoring rules, which generally does not lead to kerne l scores even if the original score was a kernel score. For binary outcomes, Buja et al. (2005) argued that scoring rules should be tailored to the rel- ative cost implications of different types of errors. In a deci sion-theoretic ... | https://arxiv.org/abs/2504.01781v2 |
with his own Pt+1and be paid2S(Pt+1,y)−S(Pt,y) if the random quantity realized is y. The idea has received renewed interest due to recent developments co ncerning cryptocurrencies. Interesting further developments include Hanson (2007),Abramowicz (2007) and Berg and Proebsting (2009). Acknowledgements The authors would... | https://arxiv.org/abs/2504.01781v2 |
scoring rules , gradients, divergences, and entropies for paths and time series. Bayesian Anal. , pages 1–32, 2024. URL https://doi.org/10.1214/24-BA1435 . Valentin De Bortoli, Alexandre Galashov, J. Swaroop Guntup alli, Guangyao Zhou, Kevin Murphy, Arthur Gretton, and Arnaud Doucet. Distributional diffusion models with... | https://arxiv.org/abs/2504.01781v2 |
proper scoring rules. Bayesian Anal. , 10:479–499, 2015. URL https://doi.org/10.1214/15-BA942 . A. Philip Dawid and Paola Sebastiani. Coherent dispersion c riteria for optimal experimental design. Ann. Statist. , 27:65–81, 1999. URL http://www.jstor.org/stable/120118 . A. Philip Dawid, Steffen Lauritzen, and Matthew Par... | https://arxiv.org/abs/2504.01781v2 |
redictive ability. Econometrica , 74: 1545–1578, 2006. URL http://www.jstor.org/stable/4123083 . GJOpen. Good Judgment Open FAQ, 2025. URL https://www.gjopen.com/faq . https://www.gjopen.com/faq , Accessed: 2025-03-03. Tilmann Gneiting. Making and evaluating point forecasts. J. Amer. Statist. Assoc. , 106:746–762, 2011... | https://arxiv.org/abs/2504.01781v2 |
sotonic distributional regression. J. R. Stat. Soc. Ser. B Stat. Method , 83:963–993, 2021. URL https://doi.org/10.1111/rssb.12450 . Hajo Holzmann and Matthias Eulert. The role of the informati on set for forecast- ing—with applications to risk management. Ann. Appl. Stat. , 8:595–621, 2014. URL http://www.jstor.org/st... | https://arxiv.org/abs/2504.01781v2 |
Proceedings of the 32nd International Conference on Machine L earning , volume 37 of Proceedings of Machine Learning Research , pages 1718–1727, Lille, France, 07–09 Jul 2015. PMLR. URL https://proceedings.mlr.press/v37/li15.html . Lina Lin, Mathias Drton, and Ali Shojaie. Estimation of high -dimensional graphical mode... | https://arxiv.org/abs/2504.01781v2 |
http://davidpfau.com/assets/generalized_bvd_proof.p df. http://davidpfau.com/assets/generalized_bvd_proof.p df, Accessed: 2025-02-25. Romain Pic, Cl´ ement Dombry, Philippe Naveau, and Maxime Ta illardat. Distributional regression and its evaluation with the CRPS: Bounds and convergence of t he minimax risk. Int. J. Fo... | https://arxiv.org/abs/2504.01781v2 |
ation through the lens of distributional regression. J. R. Stat. Soc. Ser. B Stat. Method , 2024. URL https://doi.org/10.1093/jrsssb/qkae108 . qkae108. 31 Emir H. Shuford, Arthur Albert, and H. Edward Massengill. Ad missible probability measurement procedures. Psychometrika , 31:125–145, 1966. URL https://doi.org/10.10... | https://arxiv.org/abs/2504.01781v2 |
volume 2. MIT press Cambridge, MA, 2006. Robert C. Williamson and Zac Cranko. The geometry and calcul us of losses. J. Mach. Learn. Res. , 24, 2023. URL https://www.jmlr.org/papers/volume24/22-0987/22-098 7.pdf . Hiroki Yanagisawa. Proper scoring rules for survival analy sis. In Proceedings of the 40th International Co... | https://arxiv.org/abs/2504.01781v2 |
9). Proof of Theorem 16.Suppose that the kernel score is proper. Then hmust be conditionally neg- ative definite and Berg et al. (1984, Proposition 3.3.2) implies that there exists a Hilbert spa ceH, a subset {ψy}y∈Y⊆Hand a function f:Y →R, such that h(x,y) =∝ba∇dblψx−ψy∝ba∇dbl2 H+f(x) +f(y) (21) for everyx,y∈ Y. This i... | https://arxiv.org/abs/2504.01781v2 |
get (using symmetry of K) 2[H(Q)−H(P)] =∝a\}b∇acketle{thP+hQ,Q−P∝a\}b∇acket∇i}ht =∝a\}b∇acketle{thP,Q∝a\}b∇acket∇i}ht−∝a\}b∇acketle{thP,P∝a\}b∇acket∇i}ht+∝a\}b∇acketle{thQ,Q∝a\}b∇acket∇i}ht−∝a\}b∇acketle{thQ,P∝a\}b∇acket∇i}ht =/integraldisplay /integraldisplay K(x,y) dP(x) dQ(y)−/integraldisplay /integraldisplay K(x,y)... | https://arxiv.org/abs/2504.01781v2 |
Rd/bracketleftBig 1−eiu⊤y/bracketrightBig cαdµ(u) =/integraldisplay Rd/bracketleftBig 1−eiu⊤y/bracketrightBig dµ(u/c). Again, using the evenness of µwe can write this as /integraldisplay Rd/bracketleftBig eiu⊤y−1−∝a\}b∇acketle{ty,u∝a\}b∇acket∇i}ht1{∝ba∇dblu∝ba∇dbl ≤1}/bracketrightBig cαdµ(u) =/integraldisplay Rd/bracke... | https://arxiv.org/abs/2504.01781v2 |
Estimating hazard rates from δ-records in discrete distributions Mart ´ın Alcaldea,b, Miguel Lafuentea,b, F. Javier L ´opeza,b,∗, Lina Maldonadoc, Gerardo Sanza,b aDepartment of Statistical Methods, University of Zaragoza, Spain bInstitute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza... | https://arxiv.org/abs/2504.01836v1 |
limited use of ∗Corresponding author Email addresses: malcalde@unizar.es (Mart ´ın Alcalde ),miguellb@unizar.es (Miguel Lafuente ),javier.lopez@unizar.es (F. Javier L ´opez ),lmguaje@unizar.es (Lina Maldonado ),gerardo.sanz@unizar.es (Gerardo Sanz ) Preprint submitted to arXiv April 3, 2025arXiv:2504.01836v1 [math.ST] ... | https://arxiv.org/abs/2504.01836v1 |
their distributions. The treatment of the continuous case, which can be approached through truncation and smoothing, will be addressed in future work. For the estimation of hazard rate function, we first express the likelihood of the sample of δ-records in terms of this function. This likelihood, which takes the form o... | https://arxiv.org/abs/2504.01836v1 |
is less restrictive than the standard record condition. As a result, whenδ < 0,δ-records include more observations, e ffectively addressing the issue of record scarcity—a common limitation in record-based statistical inference from i.i.d. sequences. Throughout the rest of the paper, we consider onlyδ<0. Furthermore, si... | https://arxiv.org/abs/2504.01836v1 |
been a near-record with parameter ℓ+1, plus 1 if it has been a record. The term nℓ mis 1 plus the number of records whose value lies between mandm+ℓ, plus the total number of near-records with parameter ℓ+1 associated with each of those records. We use the uppercase versions of aℓ m,vℓ mandnℓ m(Aℓ m,Vℓ mandNℓ m) for th... | https://arxiv.org/abs/2504.01836v1 |
that dj,krepresents the probability that Xiis either equal to jor greater than j+kconditioned on{Xi≥j}. In other words, dj,k=hj+F(j+k|j). Theorem 3.3. Let k∈Nand j∈Z+such that j +k≤rn. The estimator ˆhj,ktakes values in Q∩[0,1)and its probability mass function is P(ˆhj,k=q)=(dj,k−hj)∞X i=1 bi−1 ai! hai j(1−dj,k)(b−a)i−... | https://arxiv.org/abs/2504.01836v1 |
j+1, Ak j). To characterize the distribution of Aj,k, we need the joint distribution of the random vector in (12). Unfortunately, the vectors in di fferent lines of (12) are not independent. We can reorder the components of Aj,kin a di fferent way to 6 make the analysis easier. In fact, the vector may be decomposed int... | https://arxiv.org/abs/2504.01836v1 |
by Lemma 7.1, P(V1 j=a,V0 j+1=0)=P(A0 j=a,A0 j+1=0)=ha j(1−hj)(1−hj+1)=ha jF(j+1|j), which satisfies (14). Now, for b>0, by (11), we have {V1 j=a,V0 j+1=b}=[ a0+a1=a{V0 j=a0,A1 j=a1,V0 j+1=b}. To compute the probability of this event, we must consider the di fferent combinations of a0anda1such that a0+a1=a. Note that t... | https://arxiv.org/abs/2504.01836v1 |
is crucial in the proof of Theorem 3.3. Proposition 3.7. Let k∈Nand j∈Z+such that j +k≤rn. We have that a)Vk j,kX ℓ=1Vk−ℓ j+ℓ∼Geom∗(hj,1−dj,k). b) The random variable Nk jfollows a Geometric distribution (starting at 1) with parameter d j,k−hj, that is, Nk j∼Geom( dj,k−hj). Proof. Both statements follow... | https://arxiv.org/abs/2504.01836v1 |
( hj)−log (u)) (1−u)m+3du +(dj,k−hj)∞X m=0(1−dj,k)m(m+1)2 hm+1 jZhj 0um+2(log ( hj)−log (u)) (1−u)m+3du =dj,k−hj hjZhj 0u(log ( hj)−log (u)) (1−u)(1−1+hj−dj,k hju)2du+dj,k−hj h2 jZhj 0u2(log ( hj)−log (u)) (1−u)(1−1+hj−dj,k hju)3du =(dj,k−hj)(hj+dj,k−1)hj (1+hj−dj,k)3Li2(1+hj−dj,k)+(dj,k−hj)(2hj+dj,k−1)hj (1+hj−dj,k)3l... | https://arxiv.org/abs/2504.01836v1 |
case where j+k≥rn, to obtain a MLE of hjwith the distribution of Theorem 3.3, we may use ˆhj,k′with k′such that j+k′≤rn. In this scenario, all the Vl min Equation (6) are equal to the corresponding ˜Vℓ min (10). 4. Applications in Statistical Inference 4.1. Confidence intervals for h j We may use the exact distribution... | https://arxiv.org/abs/2504.01836v1 |
from F0. The p-value is then estimated as the proportion of simulations the likelihood ratio statistic obtained from the bootstrap sample is smaller than the one obtained with the original sample. If the null hypothesis is composite, we compute ˆθ, the MLE of θ∈Θ, simulate Bsamples from Fˆθand estimate the p-value in t... | https://arxiv.org/abs/2504.01836v1 |
under a monotonicity assumption is solved in Ayer et al. (1955). Using their result and the product form of the likelihood in (7), we can readily derive that the increasing and decreasing monotonic hazard rate estimators, ˆhinc. j,kand ˆhdec. j,k, are given by ˆhinc. j,k=max 0≤ℓ≤jmin j≤m≤rnmP i=ℓVk i mP i=ℓNk i, ˆhdec.... | https://arxiv.org/abs/2504.01836v1 |
1,5,4,6,9,10 6 2 1,5,3,4,3,3,3,3,6,4,9,10,8 13 3 1,5,3,2,4,3,2,3,3,3,6,4,9,6,10,7,8 17 Table 3:δ-records for the sample in Table 2. kj1 2 3 4 5 6 7 8 9 10 N 1 0.500 0.000 0.000 0.333 0.333 0.500 0.000 0.000 0.333 1 6 2 0.500 0.000 0.625 0.400 0.333 0.500 0.000 0.250 0.500 1 13 3 0.500 0.200 0.500 0.400 0.333 0.500 0.20... | https://arxiv.org/abs/2504.01836v1 |
occurrence of earthquakes with magnitudes of 7.5 and above worldwide from 1950 to 2023. United States Geological Survey (USGS). 0 2 4 6 8 100.00 0.05 0.10 0.15 0.20 Number of earthquakes per yearObserved vs. Expected rel. frequencyObserved Poisson Prob. Figure 5: Comparison between observed relative frequency of earthq... | https://arxiv.org/abs/2504.01836v1 |
well-established topic in the literature. The inclusion of near-records in the sample has been shown to be advantageous in parametric inference (Gouet et al. (2014), Gouet et al. (2020), L´opez-Bl ´azquez & Salamanca-Mi ˜no (2013)). This is the first paper to address nonparametric inference based on δ-records. Our resu... | https://arxiv.org/abs/2504.01836v1 |
26(4), 641–647. Ahsanullah, M. (2004). Record values–theory and applications . University Press of America. Ahsanullah, M., Nevzorov, V . B. (2015). Records via probability theory . Springer. Arabi Belaghi, R., Arashi, M., Tabatabaey, S. M. M. (2015). Improved estimators of the distribution function based on lower reco... | https://arxiv.org/abs/2504.01836v1 |
S., Ansari, F., Azhad, Q. J., Kabdwal, N. C. (2024). Moments and inferences of inverted topp-leone distribution based on record values. International Journal of System Assurance Engineering and Management , 15, 2623–2633. Kumar, D., Kumar, M., Saran, J. (2024). Power Generalized Weibull distribution based on record val... | https://arxiv.org/abs/2504.01836v1 |
reliability from unit-Burr III distribution under records data. Mathematical Biosciences and Engineering , 20(7), 12360–12379. Zhao, X., Wei, S., Cheng, W., Zhang, P., Zhang, Y ., Xu, Q. (2023). Upper record values from the generalized Pareto distribution and associated statistical inference. Communications in Statisti... | https://arxiv.org/abs/2504.01836v1 |
arXiv:2504.01837v1 [cs.IT] 2 Apr 20251 Cramér–Rao Inequalities for Several Generalized Fisher Information Hao Wu and Lei Yu Abstract The de Bruijn identity states that Fisher information is the half of the derivative of Shannon differential entropy along heat flow. In the same spirit, in this paper we introduce a genera... | https://arxiv.org/abs/2504.01837v1 |
Shannon entropy powe r, i.e.,N1(X) =N(X). Intuitively, Fisher information also has a natural extensi on induced by the Rényi entropy. Consider the heat equation in ( 1) again. Then, one can easily verify that ∂ ∂thα(Yt) =α 2/integraltext |∇pt|2pα−2 tdx/integraltext pα tdx. (6) Comparing this identity with the de Bruijn... | https://arxiv.org/abs/2504.01837v1 |
NOTATIONS UNDEFINED HERE IN THE CORRESPONDING THEOREMS . Type of Fisher information Dimension nand parameter α Cramér–Rao inequality Rényi–Fisher information Iα (α,n)∈ Θ∩/braceleftBig (α,n)∈R+×Z+:α >n n+2/bracerightBig e2 nhα(Kf)Iα(f)≥rα,n α-weighted Rényi–Fisher information /tildewideIα n∈Z+,α∈/parenleftBig n n+2,1/pa... | https://arxiv.org/abs/2504.01837v1 |
Iα(X) =α/integraltext(2 α)2f4 α−2|∇f|2 (/integraltextf2 αdx)2f2(α−2) α (/integraltextf2 αdx)α−2dx /integraltextf2 (/integraltextf2 αdx)αdx =4 α/integraltext |∇f|2dx/integraltext f2dx. Therefore, ( 9) can be equivalently reformulated as /parenleftbigg/integraldisplay f2dx/parenrightbigg 2 n(α−1)+1 ≤4 αrα,n/integraldispl... | https://arxiv.org/abs/2504.01837v1 |
f′2dx/parenrightbigg1−α 2α/parenleftbigg/integraldisplay f2dx/parenrightbiggα+1 2α (14) whenα∈(0,1). In the case (i), 1< α <∞, then ( 11) is equivalent to ( 13) and corresponds to ( 71) in the appendix with p= 2,γ= 2 α,q=α+1 αandβ=2(α−1) α. Therefore, by Lemma 8in the appendix,/parenleftig 4 αrα,1/parenrightigα−1 α+1... | https://arxiv.org/abs/2504.01837v1 |
Thus,p(x) =1 2e−|x|(x∈R)represents an extremizer in ( 12) forα= 0. Finally, the limit in item (i) of Theorem 1leads to the optimal constant R∞,1= lim α→∞αrα,1 = lim α→∞2απ α−1/parenleftbigg2α α+1/parenrightbiggα+1 α−1 Γ/parenleftig α+1 2(α−1)/parenrightig Γ/parenleftig α α−1/parenrightig 2 = 2π/parenleftigg Γ/... | https://arxiv.org/abs/2504.01837v1 |
|∇f|2dx/parenrightbiggθ 2/parenleftbigg/integraldisplay f2dx/parenrightbigg1−θ 2 (21) withθ= 1−αwhen0< α <1. Both inequalities enters the framework of the special case o f Gagliardo–Nirenberg’s inequalities in ( 72) in the appendix. Note that ( 20) corresponds to Gagliardo–Nirenberg’s inequalities ( 72) in the appendix... | https://arxiv.org/abs/2504.01837v1 |
we need to introduce the limiting case θ= 1 in (28), which corresponds to α=n−2 n. It amounts to the classical Sobolev inequality /parenleftbigg/integraldisplay f2n n−2dx/parenrightbiggn−2 2n ≤Cn/parenleftbigg/integraldisplay |∇f|2dx/parenrightbigg1 2 , (30) which is known to hold true with the best constant Cn=1/radic... | https://arxiv.org/abs/2504.01837v1 |
III. R ÉNYI –FISHER INFORMATION Forα∈(0,1)∪(1,∞), recall that the Rényi–Fisher information Iαis defined in ( 7). In this section, we derive the Cramér–Rao inequalities for Iαand connect these inequalities to the complete monotonicit y for Rényi entropy power. A. Cramér–Rao inequality for Rényi–Fisher information For a r... | https://arxiv.org/abs/2504.01837v1 |
Kf Γ/parenleftig α 1−α/parenrightig Γ/parenleftig 1 1−α/parenrightig /parenleftig Γ/parenleftig α+1 2(1−α)/parenrightig/parenrightig2 2 . (38) Forn= 1 andα >1, Iα(f)≥2 (3α−1)Kf/parenleftbigg2α α+1/parenrightbiggα+1 α−1/parenleftig Γ/parenleftig α+1 2(α−1)/parenrightig/parenrightig2/parenleftig Γ/pare... | https://arxiv.org/abs/2504.01837v1 |
derivative increases, i.e., (−1)k+1dk dtkh(Xt)≥0 (43) for anyk≥1. The complete monotonicity for Rényi entropy power corresp onds to replacing the differential entropyhin (43) with the Rényi entropy power Nα, for some suitbale range of α. In the following our Cramér– Rao inequality for Rényi entropy ( 37) is applied to ... | https://arxiv.org/abs/2504.01837v1 |
have proven the statement (i) about Rényi entropy power . The proofs of statement (ii) and (iii) about Rényi entropy are similar to that of Corollar y5, with some modification of replacing Nα(Xt)with N1 2α(Xt). Inspired by the concavity of Nα+1 2α(Xt)when dimension n= 1, we pose the following question. Problem 1. For an... | https://arxiv.org/abs/2504.01837v1 |
n(α−1)+2, (50) then substituting ( 50) into ( 49), we have the following. Lemma 4. Ifα >n n+2,then /parenleftbiggσ2(f) σ2(Bα)/parenrightbiggn(α−1) 2+1 ≥/tildewideNα(f) /tildewideNα(Bα)(51) with equality if the probability density function fis the generalized Gaussian Bα. Combing Lemma 2and Lemma 4, we obtain a sharp Cr... | https://arxiv.org/abs/2504.01837v1 |
refined matrix version of the Cramér–Rao inequality. Definition 1. Forn n+2< λ andλ/\e}atio\slash= 1, define the n-dimensional probability density as gλ,K(x) =Aλ(1−(λ−1)βλxTK−1x)1 λ−1 + 20 withβλ=1 2λ−n(1−λ), and normalization constants Aλ= Γ(1 1−λ)(βλ(1−λ))n 2 Γ(1 1−λ−n 2)πn 2|K|1 2ifn n+2< λ <1, Γ(λ λ−1+n 2)(βλ(λ−... | https://arxiv.org/abs/2504.01837v1 |
ˆh2(X), while,I2(X) =ˆI2(X) 1−ˆh2(X).DenoteK(X),Kf orKfor short, as the covariance of X∼f, see ( 33). In Section II, we have obtained the entropic isoperimetric inequality Nα(X)Iα(X)≥rα,n. Here we only consider the case of α= 2. Denote Θ2,nas the region of the dimension nsuch that sharp inequality N2(X)I2(X)≥r2,n holds... | https://arxiv.org/abs/2504.01837v1 |
of Proposition 3, we have the following corollaries. Corollary 6. For any random variable X∼f, such that/integraltext f2dx <1, and any j≥1, (−1)j−1 2dj−1 dtj−1I2(Xt)≥(j−1)! 512/parenleftbig Γ/parenleftbig3 2/parenrightbig/parenrightbig2/parenleftbig Γ/parenleftbig5 2/parenrightbig/parenrightbig6 135(K+t)/parenleftbig... | https://arxiv.org/abs/2504.01837v1 |
address the one-dimensional Rényi-entropic isoperimet ric inequality Nα(X)Iα(X)≥rα,1, we define the functions yp,γ=yp,γ(t)fort≥0by yp,γ(t) = (1+t)p p−γ ifp < γ, e−tifp=γ, (1−t)p p−γ1[0,1](t)ifp > γ. Defineyp,γ,β implicitly as follows. Put yp,γ,β(t) =u,0≤u≤1, with t=/integraldisplay1 u/parenleftig sγ(1−sβ)/parenrig... | https://arxiv.org/abs/2504.01837v1 |
where a2,α= (1−α 2)1 2 β/parenleftBig 1 2,3+α 2(1−α)/parenrightBig,ifα <1, 1 Γ(1 2), ifα= 1, (α−1 2)1 2 β(1 2,2 1−α), ifα >1. For dimension n≥2, define Gn(x) := b2,α/parenleftig 1−α−1 (n+2)(α+1)−2n|x|2/parenrightig2 α−1 +,ifα/\e}atio\slash= 1, b2,1e−1 2|x|2, ifα= 1,(76) 26 where b2,α= ... | https://arxiv.org/abs/2504.01837v1 |
Erwin Lutwak, Songjun Lv, Deane Yang, and Gaoyong Zhang . Extensions of Fisher information and Stam’s inequality. IEEE transactions on information theory , 58(3):1319–1327, 2012. [16] B. Nagy. Über Integralungleichungen zwischen einer Fu nktion und ihrer Ableitung. Acta Univ. Szeged. Sect. Sci. Math. , 10:64–74, 1941. ... | https://arxiv.org/abs/2504.01837v1 |
arXiv:2504.02096v1 [econ.EM] 2 Apr 2025Estimation of the complier causal hazard ratio under depend ent censoring Gilles Crommen1 gilles.crommen@kuleuven.beJad Beyhum2 jad.beyhum@kuleuven.be Ingrid Van Keilegom1 ingrid.vankeilegom@kuleuven.be 1ORSTAT, KU Leuven, Naamsestraat 69, 3000, Leuven, Belgium 2Department of Econ... | https://arxiv.org/abs/2504.02096v1 |
be identified from the conditional distribution of Tgiven (Z,X⊤), whereXrepresents a vector of observed exogenous covariates. To address the issue of en dogeneity, instrumental variable (IV) methods are commonly used ( Imbens and Angrist ,1994;Angrist and Imbens ,1995;Angrist et al. , 1996;Abadie,2003). These methods us... | https://arxiv.org/abs/2504.02096v1 |
(ATE) is not point- identified. Therefore, Imbens and Angrist (1994) andAngrist et al. (1996) proposed focusing on specific subpopulations rather than attempting to identify effects for the entire population. In par- ticular, they demonstrated that the IV framework could iden tify the local average treatment effect (LATE),... | https://arxiv.org/abs/2504.02096v1 |
the covariates requires that, given 2 distinct continuous covariates X1andX2, the conditional distribution of Tdoes not depend onX1and the conditional distribution of Cdoes not depend on X21. This type of covariate restriction was also used by Deresa and Van Keilegom (2024b) to allow for both TandCto follow a semi-para... | https://arxiv.org/abs/2504.02096v1 |
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