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eigenvalue problem Avk=λkBvk. IfBis SPD, the solution to the problem is given by the eigenvalues of the matrix B−1/2AB−1/2, where B−1/2is the matrix inverse square root of B. IfAandBare identical, B−1/2AB−1/2is the identity matrix and all the eigenvalues are 1. The farther the generalized eigenvalues are from 1, the mo... | https://arxiv.org/abs/2502.00168v2 |
the geodesic at point Ψ(t). Under the affine-invariant metric, the squared norm of Ψ′(t)is ∥Ψ′(t)∥2 Ψ(t)= Tr Ψ(t)−1Ψ′(t) Ψ(t)−1Ψ′(t) (9) This expression is equivalent (up to a factor of 2) to the Fisher information of the zero-mean Gaussian N(0,Ψ(t))in the direction Ψ′(t), which can be denoted I(t)(see the explanatio... | https://arxiv.org/abs/2502.00168v2 |
by dE(Ψi,Ψj) =∥Ψi−Ψj∥F(13) The Euclidean distance is rarely used in practice for SPD matrices because it does not take into account the geometry of SPD(m) . The symmetric KL divergence between zero-mean Gaussians is given by dKL(Ψi,Ψj) =1 2(KL(Ψ i,Ψj) + KL(Ψ j,Ψi)) (14) where KL(Ψ i,Ψj) =1 2 Tr(Ψ−1 jΨi) + logdet(Ψj) d... | https://arxiv.org/abs/2502.00168v2 |
into account the means of the classes, like in smSQFA, there is no clear way to extend the Log-Euclidean distance to take into account the means. These results show that choosing the right distance is crucial for the performance of SQFA. Furthermore, the affine-invariant distance performed better than alternatives comm... | https://arxiv.org/abs/2502.00168v2 |
by the filters). B.2. Regularization breaks invariance The equivalence of solutions above, however, is eliminated when we introduce regularization as an additive term Iσ2to each second-moment matrix. Specifically, if we regularize the second-moment matrices by adding Iσ2to each matrix, such that Ψi=FTΦiF+Iσ2, then it i... | https://arxiv.org/abs/2502.00168v2 |
the distribution assumption of the LDA classifier. The Fisher-Rao distance between two Gaussians with shared covariance matrix Σis given by (Pinele et al., 2020) dΣ(µi, µj) =q (µi−µj)TΣ−1(µi−µj) (18) The sum of pairwise squared Fisher-Rao distances between the classes is given by 1 2cX i=1cX j=1dΣ(µi, µj)2=1 2cX i=1cX ... | https://arxiv.org/abs/2502.00168v2 |
the other methods. 17 SQFA: An Information Geometry Approach to Dimensionality Reduction E. Details of the motion estimation task E.1. Dataset synthesis The motion estimation dataset consists of synthetic naturalistic videos of surfaces moving at different frontoparallel speeds, synthesized with the procedure described... | https://arxiv.org/abs/2502.00168v2 |
& Burge, 2017) and more recently in (Herrera-Esposito & Burge, 2024). Briefly, a set of linear filters is applied to each pre-processed video, and then a sample of independent noise is added to each filter output, obtaining the noisy response vector R=FTc+λ, where λ∼ N(0,Iσ2). Then, a QDA-like classifier is used to cla... | https://arxiv.org/abs/2502.00168v2 |
Right-censored models on massive data Gabriela CIUPERCA Abstract This article considers the automatic selection problem of the relevant explanatory variables in a right-censored model on a massive database. We propose and study four aggregated censored adaptive LASSO estimators constructed by dividing the observations ... | https://arxiv.org/abs/2502.00178v1 |
with a divergent number of explanatory variables and where the number of non-zero coefficients may depend on the number of observations is also considered by Fei et al. (2023). For censored models, if the quantile index of errors is unknown, the adapted LASSO composite quantile method proposed and studied by Tang et al... | https://arxiv.org/abs/2502.00178v1 |
1Ti≤Ciwhich indicates whether Tiwas observed or not. We thus have an accelerated failure time (AFT) model. We denote by PXthe probability law of the random vector Xand by EXthe expectation with respect to the distribution of X. Similarly, we denote by PC,Pε, the probability laws of Candε, respectively and byPthe joint ... | https://arxiv.org/abs/2502.00178v1 |
study the properties of the estimators proposed in the following section, for z, t∈[0, B], j= 1,···, n, consider the following random processes: y(z)≡lim n→∞1 nnX i=11 1Yi≥z, MC j(t)≡(1−δj)1 1Yj≤t−Zt 01 1Yj≥zdΛC(z), ΛC(t)≡ −log(G0(t)), where ΛCis the cumulative hazard function of the censoring variable C. Let us consid... | https://arxiv.org/abs/2502.00178v1 |
1PK k=11 1bβ(k) n,2̸=0≥w,···,1 1PK k=11 1bβ(k) n,p̸=0≥w . Based on the matrix Qwe construct Q∨ Anwhich is a matrix of dimension p×|∨ An|, containing the columns ofQcorresponding to the indices of∨ An. This way of composing∨ Anand∨ βnyields estimators that satisfy the oracle properties, i.e. sparsity and asymptotic nor... | https://arxiv.org/abs/2502.00178v1 |
group of observations) corresponding to relations (5) and (6), the follow- ing censored estimators can be calculated, respectively: bβ(k) n≡arg min β∈RpX i∈Ukδi bGn(Yi) log(Yi)−X⊤ iβ +λUkpX j=1|βj| |eβ(k) n,j| , eβ(k) n≡arg min β∈RpX i∈Ukδi bGn(Yi) log(Yi)−X⊤ iβ . Note also that the index set bA(k) nsatisfies the pro... | https://arxiv.org/abs/2502.00178v1 |
observations) the censored adaptive LASSO quantile estimators (corresponding to relations (5) and (6)) are respectively: bβ(k) n≡arg min β∈RpX i∈Ukδi bGn(Yi)ρτ log(Yi)−X⊤ iβ +λUkpX j=1|βj| |eβ(k) n,j|, eβ(k) n≡arg min β∈RpX i∈Ukδi bGn(Yi)ρτ log(Yi)−X⊤ iβ . With these elements, we can show by the following theorem t... | https://arxiv.org/abs/2502.00178v1 |
which, by assumption (A4), is bounded on [0, B], with the constant Bdefined in assumption (A4). We also define the random |A|-vector: κA(z)≡lim n→∞1 nnX i=1δi G0(Yi)1 1Yi≥zgτ(εi)XA,i. (17) The censored expectile estimator was defined and studied by Ciuperca (2025): eβn≡arg min β∈RpnX i=1δi bGn(Yi)ρτ(log(Yi)−X⊤ iβ), (18... | https://arxiv.org/abs/2502.00178v1 |
with BIC(0)≡nX i=1δi bGn(Yi)ρτ log(Yi)−X⊤ ieβn . The censored quantile estimator eβnis calculated by (11). Note that criterion (22) is different from that proposed in the paper of Tang et al. (2012) which considered: BIC(λn) = log1 JJX j=11 nnX i=1δi bGn(Yi) Yi−bbj(λn)−X⊤ ibβn(λn) + bAn(λn) logn n. 14 For the expe... | https://arxiv.org/abs/2502.00178v1 |
the expectile index τmust satisfy condition E[gτ(ε)] = 0 of assumption (A9) and then, for a given distribution of ε, we will consider an estimate for τ. Thus, the empirical estimation considered for the expectile index τis: bτ(E) n=n−1Pn i=1εi1 1εi<0 n−1 Pn i=1εi1 1εi<0−Pn i=1εi1 1εi>0. (25) Similarly, an empirical es... | https://arxiv.org/abs/2502.00178v1 |
we take J= 10 . We obtain that for the two estimators bβnand∨ βn, the percentage of false zeros is 0 for all considered values of p,K,w, for the four methods. Then the percentage of false non-zeros is what distinguishes the 17 (a)p= 10 , expectile method. (b) p= 10 , quantile method. (c)p= 100 , expectile method. (d) p... | https://arxiv.org/abs/2502.00178v1 |
200 Monte Carlo replications of a model without intercept, Xji∼ N (1,1)for any i= 1,···, nandj= 1,···, p, we first deduce that for pfixed ( p= 50 ), the bias of∨ βnandbβnis about the same for a fixed estimation method and that by the median method the bias is greater. Biases are greater for a model with intercept ( β0 ... | https://arxiv.org/abs/2502.00178v1 |
aggregated estimator∨ βnis approximately the same than that obtained on the full data by bβnand it does not depend on the number of groups Knor on the value of w(results also corroborated by those in Table 4). In Figures 3(a) and (b), for the values of w∈ {1,5}we plot the evolution of false non-zeros and ∥ ∨ βn− 20 β0... | https://arxiv.org/abs/2502.00178v1 |
2.06 1.88 . Table 4: Study of∨ βnwhen n= 105,λn=n1/2−1/10,p= 50 ,Xi∼ N(1,1). Note that by 200 Monte Carlo, when β0 0= 0, we obtain ∥(bβn−β0)A∥1= 0.33,0.48and0.35, the percentage of false zeros is 0.18, 4.98 and 0.56, using the expectile, median and quantile methods, respectively. β0 0 w K %of false non-zeros ∥(∨ βn−β0)... | https://arxiv.org/abs/2502.00178v1 |
0 or very close to 0. (Figures 5(a) and (b)). The bias of∨ βn,Ais slightly greater by the quantile method for values of ∥β0 A∥1far from 0, but then it is the same as with the expectile method if∥β0 A∥1is close to 0 (Figure 5(c)). In Figure 6 we consider A={1,2}withp= 50 andβ0 1= 1/(5j), j∈ {1,···,10},β0 2=−2. We deduce... | https://arxiv.org/abs/2502.00178v1 |
relation similar to (26): bβ(k) n−β0 A=−n K−1 Υ(k) bA(k) n−1X i∈Uk(s(1) i+s(2) i) 1 +oPn K−1/2 =−n K−1 Υ(k) A−1X i∈Uk(s(1) i+s(2) i) 1 +oPn K−1/2 . (27) With respect to the proof of Theorem 1( ii), in the present case, instead of VA, we will have the variance- covariance matrix of W1+W2defined... | https://arxiv.org/abs/2502.00178v1 |
relation similar to (26): bβ(k) n−β0 A=−n K−1 E−1 ε[hτ(ε)]EX[XAX⊤ A]−1X i∈Ukesi 1 +oP((n/K)−1/2) . With all these results we can apply a similar approach to the proof of Theorem 1, with: n1/2 ∨ βn,A−β0 AL−→ n→∞N|A| 0|A|,S . ■ References Ciuperca, G.: Variable selection in high-dimensional linear model with pos... | https://arxiv.org/abs/2502.00178v1 |
Score-Preserving Targeted Maximum Likelihood Estimation Noel Pimentel Division of Biostatistics UC Berkeley Alejandro Schuler Division of Biostatistics UC BerkeleyMark van der Laan Division of Biostatistics UC Berkeley February 4, 2025 Abstract Targeted maximum likelihood estimators (TMLEs) are asymptotically optimal a... | https://arxiv.org/abs/2502.00200v1 |
behaves like an average of IID random variables D(P)(Oi) in large samples. By the central limit theorem, the asymptotic distribution of an asymptotically linear estimator with influence function D(P) is√n(ˆΨ(Pn)−Ψ(P))⇝N(0, V[D(P)(O)]), which facilitates the construction of confidence intervals, p-values, etc. A central... | https://arxiv.org/abs/2502.00200v1 |
n) = 0 and we have exactly eliminated the plug-in bias. Generically, we say that an estimate ˆPnsolves a score equation h(·) ifPnh(ˆPn) = 0. An estimate ˆPn that solves a score equation his at a local maximum of the empirical log-likelihood in the “direction” h(ˆPn). TMLE eliminates the plug-in bias by solving the effi... | https://arxiv.org/abs/2502.00200v1 |
approaches to efficient estimation [18, 13, 8, 2]. Besides yielding asymptotic efficiency in some cases, solving a large number of scores can also improve finite-sample performance. Given the error expansion in equation 1, the finite-sample performance of a plug-in generally depends on exactly how quickly the remainder... | https://arxiv.org/abs/2502.00200v1 |
the jth basis function in the expansion Φ n. It is easy to show that the following ( ˜d+ 1)-dimensional parametric submodel for ˆQn spans each of these scores as well as the efficient influence function (when ˆPn(W) is given by the empirical distribution of the covariates): logit ˆQn(ϵ) = logit ˆQn+ϵ0A ˆgn(W)+˜dX jϵjΦn... | https://arxiv.org/abs/2502.00200v1 |
plug-in relaxed HAL estimator, a TMLE estimator using relaxed HAL as the initial fit, and the SP-TMLE estimator using relaxed HAL as the initial fit and preserving its scores. HAL is a nonparametric regression method where the fit is produced by running lasso over a set of saturated 0-order spline bases and their tenso... | https://arxiv.org/abs/2502.00200v1 |
mechanism, we see a substantial reduction in variance relative to TMLE. This is presumably attributable to approximating higher-order efficient influence functions with the span of the scores solved by SP-TMLE and thus reducing the remainder term in display 1. Furthermore, we see that in very small sample sizes the SP-... | https://arxiv.org/abs/2502.00200v1 |
with other score-solving estimators unless we had closed- form expressions for the relevant scores. Thus software for SP-TMLEs will be necessarily bespoke unless there is a way to automate or generalize the process for some larger class of initial estimators. In higher dimensions we would expect the HAL estimator to be... | https://arxiv.org/abs/2502.00200v1 |
curve, 2024. [14] Aad W. Vaart and Jon A. Wellner. Asymptotic Statistics . Cambridge University Press, Cambridge, UK, 1998. [15] Lars van der Laan, Marco Carone, Alex Luedtke, and Mark van der Laan. Adaptive debiased machine learning using data-driven model selection techniques, 2023. [16] Mark van der Laan. Higher ord... | https://arxiv.org/abs/2502.00200v1 |
Learning to Fuse Temporal Proximity Networks: A Case Study in Chimpanzee Social Interactions Yixuan He School of Mathematical and Natural Sciences Arizona State University Phoenix, AZ, United States Yixuan.He@asu.eduAaron Sandel Department of Anthropology University of Texas at Austin Austin, TX, United States David Wi... | https://arxiv.org/abs/2502.00302v2 |
a time series of proximity data is available, as in the chimpanzee case study which motivates our work. As will be described in more detail in Sec. 3, the animals form a variety of social groupings which change throughout the day. The same is true for other species with fission-fusion social dynamics [ 4,5]. A biologic... | https://arxiv.org/abs/2502.00302v2 |
groups of chimpanzees which stay in each other’s wider community for a surprisingly large amount of time. 2 Literature review Combining or fusing networks/graphs that represent different views into a unified structure is a well- studied problem relating to our proximity network fusion setting. [ 13] proposes a multi-gr... | https://arxiv.org/abs/2502.00302v2 |
but our data have a clear bias on focal individuals, which both papers overlook. [ 29] takes a spatial approach around gathering events, but following a focal male is not a gathering event. In [ 30], the network layers represent separate categories, and [ 31] assumes independent edge measurements, but in our data, the ... | https://arxiv.org/abs/2502.00302v2 |
procedure involved following one “focal" male for an hour and recording three main social interactions: (1) “party": all chimpanzees that were in social/spatial association (within roughly 100m of the focal subject during the hour session); (2) “proximity": chimpanzees within physical proximity of the focal subject (wi... | https://arxiv.org/abs/2502.00302v2 |
each single-relationship network, we construct an ancillary network to record multiple occurrences on the same day. For each ancillary network, an edge is added for each day if multiple occurrences are observed, with the number of consecutive occurrences minus one as the daily weight, while the total edge weight is the... | https://arxiv.org/abs/2502.00302v2 |
to optimize. For example, for our chimpanzee data set, TGCN [ 37] can easily require above 1000 parameters, far more than the ten combination weights that we need to learn. In general, deep neural networks have far more parameters than those of interest, making it hard to optimize the key parameters (here, the combinat... | https://arxiv.org/abs/2502.00302v2 |
the inverse of the softplus function ˜wh= log(exp( wh)−1)to the initial values of the wh’s, replacing wh= 0 bywh= 0.0001 for numerical stability. We then employ the softplus function wh= log(1 + exp( ˜ wh))to transform ˜wh∈ Rback to wh>0,and set wh= 0 for tiny whto ensure wh∈[0,∞). For wadd∈(0,1)(similarly treating 0as... | https://arxiv.org/abs/2502.00302v2 |
robust to initialization. In general, we conclude that for our synthetic data, the proposed method can perfectly recover the combination weights of interest up to one decimal point, while with α3= 0.001the final estimated values are typically closer but smaller than the actual values (due to regularization). Having a r... | https://arxiv.org/abs/2502.00302v2 |
w5= 39.7, w6=w7= 0.0, w8= 7 40.0, w9=w10= 0.0,andwadd= 0.0,resulting in W1=W2=W3=W4= 1.0, W5=W6= W7= 40.7, W8=W9=W10= 80.7,and addition parameter wadd= 0.0.This implies that the most notable proximity gap comes from being within 5m of the focal subject and from being within 2m of the focal subject. These two nontrivial... | https://arxiv.org/abs/2502.00302v2 |
over time. To quantify the two notions of similarity, we carry out a theoretical analysis on sequences of independent Bernoulli trials with different success probabilities over time. The distribution of the number of successes of independent but not necessarily identically distributed Bernoulli random variables is call... | https://arxiv.org/abs/2502.00302v2 |
above theorems, for each observed similarity value, we compute its p-value for the null hypothesis that nodes are closely related (in our case, belonging to the same community) independently and randomly across time. In other words, the null hypothesis is: a pair of nodes stays in the same community at random at each t... | https://arxiv.org/abs/2502.00302v2 |
figures [ 44]. The other pair, which is only captured by the learned networks, involved an adult male, bt, who “adopted” pp, 9 a younger male as a juvenile, and the two remained close in adulthood. Comparing the shapes of the enduring relationships from Figure 5, learned graphs typically produce fewer “tails" and more ... | https://arxiv.org/abs/2502.00302v2 |
Di Fiore, Julia Lehmann, Colleen M Schaffner, Noah Snyder-Mackler, Klaus Zuberbühler, et al. Quantifying uncertainty due to fission–fusion dynamics as a component of so- cial complexity. Proceedings of the Royal Society B: Biological Sciences , 285(1879):20180532, 2018. [6]Vedran Sekara and Sune Lehmann. The strength o... | https://arxiv.org/abs/2502.00302v2 |
Eckardt, Tara S Stoinski, Robin E Morrison, and Aaron A Sandel. Female mountain gorillas form enduring social relationships. Animal Behaviour , 213:139–147, 2024. [24] Hongchao Qin, Rong-Hua Li, Guoren Wang, Lu Qin, Yurong Cheng, and Ye Yuan. Mining periodic cliques in temporal networks. In 2019 IEEE 35th International... | https://arxiv.org/abs/2502.00302v2 |
[40] David Lusseau, Karsten Schneider, Oliver J Boisseau, Patti Haase, Elisabeth Slooten, and Steve M Dawson. The bottlenose dolphin community of doubtful sound features a large proportion of long-lasting associations: can geographic isolation explain this unique trait? Behavioral ecology and sociobiology , 54:396–405,... | https://arxiv.org/abs/2502.00302v2 |
+ a grooming individual in its “prox2". E.g., F & B. Thus, as for Type 9, they must be within roughly 2m of each other; for Type 10, they must be grooming in addition to being within roughly 2m apart. B Theorems and proofs B.1 Theorem and proof of count similarity Here we provide a theorem and the proof relating to cou... | https://arxiv.org/abs/2502.00302v2 |
time series of independent Bernoulli trials {Bt}, t∈ {0, . . . , T −1},with success rate ptforBt,denote Dt∈ {0, . . . , t}as the longest consecutive successes from the start until time t.The probability distribution satisfies for t∈ {0, . . . , T −1}, P(Dt=L) = 0 , L∈ {t+ 2, . . . , T }, (9) P(Dt=t+ 1) =tY s=0ps, P(Dt=... | https://arxiv.org/abs/2502.00302v2 |
1for the definition of the chain containing Lconsecutive successes until t; •Bt−L= 0for the definition of the “failure" point; •Dt−L−1≤LasLis defined to be the largest length until tand that this largest length could be achieved more than once (and hence we take ≤Linstead of < L). Note that the definition of Dt−L−1is v... | https://arxiv.org/abs/2502.00302v2 |
(1−pt)· P(Dt−1=L), t∈ {3, . . . , T −1}, L∈ {2, . . . , t −1}. B.3 Proposition and proof for same-community probability Proposition B.4. Suppose for a time step t,both nodes iandjexist in the network containing nt nodes and Ktcommunities Ct 0,Ct 1, . . . ,Ct Kt−1.Suppose all nodes have the same i.i.d. community assignm... | https://arxiv.org/abs/2502.00302v2 |
subgraphs G(h,t)= (N(t),E(h,t)), which share the same node set as G(t), such that E(t)=∪hE(h,t)and ∩hE(h,t)=∅.The hierarchy-level combined adjacency matrix A(h,t)is computed by normalizing the corresponding subgraph adjacency matrix: A(h,t) i,j=A(t) i,j Wh=A(t) i,jPh k=1wk for(i, j)∈ E(h,t)andA(h,t) i,j= 0for(i, j)/∈ E... | https://arxiv.org/abs/2502.00302v2 |
by our proposed method for α3= 0andα3= 0.001, respectively. Type w2 w3 w4 w5 wadd GT 0.0 1.0 0.0 1.0 0.0 α3= 0.0 0 .0±0.0 1 .0±0.0 0 .0±0.0 1 .0±0.0 0 .0±0.0 α3= 0.001 0 .0±0.0 0 .9±0.0 0 .2±0.0 0 .7±0.0 0 .0±0.0 GT 0.0 1.0 0.0 1.0 0.3 α3= 0.0 0 .0±0.0 1 .0±0.0 0 .0±0.0 1 .0±0.0 0 .3±0.0 α3= 0.001 0 .0±0.0 0 .9±0.0 0 .... | https://arxiv.org/abs/2502.00302v2 |
Fractional Cumulative Residual Entropy in the Quantile Framework and its Applications in the Financial Data Iona Ann Sebastian , S.M.Sunoj∗ Department of Statistics Cochin University of Science and Technology Cochin 682 022, Kerala, India Abstract Fractional cumulative residual entropy (FCRE) is a powerful tool for the... | https://arxiv.org/abs/2502.00349v1 |
efficient and equivalent alternative to the distribution function in modelling and analysis of statistical data (see Gilchrist (2000), Nair and Sankaran (2009)). There are several properties of QFs that are not carried out by the distribution functions. For instance, two QFs are added together to form another QF. Since... | https://arxiv.org/abs/2502.00349v1 |
complete gamma function. When α= 1,EQ(X) =b/4. (ii) If Xis following the exponential distribution with mean 1 /λand the qdf q(u) = 1/λ(1−u), then EQ α(X) =Γ(α+1) λ. When α= 1,EQ α(X) = 1 /λ. 0.800.840.88 0.00 0.25 0.50 0.75 1.00 αQFCRELambda 1.1 1.11 1.12 1.13 0.240.270.300.33 0.00 0.25 0.50 0.75 1.00 αQFCRELambda 3 3.... | https://arxiv.org/abs/2502.00349v1 |
From the properties of QFs given in Nair et al. (2022), if Q(u) =Q1(u) +Q2(u) where Q1andQ2are QFs then EQ α=Z1 0(1−p)(−log(1−p))α(q1(p) +q2(p))dp=EQ1 α+EQ2 α. For example, if Q1(u) =uandQ2(u) =−1 λ(log(1 −u)) denote the uniform and exponential distributions respectively, EQ α=Z1 0(1−p)(−log(1−p))α1 +λ−λp λ−λp dp. Se... | https://arxiv.org/abs/2502.00349v1 |
(2.6) Further, X≤rhqY=⇒1 (1−u)qX(u)≤1 (1−u)qY(u). (2.7) 8 Upon combining (2.6) and (2.7) we obtain, =⇒1 (1−u)qX(u)ζ′(QX(u))≤1 (1−u)qY(u)ζ′(QY(u)) =⇒Z1 0(1−p)(−log(1−p))αqX(p)ζ′(QX(p))dp≥Z1 0(1−p)(−log(1−p))αqY(p)ζ′(QY(p))dp =⇒ EQ α(ζ(X))≥ EQ α(ζ(Y)). The case (ii) can be proved analogously. The usefulness of weighted d... | https://arxiv.org/abs/2502.00349v1 |
or system of components operates at time t. In such situation, the residual lifeCt={x:x > t}is the set of interest. Therefore, the concept of FCRE for the residual lifetime ditribution known as dynamic fractional cumulative residual entropy (DFCRE) is required. The dynamic fractional cumulative reidual entropy (DFCRE) ... | https://arxiv.org/abs/2502.00349v1 |
1−uZ1 u(1−p)(log(1 −u)−log(1−p))αqY(p)dp =1 1−uZ1 u(1−p)(log(1 −u)−log(1−p))αqX(p)ζ′(QX(p))dp. (3.4) Since, ζis nonnegative, increasing and convex (concave) we have ζ′(QX(u)) is increasing (decreasing) and nonnegative. Hence by Lemma 3.1 of Nanda et al. (2014) EQ α(Y, u) is increasing (decreasing) in u. Hence, for 0 < ... | https://arxiv.org/abs/2502.00349v1 |
C, λ 1, λ2>0. (5.1) The model (5.1) also known as Davies distribution (Hankin and Lee, 2006), which is a flexible family for right-skewed non-negative data giving a good approximation to the exponential, gamma, Weibull and lognormal distributions. Further, when λ1=λ2=λ, it reduces to the log-logistic distribution. We h... | https://arxiv.org/abs/2502.00349v1 |
and fora <3 it shows periodic, stable behaviour. Here, we had taken the initial value of x0 as 0.1. The parameter ais chosen as 1, 1.5, 2, 2.5, 3, 3.5 and 4. 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.50.6 αQ−FCREa = 1 a = 1.5 a = 2 a = 2.5 a = 3 a = 3.5 a = 4 Figure 4: Q-FCRE of logistic map with varied parameters a, for the... | https://arxiv.org/abs/2502.00349v1 |
0.8 1 0.0020.0040.0060.0080.010 Jan 2015Apr 2015Jul 2015Oct 2015 YearQFCREα 0.2 0.4 0.6 0.8 10.0020.0040.006 Jan 2014Apr 2014Jul 2014Oct 2014 YearQFCREα 0.2 0.4 0.6 0.8 1 Figure 7: Q-FCRE of DJIA dataset From Figure 7, it can be further elucidate that during the year 2018 the value of Q-FCRE transcended the value 0.010... | https://arxiv.org/abs/2502.00349v1 |
Society , 49:457–474. May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature , 261(5560):459–467. Midhu, N. N., Sankaran, P. G., and Nair, N. U. (2013). A class of distributions with the linear mean residual quantile function and it’s generalizations. Statistical Methodology , 15:1–24. Nai... | https://arxiv.org/abs/2502.00349v1 |
Confidence intervals for intentionally biased estimators David M. Kaplan∗Xin Liu† October 31, 2023 Abstract We propose and study three confidence intervals (CIs) centered at an estimator that is intentionally biased to reduce mean squared error. The first CI simply uses an unbiased estimator’s standard error; compared ... | https://arxiv.org/abs/2502.00450v1 |
finite- sample reality. In this paper, we propose two CIs that are centered at the intentionally biased (lower MSE) estimator but attain at least the desired coverage probability, and a third CI cen- tered at a convex combination of the unbiased and biased estimators. The first CI uses the unbiased estimator’s standard... | https://arxiv.org/abs/2502.00450v1 |
results for our second CI show how it can further achieve shorter length than the benchmark. Essentially, it calibrates the critical value to achieve exact coverage probability if the two estimators have the same MSE, given the ratio of the two estimators’ standard errors. Equivalently, “same MSE” can be interpreted as... | https://arxiv.org/abs/2502.00450v1 |
the nominal confidence level. We characterize the cases for which our CI has higher CP than the CI centered at the unbiased estimator. Then in Section 4 we show that if the biased estimator has strictly lower MSE, the CP becomes even higher, specifically if we reduce the bias while keeping the variance fixed. Section 5... | https://arxiv.org/abs/2502.00450v1 |
estimator, then ˆθ1could be a highly under-smoothed estimator, although optimal bandwidths in such cases are beyond our scope. The scalar parameter θcan be a summary of an underlying vector-valued or function- valued parameter as long as (1) holds, but confidence sets or bands for non-scalar θare beyond our scope. In (... | https://arxiv.org/abs/2502.00450v1 |
There too, smoothing the indicator function in the moment conditions (estimating equa- tions) introduces bias but reduces variance, resulting in an overall MSE reduction. Both the unsmoothed QR estimator and the smoothed QR estimator are asymptotically normal; for example, see Koenker and Bassett (1978, Thm. 4.2), Angr... | https://arxiv.org/abs/2502.00450v1 |
Zou (2006) and Theorem 2 of Fan and Li (2001). However, among others, Leeb and P¨ otscher (2008) show that such oracle arguments do not hold uniformly (i.e., under all drifting sequences of data-generating pro- cesses), which in finite samples translates to portions of the parameter space where MSE is not lower than or... | https://arxiv.org/abs/2502.00450v1 |
a transformed version of the bias: b2is increasing in t, with b2= 0 when t= 0, up to b2=s1when t=π/2. The standard deviation s2moves in the opposite direction, decreasing in tfrom s2=s1at t= 0 down to s2= 0 at t=π/2. The earlier special case implicitly set t=π/2 to get s2= 0 andb2=s1. The coverage probability of CI 2de... | https://arxiv.org/abs/2502.00450v1 |
1− α=0.81 tCP(t,1.311) 0.0 0.5 1.0 1.50.60.70.80.91.0Nominal level: 1− α=0.69 tCP(t,1.015) 0.0 0.5 1.0 1.50.00.10.20.30.40.50.60.7Nominal level: 1− α=0.68 tCP(t,0.994)Figure 1: Coverage probability of CI 2at various nominal levels. 14 90.00% (rounded), increasing to 100% as|b2| →s1. Proof. The result for CI 1is well kn... | https://arxiv.org/abs/2502.00450v1 |
is 1 −αatb2= 0 and (for conventional confidence levels) increases toward 100% as |b2|increases. This CI trades some of the higher CP of CI 2for shorter length. For example, at confidence 16 level 95% when CI 2has CP strictly above 95% for any non-zero bias b2̸= 0, this alternative CI is strictly shorter than CI 2while ... | https://arxiv.org/abs/2502.00450v1 |
confidence level 90%, CI 5 is no more than 0.014% longer than CI 2, and can be significantly shorter. Proof. Given their similar structures, CI 5is shorter than CI 2if and only if ˜ z1−α/2< z 1−α/2. Consider the implicit definition of ˜ z1−α/2in (8). Given that the CP( ·,·) function is increasing in its second argument... | https://arxiv.org/abs/2502.00450v1 |
the largest possible bias of b2=√ 3s1/2, which implies MSE( ˆθ2) = MSE( ˆθ1), then the CP of both intervals is smaller. The CP of CI 2is Φ(2z0.975−√ 3)−Φ(−2z0.975−√ 3) = 0 .986, and the CP of CI 5drops to the nominal 95% level: Φ(2˜z0.975−√ 3)−Φ(−2˜z0.975−√ 3) = 0 .95. 20 In that case, CI 5trades the entire “excess” CP... | https://arxiv.org/abs/2502.00450v1 |
Proof. By construction, ˜ zw,1−α/2in (14) sets the coverage probability equal to 1 −αexactly in the equal-MSE case. By Lemma 3 (replacing s2with s3w, and b2with wb2, and bB= ±wp s2 1−s2 2), if MSE( ˆθ2)<MSE( ˆθ1) strictly, then the CP is even higher. The previous CI 5 is the special case with w= 1; given that w∗minimiz... | https://arxiv.org/abs/2502.00450v1 |
length n τ CI1 CI2 CI5 CIs 6 CI6 CI1 CI2 CI5 CIs 6 CI6 100 0.10 0.954 0.978 0.976 0.966 0.962 0.713 0.713 0.704 0.683 0.662 100 0.20 0.972 0.982 0.976 0.972 0.966 0.586 0.586 0.568 0.551 0.531 100 0.30 0.964 0.980 0.974 0.970 0.968 0.541 0.541 0.530 0.515 0.502 100 0.40 0.946 0.984 0.974 0.962 0.950 0.516 0.516 0.504 0... | https://arxiv.org/abs/2502.00450v1 |
and its square; see Section 4 of Angrist, Chernozhukov, and Fern´ andez-Val (2006) for details about the data. They use data for U.S.-born men aged 40–49, and we further restrict to Black men. Like them, we multiply the schooling coefficient by 100 to get the approximate return to schooling as a percent. As in the simu... | https://arxiv.org/abs/2502.00450v1 |
only to all the estimators cited in the 26 introduction but also any future estimators that introduce bias in order to reduce mean squared error. If the biased estimator’s standard error is not reliably estimated, then using the usual critical value (and the unbiased estimator’s standard error) also improves upon other... | https://arxiv.org/abs/2502.00450v1 |
in the current sta- tus model.” Annals of Statistics 38 (1):352–387. URL https://doi.org/10.1214/09- AOS721 . Groeneboom, Piet and Jon A. Wellner. 1992. Information Bounds and Nonparametric Max- imum Likelihood Estimation . Birkh¨ auser. Hansen, Bruce E. 2017. “A Stein-Like 2SLS Estimator.” Econometric Reviews 36 (6–9)... | https://arxiv.org/abs/2502.00450v1 |
standard normal CDF, Φ(a+d)−Φ(a−d)>Φ(b+d)−Φ(b−d). Proof. Let 0 ≤a < b without loss of generality because of the symmetry of the standard normal distribution. That is, if the original a < 0, we can replace it with −awithout changing the probability: using Φ( −z) = 1−Φ(z), Φ(a+d)−Φ(a−d) = [1 −Φ(−(a+d))]−[1−Φ(−(a−d))] = Φ... | https://arxiv.org/abs/2502.00450v1 |
arXiv:2502.00514v1 [math.PR] 1 Feb 2025A Proof of The Changepoint Detection Threshold Conjecture in Preferential Attachment Models Hang Du* Shuyang Gong†Jiaming Xu‡ February 4, 2025 Abstract We investigate the problem of detecting and estimating a cha ngepoint in the attachment function of a network evolving according ... | https://arxiv.org/abs/2502.00514v1 |
both hypotheses are O(√n). As a result, the minimum-degree test achieves strong detec tion (with vanishing Type-I and Type-II errors as n→ ∞ ) ifγ >1/2and weak detection (strictly better than random guessing) if γ= 1/2.The authors further conjecture that all tests are powerless and fail in weak detection when γ <1/2[2,... | https://arxiv.org/abs/2502.00514v1 |
changepoint, [ 3, Theorem 2.4] shows that assuming τn≥εn for some constant ε∈(0,1), there exists an estimator /tildewideτnbased onGnsuch that |/tildewideτn−τn|=O(√n) with high probability.1As an immediate consequence of Theorem 2.4, we prove that this result is order-optimal. Theorem 2.4. Assume that there exists ε∈(0,... | https://arxiv.org/abs/2502.00514v1 |
is impossible only for ∆ =o(n1/3).In more detail, given a time threshold τ′=n−∆′, Sincludes all leaf vertices vsuch that (1) the parent of varrives no later than τ′; (2) and vis the only child of its parent that arrives later than τ′.It can be shown that |S| ≈∆′, and furthermore, Scontains all vertices arriving after τ... | https://arxiv.org/abs/2502.00514v1 |
a parameter N=N(n) such that ∆2≪N≪n, and we write M=n−N. LetGMdenote the network history {Gt,i}3≤t≤M,1≤i≤mup to time M.LetPGM,Gn(resp.QGM,Gn) denote the joint distribution of network snapshots GMandGnunderP(resp.Q). Then the marginals satisfy PGn=Pn,QGn= Qn, andPGM=QGM. It follows that TV(P,Q)≤TV/parenleftBig PGM,Gn,QG... | https://arxiv.org/abs/2502.00514v1 |
medges for each of the Nvertices. Similarly, for each v∈Vn\Vn−1, |{Gn∈ G:vn=v}|=/parenleftbiggN−1 C,C(v)/parenrightbigg ×/productdisplay C∈C\{C(v)}#{admissible orders on C} ×#{admissible orders on C(v)withvbeing maximal }×(m!)N, where the counting is similar as above except that since vis the last vertex added, vmust b... | https://arxiv.org/abs/2502.00514v1 |
(t′,i′)∈ I0,(t′,i′)≺(t,i)}. We also let the multi-set /vectorEt,i=/vectorE↑ t,i∪/vectorE↓ t,i. Clearly|/vectorEt,i|= (t−1)(m−κ)+(t−3)m+i−1∆=Kt,i, and we label the edges in /vectorEt,iby/vector e1,..../vector eKt,iin an arbitrary deterministic way. Consider independent random variables {Ut,i}(t,i)∈I, such that for each ... | https://arxiv.org/abs/2502.00514v1 |
vt′,i′can only differ from /tildewidevt′,i′, ifvt′,i′is chosen as an endpoint of an edge in G\/tildewideG. However, G\/tildewideGmust be contained within C(vt). It follows that vt′,i′and consequently emust be contained in C(vt). Since e∈ C(v), this would imply v∈ C(vt), which contradicts our assumption that v /∈ C(vt).... | https://arxiv.org/abs/2502.00514v1 |
Tt−1andNt 1,...,Nt l−1, we may have already revealed some of the vl-attaching vertices. However, 11 the remaining of them are still independently sampled from t hose vertices that arrive earlier than vlwith probability proportional to current degrees plus the a dditive term δ, while conditioning on that they do not bel... | https://arxiv.org/abs/2502.00514v1 |
attachment models with Simiao Jiao, Weijia Li, and Xiaochun Niu. J. Xu also thanks the organ izers of the conference “Statistical and Probabilistic Analysis of Random Networks and Processe s”, where he first learned about the changepoint detection threshold conjecture. S. Gong is partially supported by National Key R&D ... | https://arxiv.org/abs/2502.00514v1 |
arXiv:2502.00812v2 [math.ST] 12 Feb 2025Direct Sampling from Conditional Distributions by Sequential Maximum Likelihood Estimations By Shuhei Mano The Institute of Statistical Mathematics, Japan Abstract. We can directly sample from the conditional distribution of any log-affine model. The algorithm is a Markov chain on ... | https://arxiv.org/abs/2502.00812v2 |
sample consisting of the Poisson random variables with means λj,j∈[m] withpj=λj/|λ|also follows the conditional distribution (3). 2 Since the discrete exponential family is the standard family of discre te distributions, sampling from the conditional distribution (3) has var ious sta- tistical demands. However, it also... | https://arxiv.org/abs/2502.00812v2 |
3 1221 0211 0220 1111 1120 0101 0110 1001 1010 0000 Figure 1: The Markov lattice for the matrix in Example 2.8 with the maxi- mumb=t(1,2,2,1). A sample path is shown by solid edges. algorithm . See the monograph [15] for an extensive discussion. The direct sampling algorithm is a random walk or a Markov chain on a boun... | https://arxiv.org/abs/2502.00812v2 |
the main obstacle common to using the Markov basis a nd the direct sampling algorithm for general models is the computationa l cost of Gr¨ obner bases. The Gr¨ obner basis guarantees the realizabilit y of these methods, however, Buchberger’s algorithm for computing Gr¨ obn er bases is impractical since it demands a hug... | https://arxiv.org/abs/2502.00812v2 |
of the UMVUE of the expected count to the to tal number of counts. Then, after summarizing the relevant propert ies of the MLE, we show that the UMVUE and the MLE coincide if and only if a log-linear model is a decomposable graphical model and the resulting MLE is rational. Lastly, we discuss the use of the MLE and rev... | https://arxiv.org/abs/2502.00812v2 |
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