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offers a principled functional approach to estimating the LAQTE in Definition 1, while converging to the same object in population. This connection to local Fr´ echet regression in Wasserstein space explains why the “double regression” approach in (6) is preferred over monotonizing the simple local linear estimator in ... | https://arxiv.org/abs/2504.03992v1 |
C={ω:T1(ω)> T0(ω)}. •Indefinites: I={ω:T1(ω) =T0(ω)}\{ω:T1(ω) =T0(ω)∈ {0,1}}. The treatment effects of interest are, 18 Definition 2 (Fuzzy LAQTE) .The local average quantile treatment effects for the fuzzy R3D design are, τF3D(q) :=E[QY1(q)−QY0(q)|X= 0, C]q∈[0,1]. (8) To identify these, I need the following standard a... | https://arxiv.org/abs/2504.03992v1 |
a similar setting. 20 these asymptotic results to the empirical quantile setting. 3.1 Assumptions Throughout, I work on the restricted set of quantiles [ a, b], a compact subset of (0 ,1), and letc<0<c. Also, define Yc:={Y(ω) :ω∈Ωx, X(ω)∈[c,¯c]}as the set of random cdfs that are realized in a small neighborhood around ... | https://arxiv.org/abs/2504.03992v1 |
1< q 2<1there exists an ε >0such that Yis continuously differentiable on the interval [QY(q1)−ε, Q Y(q2) +ε]with strictly positive density. Q2. There exists a sequence m=nγ, γ≥1such that min{ni:i= 1, . . . , n } ≥m. Moreover, the sample sizes for each Zijare asymptotically balanced, i.e.ni nj→ηijwith0< ηij<∞for i, j∈ {... | https://arxiv.org/abs/2504.03992v1 |
end, I use the pseudo-random samples {ξi}n i=1defined in M1 to define the estimated multiplier process, (14) ˆ ν± ξ,n(q, k) =nX i=1ξie′ 0(Γ±,p)−1ˆEk(Yi, Ti, Xi, q)rp Xi hk(q) K Xi hk(q) δ± ip nhk(q)ˆfX(0), where ˆfX(0) is any uniformly consistent estimator of fX(0), and ˆEk(Yi, Ti, Xi, q) is any uni- formly consist... | https://arxiv.org/abs/2504.03992v1 |
Corollary 1 (Convergence: Fr´ echet Treatment Effects) .Under the assumptions of Theo- rem 1 it follows that, √ nh ˆτR3D ⊕,p−τR3D ;GR3D, and under the additional Assumptions I3–I5, √ nh ˆτF3D ⊕,p−τF3D ;GF3D Corollary 2 (Bootstrap: Fr´ echet) .Under the assumptions of Theorem 3, the estimated boot- strap processes ˆ... | https://arxiv.org/abs/2504.03992v1 |
theoretical expectations, the quantile RD estimator appears to be inconsistent and suffers from large finite sample bias, with a relative bias that is at least an order of magnitude higher than the R3D estimators’, for some quantiles. As expected, the quantile RD estimator performs well at the median in DGP 1, because ... | https://arxiv.org/abs/2504.03992v1 |
on running variable Xand jump across the threshold. again both estimators quickly converge to a similarly low variance. I do not report results for the quantile RD as its inferential properties are irrelevant due to its inconsistency and bias in the R3D settings. To study the coverage properties of the confidence bands... | https://arxiv.org/abs/2504.03992v1 |
processes are described in Equations (15) and (16). All simulations used 2,500 repetitions and 5,000 bootstrap replications and estimated quantile treatment effects at the 9 deciles. Values of ∆ reflects Cohen’s d of 0, 0.5, and 1. 30 4.2 Empirical Illustration: State Governors and the Income Distribution To further il... | https://arxiv.org/abs/2504.03992v1 |
state ielected a Democratic governor in year t compared to a Republican one. The outcome variable Zijt′is real income of family jin state iin year t′=t+tj, where tjis a state-specific offset to match the income distribution in the final year of a governor’s tenure to their electoral results. Real family income is const... | https://arxiv.org/abs/2504.03992v1 |
average quantiles of within-state income (in multiples of the federal poverty threshold), calculated within bins of width 0 .01. Average quantiles were constructed by computing the weighted quantile functions of family income within each state and year, and then taking the average of the estimated quantile values for a... | https://arxiv.org/abs/2504.03992v1 |
nullity and homogeneity tests. This suggests the results are not driven by reverse causality, where the pre-existing income distribution drives the election outcomes. This aligns with the small effects of local economic conditions on voting behavior estimated in the literature on retrospective voting (Healy and Malhotr... | https://arxiv.org/abs/2504.03992v1 |
difference in average quantile functions, instead of observed ones, just above and below a treatment cutoff. This measure offers a natural and intuitive extension of the traditional RDD treatment effect to distribution-valued outcomes. To estimate the LAQTE, I propose two complementary estimators: one based on local po... | https://arxiv.org/abs/2504.03992v1 |
inference in nonlinear difference-in- differences models’, Econometrica 74(2), 431–497. Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972), Statistical Inference Under Order Restrictions , Wiley, New York. Bauschke, H. H., Combettes, P. L., Bauschke, H. H. and Combettes, P. L. (2017), Convex anal... | https://arxiv.org/abs/2504.03992v1 |
, SSRN. Chernozhukov, V., Fern´ andez-Val, I. and Galichon, A. (2010), ‘Quantile and probability curves without crossing’, Econometrica 78(3), 1093–1125. Chernozhukov, V., Fern´ andez-Val, I. and Melly, B. (2013), ‘Inference on counterfactual dis- tributions’, Econometrica 81(6), 2205–2268. Chernozhukov, V. and Hansen,... | https://arxiv.org/abs/2504.03992v1 |
with observa- tional data’, arXiv preprint arXiv:2503.07811 . Gunsilius, F. and Van Dijcke, D. (2025), ‘Free discontinuity regression: With an application to the economic effects of internet shutdowns’, arXiv preprint arXiv:2309.14630 . Hahn, J., Todd, P. and Van der Klaauw, W. (2001), ‘Identification and estimation of... | https://arxiv.org/abs/2504.03992v1 |
of Business & Economic Statistics 37(4), 625–647. Qu, Z. and Yoon, J. (2024), ‘Qte. rd: An r package for quantile treatment effects in regression- discontinuity designs with/without covariates’. Qu, Z., Yoon, J. and Perron, P. (2024), ‘Inference on conditional quantile processes in partially linear models with applicat... | https://arxiv.org/abs/2504.03992v1 |
front of Qis independent of QYi(q) and depends only on {Xi}, 43 Kh(·),h, etc., it follows: ˆm±,p(q) =nX i=1e⊤ 0 X⊤ ±W±X±−1 X⊤ ±W± :, i| {z } =:s(p) ±, in(h)QYi(q). Therefore, the one-sided local-polynomial estimator of order pcan be written as a simple weighted average: ˆm±,p(q) =nX i=1h s(p) ±, in(h)i QYi(q),where... | https://arxiv.org/abs/2504.03992v1 |
x, h) =1 σ2 0{Kh(x′−x) [µ2−µ1(x′−x)]} which corresponds to the local Fr´ echet mean, (A-4) ˜l(x) = argmin z∈RE s(X, x, h )(Z−z)2 . Just as with the definition of the classical Fr´ echet mean, this can be generalized to Y∈Ω on a general metric space as, ˜l⊕(x) = argmin ω∈Ωn ˜Ln(ω):=E s(X, x, h )d2(Y, ω)o where the d... | https://arxiv.org/abs/2504.03992v1 |
echet functional in (A-1) on (Y, dW2) as, M⊕(Ω, x) =E[d2 W2(Yi, ω)|X=x] =Z XZ1 0(QYi(q)−Qω(q))2dqdFY|X=x =Z1 0Z X QYi(q)2−2QYi(q)Qω(q) +Qω(q)2 dFY|X=xdq =C′+Z1 0(m(q)−Qω(q))2dq where C′=R1 0R XQYi(q)2dFY|X=xdq−R1 0m(q)2dqis a constant that does not depend on Qω, and the second equality follows from Fubini-Tonelli by ... | https://arxiv.org/abs/2504.03992v1 |
Λs,s+1=R Rus+1rs(u)K(u)du, Λ± s,s+1=R R±us+1rs(u)K(u)du, and Ψ± s=R R±rs(u)r′ s(u)K2(u)du. Below, I drop the R3D superscript on the treatment effect estimators ˆ τR3Dto ease notation. A-4.2.1 Local Polynomial Estimator It is well known (cf. Fan and Gijbels (1992), Calonico et al. (2014)) that for a p-th order local pol... | https://arxiv.org/abs/2504.03992v1 |
reason is that the bias term involves the second derivative of the conditional expectation. Derivatives of quantile functions are not quantile functions themselves, and hence projecting them onto the space of quantile functions lacks meaning. This approach is justified by the above derivations, since the Fr´ echet and ... | https://arxiv.org/abs/2504.03992v1 |
algorithm from Calonico et al. (2018, 2020) for optimal coverage error to these (I)MSE-optimal estimated bandwidths, hROT 1(q) =h∗ 1(q)n−s/(2s+3)(s+3) and similarly for hROT 2(q). A-4.3 Multiplier Bootstrap: Algorithm Input: •A sample {(Xi, Yi, Ti)}n i=1, where Yi∈ Y (distributional outcome), Ti∈ {0,1}, and running var... | https://arxiv.org/abs/2504.03992v1 |
by: h ˆτp(q)±1√ nhˆcB n(a, b;λ)i ,forq∈ T∗. A-4.4 Computational Details An R implementation of the package can be found at https://davidvandijcke.com/R3D . The main polynomial weights estimation was implemented with a Fortran backend, leading 56 to highly performant code, as illustrated in Figure A-1. For example, the ... | https://arxiv.org/abs/2504.03992v1 |
shattered. To see this, let ( Y1, z1) and ( Y2, z2) have Y1(z1)≤Y2(z2). The subsets ∅,{(Y2, z2)}, and{(Y1, z1),(Y2, z2)}can be realized by choosing q > Y 2(z2), q∈ [Y1(x1), Y2(z2)], and q≤Y1(z1), respectively. However, the subset {(Y1, z1)}cannot be realized, due to the monotonicity of cumulative distribution functions... | https://arxiv.org/abs/2504.03992v1 |
This part is equivalent to Assumption L2 (i). (c) For any (q, k),(q′, l)∈ [a, b]× {1,2}, it holds that σkl(q, q′|·)∈C1([c,c]\ {0})with bounded derivatives in xand σkl(q, q′|0±)<∞. (d) For each Y∈ Y,QY(q)is left- or right-continuous in q. (iii) This part is equivalent to Assumption K2. (iv) (a) K: [−1,1]→R+is bounded an... | https://arxiv.org/abs/2504.03992v1 |
holds by I5 (Chiang et al., 2019, Lemma 3). Then the functional delta method yields the result (van der Vaart and Wellner, 1996, Lemma 3.9.3). Proof of Theorem 3 Proof. The result follows from Theorem 1 and Theorem 2 in Chiang et al. (2019). The latter applies because their Assumptions 1–4 are satisfied in my setting. ... | https://arxiv.org/abs/2504.03992v1 |
x. Since ˆ m±,p→m±inL2norm with probability 1, eventually ( ˆ m±,p−m±) remains in L2\ N. Hence with probability 1, for large n, ΠQis Hadamard differentiable almost surely at ˆm±,p, and in the relevant directions ˆ m±,p−m±. Step 3. The derivative at m±is the identity map. Since m±∈ Q (Proposition A-2), I have ΠQ m± =m... | https://arxiv.org/abs/2504.03992v1 |
as required by Assumption K2. Then, define the Fr´ echet estimator with empirical distribution functions as, argmin ω∈Y1 nnX i=1s(p) ±,in(h)d2 W2 ω,ˆYi . An identical argument to the one in Proposition A-2 shows that the quantile function of this 66 estimator is, ¯m±,⊕,p= argmin h∈Q(Y)dL2(h,¯m±,p)2, that is, the proj... | https://arxiv.org/abs/2504.03992v1 |
A-6.2 Figures 68 Figure A-2: Distributional Effects of Democratic Governor Control, 1984-2010: Local Poly- nomial −3.0−2.0−1.00.01.02.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 QuantileTreatment Effect (# Poverty Thresholds) Note: local average quantile treatment effects estimates and uniform 90% confidence bands fo... | https://arxiv.org/abs/2504.03992v1 |
quantile of the (average) income distribution while Y-axis indicates the difference in average state-level income distributions, in the final year of the governor’s tenure, near the 50% vote share threshold. Income is measured as real equivalized family income in multiples of the federal poverty threshold. Sample runs ... | https://arxiv.org/abs/2504.03992v1 |
(2019) with bias correction (Qu et al., 2024), with the same bandwidth as Figure 6 and triangular kernel A-4.2. 73 A-7 Software Appendix All results in this paper were produced in Rusing RStudio . A complete reference list of packages used is provided below. References Analytics, R. and Weston, S. (2022). iterators: Pr... | https://arxiv.org/abs/2504.03992v1 |
BALANCING COMPLEXITY AND INFORMATIVENESS IN LLM-B ASED CLUSTERING : FINDING THE GOLDILOCKS ZONE Justin K. Miller School of Physics University of Sydney Camperdown, NSW 2006 justin.k.miller@sydney.edu.auTristram J. Alexander School of Physics University of Sydney Camperdown, NSW 2006 tristram.alexander@sydney.edu.au Apr... | https://arxiv.org/abs/2504.04314v1 |
overly narrow clusters fragment data, complicating interpretation and retrieval. Leveraging linguistic theory, particularly principles from information theory such as Pareto optimality, thus informs clustering strategies by suggesting optimal balances of cluster granularity and coherence. By treating each cluster as a ... | https://arxiv.org/abs/2504.04314v1 |
market research [15], and theory of mind tests, LLMs can similarly be used to emulate a human naming a cluster. It is important to note that with the way LLMs work, it is incorrect to describe them as “coming up” with a name, and more correct to suggest that they are “stochastic parrots” [16] creating content based on ... | https://arxiv.org/abs/2504.04314v1 |
encoding and decoding strategies that jointly minimize complexity and maximize accuracy (analogues for simplicity and informativeness respectively) we capture the “Goldilocks zone” in clustering, mirroring how color-naming systems in human languages achieve near-optimal compression of semantic categories [18]. In our s... | https://arxiv.org/abs/2504.04314v1 |
approach across all configurations. We provide Gemini Pro with the following prompt: “You have a large set of bios from Twitter, • You have a large set of bios from Twitter. • You have clustered them into the following groups: ¡Cluster Names¿ • The following bio belongs to only one of these groups. • Your task is to de... | https://arxiv.org/abs/2504.04314v1 |
To confirm the findings of the visualisation we then used the cosine similarities as predictors in a logistic regression, aiming to predict whether the knowledge of the cosine similarities, the number of clusters in the model, and the data source it came from were enough to predict whether the LLM was able to identify ... | https://arxiv.org/abs/2504.04314v1 |
the accuracy ranking decreases as the number of clusters increases, indicating that the language model (LLM) becomes more accurate than random assignments at higher cluster counts. Accuracy exhibits greater fluctuations than AMI, with an overall trend of improvement interspersed with periods of decline in rankings. Not... | https://arxiv.org/abs/2504.04314v1 |
Accuracy measures the proportion of bios that were correctly assigned to a cluster when given only the cluster names, representing the interpretability of the clustering process. Each data point represents the mean value across experiments for a given number of clusters, with error bars indicating the standard deviatio... | https://arxiv.org/abs/2504.04314v1 |
8.1991 0.091 89.857 0.000 Incorrect Cosine Sim. -4.9648 0.097 -50.977 0.000 Interaction Term -2.6215 0.266 -9.855 0.000 Clusters Count -0.0276 0.000 -83.584 0.000 Filename: Kavanaugh -0.0074 0.016 -0.466 0.641 Filename: Trump (06/2018) 0.0640 0.016 4.096 0.000 Filename: Trump (09/2020) 0.0061 0.016 0.392 0.695 Filename... | https://arxiv.org/abs/2504.04314v1 |
have a cosine similarity of 0.1. Of the seven datasets included in this analysis, there were statistically significant differences in the model’s predictive performance, indicating that some datasets were more conducive to creating informative cluster names than others. Although these differences reached statistical si... | https://arxiv.org/abs/2504.04314v1 |
LLMs locally loaded so that one can ensure the same results every time. Future work could incorporate human judgments, comparing them with LLMs, and testing what other factors may affect the ability to correctly identify clusters, and if humans are able to better identify them than LLMs. Second, the study’s focus on Tw... | https://arxiv.org/abs/2504.04314v1 |
Academy of Sciences , vol. 104, no. 4, pp. 1436–1441, 2007. [5] N. Zaslavsky, C. Kemp, T. Regier, and N. Tishby, “Efficient compression in color naming and its evolution,” Proceedings of the National Academy of Sciences , vol. 115, no. 31, pp. 7937–7942, 2018. [6] E. Gibson, R. Futrell, and J. Jara-Ettinger, “Color nam... | https://arxiv.org/abs/2504.04314v1 |
arXiv:2504.04518v1 [stat.ME] 6 Apr 2025On the bias of the Gini coefficient estimator for zero-truncated Poisson distributions Roberto Vila1∗and Helton Saulo1,2 1Department of Statistics, University of Brasilia, Brasili a, Brazil 2Department of Economics, Federal University of Pelotas, Pe lotas, Brazil April 8, 2025 Abstr... | https://arxiv.org/abs/2504.04518v1 |
probability mass function is given by Pλ(k) =P(X=k) =1 1−exp(−λ)exp(−λ)λk k!, k= 1,2,.... (1) It is well-known that the cumulative distribution function and the exp ected value of X∼ZTP(λ), denoted by Fλ(x) andµ=E(X), respectively, are given by Fλ(x) = 1−1 1−exp(−λ)γ(⌊x⌋+1,λ) Γ(⌊x⌋+1), x/greaterorequalslant1, 0, x ... | https://arxiv.org/abs/2504.04518v1 |
the improper integrals involved e xist. 3.1 Technical results As the determination of E(/hatwideG) relies on the explicit evaluation of R1(F),R∞(F), and E/bracketleftbig g(X∗,X)1{X=X∗}/bracketrightbig (see Theorem 3.1), this subsection presents several technical results that fa- cilitate the computation of these quanti... | https://arxiv.org/abs/2504.04518v1 |
Brychkov ,2012):Q1(a,a) = [exp( −a2)I0(a2)+1]/2, the last expression becomes =exp(−nλ) 2[1−exp(−λ)]n−1/integraldisplay1 0I0(2λy)[exp(λy)−1]n−2dy +exp(−nλ) 2[1−exp(−λ)]n−1/integraldisplay1 0exp(2λy)[exp(λy)−1]n−2dy−exp(−λ) (n−1)λ =exp(−nλ) 2[1−exp(−λ)]n−1/integraldisplay1 0I0(2λy)[exp(λy)−1]n−2dy +exp(−nλ) 2[1−exp(−λ)]n... | https://arxiv.org/abs/2504.04518v1 |
all scenarios, thereby demonstrating the effectiveness of the pr oposed correction. Figure 2presents the mean squared error (MSE) for both estimators. In general, b oth estimators exhibit similar MSE values across the configurations considered. 5 Concluding remarks In this paper, we analyzed the statistical properties of... | https://arxiv.org/abs/2504.04518v1 |
as a usefu l measure of malaria inequality among populations. Malaria Journal , 19(1):444. Brychkov, Y. A. (2012). On some properties of the marcum q func tion.Integral Transforms and Special Functions , 23(3):177–182. Damgaard, C. and Weiner, J. (2000). Describing inequality in plant siz e or fecundity. Ecology, 81:11... | https://arxiv.org/abs/2504.04518v1 |
arXiv:2504.04682v1 [math.ST] 7 Apr 2025Gaussian Mean Testing under Truncation Cl´ ement L. Canonne University of Sydney clement.canonne@sydney.edu.auThemis Gouleakis Nanyang Technological University themis.gouleakis@ntu.edu.sg Yuhao Wang National University of Singapore yuhaowang@u.nus.eduJoy Qiping Yang University of ... | https://arxiv.org/abs/2504.04682v1 |
has recently been the focus of a line of work on efficient truncated statistics, whereby one seeks to develop efficien t algorithms to efficiently estimate the parameters of a population given truncated samples: we elaborate on thi s inSection 1.2 . 1 Despite the existence of these two lines of work – one on Gauss ian mea... | https://arxiv.org/abs/2504.04682v1 |
dand treat ε, α as constants: • When ε/radicalbig log 1/ε/lessorsimilarα, i.e., the accuracy parameter is significantly larger than t he truncated probability mass, then the simple testing algorithm designed for the non-truncated version of the problem works, achieving the optimal sample complexity Θ(√ d)(Theorem 3.1 ).... | https://arxiv.org/abs/2504.04682v1 |
estimation Robust statistics [ HR11 ] considers statistical inference problems under the set- ting where samples observed could be contaminated in variou s ways. For robust estimation, the usual goal is to obtain accurate estimation of parameters for parametr ic families such as Gaussian distributions under ε-contamina... | https://arxiv.org/abs/2504.04682v1 |
defined as, /ba∇dblx−y/ba∇dblΣ=/radicalig (x−y)TΣ−1(x−y). For a matrix A∈Rm×nwith entries aij, the Frobenius norm is defined as: /ba∇dblA/ba∇dblF=/radicaltp/radicalvertex/radicalvertex/radicalbtm/summationdisplay i=1n/summationdisplay j=1|aij|2. Truncated Gaussian Distribution. LetN(µ,Σ)represent the normal distribution... | https://arxiv.org/abs/2504.04682v1 |
Lemma 12] to produce an estimate that is closer than a constant in total variation distance to the tru e distribution even for single-dimensional truncated Gaussians. Our contribution are in the first two regimes and we will elabo rate on in the following subsections. 5 3.1 When Truncation Size is Much Smaller Than Accu... | https://arxiv.org/abs/2504.04682v1 |
of Z. In the completeness case, the quantity |Z− /ba∇dblµ/ba∇dbl|2 2is small, with E[Z]<O(α2)andVar [Z]/lessorsimilarα4. In the soundness case, We can lower bound the expectation of µSforN(µ,Id, S), where /ba∇dblµ/ba∇dbl2/greaterorequalslantα, and show that E[Z]/greaterorequalslantΩ(α2), and Var[Z]/lessorsimilarE2[Z]. ... | https://arxiv.org/abs/2504.04682v1 |
an estimate ˆµthat is within αof the true mean before truncation. If ˆµis sufficiently close to zero, we return ”ACCEPT”. Otherwise, return ”REJECT”. 4 Testing under known truncation In this section, we demonstrate in Theorem 4.3 that when the truncation set is known, an alternative yet straightforward algorithm, which ... | https://arxiv.org/abs/2504.04682v1 |
for parameters (mass of truncation) 0< ε/lessorequalslant1−β, where βis a constant and (accuracy)1 4/greaterorequalslantα >0: •(Completeness) Pis a truncated Gaussian distribution N(0,Id, S)and1− N(0,Id, S)/lessorequalslantε. In this case, the algorithm will output yes with probability at least 2/3. •(Soundness) Pis a ... | https://arxiv.org/abs/2504.04682v1 |
Malik, and Li-Yang Tan. O n the power of adaptivity in statistical adversaries. In COLT , volume 178 of Proceedings of Machine Learning Research , pages 5030–5061. PMLR, 2022. [BS99] Donald R Barr and E Todd Sherrill. Mean and variance of truncated normal distributions. The American Statistician , 53(4):357–361, 1999. ... | https://arxiv.org/abs/2504.04682v1 |
Servedi o. Testing convex truncation. In SODA , pages 4050–4082. SIAM, 2023. [DRZ20] Constantinos Daskalakis, Dhruv Rohatgi, and Emman ouil Zampetakis. Truncated linear re- gression in high dimensions. Advances in Neural Information Processing Systems , 33:10338–10347, 2020. [Fis31] RA Fisher. Properties and applicatio... | https://arxiv.org/abs/2504.04682v1 |
Equation (4) to complete the computation of Equation (5) Var[Z] =E[Z2]−E2[Z] =d/summationdisplay i=1d/summationdisplay j=1/parenleftbigg1 n2Σ2 i,j+2 nΣi,jµiµj/parenrightbigg =1 n2/summationdisplay 1/lessorequalslanti,j/lessorequalslantdΣ2 i,j+2 n/summationdisplay i,jΣi,jµiµj =/ba∇dblΣ/ba∇dbl2 F n2+2 n/summationdisplay ... | https://arxiv.org/abs/2504.04682v1 |
n2+2 n(/ba∇dblΣS−Id/ba∇dblF+/ba∇dblId/ba∇dblF)(/ba∇dblµS−µ/ba∇dbl2+/ba∇dblµ/ba∇dbl2)2 /lessorsimilar(√ d+/ba∇dblId/ba∇dblF)2 n2+2 n(√ d+/ba∇dblId/ba∇dblF)/parenleftig ε·/radicalbig log(1 /ε) + 0/parenrightig2 =O/parenleftbiggd n2/parenrightbigg +O/parenleftigg√ d nε2log 1/ε/parenrightigg /lessorsimilard n2/bracehti... | https://arxiv.org/abs/2504.04682v1 |
when evaluating the gradient at µ, we have ∇¯ℓ(µ) =−Ex∼N(µ,Id,S)[x] +Ez∼N(µ,Id,S)[z] =0. So,∇¯ℓ(0)represents the difference between the truncated mean of the underlying distribution and that of the distribution with mean 0. From Lemma 2.1 , let λ0=1 213/parenleftig β C/parenrightig4 min/braceleftig 1 4,1 16/bardblµ/... | https://arxiv.org/abs/2504.04682v1 |
√ 2π(1−ε)= 0, which is equivalent to: α=exp(−(erf−1(1−2ε))2)√ 2π(1−ε)= Θ/parenleftigg ε/radicalbigg log1 ε/parenrightigg . Next, we compute an upper bound on the chi-squared divergenc e between the truncated distribution A and the standard normal distribution N(0,1). We find that χ2(A,N(0,1)) = /integraldisplayb −∞... | https://arxiv.org/abs/2504.04682v1 |
Extension of Yager’s negation of a probability distribution based on uncertainty measures Santosh Kumar Chaudhary1Pradeep Kumar Sahu2∗and Nitin Gupta2 1Department of Statistics, Central University of Jharkhand, Cheri-Manatu, Ranchi, Jharkhand, 835222, India. 2,2∗Department of Mathematics, Indian Institute of Technology... | https://arxiv.org/abs/2504.04762v1 |
this context. These entropy-based methods provide a systematic way to assess and process uncertainty, offering valuable insights into how negating a probability distribution alters the level of uncertainty represented by the data. In addition to entropy, a concept that has recently gained increasing attention is extrop... | https://arxiv.org/abs/2504.04762v1 |
of evidence, Yager (2015) demonstrated that among all possible negations, the one suggested in his work exhibits the maximal type of entropy. While there are many different measures of entropy, Yager (2015) used the following to measure the entropy of a probability distribution, H1(P) =nX i=1(1−pi)pi= 1−nX i=1p2 i. Yag... | https://arxiv.org/abs/2504.04762v1 |
H (P) =V H(¯P) =V H(¯¯P) =. . .andV J(P) =V J(¯P) =V J(¯¯P) =. . . . Figure 1: Change of H(P) and H(¯P) Figure 1, Figure 2 and Figure 3 shows the change of Shanon entropy, varentropy and varextropy, respectively, with the change of p1from 0 to 1 for n=2. Since H(¯P) =H(P), therefore gragh coincides. The entropy will be... | https://arxiv.org/abs/2504.04762v1 |
1.3751, J(¯P) = 0.8593, V H (¯P) = 0.0220, V J(¯P) =−0.4849 H(¯¯P) = 1.3851, J(¯¯P) = 0.8626, V H (¯¯P) = 0.0025, V J(¯¯P) =−0.4953 H(¯¯¯P) = 1.3862, J(¯¯¯P) = 0.8630, V H (¯¯¯P) = 0.0003, V J(¯¯¯P) =−0.4964 Example 9 LetX={x1, x2, x3}and the probability distribution P={p1, p2, p3}, where p1= 0.6,p2= 0.3, and p3= 0.1. ... | https://arxiv.org/abs/2504.04762v1 |
event space X={x1, x2, . . . x n}and the probability distribution P={p1, p2, . . . , p n},¯Prepresents the inverse of P, then V J(¯P)≥V J(P). Proof: We know from Section 3 that for n= 2, the varextropy of Pand ¯Pare equal, i.e., V J(P) =V J(¯P). Thus, inequality is present in this case. To prove the inequality V J(¯P)≥... | https://arxiv.org/abs/2504.04762v1 |
a uniform distribution, where pi=1 n, we have: V H(P) =nX i=11 n ln1 n2 − nX i=11 nln1 n!2 V H(P) = (ln n)2−(lnn)2= 0 For a uniform distribution, the value of varentropy is always 0, so: lim n→∞V H(P) = 0 Thus, varentropy does not increase with nand remains 0 for a uniform probability distri- bution. Theorem 6 Wh... | https://arxiv.org/abs/2504.04762v1 |
= 0 Solving the system of equations results in the optimal solution: pi=1 n,∀i= 1,2, . . . , n Thus, the varentropy V H(¯P) is maximized when pi=1 n. Therefore, the maximum value ofV H(¯P) occurs when the original distribution is uniform. V H(¯P) = maximized when pi=1 n Theorem 9 Assume the X={x1, x2, . . . x n}, when ... | https://arxiv.org/abs/2504.04762v1 |
Annals of Probability, 39(4), 1528–1543. 19 [4] Di Crescenzo, A., & Paolillo, L. (2021). Analysis and applications of the residual var- entropy of random lifetimes. Probability in the Engineering and Informational Sciences, 35(3), 680–698. https://doi.org/10.1017/S0269964820000133 [5] Liu, J., & Xiao, F. (2024). On the... | https://arxiv.org/abs/2504.04762v1 |
arXiv:2504.04923v1 [math.ST] 7 Apr 2025ARTICLE Truncated sequential guaranteed estimation for the Cox-Ingersoll-Ross models Mohamed BEN ALAYAa, Thi-Bao Trˆ am NG ˆOband Serguei PERGAMENCHTCHIKOVa aLaboratoire de Math´ ematiques Rapha¨ el Salem, UMR 6085 CNRS-U niversit´ e de Rouen Normandie, France;bLaboratoire de Math... | https://arxiv.org/abs/2504.04923v1 |
do not control the observation duration, which restricts their ap plications in many practical applications since, in practice, the duration of observati on is usually bounded. For example, for the portfolio optimisation problems for financ ial markets with unknown parameters in Berdjane and Pergamenshchikov (2015), it ... | https://arxiv.org/abs/2504.04923v1 |
paramete r vectorθ= (a,b)⊤in the model (1). In Section 3.2, we find conditions on the parame ters of the process (1) which provide the optimality properties in minimax sens e for the proposed se- quential procedures. Section 4 presents the concentration inequalities for the CIR process. Some important conclusions are gi... | https://arxiv.org/abs/2504.04923v1 |
defined in (9), the root of this equation can be represented as H∗ T=a∗T−(2mUm/σ)1 2m+1(H∗ T)2 2m+1Tm 2m+1. (17) Taking into account here that H∗ T<a∗Tthe parameter H∗ Tcan be estimated from below as H∗ T>a∗T−(2mUma2 ∗/σ)1 2m+1T2+m 2m+1=a∗T/parenleftBig 1−(2mUma1−2m ∗/σ)1 2m+1T−m−1 2m+1/parenrightBig . (18) Moreover, us... | https://arxiv.org/abs/2504.04923v1 |
the following result. Corollary 2.4. Assume that for some 0<δ<1/2 r= O(Tδ)asT→ ∞. (34) Then, for any m>(1−2δ)−1the optimal truncated procedure (33)posses the following asymptotic properties:. (1) the optimal parameter (32)is represented as H∗ T=µa,∗T−r2m 2m+1(2mVmµ2 a,∗/σ)1 2m+1T2+m 2m+1(1+o(1)) asT→ ∞; (35) (2) the co... | https://arxiv.org/abs/2504.04923v1 |
optimality properties for sequential pro cedures we set the local 10 class of sequential procedures defined as HT(θ0,γ) =/braceleftBigg (τ,/hatwideθτ) : sup |θ−θ0|<γEθτ≤T/bracerightBigg , (47) whereθ0∈Θ andγ >0 such that {|θ−θ0| ≤γ} ⊆Θ. Theorem 2.6. For anyθ0>0the sequential procedure (14)is pointwise optimal lim γ→0lim... | https://arxiv.org/abs/2504.04923v1 |
b >0any compact set Θ⊂]σ/2,+∞[the stopping timeτ∗ Tdefined in the procedure (33)for anyr>0satisfies the following asymptotic property lim T→∞sup θ∈ΘEθ/vextendsingle/vextendsingle/vextendsingle/vextendsingleτ∗ T T−2θ−σ 2amax−σ/vextendsingle/vextendsingle/vextendsingle/vextendsingler = 0. (55) Proof. First of all note that... | https://arxiv.org/abs/2504.04923v1 |
can show th atPθa.s. for any 14 θ∈]σ/2,+∞[×]0,∞[ the following limit equalities hold true lim t→∞1 t/integraldisplayt 0X−1 sds=2b 2a−σ:=f1and lim t→∞1 t/integraldisplayt 0Xsds=a b:=f2.(64) Therefore, setting Gt=/integraldisplayt 0X−1 sgsg⊤ sds= /integraltextt 0X−1 sds−t −t/integraltextt 0Xsds , (65) we obtain that ... | https://arxiv.org/abs/2504.04923v1 |
θ∈ΘEθ|/tildewideθH,T−θ|2≤(2u∗+ρ∗ H)σ H+Tmθmaxr2mZm (µ∗T−H)2m+24m+1v∗ 2mθmax H2m,(80) whereρ∗ H=/summationtext n>n∗ H/parenleftbig κ∗ n/parenrightbig−1,θmax= maxθ∈Θ|θ|2, the coefficient v∗ 2mis given in Lemma 3.1 and Zm= 22mLm/parenleftBigg 1 b2m min+1 σ2m/parenleftbigg4eβmax αmin+2αmaxΓmax βαmin min∧βαmax min+2αmax βmin/... | https://arxiv.org/abs/2504.04923v1 |
similarly to Corollary 2.4 we can show the following resu lt Corollary 3.3. Assume that for some 0<δ<1/2the parameter rin the definitions (79)such that r= O(Tδ)asT→ ∞. (94) Then, for any m>(1−2δ)−1the optimal truncated procedure (33)posses the following asymptotic properties:. (1) the optimal parameter (92)forT→ ∞is rep... | https://arxiv.org/abs/2504.04923v1 |
T≤T}≤Eθ/vextendsingle/vextendsingle/vextendsingle/vextendsingletH∗ T T−¯µ∗ trF/vextendsingle/vextendsingle/vextendsingle/vextendsingler 1{τH∗ T≤T}∩{υH∗ T≤n∗ H∗ T} +Pθ/parenleftBig υH∗ T>n∗ H∗ T/parenrightBig and, therefore, Eθ/vextendsingle/vextendsingle/vextendsingle/vextendsingleτ∗ T T−¯µ∗ trF/vextendsingle/vextendsi... | https://arxiv.org/abs/2504.04923v1 |
obser- vations compared with the non-sequential estimation based on the fixed observations durationT. 4. Concentration inequalities for the CIR models. In this section we study the properties of the deviation in th e ergodic theorem for the process (1). First we study the deviation problem fir this process with the fixed ... | https://arxiv.org/abs/2504.04923v1 |
are studied in Pr opositions 2.5, 2.8 and 3.5. •For the first time for continuous time statistical models, th e optimality prop- erties for the truncated sequential estimation procedures are established in the class of all possible sequential procedures with arbitrary bounded stopping times determining the duration of t... | https://arxiv.org/abs/2504.04923v1 |
D. and Lapeyre, B. 1997 “Introduction au Calcul Sto chastique Appliqu´ e ` a la Finance“, Ellipses ´Edition Marketing: Paris , 2nd edition. Liptser, R. S. and Shiryaev, Albert N. 2001 “Statistics of Random P rocesses I “, Springer- Verlag Berlin Heidelberg, Vol. 5, Stochastic Modelling and Applied Probability, 2nd edit... | https://arxiv.org/abs/2504.04923v1 |
the stochastic i ntegrals. Lemma A.4. Let(ft)0≤t≤Tbe adapted process such that for some m>1 E/integraldisplayT 0f2m tdt <∞. Then E/parenleftbigg/integraldisplayT 0ftdWt/parenrightbigg2m ≤(m(2m−1))mTm−1/integraldisplayT 0Ef2m tdt. (123) This lemma is shown in (Liptser and Shiryaev 2001, Lemma 4.12 ). A.4.Proof of Lemma ... | https://arxiv.org/abs/2504.04923v1 |
SurvSurf 1Journals of the Royal Statistical Society ,XXXX, 1–38 doi: DOI HERE Advance Access Publication Date: Day Month Year Original articlearXiv:2504.04997v1 [stat.ML] 7 Apr 2025 2 Chen et al. SurvSurf: a partially monotonic neural network for first-hitting time prediction of intermittently observed discrete and con... | https://arxiv.org/abs/2504.04997v1 |
progress through different severity grades, the first hitting time for grade k+ 1 can only occur after the first hitting time for grade k, even if the less severe state may not be explicitly observed and the person may fluctuate between severity grades. Another less obvious but directly relevant scenario is recurrent e... | https://arxiv.org/abs/2504.04997v1 |
When an event in the sequence is not observed, this does not mean a complete lack of information for the event. Observations on later and earlier events in the sequence imply whether the missing event has or has not occurred by the observation times. This ‘implicit truth’ on missing intermediate events is not accounted... | https://arxiv.org/abs/2504.04997v1 |
were replaced with 0, to represent biological sex common in medical data. To simulate a complex nonlinear relationship between the feature vectors and the ˆCIF (the predicted CIF) of different events (severity grades), a fully-connected neural network with randomly generated weights and biases (three hidden layers) and... | https://arxiv.org/abs/2504.04997v1 |
(the comparator arm) in patients with stage IV squamous non-small cell lung cancer. For each subject, the TRAE data for different organs were pooled into a single trajectory that describes the overall maximum TRAE severity (0,1,2,3,4,5) achieved by each adverse event monitoring/reporting time point. The TRAE severity i... | https://arxiv.org/abs/2504.04997v1 |
the percentage change in the yearly average price relative to that of 2015 for a specific property type (accounting for new/existing builds) in a specific local authority. There were a total of 2297 trajectories. Local and property-type specific characteristics from the 2011 UK Census data (Office for National Statisti... | https://arxiv.org/abs/2504.04997v1 |
If no events have occurred until t, then the ˆCIF for the earliest event in the sequence (and all later events) must be small. LossDyDg i,t=−[ylog(∆ˆCIF g) + (1−y)log(1−ˆCIF(t, gi,t, xi))] (3) SurvSurf 15 where ∆ˆCIF g=ˆCIF(t, gi,t, xi)−ˆCIF(t, gi,t+δg, xi) (4) For trajectory (or subject) i, at each monitoring time-poi... | https://arxiv.org/abs/2504.04997v1 |
Ciis the time of the last observation for subject ior trajectory i Additionally, the extent of monotonicity violation was defined to be the maximum positive value in ˆCIF event =k+1(t, xi)−ˆCIF event =k(t, xi) across all subjects. 3.4. Model Benchmarking For each dataset, the IBSipcw itiand the extent of monotonicity v... | https://arxiv.org/abs/2504.04997v1 |
RW-Property. All except one hyperparameter for the RSF were set to default. For all datasets, the number of trees was set to 1000. All except two hyperparameters for the GBSA were set to default. For Sim- Main, Sim-LackProg and RW-Property (i.e. the three larger datasets), the minimum number of samples per leaf was set... | https://arxiv.org/abs/2504.04997v1 |
left, Table 1). CoxPHSurvivalAnalysis had slightly worse performance than SurvSurf on Sim- Main, but had marginally better performance on Sim-LackProg (Table 1). 4.2. Correlation between IBSipcw itiand MSE In order to assess model performance on real-world data where true CIFs are not available, we propose the IBSipcw ... | https://arxiv.org/abs/2504.04997v1 |
disorders Height (cm) HISTOLG: Papillary Respiratory, thoracic and mediastinal disorders Psychiatric disorders Metabolism and nutrition disorders BMI Immune system disorders Systolic Blood Pressure (mmHg) INVESTIGATIONS Ear and labyrinth disorders HISTOLG: Other Body Surface Area (m2) Hepatobiliary disorders HISTOLG: S... | https://arxiv.org/abs/2504.04997v1 |
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