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observations yR`nforn“1,...,T ´Rused to evaluateCnpQ,P q “śn m“1RmpQ,P qand implement SSRE using boundary values kland ku. If SSRE terminates for nďT´Rthen they will use the method selected to produce their forecast. If the event CQXP T´Roccurs, however, an adjustment to the decision process must be applied, assuming t... | https://arxiv.org/abs/2505.09090v1 |
strength of evid ence for, or against, HQcan be gauged by the magnitude of EPrexpt´ω¨signtEPr∆npQ,P qsu∆npQ,P qus, for some unknown values of ωP r0,8q. A formal statement that clarifies the nature of this relationship is given in the following lemma. LEMMA 2.Under the conditions of Lemma 1, and for any ωP r0,|hP|s, # EP... | https://arxiv.org/abs/2505.09090v1 |
errors –klcontrols the probability of terminating by selecting metho d Q when HPholds, andkucontrols the probability of terminating by selecting metho d P when HQholds. 4.2. Implementation and False Discovery. Consider again the scenario introduced in Section 3.4, where a sample of T`τvalues ofY,y1,...,yT`τ, is partiti... | https://arxiv.org/abs/2505.09090v1 |
which shows that the error of falsely rejecting a tru eHn Qamong the H1 Q,...,HN Qcan be controlled. THEOREM 1.If all ofHp1q Q,...,HpNq Qare true, then, for any πP p0,1q, andNě2, PrP! Epωq Něπ´1) ďπ PROOF . From Markov’s inequality, PrP! Epωq Něπ´1) ďEPrEpωq Nsπ“π1 NNÿ j“1j NEP” Cpωq pjqpQ,P qı “πNpN`1q 2N2 ďπ, where t... | https://arxiv.org/abs/2505.09090v1 |
If all the statistics behave similarly, this is meaningful eviden ce that we have not inadvertently chosenωąωP. The behavior of the statistics can be plotted visually acro ss the in-sample set and the resulting figures compared to determine that their be havior is similar. Assuming that the values of ωchosen deliver sta... | https://arxiv.org/abs/2505.09090v1 |
to selectkuto control the probability of falsely favouring P when the be nchmark Q holds. To this end, define the event TP N´j:“! EAvg N´jěku) , kuą1, j “0,1...,K which represents termination of SSRE and selection of metho d P at timeN´jďN. We can control the behavior of Pr´ TP N|H1 Q,...,HN Q¯ - the probability of sele... | https://arxiv.org/abs/2505.09090v1 |
deliver accurate results in exactly the situations where standard testing methods will fail. Across the experiments that follow, we compare the accuracy of the different models in terms of log-scores and root-mean-squared error. We compar e the accuracy of our SSRE scheme and the DM test across M“1000 replications from... | https://arxiv.org/abs/2505.09090v1 |
and con- siderρP t1,0.90u. The case where ρ“1corresponds to model Q being more accurate than modelP, while the case ρ“0.90covers the converse. Under the stipulated choice of ku, our theoretical results imply that the empirical rejection fre quency should be less than 11% in the first case pρ“1q, while in the second ( ρ“... | https://arxiv.org/abs/2505.09090v1 |
ω“1{2. Given this, we consider two scenarios: in the first case we generate data so that method Q is approx- imately correct, which corresponds to setting ω“1{2; in the second, we set ω“1{4so that the forecast model Q is less accurate than model P. 5.2.4. Results. We give the results across the different examples in Tab... | https://arxiv.org/abs/2505.09090v1 |
The av erage rejection probabilities under HQandHPfor SSRE and DM are presented in columns SSRE and DM respectiv ely. Example 1 Example 2 Example 3 SSRE DM SSRE DM SSRE DM MSEHQ 0.010 0.003 0.007 0.001 0.050 0.020 HP 0.950 0.810 0.012 0.003 0.470 0.460 Log ScoreHQ 0.001 0.001 0.050 0.001 n/a n/a HP 1.000 1.000 0.001 0.... | https://arxiv.org/abs/2505.09090v1 |
nondegenerate and that the probability of at least one of EQ morEP mis non-zero, m“ 1,2,.... Letpm“PrpklăCmpQ,P q ăkuqq. ThenPrpNąnq ď̺n“śn m“1pmwhere hmă1,m“1,...,n , and the sequence ̺n,n“1,2,..., converges as nincreases since it is monotonically nonincreasing and bounded below by zero. Suppose that limnÑ8śn m“1pmą0.... | https://arxiv.org/abs/2505.09090v1 |
B. M. (1969). Moments of a stopping rule related to the centr al limit theorem. Annals of Mathemetical Statistics 401236-1249. DEGROOT , M. H. (1970). Optimal Statistical Decisions . McGraw-Hill, New York. DEY, N., M ARTIN , R. and W ILLIAMS , J. P. (2024a). Anytime-Valid Generalized Universal Infer ence on Risk Minimi... | https://arxiv.org/abs/2505.09090v1 |
W., F RAZIER , D. T. and R AMÍREZ -HASSAN , A. (2022). Optimal probabilistic forecasts: When do they work ?International Journal of Forecasting 38384– 406. NOCEDAL , J. and W RIGHT , S. (2006). Numerical Optimization , 2nd ed. Springer. PATTON , A. J. (2020). Comparing possibly misspecified forecasts. Journal of Busines... | https://arxiv.org/abs/2505.09090v1 |
arXiv:2505.09098v1 [cs.IT] 14 May 20251 Statistical Mean Estimation with Coded Relayed Observations Yan Hao Ling, Zhouhao Yang, and Jonathan Scarlett Abstract We consider a problem of statistical mean estimation in which the samples are not observed directly, but are instead observed by a relay (“teacher”) that transmi... | https://arxiv.org/abs/2505.09098v1 |
we focus on the large deviations regime; our reason- ing/motivation for this is discussed in Section I-B. Specifically, for some constant ε >0(not depending on n), we seek to minimize P(|ˆθn−θ∗|> ε) (1) and analyze how quickly this quantity decays as a function of n. We will generally assume that εandPZ|Ware known thro... | https://arxiv.org/abs/2505.09098v1 |
tools from binary hypothesis testing (e.g., see [14], [15], [16]), and our strategy will be to combine the results of several hypothesis tests to produce the final mean estimate. In Remark 15, we will discuss how this hypothesis testing approach can outperform approaches based on the sample mean, even in the setting of... | https://arxiv.org/abs/2505.09098v1 |
to write the results in terms of two error exponents associated with the source and the channel individually, defined as follows. Definition 1. For a given class PXof source distributions and an accuracy parameter ε >0, we define the source exponent (ordirect observation exponent )Esrc ε(PX)as the highest Esuch that th... | https://arxiv.org/abs/2505.09098v1 |
more formally in the proof of Lemma 4) can be analyzed in a similar manner to the non-causal setting, giving the following. Lemma 4. For any distribution class PXwhose set of possible means is Θ∗= [0,1], the one-shot estimate-and- forward protocol described above with parameters (λ, δ)attains an error exponent of min{(... | https://arxiv.org/abs/2505.09098v1 |
block, we simply have the teacher send information about the sum or average of its received symbols using a general codebook (whose codewords need not be of the form 1. . .10. . .0). We then consider a decoder that performs binary hypothesis tests on a number of (θ, θ′)pairs, constructs a set Sofθvalues that are favore... | https://arxiv.org/abs/2505.09098v1 |
the same length over W, we adopt the shorthand ρ(⃗W,⃗W′, PZ|W) =ρ P(·|⃗W),P(·|⃗W′) (8) withP(·|⃗W)being the distribution of the output string ⃗Zwhen ⃗Wis sent over the memoryless channel PZ|W, and similarly for P(·|⃗W′). Similarly, we write dB(⃗W,⃗W′, PZ|W) =−logρ(⃗W,⃗W′, PZ|W). The following tensorization property o... | https://arxiv.org/abs/2505.09098v1 |
have the following as M→ ∞ : cM≥M M−1−O1 M2 , c M≤1 +O1√log log M , (16) and hence limM→∞cM= 1. The following corollary for low rates will also be useful; this serves as a natural counterpart to Lemma 8. Corollary 12. For any DMC PZ|Wand any C > 0, there exists a length- kcodebook over PZ|Wcontaining kC codewords s... | https://arxiv.org/abs/2505.09098v1 |
from Section I-C in two ways: •In Figure 2, we fix the BSC parameter p= 0.1and vary ε∈ 0,1 2 ; •In Figure 3, we fix the accuracy parameter ε= 0.1and vary p∈ 0,1 2 . We see that the the achievable error exponents for simple forwarding and one-shot estimate-and-forward strategies are mostly highly suboptimal, though ... | https://arxiv.org/abs/2505.09098v1 |
(Zjk+1, Zjk+2, . . . , Z (j+1)k) (19) to be the j-th block received by student (excluding the ignored one), and let ⃗Zrefer to a generic block. 2) Teacher strategy: The teacher and student first agree on a length- kcodebook over BSC( p) with k+1codewords such that any two codewords differ in at leastk 2−o(k)bits (see L... | https://arxiv.org/abs/2505.09098v1 |
leads to the following: g(θ′, α,⃗Z) g(θ∗, α,⃗Z)=s P(α|θ′) P(α|θ∗), (28) which implies X αP(α|θ∗)g(θ′, α,⃗Z) g(θ∗, α,⃗Z)=X αP(α|θ∗)s P(α|θ′) P(α|θ∗)= exp( −k·dB(θ, θ′)). (29) •When α̸=α′, substituting the definition of gand applying P(α|θ∗)≤1gives P(α|θ∗)g(θ′, α′,⃗Z) g(θ∗, α,⃗Z)≤P(α|θ∗)s P(⃗Z|⃗Wα′) P(α|θ∗)P(⃗Z|⃗Wα)≤s P(... | https://arxiv.org/abs/2505.09098v1 |
the general converse result for non-causal protocols from Lemma 3, and then rewrite the channel term using cMfrom Definition 10 and its characterization for the BSC from Lemma 11. It then only remains to show that the source term simplifies as Esrc ε=D 1 2∥1 2+ε . It follows immediately from Theorem 14 (e.g., paired w... | https://arxiv.org/abs/2505.09098v1 |
E∗≤minε2 2σ2, c⌈1/(2ε)⌉Echan(0) , (46) where c⌈1/(2ε)⌉is defined in Definition 10, and satisfies c⌈1/(2ε)⌉→1asε→0(by Lemma 11). Thus, the upper and lower bounds match whenever the min{·,·}in (45) is achieved by the first term. Moreover, even the second terms match to within a factor that approaches 1 as ε→0. (Unlike ... | https://arxiv.org/abs/2505.09098v1 |
we let ⃗Wαbe the corresponding codeword sent by the teacher. We claim that αsatisfies similar sub-Gaussian style bounds as the sample mean ¯X(see (44)); specifically, the bound that will be useful for our purposes is P(α|θ∗)≤exp −k(θ∗−α)2 2σ2 , (48) where here and subsequently, P(α|θ∗)represents the PMF of αunder an ... | https://arxiv.org/abs/2505.09098v1 |
cases α=α′andα̸=α′: •When α=α′, theP(⃗Z|⃗Wα)terms from (50) cancel out, and we obtain exp −k(θ∗−α)2 2σ2g(θ′, α,⃗Z) g(θ∗, α,⃗Z)= exp −k(θ′−α)2 4σ2−k(θ∗−α)2 4σ2 (58) By a simple differentiation exercise, the maximum value of −k(θ′−α)2 4σ2−k(θ∗−α)2 4σ2 occurs at α=1 2(θ∗+θ′) and achieves the value −k(θ′−θ∗) 8σ2. Hence... | https://arxiv.org/abs/2505.09098v1 |
(81) applies (75) and the fact that |Tε|has polynomial size. Equation (82) upper bounds the probability that θ∗∗/∈S, and recalling that this in turn upper bounds the error probability, it follows that the error exponent min ε2 2σ2, Echan(0) is achievable, as desired. D. Proof of Corollary 21 (Converse for Gaussian sou... | https://arxiv.org/abs/2505.09098v1 |
only polynomially many possible outcomes. The rest of the analysis is essentially unchanged except for the polynomial pre-factors. We also note that our converse results are based on Lemma 3, and if we generalize from θ∗∈[0,1]toθ∗∈[−C, C] then we should replace Echan ⌈1/(2ε)⌉byEchan ⌈C/ε⌉therein. C. Vector-valued sourc... | https://arxiv.org/abs/2505.09098v1 |
since there are n/k−1values of j. •Hence, we can compute the set S(and thus ˆθn) in time O(n2k)since there are O(n)values of θto consider. In the sub-Gaussian case, a similar argument applies, but there are O(k3)values of αandO(n2) values of θ, so the computation time is O(n3k3). This is perhaps higher than ideal, but ... | https://arxiv.org/abs/2505.09098v1 |
following protocol for the non-causal setting: •The teacher uses an optimal estimation procedure for direct observations to estimate θ∗to within accuracy (1−δ)ε; this estimate is denoted by ˜θ. •The teacher rounds ˜θto the nearest point in a set Iof points in [0,1]. We design this set such that every θ∈[0,1]has distanc... | https://arxiv.org/abs/2505.09098v1 |
small in cby Hoeffding’s inequality [10, Sec. 2.6]. Since there are polynomially many codewords, by the union bound, it holds with high probability that dB(⃗W,⃗W′, PZ|W)≥ (k−o(k))Echan(0). In the case that Echan(0) =∞, there must exist two symbols w, w′withdB(w, w′, PZ|W) =∞andPW(w)PW(w′)> 0. Moreover, we have dB(⃗W,⃗W... | https://arxiv.org/abs/2505.09098v1 |
Theory of Independence . OUP Oxford, 2013. [11] R. Vershynin, High-dimensional probability: An introduction with applications in data science . Cambridge University Press, 2018, vol. 47. [12] A. Dembo and O. Zeitouni, Large deviations techniques and applications . Springer Science & Business Media, 2009, vol. 38. [13] ... | https://arxiv.org/abs/2505.09098v1 |
arXiv:2505.09152v1 [math.ST] 14 May 2025Nelson-Aalen kernel estimator to the tail index of right censored Pareto-type data Nour Elhouda Guesmia, Abdelhakim Necir∗, Djamel Meraghni Laboratory of Applied Mathematics, Mohamed Khider Univers ity, Biskra, Algeria On the basis of Nelson-Aalen product-limit estimator of a ran... | https://arxiv.org/abs/2505.09152v1 |
p:=γ2/(γ1+γ2). (1.3) 3 The integer sequence k=knrepresents the number of top order statistics satisfying k→ ∞ andk/n→0 asn→ ∞.The sequence of rv’s Z1:n≤...≤Zn:nrepresent the order statistics pertaining to the sample Z1,...,Znandδ[1:n],...,δ[n:n]denote the corresponding concomitant values, satisfying δ[j:n]=δiforisuch t... | https://arxiv.org/abs/2505.09152v1 |
substituting Fby Nelson-Aalen estimator F(NA) n,we derive a kernel estimator to the tail index γ1as follows /hatwideγ1,K:=/integraldisplay∞ Zn−k:ng′ K/parenleftBigg F(NA) n(x) F(NA) n(Zn−k:n)/parenrightBigg logx Zn−k:ndF(NA) n(x) F(NA) n(Zn−k:n). (1.7) Let us rewrite the previous integral into a more friendly form as a... | https://arxiv.org/abs/2505.09152v1 |
that is when p= 1, µKandσ2 Krespectively coincide with the asymptotic mean and variance of CDM’s esti mator (Cs¨ org˝ o et al. ,1985) defined by/hatwideγ(CDM) K:=/summationtextk i=1ϕK(i/k)log(Zn−i+1:n/Zn−k:n). Remark 2.3. It is worth remembering that kernel estimation procedure re duces the asymp- totic bias. However it... | https://arxiv.org/abs/2505.09152v1 |
1500 observations, 34 of which are right censored. In insurance, censoring occurs when the loss exceeds a policy limit representing the maximum amount a company can compensate. Each loss has its own policy limit which varies from one contract to anot her. We apply our estimation methodology to this dataset using the bi... | https://arxiv.org/abs/2505.09152v1 |
>1/2 (see Remark ( 2.4)), then the integral/integraltext∞ 1x(2η−p)/γ±ǫ0−1dxis finite for any smallη,ǫ0>0.On the other hand, by using the properties of the Wiener process W(t),we deduce that E|Wn,1(s)| ≤(ps)1/2andE|Wn,1(s)| ≤(qs)1/2.Thus we may readily check thatE|Jn(x)| ≤Mx−1/γ1/parenleftbig x1/(2γ)−1/parenrightbig ,for... | https://arxiv.org/abs/2505.09152v1 |
( 5.12) and (5.16).In other words, Tnis expressed in terms of two independent Wiener processes Wn,1andWn,2.Consequently, Tnis itself Gaussian. By squaring Tn,we write E[T2 n] as the sum of the following three terms S1:=γ2 1E/parenleftbigg/integraldisplay1 0t−1Jn1/parenleftbig t−γ1/parenrightbig d{tK(t)}/parenrightbigg2... | https://arxiv.org/abs/2505.09152v1 |
product-limit proc ess for truncated data with application to extreme value index estimation. Extremes 19, no. 2, 219-251. Benchaira, S., Meraghni, D., & Necir, A., 2016. Kernel estimation of t he tail index of a right-truncated Pareto-type distribution. Statist. Probab. Lett. 119, 186-193. Bladt, M., & Rodionov, I., 2... | https://arxiv.org/abs/2505.09152v1 |
Robust Representation and Estimation of Barycenters and Modes of Probability Measures on Metric Spaces Washington Mio and Tom Needham Department of Mathematics, Florida State University, Tallahassee, FL, USA Abstract This paper is concerned with the problem of defining and estimating statistics for distribu- tions on s... | https://arxiv.org/abs/2505.09609v2 |
probability measures on ( X, d X) with finite p-moments is denoted B(X, d X;p). For µ∈ B(X, d X;p), theFr´ echet variance function of order pofX, denoted Vp:X→R, is defined by Vp(x):=Eµ[dp X(x,·)] =Z Xdp X(x, y)dµ(y). (2) The set of global minima of Vp(the global minimum may not be unique—see Example 1.1) is known as t... | https://arxiv.org/abs/2505.09609v2 |
of global minima of σ2. Take the underlying metric space to be the unit circle endowed with geodesic distance and let µ 2 (a) (b) (c) (d) Figure 2: Instability of global minima of σpon the circle. (a)A pdf on the circle, its corresponding deviation function σ2, and the global minimizers of σ2. The deviation function an... | https://arxiv.org/abs/2505.09609v2 |
analogous to their counterparts for barycenters, as modes can 3 (a) (b) (c) Figure 3: Stability of Barycentric Merge Trees on the circle. (a)A circle with a bimodal distribution, its deviation function σ2and the Fr´ echet mean set, as in Figure 2. (b)The BMT for the distribution in (a); the leaves of the tree correspon... | https://arxiv.org/abs/2505.09609v2 |
with respect to diffusion distances depending on an exponential kernel that has a scale parameter t >0. The results shown are for various values of this parameter. a distribution on the real line. The variance function V2is calculated with respect to diffusion distances derived from the heat kernel for several values o... | https://arxiv.org/abs/2505.09609v2 |
analyze variation 5 in barycentric merge trees in a (functional) Gromov-Wasserstein framework [32, 36, 3], instead of taking a purely metric approach. Our main stability result, obtained in Theorem 4.5, asserts that ifµandµ′are probability measures on X, then the distance between their barycentric merge trees, viewed a... | https://arxiv.org/abs/2505.09609v2 |
Fr´ echet variance 6 with respect to more general pseudo metrics θ:X×X→R, as this lets us frame modes as special cases of barycenters. A discussion of pseudo metrics derived from kernel functions and well suited to mode analysis is presented in Section 5. Definition 2.1. LetX= (X, d X, µ) be an mm-space with µ∈ B(X, d ... | https://arxiv.org/abs/2505.09609v2 |
a connec- tivity constant for (X, d X), then KLis aθ-connectivity constant for (X, d X); that is, rθ(x, y)≤ KLd X(x, y). Proof. By assumption, for any x, y∈Xandϵ >0, there exists a path γ∈Γ(x, y) such that supt∈IdX(x, γ(t))∨dX(γ(t), y)≤ϵ+KdX(x, y). Then, sup t∈Iθ(x, γ(t))∨θ(γ(t), y)≤Lsup t∈I dX(x, γ(t))∨dX(γ(t), y) ≤... | https://arxiv.org/abs/2505.09609v2 |
Definition 3.3. Fora∈Tp(X), let Ca⊆Xbe the connected component of the sublevel set of σp represented by a. (i) The partial order ⪯is defined by a⪯bifCa⊆Cb. (ii) A point a∈Tp(X) is a leafif it is a minimum point of the poset; that is, if b⪯a, then a=b. (iii) Given a, b, c∈Tp(X),cis amerge point foraandbifa⪯candb⪯c. The ... | https://arxiv.org/abs/2505.09609v2 |
whose image is contained in Cϵ. This implies that mp,X(x, x′)≤ρ(γ)< ϵ. Since ϵ >0 is arbitrary, mp,X(x, x′) = 0. (iii) By (i) and (ii), mp,X(x, y)≤mp,X(x, x′) +mp,X(x′, y′) +mp,X(y′, y) =mp,X(x′, y′). (16) Similarly, mp,X(x′, y′)≤mp,X(x, y). This concludes the proof. Proposition 3.8 implies that mp,Xinduces a pseudo me... | https://arxiv.org/abs/2505.09609v2 |
so that |σp(β(s))−σp(β(t))| ≤ |σp(β(s))−σp(x)|+|σp(β(t))−σp(x)| ≤LdX(β(s), x) +LdX(β(t), x)< ϵ,(20) for any s, t∈I. In particular, ρ(β) = supt∈I σp(β(t))−σp(x) ∨ σp(β(t))−σp(y) < ϵ. This implies that ˆdp,X(a, αp(y)) = ˆdp,X(αp(x), αp(y)) = inf γ∈ρ(γ)≤ρ(β)< ϵ . (21) By Proposition 3.9, dp,X(a, αp(y))< ϵ, showing tha... | https://arxiv.org/abs/2505.09609v2 |
LetZ= (Z, d, µ ) and Z′= (Z′, d′, µ′) be (pseudo) mm-spaces, where µandµ′have finite p- moments. Let A⊆ZandA′⊆Z′be the supports of µandµ′, respectively. A metric coupling between ZandZ′is a (pseudo) metric δ: (Z⊔Z′)×(Z⊔Z′)→Ron the disjoint union Z⊔Z′ with the property that δ|A×A=d|A×Aandδ|A′×A′=d′|A′×A′. The set of all... | https://arxiv.org/abs/2505.09609v2 |
Lemma) .If the p-deviation functions satisfy |σp(x)−σ′ p(x)| ≤r, ∀x∈X, then δrdefines a pseudo metric on Tp⊔T′ p. In particular, δryields a metric coupling between TpandT′ p. Proof. By Proposition 4.3, it suffices to check that dis(R)≤2r; that is, for any x, y∈X, |dp(αp(x), αp(y))−d′ p(α′ p(x), α′ p(y))| ≤2r . (29) We ... | https://arxiv.org/abs/2505.09609v2 |
y)1/p =L(1 +K)wp(µ, µ′),(42) as claimed. For a geodesic space ( X, d X) and θ=dX, we can choose L= 1 and K= 1, as shown in Proposition 2.2. 16 Figure 7: Quantitative stability of BMTs (see Example 4.6) Top Row: A summary of the exper- iment. We begin with a smooth bimodal distribution on the circle, which is then samp... | https://arxiv.org/abs/2505.09609v2 |
for barycentric merge trees and the facts that µn→µweakly a. s. and wpmetrizes weak convergence of probability measures. (ii) This follows from the Stability Theorem and the estimate E[wp(µ, µn)]≤C′n−1/sby Weed and Bach under the hypothesis that diam( X)≤1, where C′depends only on sandp[38]. 5 Modes We begin our discus... | https://arxiv.org/abs/2505.09609v2 |
with y7→exp(−d(x0, y)2) + exp( −d(x1, y)2), then renormalizing by total volume. Intuitively, the modes of the resulting distribution should be located at the points x0andx1; this intuition is borne out computationally, as is shown in Figure 8. 2. Second, we define a pdf by renormalizing the function x7→(TC( x)−cTC)4, w... | https://arxiv.org/abs/2505.09609v2 |
For q= 2 and t >0, denote the corresponding diffusion distance (with νthe Lebesgue measure) by θt:Rd×Rd→R. A routine calculation (see [15, Eq. 6]) shows that θ2 t(x, y) = 2 1 (8πt)d/2−k2t(x, y) . For a fixed pair ( x, y), parameterize the segment from xtoyby β(s) = (1 −s)x+sy,s∈I, and let g(s):=θ2 t(x, β(s)) =2 (8πt)d... | https://arxiv.org/abs/2505.09609v2 |
edge set is defined by {[v],[w]} ∈Epif{v, w} ∈Eand [ v]̸= [w]. Here, [·] denotes equivalence class. The quotient graph Tpis the p-barycentric merge tree ofV. Clearly, σpinduces a function κp:Wp→Rsuch that σp=κp◦αp, where αp:V→Wpis the quotient map. As in the continuous case, we introduce a poset structure and a pseudo-... | https://arxiv.org/abs/2505.09609v2 |
ψ:Xn→X×Xbe given by ψ(xi) = (xi, ı◦ϕ(xi)) and h=ψ♯(pn). Then, h∈C(µn, νn) and wp(µn, νn)≤Z X×Xdp X(x, y)dh(x, y)1/p =1 nnX i=1dp X(xi, ϕ(xi))1/p ≤δ. (51) Therefore, under the assumptions of Theorem 4.5, the functional barycentric merge trees for Xn andWnsatisfy DKS,p Fp(Xn),Fp(Wn) ≤L(1 +K)δ . (52) As such, to est... | https://arxiv.org/abs/2505.09609v2 |
αp,Wn(v)) =r. Let ∆: V→V×Xbe given by ∆( v) = (v, v). Since νn=ı♯(ωn),h= ∆ ♯(ωn) gives a coupling between ωnandνn, which in turn, induces a coupling ¯h:= (σp,Vn×σp,Wn)♯(h)∈C(ωp,n, νp,n). Since the coupling his diagonal, by Proposition 3.12 and (6.2), the structural offset of ( δr,¯h) satisfies Z Tp(Vn)×Tp(Wn)δp r(¯v,¯... | https://arxiv.org/abs/2505.09609v2 |
The merge tree provides a sketch of the interrelationships between the various sublevel sets of the p-deviation function, in particular, the merging patterns of the barycenters or modes of µ, thus providing a more complete summary of ( X, d X, µ) than the barycenters alone. A unifying framework was devel- oped for the ... | https://arxiv.org/abs/2505.09609v2 |
T. Needham, C. Shonkwiler, and G. Stewart. Random triangles and polygons in the plane. The American Mathematical Monthly , 126(2):113–134, 2019. [11] P. Chaudhuri and J. S. Marron. Scale space view of curve estimation. Ann. Statist. , 28:408–428, 2000. [12] R. R. Coifman and S. Lafon. Diffusion maps. Appl. Comput. Harm... | https://arxiv.org/abs/2505.09609v2 |
D. Morozov, K. Beketayev, and G. Weber. Interleaving distance between merge trees. Discrete Comput. Geom. , 49:22–45, 2013. [34] R. R. Sokal and F. J. Rohlf. The comparison of dendrograms by objective methods. Taxon , 11:33–40, 1962. [35] K.-T. Sturm. On the geometry of metric measure spaces. Acta Mathematica , 196(1):... | https://arxiv.org/abs/2505.09609v2 |
arXiv:2505.09647v1 [cs.DS] 12 May 2025On Unbiased Low-Rank Approximation with Minimum Distortion Leighton Pate Barnes, Stephen Cameron, and Benjamin Howard Center for Communications Research – Princeton May 2025 Abstract We describe an algorithm for sampling a low-rank random matrix Qthat best approximates a fixed targ... | https://arxiv.org/abs/2505.09647v1 |
5# , with probability 1 /5, whereas the algorithm in [2] generates the random variable Q′=4±2 ±2 1 each with equal probability 1/2. Both Q, Q′achieve the minimal distortion E∥P−Q∥2 F=E∥P−Q′∥2 F= 8, as does the mixture Qtwhich takes the values Qt=Qwith probability tandQt=Q′with probability 1 −t for any 0 ≤t≤1. This va... | https://arxiv.org/abs/2505.09647v1 |
that (r−k)dk+1PN i=k+1di<1⇐⇒ (r−k)dk+1<NX i=k+1di, and these components are automatically included in every realization of the diagonal Q′. This notion of the firstkindices being heavy (where kis potentially zero) is well-defined in the sense that (r−k′)dk′+1<NX i=k′+1di=⇒(r−(k′+ 1)) dk′+2<NX i=k′+2di. Of the remaining... | https://arxiv.org/abs/2505.09647v1 |
the unbiased approximations. 4 Figure 2: Upper-left is an image of rank ≈700, and the lower-right is the average of 16 unbiased low-rank approximations, all instances of Algorithm 1 with rank r= 3. The images in the middle are the unbiased approximations. Algorithm 1 Unbiased Low-Rank Approximation with Minimum Distort... | https://arxiv.org/abs/2505.09647v1 |
ITMM – 2024 INFORMATION SPREADING IN RANDOM GRAPHS EVOLVING BY NORROS-REITTU MODEL N. M. Markovich1, D. V. Osipov2 1V.A. Trapeznikov Institute of Control Sciences Russian Academy of Sciences Profsoyuznaya Str. 65, 117997 Moscow Russia 2The Department of Discrete Mathematics Moscow Institute of Physics and Technology, M... | https://arxiv.org/abs/2505.09713v1 |
the ticks of a Poissonian clock, which drive the evolution of the graph, followed by the propagation of the message to the new node. We assume that message transmissions occur for a fixed time 𝑇*.𝐾*is the number of ticks of the clock (or evolution steps) up to 𝑇*. We aim to consider the dynamics of 𝑁𝑘and𝑆𝑘over t... | https://arxiv.org/abs/2505.09713v1 |
𝑁+ 1)] =Λ𝑖Λ𝑁+1 𝐿𝑁+1. (5) New edges are added independently of the existing graph structure. 4 N. M. Markovich, D. V. Osipov (ii) Each old edge of 𝐺𝑁is independently deleted with probability 𝑃𝑁+1:= 1−𝐿𝑁 𝐿𝑁+1. The evolution model can play a double role: it serves for the growing net- work (evolution) and for... | https://arxiv.org/abs/2505.09713v1 |
message propagation, while the higher 𝜏(e.g., 𝜏= 3.5) results inhomogeneous capacities, slowing the spread. 3.1. S u c c e s s p r o b a b i l i t y 𝑝𝑘 Lemma 2. Under the conditions of graph evolution according to the Norros-Reittu PAM, let # 𝑆0= 1, and Λ be distributed as in (4). Then the success probability 𝑝𝑘... | https://arxiv.org/abs/2505.09713v1 |
creation is independent across steps, the sequence of successes {𝑝𝑘}forms a Poisson binomial distribution model. Further analysis is based on the arguments of [11]. 8 N. M. Markovich, D. V. Osipov From Lemma 1 we have that the pmf of 𝑆𝑘,𝑘≥1, for the maximum number of evolution steps 𝐾*within a fixed time 𝑇*is de... | https://arxiv.org/abs/2505.09713v1 |
𝑆𝑘/𝑁𝑘against 𝑁𝑘for 20 repetitions of graph evolution by the Norros-Reittu PAM: on the left side −𝑁0= 10 and # 𝑆0∈ {1,5,10}; in the middle−𝑁0= 50 and # 𝑆0∈{1,5,10}; on the right−𝑁0= 100 and # 𝑆0∈{1,5,10}, where the first line is responsible for 𝜏= 1.5, the second line for 𝜏= 2.5, and the third line for 𝜏=... | https://arxiv.org/abs/2505.09713v1 |
ness. arXiv:2311.13856v1 [physics.soc-ph] (2023) 8. Pang, G., Pardoux, E., Velleret, A.: Stochastic SIR model with indi- vidual heterogeneity and infection-age dependent infectivity on large non- homogeneous random graphs. arXiv:2502.04225v1 (2025) 9. Shah, D.: Gossip algorithms. Foundations and Trends in Networking 3(... | https://arxiv.org/abs/2505.09713v1 |
arXiv:2505.09860v1 [stat.ME] 14 May 2025Robust and Computationally Efficient Trimmed L-Moments Estimation for Parametric Distributions Chudamani Poudyal1 University of Central Florida Qian Zhao2 Robert Morris University Hari Sitaula3 Montana Technological University © Copyright of this Manuscript is held by the Authors! ... | https://arxiv.org/abs/2505.09860v1 |
as a practical and computationally efficien t alternative to likelihood-based esti- mation. MTM replaces classical moments with trimmed moment s computed by excluding extreme order statistics, yielding estimators that are finite, stab le, and robust to outliers. These estimators have been developed for various distributio... | https://arxiv.org/abs/2505.09860v1 |
general L-estimator framework and derives its theoretical properties, specific ally focusing on MTM. Section 3is the section where we implement the designed methodology for spe cific parcomprehensive simulation study, and Section 5provides real-data applications. Section 6concludes with a discussion and future directions... | https://arxiv.org/abs/2505.09860v1 |
triesσ2 ijof the variance-covariance matrix ΣTdirectly from Eq. ( 13) using a numerical bivariate trapezoidal rule ( Brazauskas and Kleefeld ,2009, Appendix A.2), our approach derives the simplified closed-form expression in Eq. ( 14), as presented in Theorem 1. Note 1. If the trimming inequality (8)is replaced by 0≤ai≤... | https://arxiv.org/abs/2505.09860v1 |
we are estimating two unknown parameters, θandσ, we equate the first two sample trimmed- moments with their corresponding population trimmed-mome nts. Further, knowing −∞< θ <∞ andσ >0, we choose h1(x) =xandh2(x) =x2. (20) From Eq. ( 6), we note that Hj:=hj◦F−1. Then, from Eq. ( 19) and Eq. ( 20), we have H1(u) =h1/pare... | https://arxiv.org/abs/2505.09860v1 |
condition 0≤a1≤a2<b1≤b2≤1, (31) which allows greater flexibility in capturing potential dis tributional asymmetries and tail behaviors. Therefore, our motivation for adopting the trimming inequa lity(25)is to ensure consistent trimming across both moments when the underlying distribution is sym metric about zero, wherea... | https://arxiv.org/abs/2505.09860v1 |
Eq. ( 36); 7: Set/hatwideσ+←/hatwideσFT+/hatwideσSTand/hatwideσ−←−/hatwideσFT+/hatwideσST; 8: ifa1=a2andb1=b2then 9:/hatwideσST←0, resulting in/hatwideσT←/hatwideσFT; 10: return /hatwideσT; 11: end 12: else if (a2≤a1andb1≤b2)or(a1≤a2andb2≤b1)then 13: if max{/hatwideσ−,/hatwideσ+}≤0then /* Exit and update trimming propo... | https://arxiv.org/abs/2505.09860v1 |
in Eq. ( 17), whereΣC= DTΣTD′ T.Since both matrices DTandΣTare2×2square matrices, it follows from Schott (2017, Theorem 1.7) that det (DTΣTD′ T) = (det(DT))2det(ΣT).Therefore, by the property established in Lemma 1, choosing either D− TorD+ Tfor the Jacobian matrix DTdoes not affect the final ARE calculation. Thus, witho... | https://arxiv.org/abs/2505.09860v1 |
observations for specific moments , leading to higher efficiency. However, when|θ|becomes large, trimming inequalities ( 25) and ( 31) do not trim the same data points across moments, unlike equal trimming, which can lea d to reduced efficiency and lower AREs. Table 1also illustrates that the ARE generally decreases with in... | https://arxiv.org/abs/2505.09860v1 |
heavier tail. This is becauseS(x;σ,β1)≤S(x;σ,β2)for allx≥0andβ1≤β2, whereSdenotes the survival function. With the trimming proportions (a1,b1)and(a2,b2)as given in ( 25), then from Eq. ( 1), the first two sample trimmed-moments are given by: /hatwideT1=1 n−⌊na1⌋−⌊nb1⌋n−⌊nb1⌋/summationdisplay i=⌊na1⌋+1h1... | https://arxiv.org/abs/2505.09860v1 |
59); 6:/hatwideβMLEfrom Eq. ( 60); 7: Set/hatwideβ+←/hatwideβFT+/hatwideβSTand/hatwideβ−←−/hatwideβFT+/hatwideβST; 8: ifa1=a2andb1=b2then 9:/hatwideβST←0, resulting in /hatwideβT←/hatwideβFT; 10: return /hatwideβT; 11: end 12: else if (a2≤a1andb1≤b2)or(a1≤a2andb2≤b1)then 13: if max/braceleftig /hatwideβ−,/hatwideβ+/br... | https://arxiv.org/abs/2505.09860v1 |
from Eq. ( 17) and Eq. ( 69), we have ARE/parenleftig/parenleftig /hatwideβT,/hatwideσT/parenrightig ,/parenleftig /hatwideβMLE,/hatwideσMLE/parenrightig/parenrightig = (det(SMLE)/det(ST))0.5. (72) Table 2: From Frechet (β,σ= 2), and we vary the tail index β, reciprocal of the shape parameter. A larger β(i.e., a ... | https://arxiv.org/abs/2505.09860v1 |
es (REs) in relation to their corresponding asymptotic relative efficiencies (AREs). To compute the RE of MTM estimators, we use the MLE as the benchmark. Accordingly, the definition of ARE in Eq. ( 17) is adapted for finite-sample performance as follows: RE(MTM, MLE ) =asymptotic variance of MLE estimator small-sample var... | https://arxiv.org/abs/2505.09860v1 |
( 25) is more likely to remove the same observations from both moments, preserving balanc e. In contrast, under inequality ( 31), whereb1≤b2, fewer large values are excluded from the second moment, pot entially inflating the location estimate. The finite relative efficiencies (FREs) al so approach their asymptotic limits, ... | https://arxiv.org/abs/2505.09860v1 |
hurricanes in the United St ates between 1925 and 1995, as reported by Pielke and Landsea (1998). This dataset has previously been examined using the same trimming proportions for all moments by Brazauskas et al. (2009), and using the same winsorizing proportions for all moments by Zhaoet al. (2018). The damage figures ... | https://arxiv.org/abs/2505.09860v1 |
0.1026 1446 1449 0.66 5.73 0.1114 1447 1449 T10(7/30,15/30) (15 /30,7/30) 22.78 0.78 0.1108 1447 1449 0.67 6.00 0.1250 1447 1450 Modified Data MLE – 22.88 1.10 0.2932 1467 1470 0.77 5.47 0.1896 1456 1459 T1 (0,0) (0 ,0) 22.88 1.10 0.2932 1467 1470 0.86 5.26 0.2121 1457 1460 T2(0,1/30) (0 ,1/30) 22.79 0.80 0.1838 1475 14... | https://arxiv.org/abs/2505.09860v1 |
to different moments enhance s adaptability to the data structure and contamination patterns. This advantage is especially evid ent in asymmetric or heavy-tailed settings, where the general MTM consistently outperforms classical a nd symmetric-trimming methods in terms of bias and mean squared error. The practical utili... | https://arxiv.org/abs/2505.09860v1 |
Fréchet distribution. Journal of Computational and Applied Mathematics ,457, Paper No. 116293, 9. Opdyke, J. and Cavallo, A. (2012). Estimating operational r isk capital: the challenges of truncation, the hazards of MLE, and the promise of robust statistics. Journal of Operational Risk ,7(3), 3–90. Pielke, Jr., R.A. an... | https://arxiv.org/abs/2505.09860v1 |
θis positive, then this inequality again follows immediately from (iv). But in general, it follows that η/parenleftbig ai,bj/parenrightbig T2−η/parenleftbig aj,bj/parenrightbig T2 1 =(c2/parenleftbig aj,bj/parenrightbig σ−c1/parenleftbig ai,bi/parenrightbig c1/parenleftbig aj,bj/parenrightbig σ+c1/parenleftbig aj,bj/pa... | https://arxiv.org/abs/2505.09860v1 |
arXiv:2505.09976v1 [math.PR] 15 May 2025Results related to the Gaussian product inequality conject ure for mixed-sign exponents in arbitrary dimension Guolie Lana, Fr´ ed´ eric Ouimetb, Wei Sunc aSchool of Economics and Statistics, Guangzhou University, Guangzhou, China bDepartment of Mathematics and Statistics, McGill... | https://arxiv.org/abs/2505.09976v1 |
Ge nest & Ouimet (2023); Genest et al.(2024, 2025). In the Gaussian setting, the mixed-sign case, where both positive a nd negative exponents appear, remains largely unresolved. The two-dimensional situation was settled by Russell & Sun (2022a) (tighter quantitative bounds have also been proved by Hu et al.(2023)), and... | https://arxiv.org/abs/2505.09976v1 |
the following corollary is immediate. Corollary 2. LetX∼ Nd(0d,Σ)for some positive definite matrix Σ, and let ∅ /\e}atio\slash=J ⊆ {1,...,d}be given. Assume that νj∈[0,1/2)for allj∈ Jandνi∈[1/2,∞)for alli∈ J∁. Then, we have E /productdisplay j∈J|Xj|−2νj×/productdisplay i∈J∁|Xi|2νi ≤E /productdisplay j∈J|Xj|−2νj ... | https://arxiv.org/abs/2505.09976v1 |
componentwise convex order, written Y(s)≤ccxX, by Lemma 2, so that E/bracketleftBig ψ((Y(s) i)i>k)/bracketrightBig ≤E[ψ((Xi)i>k)]. Combining the last two equations and applying (5) again together with Fubini’s theorem, we find that E k/productdisplay j=1|Xj|−2νj×d/productdisplay i=k+1|Xi|2νi ≤E k/productdisplay j=... | https://arxiv.org/abs/2505.09976v1 |
MacKay). Wei Sun’s research is funded through the Natural Sciences and Engineering Research Council of Canada (Grant RGPIN-2025-06779). References Ben´ ıtez, C., Sarantopoulos, Y., & Tonge, A. 1998. Lower bounds f or norms of products of polynomials. Math. Proc. Cambridge Philos. Soc. ,124(3), 395–408. Edelmann, D., Ri... | https://arxiv.org/abs/2505.09976v1 |
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