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study of the functional theory of randomne ss started in Vovk (2020). While its statements are somewhat less intuitive than t hose of the algorithmic theory of randomness, they are more precise (do n ot involve unspecified constants) and simpler in an important respect: e.g., th e analogue of (4) in the algorithmic theo... | https://arxiv.org/abs/2501.11689v2 |
N. Kolmogorov. Three approaches to the quantitative defi nition of in- formation. Problems of Information Transmission , 1:1–7, 1965. Andrei N. Kolmogorov. Logical basis for information theory and pr obability theory.IEEE Transactions on Information Theory , IT-14:662–664, 1968. Ilia Nouretdinov, Vladimir V’yugin, and A... | https://arxiv.org/abs/2501.11689v2 |
of each zito a different label leads to a random sequence that is still IID. And to check that G3∈ EX, notice that the average of G3over all permutations of its arguments is 1 by the definition of G1: 1 (n+1)!/summationdisplay πG3(zπ(1),...,z π(n+1)) =1 (n+1)!(|Y|−1)n+1/summationdisplay i=1/summationdisplay y∈Y\{yi}n!F(/... | https://arxiv.org/abs/2501.11689v2 |
n) = (0,...,0) 17 (all the objects are x0and omitted, and we assume, without loss of generality, {0,1} ⊆Y); weareaskedtopredict yn+1. Thenatural(universal)exchangeabil- itye-valueforthepotentiallabel yn+1= 1willbe n+1undertheexchangeability model, and the natural randomness e-value for the potential labe lyn+1= 1 will ... | https://arxiv.org/abs/2501.11689v2 |
does not involve Zn+1). Since Y1,...,Y n+1are dis- tributed according to the Bernoulli model, the maximum probability, a gain by the local limit theorem, is asymptotically equivalent to 2k+1/radicalbig 2πn/4∼2√ 2k√πn; it is attained at θ:= 1/2. Notice that now the assumption 1 ≪k≪√nis essential in order for the exponen... | https://arxiv.org/abs/2501.11689v2 |
a randomness e-variable Gsuch that, for all z1,...,z n,zn+1= (xn+1,yn+1), andy/\e}atio\slash=yn+1, ¯E(z1,...,z n,xn+1,y)≥1 |Y|−1E(z1,...,z n,xn+1,y) G(z1,...,z n,zn+1). A significant weakness of the following inverse to Corollary 11 in the bin ary case will be discussed after its proof. Proposition 12. Let|Y|= 2. For ea... | https://arxiv.org/abs/2501.11689v2 |
which is impossible. The maximum Bn θ-probabilityofthe set Y′will be asymptoticallyequivalent to|Y|2−n(which is alsoasymptoticallyequivalent to Bn 1/2(Y′)). The analogous statement will also be true about the set Y′×{0′,1′}of one-bit extensions of the elements of Y′: max θBn+1 θ(Y′×{0′,1′})∼ |Y|2−n. Therefore, there ex... | https://arxiv.org/abs/2501.11689v2 |
23 level) will be at least b2|Y|2−n−1/radicalbig πn/2 asymptotically. Therefore, we have G≤b−22n+1/(/radicalbig πn/2|Y|) on some data sequence with labels ( y1,...,y n)∈Y′ andj(and with all objects being x0). Replacing the old lower bound on Gby this new one (with bsufficiently close to 1), we will obtain (26). To get ri... | https://arxiv.org/abs/2501.11689v2 |
Can Bayesian Neural Networks Make Confident Predictions? Katharine Fisher Massachusetts Institute of Technology Cambridge, MA 02139 kefisher@mit.eduYoussef Marzouk Massachusetts Institute of Technology Cambridge, MA 02139 ymarz@mit.edu Abstract Bayesian inference promises a framework for principled uncertainty quantifi... | https://arxiv.org/abs/2501.11773v1 |
Such priors allow us to identify cases where different posterior modes map to distinct modes in the predictive distribution. Then we can determine when predictions based on a single posterior mode will fail. To the authors’ knowledge, this approach to analyzing multimodality is unique, though the different treatment of... | https://arxiv.org/abs/2501.11773v1 |
we may abbreviate L(XL(Θ(j)))as L(Θ(j)). Assuming that ρj= 1/Jfor all j∈[J], thejthmode of the posterior predictive will have a large weight only if L(Θ(j))is large compared with the marginal likelihood of all other candidate Θvalues. 3 Multimodality under a discretized Gaussian prior At this stage, it is not obvious w... | https://arxiv.org/abs/2501.11773v1 |
the posterior predictive distribution. When d= 10 we see that for several of the nandpvalues considered, up to 98% of the candidate inner layer parameter values make significant contributions to the posterior 2In this paper, we focus only on marginal predictive distributions. It is straightforward, however, to characte... | https://arxiv.org/abs/2501.11773v1 |
constructed to achieve (11). The full distribution is plotted in black and components are shaded according to their weight in indigo. We consider 10rotations, 10preimage samples, and 10column space samples to construct the distribution — a total of 1000 samples. Top right: The scale of predictive distributions for sele... | https://arxiv.org/abs/2501.11773v1 |
Using these constructions, we will demonstrate that not all elements of [Θ]L correspond to identical predictive distributions. Figure 2 describes posterior predictive distributions for a two-layer network with b1= 0and a prior on the inner layer parameters that puts mass on elements of [Θ]L. For a comparison of this pr... | https://arxiv.org/abs/2501.11773v1 |
for neural networks have proven effective [48,19,34,38,15], though prediction calibration is often used to evaluate the “effectiveness” of uncertainty quantification. Our results suggest that the full Bayesian predictive distribution may 6 be poorly calibrated, and further work is necessary to understand how the distri... | https://arxiv.org/abs/2501.11773v1 |
deep linear models and demonstrate that this distribution is Gaussian in certain asymptotic regimes. While deep linear models provide useful insight into extrapolation and feature learning [ 41,62], these networks will be misspecified for many data-generating processes. Thus, we can expect that overparameterization has... | https://arxiv.org/abs/2501.11773v1 |
distributions produced by partially stochastic networks diverge significantly from the full Bayesian predictive distribution. An additional piece of this puzzle is the success of cold posteriors. [ 61] find that sharpened posteriors perform well, while full Bayesian posteriors yield systematically worse predictions tha... | https://arxiv.org/abs/2501.11773v1 |
symmetries, 2023. URL https://arxiv.org/abs/2209.04836 . [3]Francesca Albertini and Eduardo D. Sontag. For neural networks, function determines form. Neural Networks , 6(7):975–990, 1993. ISSN 0893-6080. doi: https://doi.org/10.1016/ S0893-6080(09)80007-5. URL https://www.sciencedirect.com/science/article/ pii/S0893608... | https://arxiv.org/abs/2501.11773v1 |
In Proceedings of the 32nd International Conference on Neural Information Processing Systems , NIPS’18, page 8803–8812, Red Hook, NY , USA, 2018. Curran Associates Inc. [21] Jakob Gawlikowski, Cedrique Rovile Njieutcheu Tassi, Mohsin Ali, Jongseok Lee, Matthias Humt, Jianxiang Feng, Anna Kruspe, Rudolph Triebel, Peter ... | https://arxiv.org/abs/2501.11773v1 |
via local linearization. In International Conference on Artificial Intelligence and Statistics , 2021. URL https://api.semanticscholar.org/CorpusID:221172984 . [35] Pavel Izmailov, Sharad Vikram, Matthew D Hoffman, and Andrew Gordon Gordon Wilson. What are bayesian neural network posteriors really like? In Marina Meila... | https://arxiv.org/abs/2501.11773v1 |
Platt, D. Koller, Y . Singer, and S. Roweis, editors, Advances in Neural Information Processing Sys- tems, volume 20. Curran Associates, Inc., 2007. URL https://proceedings.neurips.cc/ paper_files/paper/2007/file/013a006f03dbc5392effeb8f18fda755-Paper.pdf . [53] Tommy Rochussen. Structured partial stochasticity in baye... | https://arxiv.org/abs/2501.11773v1 |
1< ℓ≤L, (13) where sk:R→[0,1]. The predictive class kbelongs to {0,1}in the binary setting and [K]in the multi-class case. Let Θrepresent all interior parameters. Under the prior assumption, w∼ N(0, p−1Ip),P(Θ = Θ(j)) =ρj,JX j=1ρj= 1, (14) we can write P(˜y=k|X1, Y,˜x1) =JX j=1P(Θ(j)|X1, Y)P(˜y=k|X1, Y,˜x1,Θ(j)) =JX j=... | https://arxiv.org/abs/2501.11773v1 |
black line shows the pdf which is a mixture of Gaussians. Each shaded distribution is a component of this mixture with transparency corresponding to its weight. 18 Figure 6: Posterior predictive distributions at test point ex(3) 1for input dimension d= 1000 at select training set sizes nand final layer widths p, as ind... | https://arxiv.org/abs/2501.11773v1 |
Figure 9: The variance of the posterior predictive distribution averaged over 100realizations of ex1 plotted against the number of parameter candidates, J. The prior details are specified in Section 3 and the network and training set size are given in the plot title. 21 J 1INumber of ModesFigure 10: A comparison of the... | https://arxiv.org/abs/2501.11773v1 |
and ratios p/d andn/d ranging from 0.5to2. As expected, we find that the conjectured optimum is at least as large as the empirically determined maximum for n≤d. In cases where d < n , enforcing (22) leads to n−1logL(XL)considerably smaller than our empirically located maximum. It is interesting to note that the distanc... | https://arxiv.org/abs/2501.11773v1 |
the distribution of the parameters drawn from the prior distributions considered in Section 3 (gold). We see that the distributions are close, but for both Θandw, the variance of the distributions on the constructed parameters is slightly wider. Note that these comparisons do not capture the correlation between element... | https://arxiv.org/abs/2501.11773v1 |
Automatic Debiased Machine Learning for Smooth Functionals of Nonparametric M-Estimands Lars van der Laan∗1,2, Aur´ elien Bibaut2, Nathan Kallus2,3, and Alex Luedtke1 1Department of Statistics, University of Washington 2Netflix Research 3Cornell Tech, Cornell University January 22, 2025 Abstract We propose a unified fr... | https://arxiv.org/abs/2501.11868v1 |
estimation of the nuisance function using flexible machine learning techniques, such as regularized empirical risk minimization; and second, a debiasing step, which facilitates valid uncertainty assessment. The debiasing step often requires estimating additional nuisance functions specific to the parameter of interest.... | https://arxiv.org/abs/2501.11868v1 |
a unified and generalized autoDML framework for inference on smooth functionals of M- estimands , defined as population risk minimizers over generic Hilbert spaces. This framework extends the prior work of Chernozhukov et al. (2022b) and Chernozhukov et al. (2024) on regression functionals to broader classes of paramet... | https://arxiv.org/abs/2501.11868v1 |
that the M-estimand θP∈ His uniquely identified for each P∈ M as the solution to the following risk minimization problem: θP= argmin θ∈HLP(θ, ηP), (1) with LP: (θ, η)7→EP[ℓη(Z, θ)] a risk functional corresponding to a loss function ℓη:Z × H → R. Our framework supports general loss functions and models parameterized by ... | https://arxiv.org/abs/2501.11868v1 |
leading order, equal to ˙ψ0(θ0)(θn−θ0) =∂2 θL0(θ0, η0)(θn−θ0, α0) for a Hessian Riesz representer α0. To develop a bias correction, we further derive the Hessian-gradient relation: ∂2 θL0(θ0, η0)(θn−θ0, α0) =∂θL0(θn, η0)(α0)+Op(∥θn−θ0∥2 H,P), based on the first-order optimality conditions of the M-estimand θ0, which im... | https://arxiv.org/abs/2501.11868v1 |
θ7→bTθatθn, noting that bTθ=αT n 1 nPn i=1∇2ℓ(Zi, θn) θ. Consequently, the one-step estimator bTbθnof the linear functional bTθ0is a special case of our autoDML estimator, corresponding to the case where His the Euclidean space Rd. 2.3 Examples In this section, we provide examples illustrating how various parameters o... | https://arxiv.org/abs/2501.11868v1 |
x) + (1 − πP(x)) logit eµP(0, x), and νP: (a, x)7→eµP(a, x){1−eµP(a, x)}. The corresponding loss for the Hessian Riesz representer α0involves overlap weights (Li et al., 2019) and is given by ( z, α)7→ {a−πP(x)}2α(x, a)2− 2 ˙mθ0(a, x)].□ Our framework also supports smooth functionals of pseudo-outcome regressions, whic... | https://arxiv.org/abs/2501.11868v1 |
the development of novel autoDML estimators for functionals of survival and longitudinal models, such as the beta-geometric model in Section 6.1, as well as smooth functionals of causal contrasts, such as the CATE and conditional relative risk (van der Laan et al., 2024a). Another novel contribution of this work is the... | https://arxiv.org/abs/2501.11868v1 |
key requirement of our approach is that the population risk function admits a Neyman-orthogonal loss (Foster and Syrgkanis, 2023), ensuring the construction of√n-consistent, regular estimators of the population risk (van der Vaart, 1991). In statistical learning, Neyman-orthogonal loss functions enable fast empirical r... | https://arxiv.org/abs/2501.11868v1 |
trivially satisfied when ψPis a bounded linear functional and is satisfied by the linear functionals of the outcome regression studied in Chernozhukov et al. (2018b) and Hirshberg and Wager (2021). Conditions C2-C3 are standard regularity conditions for loss functions in the context of empirical risk minimization invol... | https://arxiv.org/abs/2501.11868v1 |
corrections based on the efficient influence function. The following additional conditions will be required. C4) (Existence and uniqueness): argminθ∈HLP(θ, ηP) is nonempty and contains exactly one element. C5) (Locally quadratic risk): Both of the following hold: i)∂2 θLP(θP, ηP) :˚H × ˚H →Ris symmetric and positive-de... | https://arxiv.org/abs/2501.11868v1 |
Chen and Liao (2014), and Hahn et al. (2018). Condition C5i ensures ∂2 θLP(θP, ηP)(·,·) defines a positive-definite inner product onH, which is known as the generalized Fisher inner product (e.g., Section 4.1.1 of Chen et al. 2014; Chen and Liao 2014). C5ii is trivially satisfied when ⟨·,·⟩H,Pis taken to be ∂2 θLP(θP, ... | https://arxiv.org/abs/2501.11868v1 |
minimum loss-based estimation, the Hessian representer αPassumes the role of the efficient influence function χPin targeted maximum likelihood estimation for 15 determining the least favorable submodel through θP. Interestingly, when His nonparametric, different loss functions may identify the same parameter Ψ, and The... | https://arxiv.org/abs/2501.11868v1 |
estimator, bψdml n, for Ψ( P0) is a one-step debiased estimator (Bickel et al., 1993) based on the efficient influence function of Ψ, as provided in Theorem 2. Specifically, bψdml nis obtained by adding an influence function-based bias correction to the plug-in estimator ψn(θn) and is defined as bψdml n:=1 nnX i=1m(Zi,... | https://arxiv.org/abs/2501.11868v1 |
procedure will typically not have a closed-form expression. However, in the localized regime where the initial estimator θnis sufficiently close to θ0, we can approximate the loss function ε7→ℓηn(·, θn+εαn) locally around θnby the quadratic loss function ε7→ε˙ℓηn(θn)(αn) + 1 2ε2¨ℓηn(θn)(αn, αn) to find, in first order,... | https://arxiv.org/abs/2501.11868v1 |
n is automatically debiased, corresponding to a one-step estimator with the nuisance estimates (ηn, θn,k(n), α0,k(n)). Forbψsieve n to be asymptotically linear and efficient for Ψ( P0), the sieve dimension k(n) must grow quickly 19 enough to ensure that both θn,k(n)andα0,k(n)are consistent for the true parameters θ0and... | https://arxiv.org/abs/2501.11868v1 |
. . . , J do 4: get estimator ηn,s:=Aη(Dn\T(s)) ofη0usingAηfromDn\T(s); 5:end for # Cross-fit M-estimand. 6:fors= 1, . . . , J do 7: get estimator θn,s ofθ0from Dn\T(s)using the cross-fitted orthogonal risk: θ7→P i∈[n]\I(s)ℓηn,j(i)(Zi, θ); 8:end for # Cross-fit Riesz representer 9:fors= 1, . . . , J do 10: get estimato... | https://arxiv.org/abs/2501.11868v1 |
result to study the autoTML and autoSieve estimators, bψtmle n andbψsieve n. Denote the EIF estimate corresponding to θn,ηn, and αnbyχn:=m(·, θn)−Pnm(·, θn)−˙ℓηn(θn)(αn). Our main theorem, which we now present, will make use of the following conditions. D1) Bounded estimators: ∥ηn∥N,⋄,∥θn∥H,⋄, and∥αn∥H,⋄areOp(1). D2) N... | https://arxiv.org/abs/2501.11868v1 |
correspond to the class of orthogonal squared error losses in Example 3. Condition D5 is a standard empirical process condition that holds when ∥χn−χ0∥P0=op(1) and the possible realizations of χn−χ0are not too complex, such that they fall within a fixed Donsker class. In 23 the context of debiased machine learning, sam... | https://arxiv.org/abs/2501.11868v1 |
more complex — e.g., higher dimensional and less smooth — than the M-estimand θ0, and therefore less amenable to estimation at sufficiently fast rates to satisfy D2 using the method of sieves. Given a finite-dimensional model H, plug-in M-estimators of the form1 nPn i=1m(Zi, θn) with θn:= argminθ∈HPn i=1ℓηn(Zi, θ) are ... | https://arxiv.org/abs/2501.11868v1 |
or a data-dependent feature embedding, with H0as a limiting oracle embedding. Given the selected model Hn, our proposed ADML estimator of Ψ( P0) is the debiased machine learning estimator bψn,Hnof the data-adaptive working parameter Ψ Hn:M → R, obtained using any of the methods proposed in Section 4.1. In this section,... | https://arxiv.org/abs/2501.11868v1 |
the validity of the debiased machine learning estimator. Condition C2 is an asymptotic stability condition that informally requires the learned model Hnto asymptotically stabilize to a fixed oracle submodel H0. This condition generally requires that ∥χ0,Hn−χ0,H0∥P=op(1), which necessitates that the nuisance functions α... | https://arxiv.org/abs/2501.11868v1 |
exp(aP(x)) + exp( bP(x)) +t−1. Notably, the hazard probabilities under this model approach zero as time increases, ensuring that failure events are more likely to occur early in time. This semiparametric hazard model enforces a parametric dependence on time, allowing for the forecasting of survival probabilities beyond... | https://arxiv.org/abs/2501.11868v1 |
the interaction constraints ar- gument in xgboost . We consider three autoDML estimators for the estimand ψ0(θ0), as proposed in Section 4.1. Specifically, we evaluate two variants of the one-step autoDML estimator: one that employs the TMLE- inspired stabilization (see (5)) from Algorithm 2 to calibrate the estimated ... | https://arxiv.org/abs/2501.11868v1 |
function of sample size ( n) for different estimators: (Top Left) Absolute Bias, (Top Right) Standard Error, and (Bottom) Coverage (95%). the loss function and a linear approximation of the functional. In settings where these approximations are in- adequate, the one-step autoDML estimator may suffer from excessive bias... | https://arxiv.org/abs/2501.11868v1 |
Herv´ e Abdi, Dominique Valentin, and Betty Edelman. Neural networks . Number 124. Sage, 1999. Heejung Bang and James M Robins. Doubly robust estimation in missing data and causal inference models. 32 Biometrics , 61(4):962–973, 2005. David Benkeser, Marco Carone, MJ Van Der Laan, and Peter B Gilbert. Doubly robust non... | https://arxiv.org/abs/2501.11868v1 |
J Foster and Vasilis Syrgkanis. Orthogonal statistical learning. The Annals of Statistics , 51(3): 879–908, 2023. Constantine E Frangakis, Tianchen Qian, Zhenke Wu, and Ivan Diaz. Deductive derivation and turing- computerization of semiparametric efficient estimation. Biometrics , 71(4):867–874, 2015. Yoav Freund and R... | https://arxiv.org/abs/2501.11868v1 |
learning tools, 03 2020. Hongxiang Qiu, Alex Luedtke, and Marco Carone. Universal sieve-based strategies for efficient estimation using machine learning tools. Bernoulli: official journal of the Bernoulli Society for Mathematical Statistics and Probability , 27(4):2300, 2021. Louis B Rall. Automatic differentiation: Te... | https://arxiv.org/abs/2501.11868v1 |
for selection among estimators and a general cross-validated adaptive epsilon-net estimator: Finite sample oracle inequalities and examples. 2003. 37 Mark J van der Laan and Susan Gruber. Targeted minimum loss based estimation of an intervention specific mean outcome. 2011. Mark J van der Laan and Alexander R Luedtke. ... | https://arxiv.org/abs/2501.11868v1 |
θLP(θ), ηP)d dεθPε ε=0, h =⟨−˙ℓηP(·, θP)(h), S⟩P. (9) 39 By C1, we have that d dεψPε(θPε) ε=0=d dεψP(θPε) ε=0+d dεψPε(θP) ε=0 =˙ψP(θP)d dεθPε ε=0 +⟨m(·, θP)−Ψ(P), S⟩P. By C5 and the definition of αP, we have the Riesz representation property: ˙ψP(θP) (h) =∂2 θLP(θP, ηP)(h, αP). Taking h=d dεθPε ε=0in the above disp... | https://arxiv.org/abs/2501.11868v1 |
Taking h=ε−1{θPε−θP}in the above display, we find that ψPε(θPε)−ψP(θP) =ε∂2 θLP(θP, ηP) ε−1{θPε−θP}, αP +ε⟨m(·, θP), S⟩P+o(ε). Substituting (12) into the above expression, this implies that ψPε(θPε)−ψP(θP) =ε⟨−˙ℓηP(·, θP)(h), S⟩P+ε⟨m(·, θP), S⟩P+o(ε). Hence, lim ε→0ψPε(θPε)−ψP(θP) ε ε=0=⟨−˙ℓηP(·, θP)(αP), S⟩P+⟨m(·, θ... | https://arxiv.org/abs/2501.11868v1 |
of one-step estimator) .Assume C1-C5 hold at P:=P0. Then, bψdml n−Ψ(P0) =Pnχ0+ (Pn−P0)(χn−χ0) + Rem P0(ηn, θn, αn). Proof of 2. Under conditions C1-C5, Ψ is a pathwise differentiable parameter at P0with EIF χ0provided in Theorem 2. By definition of Rem P0(ηn, θn, αn), we have the following von Mises expansion of the bi... | https://arxiv.org/abs/2501.11868v1 |
parameter of interest Ψ( P) =ψP(θP) is the mean survival probability P(T > t 0) at some time t0∈N.We can write the functional ψP:H →Ras θ7→ψP(θ) :=Pθ(T > t 0) =EP[m(Z, θ)], where m: (z, θ)7→Pθ(T > t 0|X=x) is a nonlinear functional. The derivative of this functional at θin the direction of h∈ Hsatisfies the expression:... | https://arxiv.org/abs/2501.11868v1 |
synthetic experiment The data structure Z= (X, A,eT,∆) is generated as follows. The covariate vector Xis drawn from a uniform distribution on [ −1,1]3. Given X, the treatment assignment Ais sampled from a Bernoulli distribution with success probability π0(X) := expit(2√ 3(X1+X2+X3)). The uncensored survival time Tis dr... | https://arxiv.org/abs/2501.11868v1 |
Block Adaptive Progressive Type-II Censored Sampling for the Inverted Exponentiated Pareto Distribution: Parameter Inference and Reliability Assessment Rajendranath Mondala∗, Aditi Kar Gangopadhyaya†, Raju Bhaktaa‡and Kousik Maitib§ aDepartment of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, Utt... | https://arxiv.org/abs/2501.12179v1 |
censoring scheme and the computaions for expected total test time were performed. Sobhi and Soliman (2016) studied the estimation of the unknown parameters and the reliability and hazard functions of the two-parameter exponentiated Weibull distribution under the adaptive progressive Type-II censoring. Here, both the ma... | https://arxiv.org/abs/2501.12179v1 |
the occurance of first failure in each test group. Wu and Ku¸ s (2009) com- bined the progressive Type-II censoring scheme with FFCS to introduce the scheme named, progressive first-failure censoring scheme (PFFCS). Under this new censoring scheme, maximum likelihood estimation, exact and approximate confidence interva... | https://arxiv.org/abs/2501.12179v1 |
(1−2−1/α)−1/β−1 −1;α, β > 0. (1.5) Maurya et al. (2019) estimated the parameters of the IEP distribution by utilizing both the MLE method and Bayesian estimation method under the progressive Type-II censoring. Here, MLEs are derived by using the expectation-maximization algorithm and the Bayesian estimation was carried... | https://arxiv.org/abs/2501.12179v1 |
Maximum Likelihood Estimation Suppose, in a block censoring setup, nnumber of life testing units are divided into knumber of groups (or facilities) and the i-th group contains ninumber of units ( i= 1,2, . . . , k ) such thatPk i=1ni=n. Let each facility is tested under the adaptive progressive model introduced by Ng e... | https://arxiv.org/abs/2501.12179v1 |
αandβ. Consequently, the MLEs of αiandβare ˆαiand ˆβ, respectively ( i= 1,2, . . . , k ). It should be noted that, (2.4) cannot be solved in closed form expressions and so, we have to solve it numerically. Further, due to the invariance property of MLE, the MLE of any parametric function ϕ(α, β) isϕ(ˆα,ˆβ). Therefore, ... | https://arxiv.org/abs/2501.12179v1 |
should make use of the delta method (see Beutner (2024)). According to this method, ϕ(ˆβ,ˆα1,ˆα2, . . . , ˆαk) also has asymptotic normal distribution with mean ϕ(β, α 1, α2, . . . , α k) and variance Vϕ= [∇ϕ(β, α 1, α2, . . . , α k)]TI−1[∇ϕ(β, α 1, α2, . . . , α k)], where I−1is the inverse of the Fisher information m... | https://arxiv.org/abs/2501.12179v1 |
rij=Rij,forj= 1,2, . . . , m iand i= 1,2, . . . , k. Again, the ratios Qij=Wij/Wimi; (j= 1,2, . . . , m i−1) form an order statistic of size ( mi−1) from the standard uniform distribution. Moreover, Qij’s are independent of Wimi=−αimi−1X l=1(ril+ 1) log" 1−Til 1 +Tilβ# −αi ni−mi−1X l=1(ril+ 1)! log" 1−Timi 1 +Timiβ... | https://arxiv.org/abs/2501.12179v1 |
the function Φ iis defined as Φi(β) =−mi−1X l=1(ril+ 1) log" 1−Til 1 +Tilβ# − ni−mi−1X l=1(ril+ 1)! log" 1−Timi 1 +Timiβ# (4.3) andR∗ iis an unit sample drawn from the χ2(2mi) distribution. Furthermore, pivotal estimates for the unknown parameters αandβas well as for the reliability and hazard functions, and MTF ca... | https://arxiv.org/abs/2501.12179v1 |
statistics and records are found in Barter (1988). Order statistics are used in many different domains, including wireless communication, signal processing, quality control, survival analysis, life testing, relia- bility, and classification analysis. For more details on the concept of order statistics and its applicati... | https://arxiv.org/abs/2501.12179v1 |
Here, the notations ⌊.⌋and⌈.⌉represent the floor function and the ceiling function, respectively. For each BAPCS setup, to maintain simplicity, the threshold times Ti’s are taken equal for every block or facility. All the estimated values, approximate biases and variances , ACIs, GCIs, and the lengths (denoted by L in ... | https://arxiv.org/abs/2501.12179v1 |
2.5629 4.6115 2.0486 3.5019 0.0019 0.2722 2.5962 4.5999 2.0037 R(t) 0.5805 0.0110 0.0008 0.5238 0.6372 0.1134 0.5778 0.0084 0.0008 0.5200 0.6399 0.1139 h(t) 1.0323 -0.0150 0.0075 0.8637 1.2009 0.3372 1.0242 -0.0231 0.0075 0.8620 1.1985 0.3365 MTF 0.8963 0.0226 0.0033 0.7842 1.0083 0.2241 0.8958 0.0221 0.0035 0.7859 1.0... | https://arxiv.org/abs/2501.12179v1 |
0.3184 1.0271 -0.0202 0.0066 0.8734 1.1911 0.3177 MTF 0.8935 0.0199 0.0029 0.7884 0.9987 0.2102 0.8930 0.0194 0.0031 0.7899 1.0055 0.2157 15 Table 4: Point and interval estimates under the BAPCS setup 3. ML Estimates Pivotal Estimates Est. Bias Variance ACIs L Est. Bias Variance GCIs L nCP Pars. Lower Upper Lower Upper... | https://arxiv.org/abs/2501.12179v1 |
2.5315 4.6397 2.1082 R(t) 0.5836 0.0142 0.0008 0.5278 0.6394 0.1116 0.5801 0.0107 0.0008 0.523 0.6354 0.1124 h(t) 1.0272 -0.0201 0.0078 0.8554 1.1991 0.3437 1.0160 -0.0313 0.0077 0.8514 1.1932 0.3418 MTF 0.9023 0.0286 0.0033 0.7910 1.0136 0.2227 0.9011 0.0274 0.0035 0.7913 1.022 0.2307 200 2 β2.3139 0.0639 0.0330 1.958... | https://arxiv.org/abs/2501.12179v1 |
0.2390 0.5662 2.3026 5.1754 2.8728 3.6396 0.1396 0.5445 2.4038 5.2054 2.8016 α3.5725 0.0725 0.2502 2.6066 4.5383 1.9317 3.4778 -0.0222 0.2397 2.6215 4.5043 1.8828 R(t) 0.5817 0.0122 0.0007 0.5283 0.6351 0.1068 0.5787 0.0093 0.0008 0.5242 0.6316 0.1074 h(t) 1.0287 -0.0187 0.0066 0.8699 1.1874 0.3175 1.0205 -0.0268 0.006... | https://arxiv.org/abs/2501.12179v1 |
2.7779 3.6384 0.1384 0.5127 2.4297 5.1479 2.7182 α43.6814 0.1814 0.4584 2.3847 4.9780 2.5933 3.6044 0.1044 0.4437 2.4678 5.0095 2.5416 α53.7198 0.2198 0.5250 2.3339 5.1058 2.7720 3.6390 0.1390 0.5083 2.4314 5.1459 2.7145 α3.5512 0.0512 0.1893 2.7091 4.3933 1.6843 3.4771 -0.0229 0.1825 2.7172 4.366 1.6488 R(t) 0.5791 0.... | https://arxiv.org/abs/2501.12179v1 |
parameters using the carbon fibre test data and AIC, BIC, CAIC, HQIC measures, K-S statistics and p-value. Model Pars. MLE AIC BIC CAIC HQIC K-S statistic p-value IEP (α, β)(43.8478, 7.6876) 108.6455 113.1138 115.1138 110.4182 0.0755 0.8266 GP (k, σ) (0, 1.4510) 193.3750 197.8433 199.8433 195.1477 0.3623 2.716×10−8 EP ... | https://arxiv.org/abs/2501.12179v1 |
India), is sincerely acknowledged with thanks by Rajendranath Mondal. Dr. Raju Bhakta gratefully acknowledges the financial support for this research work under the project grant IITR/SRIC/2301/DRD-2236-CSE/23- 24/SR-03. Conflict of Interest All the authors have no conflict of interest. References Abu-Moussa, M., El-Di... | https://arxiv.org/abs/2501.12179v1 |
sampling plans for inspection by variables. Handbook of Statistics , 17:497–511. Sewailem, M. F. and Baklizi, A. (2019). Inference for the log-logistic distribution based on an adaptive progressive type-II censoring scheme. Cogent Mathematics and Statistics , 6(1):1684228. Shrahili, M. and Kayid, M. (2023). Excess life... | https://arxiv.org/abs/2501.12179v1 |
arXiv:2501.12212v1 [stat.ML] 21 Jan 2025Quantitative Error Bounds for Scaling Limits of Stochastic Iterative Algorithms Xiaoyu Wang1, Miko/suppress laj J. Kasprzak2, Jeffrey Negrea3, Solesne Bourguin1, and Jonathan H. Huggins1,4 1Department of Mathematics & Statistics, Boston University , e-mail: shawnwxy@bu.edu ;bourgu... | https://arxiv.org/abs/2501.12212v1 |
Gk:=∇Lk. The SGD algorithm con- structs a sequence of estimates θ1,θ2,...of a local critical point θ⋆that satisfies G(θ⋆) = 0. Given an initial value θ0and step size sequence h1,h2,..., the estimates are defined recur- sively by the update equation θk+1=θk+hkGk(θk). (1.1) More recently, SGD and variants such as stochasti... | https://arxiv.org/abs/2501.12212v1 |
a Markov c hain Monte Carlo (MCMC) sampler by analyzing its mixing time. In addition, they can be us ed to determine the asymptotic stationary distribution of the sampler and establish Bernstein-von Mises- type theorems for the stationary distribution ( Kushner and Yin ,2003;Mandt, Hoffman and Blei,2017;Negrea et al. ,2... | https://arxiv.org/abs/2501.12212v1 |
into three components: the error between YandZ, the linear approximation error, and the discretization error, with the latter two being straightforward to bound. To bound the first error, we construct an exchangeable pair ( Y,Y′) by randomly replacing the batch in one of the SG(L)D stochastic gradients. This approach en... | https://arxiv.org/abs/2501.12212v1 |
the univariate case, we can use the following functional version of the method of exchangeable pairs. We recall that ( Y,Y′) is anexchangeable pair ofD([0,1])-valued random variables if ( Y,Y′) and (Y′,Y) are equal in distribution. Proposition 1 (D¨ obler and Kasprzak (2021, Theorem 4.1 and Proposition 3.2)) .Assume th... | https://arxiv.org/abs/2501.12212v1 |
Al gorithms and Ornstein–Uhlenbeck Processes 3.1. Assumptions Our main result relies on the following assumptions, all of which are sta ndard or weaker thanthosetypicallyusedinnon-asymptoticanalysesofSG(L)D( Brosseetal. ,2017;Dieuleveut, Durmus and Bach ,2020;Kushner and Yin ,2003;Moulines and Bach ,2011;Pflug,1986) Ass... | https://arxiv.org/abs/2501.12212v1 |
b. We are interested in the dependence on the sample size n, batch size b, and number of epochs m. In particular, for the batch size, we are interested in selecting bto minimize the error bound. Numerical setting For positive constants c1,c2andc3, setα=c1h−1,w=c2{(b/h)1/2∧ β1/2}and assume b≤c3h−4. Assumeh≤1,β≥1. This s... | https://arxiv.org/abs/2501.12212v1 |
two sources: that the second derivat ive matrices, σi, that appear in the Taylor expansion ∇ℓi(θ) =ψi+σi(θ−θ⋆)+Ri(θ) do not commute; furthermore, the remainder term Ri(θ) is non-constant. In the present work, we handle the remainder term, while deferring analysis of non- commutativity of the second derivative matrices.... | https://arxiv.org/abs/2501.12212v1 |
where Σ(n):=−n−1/summationtextn i=1∇⊗2ℓi(θ(n) ⋆) and Ω(n):=n−1/summationtextn i=1∇ℓi(θ(n) ⋆)⊗2. While Theorem 1bounds the difference between Y(n)andZ(n), it remains to show that Z(n)converges to a population process Z(∞)asn→ ∞. In fact, Negrea et al. (2022) show thatY(n)convergences to Z(∞)= (Z(∞) t)t∈[0,1]defined by the... | https://arxiv.org/abs/2501.12212v1 |
0≤t≤1/vextendsingle/vextendsingle/vextendsingle/vextendsinglee−Bt/integraldisplayt 0eBs/parenleftBig√ A−/radicalbig ˜A/parenrightBig dWs/vextendsingle/vextendsingle/vextendsingle/vextendsinglep/parenrightBigg +22(p−1)E/parenleftBigg sup 0≤t≤1/vextendsingle/vextendsingle/vextendsingle/vextendsinglee−Bt/integraldisplayt ... | https://arxiv.org/abs/2501.12212v1 |
Limits of Stochastic Iter ative Algorithms 16 Therefore, sup u∈D([0,1])|g2(u)| 1+∝⌊a∇d⌊lu∝⌊a∇d⌊l3≤sup u∈D([0,1])∝⌊a∇d⌊lu∝⌊a∇d⌊l2 1+∝⌊a∇d⌊lu∝⌊a∇d⌊l3<0.53, sup u∈D([0,1])∝⌊a∇d⌊lDg2(u)∝⌊a∇d⌊l 1+∝⌊a∇d⌊lu∝⌊a∇d⌊l2≤sup u∈D([0,1])2∝⌊a∇d⌊lu∝⌊a∇d⌊l 1+∝⌊a∇d⌊lu∝⌊a∇d⌊l2= 1, sup u∈D([0,1])∝⌊a∇d⌊lD2g2(u)∝⌊a∇d⌊l 1+∝⌊a∇d⌊lu∝⌊a∇d⌊l≤sup ... | https://arxiv.org/abs/2501.12212v1 |
→0 and /vextendsingle/vextendsingle/vextendsingleΩ(n)−Ω(∞)/vextendsingle/vextendsingle/vextendsinglen→∞− −−− →0 forΩ(∞)>0, /vextendsingle/vextendsingle/vextendsingleΣ(n)−Σ(∞)/vextendsingle/vextendsingle/vextendsinglen→∞− −−− →0 forΣ(∞)>0. Then, the law of Y(n)converges weakly to the law of Z(∞)asn→ ∞, in D([0,1]), with... | https://arxiv.org/abs/2501.12212v1 |
1. Proof of Proposition 4.Using the integral form of Taylor’s remainder theorem, ∇ℓi(θ) =ψ(n) i+σ(n) i(θ−θ(n) ⋆)+/integraldisplayθ θ(n) ⋆∇⊗3ℓi(ϑ)(θ−ϑ)dϑ. Thus, |∇R(n) i(θ)| ≤|θ−θ(n) ⋆|2 2∝⌊a∇d⌊l∇⊗3ℓi(·)∝⌊a∇d⌊lL∞ and the result follows from Lemma 1. Remark 8.The conditions for Propositions 3and4can be slightly relaxed t... | https://arxiv.org/abs/2501.12212v1 |
be found in Appendix Cof the supplementary material. Remark 9.Suppose that the assumptions of Theorem 1hold. Assume additionally that n/b→ ∞andm∈O(1) forn→ ∞, and thatr >1 is such that E|ΨX|4r<∞. Moreover, suppose that Assumption 4also holds for p= 4r. Assume that CR≥1 and thatL6+C2 R n∈ Wang et al. / Quantitative Scal... | https://arxiv.org/abs/2501.12212v1 |
Limits of Stochastic Iter ative Algorithms 23 The proof of Lemma 4is a straightforward induction, so we defer it to Appendix F.1in the supplementary material. Under Assumptions 1and3, the following three lemmas give the bound for moments of ηkand moments of the sup-norms of ηk. These bounds are useful in bounding the e... | https://arxiv.org/abs/2501.12212v1 |
Next, we bound the exchange error term ǫexch=∝⌊a∇d⌊lg∝⌊a∇d⌊lM 6ΛE∝⌊a∇d⌊lY −Y′∝⌊a∇d⌊l3. Lemma 9. The following bound on ǫexchholds: ǫexch/lessorsimilar∝⌊a∇d⌊lg∝⌊a∇d⌊lMw3/braceleftBigg L3α5/2h6 b3/2/bracketleftBigg Eψ4 I(1,1)+Ω2+√ Ω(1+L3)/bracketleftbigg/parenleftBig Eψ4 I(1,1)+Ω2/parenrightBig3/4 +1/bracketrightbigg +ΩL... | https://arxiv.org/abs/2501.12212v1 |
bound has not been obtained in t he literature in any case in which the target process has non-independent increments, as e xplained in Remark 5. The reasonis that previousworkson functional Stein’s method, includin g (D¨ obler and Kasprzak , 2021;Kasprzak ,2020a,b) did not consider quantities of the form YkΨ t, as inE... | https://arxiv.org/abs/2501.12212v1 |
due to its length. Acknowledgments XW and JHH were partially supported by National Science Foundation CAREER award IIS-2340586.MK was partially supported by the France 2030 prog ram and by the European Wang et al. / Quantitative Scaling Limits of Stochastic Iter ative Algorithms 30 Union’s Horizon 2020 research and inn... | https://arxiv.org/abs/2501.12212v1 |
By plugging in the parameters into the error bound from Theorem 5, we obtain a bound expressed in terms of m,n,b: Wang et al. / Quantitative Scaling Limits of Stochastic Iter ative Algorithms 33 ǫR/lessorsimilarm n1/2K1CR(Ω+1) +m3 n3/2C3 RK3 3/bracketleftBig/parenleftBig Ω3+(Eψ4 I(1,1))3/2+Eψ4 I(1,1)Ω+Eψ6 I(1,1)/parenr... | https://arxiv.org/abs/2501.12212v1 |
/ Quantitative Scaling Limits of Stochastic Iter ative Algorithms 36 Thirdly, let us upper-bound the term P(∝⌊a∇d⌊lYǫ−Y∝⌊a∇d⌊l ≥θ). Let us fix r>1. Note that E|Yt−Ys|2r =w2rE/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle⌊αt⌋/summationdisplay l=⌊αs⌋+1hGl(ηl)+/radicalBigg 2h βξl/vexte... | https://arxiv.org/abs/2501.12212v1 |
+L2r/bracerightBig . Wang et al. / Quantitative Scaling Limits of Stochastic Iter ative Algorithms 39 C.3. Proof of Lemma 11 For alli,j, letRI(i,j)(θ) =∇ℓI(i,j)(θ)−/hatwide∇ℓI(i,j)(θ) Note that, for all k∈ {1,...,α}, θ2 k=θ2 k−1+2θk−1/parenleftbigg −h bb/summationdisplay i=1σI(k−1,i)θk−1+h bb/summationdisplay i=1ψI(k−1... | https://arxiv.org/abs/2501.12212v1 |
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