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in high-dimensional graphical models. arXiv preprint arXiv:2309.11420 . Moitra, A. and Valiant, G. (2010). Settling the polynomial l earnability of mixtures of gaussians. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Scienc e, pages 93–102. IEEE. Oko, K., Akiyama, S., and Suzuki, T. (2023). Diffusion mod... | https://arxiv.org/abs/2504.05300v1 |
A Generalized Tangent Approximation Framework for Strongly Super-Gaussian Likelihoods Somjit Roy1 *Pritam Dey1 *Debdeep Pati2Bani K. Mallick1 Abstract Tangent approximation form a popular class of variational inference ( VI) techniques for Bayesian analysis in intractable non-conjugate models. It is based on the princi... | https://arxiv.org/abs/2504.05431v1 |
variational inference (CAVI ) which updates the single block components at a time, keeping the others fixed. Corresponding algorithmic convergence guarantees have been studied by Bhattacharya et al., 2023. CAVI ’s widespread application in VIextend out to popular modeling structures spanning *Equal contribution1Departm... | https://arxiv.org/abs/2504.05431v1 |
1978; Yu & Moyeed, 2001; Wang et al., 2012), which can be interpreted as a case of skewed heavy-tailed likelihood modeling. Moreover, TAVIE extends to count data models which are prevalent in various biological applications including genomics (Anders & Huber, 2010), genetics (Zhang et al., 2020) and microbiome studies ... | https://arxiv.org/abs/2504.05431v1 |
obtained is agnostic to the choice of SSGlikelihoods and (ii) the computational complexity of each iteration is O(n), as the multivariate optimization of the variational parameters decomposes into n univariate optimizations at each iteration, making TAVIE embarrassingly parallelizable. Consequently for the aforemention... | https://arxiv.org/abs/2504.05431v1 |
. . . , n b i di LAPLACE τexp −τ|yi−x⊤ iβ| 0 — TYPE I QUANTILE REGRESSION (ALD) τu(1−u) exp −2τρu(yi−x⊤ iβ) 1−2u — STUDENT ’S-t 1 +τ2(yi−x⊤ iβ)2/ν−(ν+1)/20 — NEGATIVE -BINOMIAL exp yix⊤ iβ / 1 + exp x⊤ iβ yi+m(yi−m)/2yi+m TYPE II BINOMIAL (LOGISTIC REGRESSION ) exp yix⊤ iβ / 1 + exp x⊤ iβ myi−(m/2) m Table ... | https://arxiv.org/abs/2504.05431v1 |
4 TA VIE for SSG Likelihoods et al., 2019). Given a Gaussian prior distribution π(β) =Np(β|µβ,Σβ), we denote the joint fractional posterior distribution aspα(y, β|X) =pα(y|X, β)π(β)by slight abuse of notation. Using these notations and the minorant form of the joint likelihood in (4): pα(y, β|X)∝π(β)(nY i=1p(yi|xi, β))... | https://arxiv.org/abs/2504.05431v1 |
risk obtained by integrating theα−R´enyi divergence with respect to the optimal variational solution. Consider ϕp(x;β,Σ)to be the p−dimensional multivariate Gaussian density evaluated at x∈Rpwith mean vector µand variance-covariance matrix Σ. Let∥X∥2,∞= max {∥xi∥, i= 1,2, . . . , n }and∥X∥∞= max {|xij|, i= 1,2, . . . ,... | https://arxiv.org/abs/2504.05431v1 |
1,2, . . . , n . In each of the following simulation examples, the ℓ2norm between the estimates and the true regression parameter βois used a measure of discrepancy to analyze the accuracy of the resultant estimates. 6.1. TAVIE for Laplace Regression Fixing the scale parameter τ= 0.5, the errors are generated independe... | https://arxiv.org/abs/2504.05431v1 |
dispersion parameter τ= 0.5,βo={1,2, . . . , p }andp= 20 . as the sample size increases. However, TAVIE offers significant computational efficiency by being 103times faster than the Gibbs sampling algorithm. 7. Application of TAVIE in Bayesian Quantile Regression In context of application to robust regression, the real... | https://arxiv.org/abs/2504.05431v1 |
impact on income quantiles and (v) The difference in age does not have significant effect on lower income quantiles, but becomes fairly pronounced in higher income quantiles. Now we turn to comparing the TAVIE QR estimates in Table 4 with competing estimates provided in Yang et al., 2013 for analyzing the U.S. 2000 Cen... | https://arxiv.org/abs/2504.05431v1 |
values of the prior hyper-parameters are taken as, µβ=0pandΣβ=Ip. This setting has also been applied when comparing the TAVIE QR algorithm with other competing methods. 9 TA VIE for SSG Likelihoods Figure 2. Comparison of the TAVIE QR coefficient estimates with estimates obtained from competing methods: FAST QR ,SPC3,L... | https://arxiv.org/abs/2504.05431v1 |
1984. doi: 10.1109/TPAMI.1984. 4767596. Ghosh, I., Bhattacharya, A., and Pati, D. Statistical optimality and stability of tangent transform algorithms in logit models. Journal of Machine Learning Research , 23(184):1–42, 2022. URL http://jmlr.org/papers/v23/21-0190. html . Graves, A. Practical variational inference for... | https://arxiv.org/abs/2504.05431v1 |
Weiss, Y ., Sch ¨olkopf, B., and Platt, J. (eds.), Advances in Neural Information Processing Systems , volume 18. MIT Press, 2005. Pati, D., Bhattacharya, A., and Yang, Y . On statistical optimality of variational bayes. In Storkey, A. and Perez-Cruz, F. (eds.), Proceedings of the Twenty-First International Conference ... | https://arxiv.org/abs/2504.05431v1 |
Bioinformatics , 21(1):474, 2020. URL https://doi.org/10.1186/s12859-020-03758-1 . 13 TA VIE for SSG Likelihoods A. Lemmas Lemma A.1 (General TAVIE variational update for strongly super-Gaussian function) .Consider a strongly super-Gaussian function f(.)andκbe any positive constant. Suppose g(x) = log f(x)−bx, as defin... | https://arxiv.org/abs/2504.05431v1 |
to show that, the solution of (30) satisfies the fixed point update in (12), we use the first order stationarity condition for maximizing the E-step objective function Qα(ξt+1|ξt)in (7) with respect to ξt+1, given by: d dξt+1Qα(ξt+1|ξt) =Eβ|y,X,ξtd dξt+1logpα l(y, β|ξt+1,X) = 0 (31) which is essentially equivalent to... | https://arxiv.org/abs/2504.05431v1 |
log-prior mass, −log(π(Bn(βo, ε))). Therefore, what remains, is to provide a high-probability bound for the first term in (39) and at the same time develop an upper bound for the negative log-prior concentration term ,−log(π(Bn(βo, ε))). High-probability upper bound for the first term in (39):From (39), using Fubini’s ... | https://arxiv.org/abs/2504.05431v1 |
i=1{x⊤ i(β−βo)}2 −x⊤ i(β−βo) + 2( yi−x⊤ iβo) 2 ≤Mτ4 2nX i=1{x⊤ i(β−βo)}2n 2 x⊤ i(β−βo) 2+ 8(yi−x⊤ iβo)2o ,using the inequality (a+b)2≤2a2+ 2b2 ≤Mnτ4" ∥X∥4 2,∞∥β−βo∥4 2+ 4∥X∥2 2,∞∥β−βo∥2 2( 1 nnX i=1(yi−x⊤ iβo)2)# (52) We obtain a probability bound for the quantity n−1Pn i=1(yi−x⊤ iβo)2in(52) above. Under Pβo,ϵi=yi−x⊤... | https://arxiv.org/abs/2504.05431v1 |
iβo (x⊤ iβ)2−(x⊤ iβo)2 #) ≤nX i=1yi+m 2d2g(x) dx4 ˜s (x⊤ iβ)2−(x⊤ iβo)2 2 ,by second order Taylor expansion ofg(x)w.r.tx2 ≤nX i=1yi+m 2h x⊤ i(β−βo) 2 x⊤ i(β−βo) + 2x⊤ iβo 2i(64) 20 TA VIE for SSG Likelihoods where the last inequality in (64) above follows from the fact that, 0< d2g(x)/dx4<1for all x∈R, sinc... | https://arxiv.org/abs/2504.05431v1 |
ξt+1 iyix⊤ iµα(ξt)−y2 i 2ξt+1 i−r(ξt+1 i) 2 =nX i=1a+n bα(ξt) −1 2ξt+1 in x⊤ iΣα(ξt)xi+ yi−x⊤ iµα(ξt)2o −r(ξt+1 i) 2 (71) which can be maximized as in Section 3 to obtain the updates as: ξt+1 i=−p κi(ξt) g′(p κi(ξt))(72) fori= 1,2, . . . , n , where: κi(ξ) =a+n bα(ξ)n x⊤ iΣα(ξ)xi+ yi−x⊤ iµα(ξ)2o Thus, (70) an... | https://arxiv.org/abs/2504.05431v1 |
QR algorithm with point-wise 95% confidence intervals. The response is the logof annual salary. Except for the intercept and the education covariates, all the other covariates are 0−1binary indicators. 24 TA VIE for SSG Likelihoods E.2. Comparison of Remaining TAVIE QR Estimates with FAST QR ,SPC3,LSand Benchmark Quant... | https://arxiv.org/abs/2504.05431v1 |
Debiasing Continuous-time Nonlinear Autoregressions⋆ Simon Kuanga, Xinfan Lina, aDepartment of Mechanical and Aerospace Engineering, University of California, Davis; Davis, California, USA Abstract We study how to identify a class of continuous-time nonlinear systems defined by an ordinary differential equation affine ... | https://arxiv.org/abs/2504.05525v1 |
methods and online estimation, are time- and memory-efficient, and are insensitive to algorithm initialization. It has been observed that noise degrades identification accuracy [8, Fig. 6], and if the data is noisy, an initial smoothing pass on the state and/or deriva- tives improves the regression. Continuous-time SIN... | https://arxiv.org/abs/2504.05525v1 |
noisy measurements zi=x(ih) +ϵi, i∈[1. . . n], (2) where {ϵi}i∈[1...n]are independent random variables sat- isfying Eϵi= 0 and Eϵ4 i<∞independent of n, h. We assume that Eϵiϵ⊺ i= Σ ϵis known or reliably estimated. The following two assumptions are necessary for dis- cretizing (1). Assumption 1 (Space regularity of ϕ)Fo... | https://arxiv.org/abs/2504.05525v1 |
(4b) ˆyj= ˆx(m) j (4c) The smoothed derivatives ˆ xjare estimated using a linear filter, where the coefficients may be taken from local polynomial regression ( §D). 4.3 Estimating θ: three ways The simplest way to recover θ0from (4) is by least squares, given by the normal equations ˆΦ⊺ˆY=ˆΦ⊺ˆΦˆθLS. This estimator is a... | https://arxiv.org/abs/2504.05525v1 |
us examine the three terms of Lemma 6, Op(hβ+ n−1/2h−1 2α+γ+h2γ). The first, hβ, is the Taylor expan- sion error of yand would matter if ywere measured with- out noise. The second, n−1/2h−1 2α+γ, refers to how the fluctuation induced by noise in ycancels out over large- sample averaging. The third, h2γ, which has no ca... | https://arxiv.org/abs/2504.05525v1 |
effect of obser- vation noise. The third term is the nonlinear effect of observation noise. In the BC estimator, this term is an order of magnitude smaller in h. Remark 14 The matrices ˆΣϕϕ,ˆΣϕycan be viewed as a perturbative nonlinear generalization of “bias compen- sation” in least squares linear system identificatio... | https://arxiv.org/abs/2504.05525v1 |
let us summarize the leading order terms in order of polynomial degree in ηjandεj. (0) By Lipschitz continuity, ϕν j(Eˆxj)ϕν′ j(E˜xj) = ϕν j(xj)ϕν′ j(x) +O(hβ) (1)ϕν j(Eˆxj)∂µ1ϕν′ j(E˜xj)εµ1 jand its counterpart are Op(hγ) with zero mean. (2) (a) ϕν j(Eˆxj)1 2∂µ1∂µ2ϕ(E˜xj)h εµ1 jεµ2 j−Eεµ1 jεµ2 ji and its counterpart a... | https://arxiv.org/abs/2504.05525v1 |
the smaller bias of the BCLS and IV estimators to the reduction of the Op(h2γ) term to Op(h3γ). 11 Numerical example: Lorenz system This section applies all three estimators to the three- dimensional Lorenz system for x:[0, T]→R3, specified 7 Regression bias (%) std (%) RMSE (%) LS -8.22 2.01 8.46 BCLS 0.10 2.02 2.02 I... | https://arxiv.org/abs/2504.05525v1 |
A3,1 BC A3,1 IV 0.003162 0.001581 0.000000 0.001581 0.003162A3,2 LS A3,2 BC A3,2 IV 2.670690 2.668678 2.666667 2.664655 2.662643 A3,3 LS A3,3 BC A3,3 IV 0.001033 0.000516 0.000000 0.000516 0.001033A4,1 LS A4,1 BC A4,1 IV 0.001012 0.000506 0.000000 0.000506 0.001012A4,2 LS A4,2 BC A4,2 IV 0.998396 0.999198 1.000000 1.00... | https://arxiv.org/abs/2504.05525v1 |
bias when the estimator is nonlinear in the measurements. For example, if the estimator is f(z) =z2, then Ef(z)−f(Ez) =σ2for z∼ N(0, σ2). Abstract principles of our bias correction can be found in the statistics and econometrics literature [55, Chapter 10], [47,30]. [42] and [35] work out bias-corrected least squares (... | https://arxiv.org/abs/2504.05525v1 |
not the first to disregard this advice. [53] is a practically-oriented review of algorithms for differentiating noisy data. In the chemistry litera- ture, the use of local polynomial fits for smoothing and differentiation is called Savitzky-Golay filtering [3]. State Variable Filtering (SVF) [19, Chapter 1] precon- dit... | https://arxiv.org/abs/2504.05525v1 |
consistent filtering, f(¯xi) = f(xi) +O(hβ). This settles the first sum (I). The third term (III) is Op(h2γ) by the fluctuation hypothesis of consistent filtering. The sum (II) has n′∼nwith a local dependence structure: summands iandjare dependent if|i−j| ≤N∼h−α. We split the sum into Ndifferent sums of O(n/N) independ... | https://arxiv.org/abs/2504.05525v1 |
gives an explicit formula for a certain version of D,A⊺A contains the infamously ill-conditioned Hilbert matrix (D.6) . For numerical stability, we solve for Dby rewrit- ing the natural conditions (D.2) in a basis of Legendre polynomials. Lemma 20 (Row-by-row bound on D)The solu- tion to (D.1) using f(D) =∥D∥Fis D=B(A⊺... | https://arxiv.org/abs/2504.05525v1 |
L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences , 113(15):3932–3937, April 2016. Publisher: Proceedings of the National Academy of Sciences. [9] Sebastian Calonico, Matia... | https://arxiv.org/abs/2504.05525v1 |
S. J. Sheather. A Brief Survey of Bandwidth Selection for Density Estimation. Journal of the American Statistical Association , 91(433):401–407, March 1996. [28] V. Katkovnik. On adaptive local polynomial approximation with varying bandwidth. In Proceedings of the 1998 IEEE 14 International Conference on Acoustics, Spe... | https://arxiv.org/abs/2504.05525v1 |
the American Statistical Association , 1995. [47] Susanne M. Schennach. Recent Advances in the Measurement Error Literature. Annual Review of Economics , 8(Volume 8, 2016):341–377, October 2016. Publisher: Annual Reviews. [48] T. Soderstrom, H. Fan, B. Carlsson, and S. Bigi. Least squares parameter estimation of contin... | https://arxiv.org/abs/2504.05525v1 |
Online Bernstein–von Mises theorem Jeyong Lee∗Junhyeok Choi∗Minwoo Chae∗ April 9, 2025 Abstract Online learning is an inferential paradigm in which parameters are updated incrementally from sequentially available data, in contrast to batch learning, where the entire dataset is processed at once. In this paper, we assum... | https://arxiv.org/abs/2504.05661v1 |
Lin, 2013, Nguyen et al., 2018). In the Bayesian online learning framework described above, the posterior distribution must be repeatedly approximated as new data arrive. While the approximation error at each step may be negligible, the cumulative error over multiple updates may not be. Thus, a central concern is wheth... | https://arxiv.org/abs/2504.05661v1 |
total variation distance be- tween the full posterior Π( · |D) and the sequentially updated posterior Π T(·) is sufficiently small with high probability. When the dimension pofθ0is fixed, the required condition boils down to n≫(logN)4. The main results are formulated in a non-asymptotic framework and also cover the cas... | https://arxiv.org/abs/2504.05661v1 |
of the sequentially updated posterior and the penalized M-estimation, 3 respectively. Section 5 provides a non-asymptotic analysis of several regularity quantities. Section 6 establishes the BvM theorem for the full posterior. Our main results concerning the online BvM theorem are presented in Section 7. A concrete exa... | https://arxiv.org/abs/2504.05661v1 |
process mixture 4 models, while Jeong et al. (2023) employed an assumed density filtering (ADF) approach for similar tasks. More recently, Lambert et al. (2022) and Lambert et al. (2023) proposed computationally efficient online VB approximations using the Gaussian variational family. These works demonstrate that proje... | https://arxiv.org/abs/2504.05661v1 |
Y Nt),and D1:t= (D1,D2, ...,Dt) = (Yi)i∈[Nt]. Fori∈[N], let pθ,i(·) be the probability density function for Yiparametrized by θ∈Θ⊂Rp, and letℓθ,i(y) = log pθ,i(y) be the log density. For t∈[T], let Lt(θ) =Lt(θ;Dt) =NtX i=Nt−1+1ℓθ,i(Yi). LetP(N) θdenote the joint probability measure corresponding to the product density ... | https://arxiv.org/abs/2504.05661v1 |
where f′′andf′′′denote the second and third derivative of f(·), respectively. Then, one can prove that 1−δ(θ, η) f′′(θ)≤f′′(η)≤1/ 1−δ(θ, η)2f′′(θ),∀θ, η∈Rwith δ(θ, η)≤1 where δ(θ, η) =|f′′(θ)1/2(η−θ)|; see Theorem 5.1.7 in Nesterov et al. (2018) for a general statement. Intuitively, this condition ensures that the... | https://arxiv.org/abs/2504.05661v1 |
the KL divergence, we significantly improve the convergence rate presented in Theorem 2.6 of Spokoiny (2023); see the discussion following Theorem 3.2. Since the variational posterior Π tminimizes the KL divergence, obtaining a sharp upper bound for the KL divergence between ΠLA tandeΠt(· |Dt) is a crucial step in boun... | https://arxiv.org/abs/2504.05661v1 |
it is important to emphasize that the critical dimension for the Laplace approximation—or more broadly, the asymptotic normality of the posterior distribution (i.e., the BvM assertion)—depends on the specific statistical model under consideration. For instance, Panov and Spokoiny (2015, Section 4.1) demonstrated that t... | https://arxiv.org/abs/2504.05661v1 |
under certain conditions, that K(ΠLA t(·);eΠt(· |Dt)) is bounded by ( p3 eff/n)1/2 up to a constant factor. Here, peffdenotes an effective dimension satisfying peff≤p, and we have peff≍punless λmin(Ω0)≫1. In this sense, our result represents a substantial improvement over existing ones in terms of dependence on nandp. ... | https://arxiv.org/abs/2504.05661v1 |
a∈Rpandb∈R. Note that stochastic linearity implies that ∇kLt(θ) is non-random for all t∈[T] and k∈ {2,3,4}. Also, since ∇ξt(θ) does not depend on θ, we hereafter denote this random vector by ∇ξt. The stochastically linear framework encompasses many important statistical models, such as the logistic regression, Poisson ... | https://arxiv.org/abs/2504.05661v1 |
logistic regression model is one of the popular cases. In Section 8, we present a theoretical verification of ( A1) under a simple random design setting. For the logistic regression model, one can show (4.1) and (4.2) are satisfied with Vt=X⊤ tXt/4 and Mn=O(1), where Xt∈Rn×pis the design matrix for the t-th mini-batch.... | https://arxiv.org/abs/2504.05661v1 |
size . Therefore, we need to show that the effective sample size is proportional to the accumulated (actual) sample size nt. Once this step is established, it is not too difficult to bound the other quantities (e.g., bτ3,t,bτ4,t andτ∗ 3,t). However, unless the log-likelihood Lt(·) is strongly concave in the sense that ... | https://arxiv.org/abs/2504.05661v1 |
that eF1/2 t,bθt µt−bθt 2∨ Ω−1/2 teFt,bθtΩ−1/2 t−Ip F≲ϵn,t,KL=O(t−1n−1/2p∗). (5.5) Note that, due to the symmetry of the total variation distance, the above bound remains invariant under interchange of ( bθt,eFt,bθt) and ( µt,Ωt). Importantly, (5.5) plays a crucial role in establishing the online BvM assertion, namel... | https://arxiv.org/abs/2504.05661v1 |
N(bθML 1:t,F−1 1:t,θ0) remains unresolved. A standard BvM theorem states that N(bθML 1:t,F−1 1:t,θ0)≈Π (· |D1:t) in the TV sense. Importantly, N(bθML 1:t,F−1 1:t,θ0) does not exhibit the prior effects. Hence, we impose an additional assumption ( P∗) for ( µ0,Ω0) so that the prior effects become asymptotically negligibl... | https://arxiv.org/abs/2504.05661v1 |
score equations and (7.4), we have 0 =∇eL1:t(bθ1:t) =∇eLt(bθ1:t) +t−1X s=1∇ηs(bθ1:t) =∇eLt(bθ1:t)− ∇eLt(bθt) +t−1X s=1∇ηs(bθ1:t) =−eFt,θ◦ t bθ1:t−bθt +t−1X s=1∇ηs(bθ1:t) for some θ◦ ton the line segment between bθ1:tandbθtby Taylor’s theorem. If bθ1:tandbθtare sufficiently close, one can replace θ◦ twithbθtin the last... | https://arxiv.org/abs/2504.05661v1 |
1:t,θ0 ≤KM2 np3 ∗ n1/2 for all t∈[T]! ≥1−3n−1. Here, K=K(Kmin, Kmax). When pis fixed, Theorem 7.3 guarantees that the online BvM theorem holds even with a very small mini-batch size n. Specifically, if n≫(logN)4(as required by condition ( S)), the online BvM theorem holds. Moreover, Theorem 7.3 further ensures that... | https://arxiv.org/abs/2504.05661v1 |
impose the following conditions. (EX) The true parameter θ0and the initial prior parameters µ0andΩ0satisfy ∥θ0∥2≤K1, Ω1/2 0 θ0−µ0 2≤K2p1/2 ∗,∥Ω0∥2≤K3p∗, for some universal constants K1, K2, K3>0. Furthermore, for a large enough constant C= C(K1, K2, K3)>0, n≥C plog6(T∨n) log6(2n/p) ∨ p2log4T . 23 Under the assu... | https://arxiv.org/abs/2504.05661v1 |
4 0.982 0.346 µt,n= 6 0.980 0.310 µt,n= 8 0.968 0.290 µt,n= 10 0.958 0.277 µt,n= 20 0.956 0.273 µt,n= 50 0.950 0.259 µt,n= 200 0.944 0.254 µt,n= 1000 0.940 0.248 consistent regardless of the mini-batch size n. For further comparison, we also computed the relative efficiency , defined as REt=PM m=1|µt,m−θ0|2 PM m=1|bθML... | https://arxiv.org/abs/2504.05661v1 |
J. D., Tong, X. T., and Zhang, Y. (2020). Statistical inference for model parameters in stochastic gradient descent. Ann. Statist. , 48(1):251 – 273. Chen, X., Liu, W., and Zhang, Y. (2019). Quantile regression under memory constraint. Ann. Statist. , 47(6):3244–3273. Choi, J., Lee, J., Kim, Y., and Chae, M. (2025). On... | https://arxiv.org/abs/2504.05661v1 |
11(1). Nesterov, Y. et al. (2018). Lectures on Convex Optimization , volume 137. Springer. 27 Nesterov, Y. and Nemirovskii, A. (1994). Interior-point Polynomial Algorithms in Convex Program- ming. Society for Industrial Mathematics. Nguyen, C. V., Li, Y., Bui, T. D., and Turner, R. E. (2018). Variational continual lear... | https://arxiv.org/abs/2504.05661v1 |
H., Yan, J., and Schifano, E. D. (2020). An online updating approach for testing the proportional hazards assumption with streams of survival data. Biometrics , 76(1):171–182. Yano, K. and Kato, K. (2020). On frequentist coverage errors of Bayesian credible sets in moderately high dimensions. Bernoulli , 26(1). Zhang, ... | https://arxiv.org/abs/2504.05661v1 |
For A⊆Θ, let 1A(θ) : Θ→ {0,1}be the indicator function, defined as 1 if θ∈Aand 0 otherwise. 30 B Proofs for Section 3 Throughout this section, let Θ n,t= Θ(eFt,bθt,4rLA). (Note that the notation Θ n,tis reserved for defining other local sets in subsequent sections.) B.1 Laplace approximation For a function g: Θ→R, let ... | https://arxiv.org/abs/2504.05661v1 |
+ ∆ tail,LA,t(g) .(B.8) Next, we will obtain an upper bound of (ii). Let eZ∼ N(0,eF−1 t,bθt) and EeZdenote the expectation with respect to the law of eZ. Note that |(ii)| ≤ Z g(u) exp −1 2 eF1/2 t,bθtu 2 2 du Z exp −1 2 eF1/2 t,bθtu′ 2 2 du′ × Z exp −1 2 eF1/2 t,bθtu 2 2 du Z eft(u′;bθt)du′−1 = EeZg(eZ) × Z exp... | https://arxiv.org/abs/2504.05661v1 |
eF1/2 t,bθtu 2 2#exp −1 2 eF1/2 t,bθtu 2 2 Z exp −1 2 eF1/2 t,bθtu′ 2 2 du′du =EZ exp −2 1−4bτ3,trLA rLA∥Z∥2+1 2∥Z∥2 2 1{∥Z∥2≥4rLA}! ≤EZ exp −rLA∥Z∥2+1 2∥Z∥2 2 1{∥Z∥2≥4rLA}! , where Z∼ N(0,Ip) and the inequality holds by the assumption bτ3,trLA≤1/8. The right-hand side of the last display is equal toZ∞ 4rLAex... | https://arxiv.org/abs/2504.05661v1 |
exp −1 2 eF1/2 t,bθtu′ 2 2 du′du =E exph An eF1/2 t,bθteZ 2i 1{∥Z∥2≥4rLA}! =E eAn∥Z∥21{∥Z∥2≥4rLA}! . As in the proof of Lemma B.2, let p(·) be the density of ∥Z∥2, which is the derivative of the map ω7→S(ω) =−P(∥Z∥2> ω). Then, the last display equals Z∞ 4rLAeAnωp(ω)dω= exp(4 δnrLA)P(∥Z∥2>4rLA) +Z∞ 4rLAAnexp (Anω)P(∥Z... | https://arxiv.org/abs/2504.05661v1 |
0 (B.28) by the symmetry of Θ n,t. It follows that (i) =EΘn,th R3,t(eZ)i(B.22)=EΘn,t1 6⟨∇3eLt(bθt),eZ⊗3⟩+R4,t(eZ) (B.28)=EΘn,th R4,t(eZ)i ≤q EΘn,tR2 4,t(eZ).(B.29) By Taylor’s theorem, for u∈Θn,t, there exists eu∈Θn,tsuch that R4,t(u) =1 24⟨∇4eLt(bθt+eu), u⊗4⟩(3.1) ≤bτ4,t 24 eF1/2 t,bθtu 4 2. Hence, EΘn,tR2 4,t(eZ)≤E... | https://arxiv.org/abs/2504.05661v1 |
2g(eZ)# +c1bτ3 3,tr6 LA ≤EΘn,t Rt,4(eZ) +1 2EΘn,tR2 t,3(eZ) +c1bτ3 3,tr6 LA ≤q EΘn,tR2 t,4(eZ) +1 2EΘn,tR2 t,3(eZ) +c1bτ3 3,tr6 LA (B.30) (B.34) ≲bτ4,tp2+bτ2 3,tp2+c1bτ3 3,tr6 LA≤c4 bτ2 3,t+bτ4,t p2+bτ3 3,tr6 LA(B.36) for some universal constant c4=c4(c1)>0, where the second equality holds by the symmetry of Θ n,t ... | https://arxiv.org/abs/2504.05661v1 |
n,th −Rt,3(eZ)i (B.26)=EΘn,th −Rt,4(eZ)i +EΘc n,th −Rt,3(eZ)i For the first term in the right-hand side of the last display, note that EΘn,th −Rt,4(eZ)i ≤EΘn,t Rt,4(eZ) ≤q EΘn,tR2 t,4(eZ)(B.31) ≤1 24bτ4,t(p+ 3)2. (B.38) Also, for u∈Θc n,t, we have −Rt,3(u) =− eLt(bθt+u)−eLt(bθt)− ⟨∇eLt(bθt), u⟩+1 2 eF1/2 t,bθtu 2 2 =... | https://arxiv.org/abs/2504.05661v1 |
K2/2, we complete the proof. Fort∈[T], let ∆t= Ω1/2 t µt−bθt 2∨ eF1/2 t,bθt µt−bθt 2 ∨ Ω−1/2 teFt,bθtΩ−1/2 t−Ip F∨ eF−1/2 t,bθtΩteF−1/2 t,bθt−Ip F.(B.42) Corollary B.5. Suppose that conditions in Theorem 3.3 hold. Also, assume that ϵn,t,KL≤(1200 K)−1,∀t∈[T] onE1defined in Theorem 3.3, where Kis the universal consta... | https://arxiv.org/abs/2504.05661v1 |
Section 5 Fort∈[T], let bn,t= eF−1/2 t,θ∗ tΩt−1(θ0−µt−1) 2, τ3,t,bias= inf τ3∈R+: sup u∈Θ(eFt,θ∗ t,4bn,t)sup z∈Rp ⟨∇3EteLt(θ∗ t+u), z⊗3⟩ eF1/2 t,θ∗ tz 3 2≤τ3 , τ4,t,bias= inf τ4∈R+: sup u∈Θ(eFt,θ∗ t,4bn,t)sup z∈Rp ⟨∇4EteLt(θ∗ t+u), z⊗4⟩ eF1/2 t,θ∗ tz 4 2≤τ4 .(D.1) Lemma D.1. Suppose that ( A0) a... | https://arxiv.org/abs/2504.05661v1 |
t+ϕt) +eF1/2 t,θ∗ tϕt−1 2eF−1/2 t,θ∗ tD ∇3eLt(θ∗ t),(ϕt)⊗2E 2 + 1 2eF−1/2 t,θ∗ tD ∇3eLt(θ∗ t),(ϕt)⊗2E −1 2eF−1/2 t,θ∗ t ∇3eLt(θ∗ t), eF−1 t,θ∗ tφt⊗2 2. To bound (ii), we need to obtain an upper bounds of the following quantities: (iii) = eF−1/2 t,θ∗ t∇EteLt(θ∗ t+ϕt) +eF1/2 t,θ∗ tϕt−1 2eF−1/2 t,θ∗ tD ∇3eLt(θ∗ t),(ϕt)⊗... | https://arxiv.org/abs/2504.05661v1 |
t,θ∗ tϕt 2 (D.9) ≤τ3,t,bias 1 +1 2τ3,t,biasbn,t bn,t. Also, (D.8) implies that eF1/2 t,θ∗ t ϕt−eF−1 t,θ∗ tφt 2≤1 2τ3,t,biasb2 n,t. By the last two displays, we complete the proof of the second inequality in (D.11). The quantities ( bτ3,t,bτ4,t),bτ3,t,r,τ∗ 3,tand ( τ3,t,bias, τ4,t,bias), which appear in the followi... | https://arxiv.org/abs/2504.05661v1 |
τ3,t+1,biasbn,t+1 1 + 2 τ3,t+1,biasbn,t+1 b3 n,t+1 (D.19) ≤1 6τ4,t+1,bias 1 +1 2 1 +1 8 b3 n,t+1≤τ4,t+1,biasb3 n,t+1. Similarly, we have (b) =1 2τ2 3,t+1,bias 1−4τ3,t+1,biasbn,t+1−1 1 +1 2τ3,t+1,biasbn,t+1 b3 n,t+1 ≤1 2τ2 3,t+1,bias 1 + 8 τ3,t+1,biasbn,t+1 1 +1 2τ3,t+1,biasbn,t+1 b3 n,t+1 (D.19) ≤1 2τ2 3... | https://arxiv.org/abs/2504.05661v1 |
n≥CM2 nt4α−2p2 ∗(D.24) for a large enough constant C=C(Kmin, Kmax, α, C 5). Then, on Eest,1∩E, eF1/2 t,θ∗ t bθt−θ∗ t 2≤KM np t−1p∗, τ∗ 3,t∨bτ3,t≤Kt−3/2n−1/2, bτ4,t≤Kt−2n−1, ∆t∨ϵn,t,KL≤Kt−1n−1/2p∗, bθt, θ∗ t∈Θ θ0,Ip,1/2 ,(D.25) where K=K(Kmin, Kmax)is a large enough constant, and λmin eFt+1,θ∗ t+1 ∧λmin eFt+1,bθt+... | https://arxiv.org/abs/2504.05661v1 |
λmin(Ωt),λmax(Ωt) By (D.38), we have eF−1/2 t,bθtΩteF−1/2 t,bθt−Ip 2(B.42) ≤∆t≤c2t−1n−1/2p∗. It follows that λmin(Ωt)≥(1−∆t)λmin(eFt,bθt)(D.23) ≥C3(1−∆t)nt, λmax(Ωt)≤(1 + ∆ t)λmax(eFt,bθt)(D.23) ≤C4(1 + ∆ t)nt.(D.39) Step 5: ∥θ∗ t+1−θ0∥2,λmin(eFt+1,θ∗ t+1),λmax(eFt+1,θ∗ t+1),λmin(eFt+1,bθt+1),λmax(eFt+1,bθt+1) The resu... | https://arxiv.org/abs/2504.05661v1 |
1)−3/2n−1/2.(D.44) Similarly, τ4,t+1,bias≤(Kmaxn)λ−2 min(eFt+1,θ∗ t+1)≤(Kmaxn) c3,tn(t+ 1)−2 =Kmax(c3,t)−2(t+ 1)−2n−1.(D.45) Step 7: ∥eF1/2 t+1,θ∗ t+1 θ0−θ∗ t+1 ∥2 To complete the proof of the last inequality in (D.26), we will show that the assumptions in Lemma 61 D.2 are satisfied. By the results in Step 1-6 , we... | https://arxiv.org/abs/2504.05661v1 |
1∥2≤λ−1/2 min F1,θ0 (√ 2δ)n1/2(A2) ≤(Kminn)−1/2(√ 2δ)n1/2(P) ≤1/8, λmin(eF1,θ∗ 1)≥λmin(F1,θ∗ 1) +λmin(Ω0)(A2) ≥Kminn, λmax(eF1,θ∗ 1)≤λmax(Ω0) +λmax(F1,θ∗ 1)(A2),(P) ≤ Kmaxp∗+Kmaxn(S) ≤4 3Kmaxn.(D.49) Step 3: ∥eF1/2 1,θ∗ 1(θ0−θ∗ 1)∥2 Note that bn,1= eF−1/2 1,θ∗ 1Ω0 θ0−µ0 2≤λ−1/2 min eF1,θ∗ 1 ∥Ω0∥1/2 2 Ω1/2 0 θ0−µ0... | https://arxiv.org/abs/2504.05661v1 |
{2,3, ..., T}. (D.60) Then, for any t0∈ {2,3, ..., T}, we have t0−1X s=1 " 1−2t0−1X r=1∆t0−r# Kminn! +Kminn≥Kmin 2nt0. Hence, we need to show (D.60). Note that t0−1X s=1∆s(D.58) ≤t0−1X s=1K1p∗n−1/2s−1=K1p∗n−1/2t0−1X s=1s−1≤K1p∗n−1/2TX s=1s−1 ≤K1p∗n−1/2(logT+ 1)≤2K1p∗n−1/2log(T∨3)(S) ≤1/4. As in (D.59), for any t∈[T], w... | https://arxiv.org/abs/2504.05661v1 |
Ω1/2 0 θ0−µ0 2 (P) ≤ Kminnt−1/2 Kmaxp∗1/2 δn1/2 = K−1/2 minK1/2 maxδ t−1/2p1/2 ∗ ≤t−1/2p1/2 ∗,(E.6) where the last inequality holds by a small enough δ. For θ∈Θt,bias, note that ∥θ−θ∗ 1:t∥2≤(Kminnt)−1/24ρn,t≤ 4K−1/2 min n−1/2t−1p1/2 ∗(S) ≤1 4. 68 Also, by θ∗ 1:t∈Θn,t, we have ∥θ∗ 1:t−θ0∥ ≤1/4. It follows th... | https://arxiv.org/abs/2504.05661v1 |
of Theorem 6.2. In this proof, we work on the event Eest,2without explicitly mentioning it. Let t∈[T]. To complete this proof, we utilize Lemma H.4. Hence, we need to obtain upper bounds of the following quantities: (i) = eF1/2 1:t,bθ1:t bθML 1:t−bθ1:t 2,(ii) = F−1/2 1:t,θ0eF1:t,bθ1:tF−1/2 1:t,θ0−Ip F. Step 1: (i) Fi... | https://arxiv.org/abs/2504.05661v1 |
1:t,θ∗ 1:t θ0−bθ1:t 2 Lemma H.1 ≤ 1 +KmaxK−3/2 min(nt)−1/2(4Mnp1/2 ∗)1/2 eF1/2 1:t,θ∗ 1:t θ0−bθ1:t 2 (E.15) ≤ 1 +KmaxK−3/2 min(nt)−1/2(4Mnp1/2 ∗)1/2 8Mnp1/2 ∗(S) ≤10Mnp1/2 ∗, which implies that θ0∈Θ bθ1:t,eF1:t,bθ1:t,10Mnp1/2 ∗ ⊂Θ bθ1:t,F1:t,bθ1:t,10Mnp1/2 ∗ ⊆Θ (θ0,Ip,1/2). Consequently, by Lemma H.3, L1:t(... | https://arxiv.org/abs/2504.05661v1 |
2≤ 8K−1/2 minK1/2 up+ 1 Mn√p∗,∀t∈[t0−1]. (F.3) Based on (F.3), we will show that eF1/2 t0,bθt0 bθ1:t0−bθt0 2≤D1M2 np3 ∗ n1/2 , eF1/2 t0,bθt0 θ0−bθt0 2≤ 8K−1/2 minK1/2 up+ 1 Mn√p∗,(F.4) where D1=D1(Kmin, Kmax, Klow, Kup). It then follows by induction that (F.2) holds for all t∈[T]. The proof is divided into sev... | https://arxiv.org/abs/2504.05661v1 |
t0 eF−1/2 t,bθt˙Rt,3(bθt,bθ1:t0−bθt) 2.(F.8) Also, by bθ1:t0∈Θn,tand Taylor’s theorem, there exists some eu∈Θn,tsuch that eF−1/2 t,bθt˙Rt,3(bθt,bθ1:t0−bθt) 2 =1 2sup u∈Rp:∥u∥2=1 ∇3eLt(bθt+eu), bθ1:t0−bθt⊗2 ⊗ eF−1/2 t,bθtu ≤1 2sup u∈Rp:∥u∥2=1sup eu∈Θn,t ∇3eLt(bθt+eu), bθ1:t0−bθt⊗2 ⊗ eF−1/2 t,bθtu (F.7) ≤1 2 Kma... | https://arxiv.org/abs/2504.05661v1 |
0n−1/2,eFt0,bθt0,4ϱn,t0 . (F.10) Step 7: (F.4) and (7.6) ByStep 1-6 , we are ready to prove (F.4). Note that KmaxK−3/2 lowt−3/2 0n−1/2ϱn,t0≤(KmaxK−3/2 lowc4)M2 np3/2 ∗n−1t−3/2 0(S) ≤1/16, (F.11) which allows us to apply Lemma H.2. By Lemma H.2 with τ3=KmaxK−3/2 lowt−3/2 0n−1/2, f(θ) =eLt0(θ), θ =bθt0,eθ=bθ1:t0, β=t0−1... | https://arxiv.org/abs/2504.05661v1 |
n1/2t−1X s=11≤c1p2 ∗ n1/2 , where c1=K2 upK−1 low. Step 3: (ii) Lets∈[t]. Note that eF−1/2 t,bθtF1/2 s,bθs F−1/2 s,bθsFs,bθ1:tF−1/2 s,bθs−Ip F1/2 s,bθseF−1/2 t,bθt F ≤λ−1 min eFt,bθt λmax Fs,bθs F−1/2 s,bθsFs,bθ1:tF−1/2 s,bθs−Ip F ≤Kmax Klowt−1√p F−1/2 s,bθsFs,bθ1:tF−1/2 s,bθs−Ip 2.(F.14) where the last inequ... | https://arxiv.org/abs/2504.05661v1 |
, where K4= (K2+K3+ 2K1)/2. If we further assume ( P∗), we can employ Theorem 6.2. For all t∈[T], it holds that dV N bθML 1:t,F−1 1:t,θ0 ,Π (· |D1:t) ≤K5Mnp2 ∗ nt1/2 , where K5=K5(Kmin, Kmax) is the constant specified in Theorem 6.2. It follows that dV Πt,N bθML 1:t,F−1 1:t,θ0 ≤dV Πt(·),Π (· |D1:t) +dV N ... | https://arxiv.org/abs/2504.05661v1 |
( A0), (A1∗),(A2),(S)and (P∗) hold. Then, on Eest,1∩Eest,2, the following inequalities holds uniformly for all t∈[T]: λmin Ωt ∧λmin F1:t,θ0 ≥K1nt, λmax eF1:t,bθ1:t ∧λmax F1:t,θ0 ≤K2nt, F−1/2 1:t,θ0eF1:t,bθ1:tF−1/2 1:t,θ0−Ip F≤K2Mnp2 ∗ nt1/2 , F1/2 1:t,θ0 bθML 1:t−µt 2≤K2M2 np3 ∗ n1/2 , where K1andK2are posi... | https://arxiv.org/abs/2504.05661v1 |
Ω1/2 t θ0−µt 2− F1/2 1:t,θ0 θ0−bθML 1:t 2≤5ϵn,3. Also, Ω1/2 t θ0−µt 2≥ Ω1/2 t θ0−bθML 1:t 2− Ω1/2 t bθML 1:t−µt 2 ≥ 1−ϵn,21/2 F1/2 1:t,θ0 θ0−bθML 1:t 2− Ω1/2 t bθML 1:t−µt 2 ≥ 1−ϵn,2 F1/2 1:t,θ0 θ0−bθML 1:t 2− Ω1/2 t bθML 1:t−µt 2 ≥ F1/2 1:t,θ0 θ0−bθML 1:t 2−ϵn,2 4Mnp1/2 ∗ −ϵn,3, which implies th... | https://arxiv.org/abs/2504.05661v1 |
i∈Iω(bθ,t)b′′(X⊤ ibθ)XiX⊤ i ≥exp −3ϵ·max i∈[N]∥Xi∥2 b′′ ω(τ+ 1) λmin X i∈Iω(bθ,t)XiX⊤ i (G.8) where the last inequality holds by the symmetry and monotonicity of b′′(·) in the logistic regression model. 91 First, for bθ∈bΘϵ,τandt∈[T], we will prove that |I2(bθ, t)| ≥n/6 with high probability. Since X⊤ ibθ∼ N... | https://arxiv.org/abs/2504.05661v1 |
4 = 2430 e2K1+3= 27√ 30eK1+3/22 9. The last two displays complete the proof of (G.11) by taking M= 27√ 30eK1+3/2. Lemma G.6. For any ω≥0andt∈[T], we have P0,t eF−1/2 t,θ∗ t∇ζt 2≥s tr eF−1 t,θ∗ tX⊤ tXt 4 +s 2 eF−1 t,θ∗ tX⊤ tXt 4 2ω X ≤e−ω, Proof. Lett∈[T] andEt= (ϵi)i∈It∈Rn, where ϵi=Yi−Et(Yi|X). Note that ∇ζt... | https://arxiv.org/abs/2504.05661v1 |
2q tr eB1:t +q 2 eB1:t 2(logn+ log T) for some t∈[T] X ≤T·e−logn−logT=n−1. 95 By integrating over the values of X, we have P eF−1/2 1:t,θ∗ 1:tX⊤ 1:tE1:t 2≥1 2q tr eB1:t +q 2 eB1:t 2(logn+ log T) for some t∈[T] ≤n−1, which completes the proof of the second assertion. Next, we will prove the third assertion, whi... | https://arxiv.org/abs/2504.05661v1 |
least 1 −2n−1. By taking eK′ max= (3 + 2p 2/π)∨18 = 18, we complete the proof. Proof of Proposition 8.1. Let Ω1= The three assertions in Proposition G.7 hold uniformly for all t∈[T] , Ω2= (G.10) and (G.11) hold with the constants eKmin,eKmaxandM , Ω3= (G.12) holds with the constant eK′ max . By Propositions G.5, G.7... | https://arxiv.org/abs/2504.05661v1 |
sup u∈Θ(F,r)sup z∈Rp ⟨∇3f(θc+u), z⊗3⟩ F1/2z 3 2= sup u∈Θ(F,r)sup z∈Rp * ∇3f(θc+u),z⊗3 F1/2z 3 2+ ≤λ−3/2 min(F) sup u∈Θ(F,r)sup z∈Rp:∥z∥2=1 ⟨∇3f(θc+u), z⊗3⟩ ≤λ−3/2 min(F) sup θ∈Θ(θc,F,r) ∇3f(θ) op. Also, sup u∈Θ(F,r)sup z∈Rp ⟨∇4f(θc+u), z⊗4⟩ F1/2z 4 2= sup u∈Θ(F,r)sup z∈Rp * ∇4f(θc+u),z⊗4 F1/2z 4 2+ ≤λ−2 min(F) sup u∈Θ(... | https://arxiv.org/abs/2504.05661v1 |
have Et+1Lt+1(θ∗ t+1)−Et+1Lt+1(θ0)<−1 2 Ω1/2 t(θ0−µt) 2 2, by the concavity of the map θ7→Et+1Lt+1(θ), which contradicts to (H.5). This completes the proof of (H.4). 102 Remark. The constant Cin(H.3) can be chosen as C= 8K−1 min∨ 8K2 maxK−3 min+ 1 . Lemma H.7. Suppose that ( A0)-(A2) hold. Also, on an event E, assume... | https://arxiv.org/abs/2504.05661v1 |
2≤ Ω1/2 t(bθt−θ∗ t) 2+ Ω1/2 t(bθt−µt) 2 ≤ Ω1/2 teF−1/2 t,bθt 2 eF1/2 t,bθteF−1/2 t,θ∗ t 2 eF1/2 t,θ∗ t(bθt−θ∗ t) 2+ Ω1/2 t(bθt−µt) 2 ≤ 1 + eF−1/2 t,bθtΩteF−1/2 t,bθt−Ip 21/2 1 + eF−1/2 t,θ∗ teFt,bθteF−1/2 t,θ∗ t−Ip 21/2 eF1/2 t,θ∗ t(bθt−θ∗ t) 2 + Ω1/2 t(bθt−µt) 2 (H.11) ≤3 2 eF1/2 t,θ∗ t(bθt−θ∗ t) 2+ Ω1/2 t(bθt−µt)... | https://arxiv.org/abs/2504.05661v1 |
Z⟩2 = 6∥T∥2 F,ET2(Z) = 6∥T∥2 F+ 9∥M∥2 2. 106 Proof. See Lemma B.32 in Spokoiny (2024). Lemma H.14. For a symmetric 3-order tensor T∈Rp×p×p, suppose that there exist some F∈ Sp ++ andτ3≥0such that T(u) =⟨T, u⊗3⟩ ≤τ3∥Fu∥3 2,∀u∈Rp. LeteZ∼ N(0,D−1)for some D∈ Sp ++andV=D−1/2FD−1/2. Then, E[{T(eZ)}2]≤15τ2 3∥V∥2tr2(V) Proof... | https://arxiv.org/abs/2504.05661v1 |
Revisiting poverty measures using quantile functions N. Unnikrishnan Nair, S.M.Sunoj∗ Department of Statistics, Cochin University of Science and Technology, Cochin 682 022, Kerala, India Abstract In this article we redefine various poverty measures in literature in terms of quan- tile functions instead of distribution ... | https://arxiv.org/abs/2504.05713v1 |
the income law and conditions under which they can be chosen. The relationship between Lorenz curve and the poverty measures are also found. Some highly flexible quantile function that represent the gener- alized lambda, Wakeby, Kappa and Govindarajulu distributions are proposed as prospective income models and their p... | https://arxiv.org/abs/2504.05713v1 |
the empirical form of A1(u), then the income distribution can be found directly from (2.6)(see Section 5). The expression of A1(u) for several quantile functions are exhibited in Table 1 for easy reference to the corresponding distribution. 2. A large number of distributions prescribed in literature as models of income... | https://arxiv.org/abs/2504.05713v1 |
=αu1/β. Then for this distribution A1(u) =u β+ 1, proportional to the head count ratio. 5 A third special case of (2.3) is A2(t) =AF(t,2) =Zt 0 1−x t2 dF(x) (2.9) which measures depth of poverty while AF(t,1) indicates its severity. Correspond- ing to AF(t,2), we have A2(u) =AQ(u,2) =Zu 0 1−Q(p) Q(u)2 dp. (2.10) Th... | https://arxiv.org/abs/2504.05713v1 |
distribution function of Xor through parametric forms. A large number of parametric forms of LC have been proposed in literature. Some important forms, their PGR’s and their quantile functions are exhibited in Table 1 for easy reference. As a weakness of parametric LC’s it is pointed out that they provide distri- butio... | https://arxiv.org/abs/2504.05713v1 |
d dulogZu 0Q(p)dp =B(u) and Zu 0Q(p)dp= exp −Z1 uB(p)dp (3.6) The proof is completed by noting that (3.5) is the derivative of the last expression (3.6). Corollary 3.1. All the basic functions associated with poverty can be expressed in terms of the PGR, A1(u). (i) the IGR, I(u) =A1(u) u (ii) mean income of the poo... | https://arxiv.org/abs/2504.05713v1 |
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