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1,∥X−x0∥ ≤h >0. Call this infimum δ >0.∥vTU∥is bounded, so δis finite. Define: λn≡vT nEP(n)" UX−x0,n hn UX−x0,n hnT |D= 1, X∈An# vn =EP(n)" vT nUX−x0,n hn vT nUX−x0,n hnT |D= 1, X∈An# =EP(n)" vT nUX−x0,n hn2 |D= 1, X∈An# ≥ε2P(n) vT nUX−x0,n hn ≥ε|D= 1, X∈An ≥ε2δ >0. Therefore, for all ¯h≤h′,λ(¯h)≥ε2δ... | https://arxiv.org/abs/2504.13273v2 |
that max jPDiK∥Xi−xj∥ hn,j nEh DK∥X−xj∥ hn,ji≤2. I now apply the Hoeffding inequality to thePDK ∥X−xj∥ hn,j ≤2Eh DK ∥X−xj∥ hn,ji elements of Vn,j conditional on A, for all k, n, j P(n) (|Vn,j,k| ≥εn)≤2exp −2 nEP(n)h DK ∥X−xj∥ hn,ji2 ε2 nPDiK ∥Xi−xj∥ hn,j b2 ≤2exp −Rnε2 n . Then: P(n) knmax k=1m... | https://arxiv.org/abs/2504.13273v2 |
max j 1−PDiK ∥Xi−xn,j∥ h∗ n,j PDiK ∥Xi−xn,j∥ hn,j +O max jPDi1 ∥Xi−xn,j∥ ∈[h¯n,j,¯hn,j] PDi1 ∥Xi−xn,j∥ ≤h¯n,j ! =oP(n)(1) + OP(n) ε(a) n +o 1 +ε(a) n2 (1 +ε(b) n) =oP(n)(1). Therefore max j∥An,j−Bn,j∥(op)=o(1). Then, by well-known arguments (Horn and Johnson, 2013, p. 381): max j|λmin(Bn,j)−λmin(A,n... | https://arxiv.org/abs/2504.13273v2 |
2βµ+dγ0 γ0−1. Then: KL(Pn,1, Pn,2) =nPn,1 logdPn,1 dPn,2 ≤nPn,1 ∥X−x0∥ ≤hn 4 Pn,1 D= 1| ∥X−x0∥ ≤hn 4 L′ exp(−4 3)hβµnexp −1 1−1/42 2 =nk2−2d−1(L′)2k C 1 γ0−1 hd+d γ0−1+2βµ n =k2−2d−1(L′)2k C 1 γ0−1 = “α.” Proof of Proposition 6. Letδ¯≥1 solve: δ¯log(δ¯)2βµ+dγ0 γ0−1 2βµ = (δ¯)(δ¯log(δ¯))2βµ+dγ0 γ0−1 4−1.... | https://arxiv.org/abs/2504.13273v2 |
define the estimator through a series of steps. At a high level, a first step finds the grid width through a 72 minimum level of implied overlap within grid regions. A second step chooses the local polynomial bandwidth for each gridpoint based on the number of treated observations nearby, and then conducts the associat... | https://arxiv.org/abs/2504.13273v2 |
19, Lemma 22, and Lemma 24 apply immediately, so that the eigenvalues at 73 hn,jare nondegenerate. Now consider the tuples ( xn,j, h(k∗ n)) and the kn=k∗ n+ 1 functions: first, fk(v) = 1n h(k∗ n)v≤h(k) no , and fk∗n+1(v) = 1. Define h¯∗ n=h¯nlog(n)2(γ0−1)/d. By Lemma 19 and construction, h(k∗ n)≥h¯∗ nfor large enough n... | https://arxiv.org/abs/2504.13273v2 |
gridpoint squared difference overall is bounded above by EP gnmax j=1(ˆµ(xn,j)−E[ˆµ(xn,j)|Z])2|Z ≤k∗ nX k=1EP gnmax j=1(ˆµ(xn,j)−E[ˆµ(xn,j)|Z])2|Z ≤k∗ nX k=1EP" max hn,j=h(k) n(ˆµ(xn,j)−E[ˆµ(xn,j)|Z])2|Z# =Op k∗ n−1X k=1log m(j) n h(k) n2βµ +Op log(gn) h(k∗ n) n2βµ =Op δk∗ nX k=12−k h(k) n h(1) n!−2βµ... | https://arxiv.org/abs/2504.13273v2 |
arXiv:2504.13322v1 [math.PR] 17 Apr 2025FOUNDATIONS OF LOCALLY-BALANCED MARKOV PROCESSES Samuel Livingstone Department of Statistical Science, University College Lon don, U.K. samuel.livingstone@ucl.ac.uk Giorgos Vasdekis School of Mathematics, Statistics and Physics, Newcastle U niversity, U.K. giorgos.vasdekis@newcas... | https://arxiv.org/abs/2504.13322v1 |
[2020], thecontinuous-timeformulationinfinitestatespaceswasi ntroducedbyPower and Goldman[2019]. Many approaches for both discrete and continuous sampling p roblems have since been motivated by these processes, such as discrete state-space algorithm s Zhou et al. [2022], Chang and Zhou [2024],Grathwohl et al.[2021],Zhan... | https://arxiv.org/abs/2504.13322v1 |
for any two functions f,hsuch that the range of his within the domain of f FOUNDATIONS OF LOCALLY-BALANCED MARKOV PROCESSES 3 2.Definition and basic properties 2.1.Definition of the process. Consider a Borel measure space ( E,E) whereEis Polish. Con- sider also an underlying probability space (Ω ,F,P). Letπbe a measure ... | https://arxiv.org/abs/2504.13322v1 |
is well-posed and non-explosive, meaning that λisπ-almost surely (a.s.) finite and the process will never reach the graveyard state ∂. The jump kernel associated to ( Yt)t≥0is defined for any x∈Eand any A∈EasJ(x,A) := λ(x)Γg(x,A). This can also be written J(x,dy) =g◦t(x,y)γ(x,dy), (5) from which the below proposition is ... | https://arxiv.org/abs/2504.13322v1 |
To ensure non-explosivity of the process, we make th e following assumption on the kernel Γg, which is connected with the concept of φ−irreducibility (see e.g. Meyn and Tweedie [1993]). While irreducibility is not in general necessary for non-ex plosivity, it allows for a simpler treatment and it is a natural requireme... | https://arxiv.org/abs/2504.13322v1 |
nn/summationdisplay i=1τif(Xi−1) =Eˆπ[g] =/integraldisplay E/integraldisplay+∞ 0f(x)s1 Zλλ2(x)exp{−λ(x)s}ds π(dx) =1 Zλ/integraldisplay Ef(x)π(dx), (11) 8 SAMUEL LIVINGSTONE, GIORGOS VASDEKIS AND GIACOMO ZANELLA and lim n→∞1 nTn= lim n→∞1 nn/summationdisplay i=1τi=Eˆπ[h] =/integraldisplay E/integraldisplay+∞ 0s1 Zλλ2(x... | https://arxiv.org/abs/2504.13322v1 |
as in Theorem 2.1(b), we get that th e process is π-reversible and invariant. Finally, (14) follows by the sam e arguments of the proof of Theorem 2.1(c) by appealing to Theorem 2 of Asmussen and Glyn n [2011]. /square Remark 2.7. Asλ∈L1(π)(by Proposition 2.2), and λ(Yt)<∞a.s. for all t≥0(by Theo- rem 2.2), we can alwa... | https://arxiv.org/abs/2504.13322v1 |
If we have that then for any f∈Cb(E), we have lim y→xf(Yy t) =f(Yx t) a.s. and from bounded convergence, Ptf(y) =Ey[f(Yt)]y→x−−−→Ex[f(Yt)] =Ptf(x) which proves that Ptfis continuous. We now turn on proving that a.s. Yy ty→x−−−→Yx t. Due to non-explosivity of the process, a.s. there existsn∈Nsuch that t∈[Tx n,Tx n+1) an... | https://arxiv.org/abs/2504.13322v1 |
ZANELL A therefore for any T≥0 sup t≤T/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt 0Lf(Ys)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤2∝ba∇dblf∝ba∇dbl∞/integraldisplayT 0λ(Ys)ds which has finite expectation since x∈Nand from Lemma 2.2. Since fis bounded, we get that M is a tru... | https://arxiv.org/abs/2504.13322v1 |
NotethatbyProposition2.4, point-wisethederivativeof Ptf(x)2withrespectto tis2Ptf(x)LPtf(x). Furthermore, usingthe same argument as in theProof of Theor em 2.3, we have|2Ptf(x)LPtf(x)|≤ 4∝ba∇dblf∝ba∇dbl2 ∞λ(x), which is π-integrable using Proposition 2.2(b) and recalling that f∈B(E). Therefore we can switch the integral... | https://arxiv.org/abs/2504.13322v1 |
same γ, bounded gand initialised from µis exponentially ergodic. (ii)If two LBMJPs with bounded and non-decreasing g1andg2have the same γ, then either both have positive spectral gap or neither does. Proof.(i) Using Theorem 2.1 of Roberts and Rosenthal [1997] then if Pis geometrically ergodic andχ2(µ||π)<∞then∝ba∇dblP∝... | https://arxiv.org/abs/2504.13322v1 |
setCis called smallifαcan be taken as the Dirac measure α=δnfor some n∈N. These definitions naturally extend to continuous time processes. We begin with a result that guarantees that the class of small sets contains all the compact subsets ofE. This will be helpful later when we establish uniform ergodi city for particu... | https://arxiv.org/abs/2504.13322v1 |
>1, andk >0such that for all t≥1, g(t)≥t˜a, and for all n∈N,π(n)>0, and for all n≥k, π(n) π(n+1)≥exp/braceleftBig aβnβ−1/bracerightBig . Under this assumption we have the following. Theorem 3.1. Assume that γis of the form (25)and Assumption 3.1 holds. Then the LBMJP is uniformly ergodic. Proof of Theorem 3.1. The proo... | https://arxiv.org/abs/2504.13322v1 |
sup x∈CEγ/bracketleftbig e(x,σn)8/bracketrightbig to control the higher order terms (such as the term A(x,σn)in the proof) In particular, we observe that kernels γsuch as those considered in Section 3.2 satisfy this propert y. 5.Discussion 5.1.Use in Monte Carlo simulation. There are two natural ways to use locally-bal... | https://arxiv.org/abs/2504.13322v1 |
We leave a thorough exploration of these non-rev ersible locally-balanced processes to future work. 20 SAMUEL LIVINGSTONE, GIORGOS VASDEKIS AND GIACOMO ZANELL A Acknowledgements The authors would like to thank the Isaac Newton Institute fo r Mathematical Sciences, Cambridge, for support and hospitality during the progr... | https://arxiv.org/abs/2504.13322v1 |
Exponential and uniform ergodicity of Markov processes. The Annals of Probability , 23(4):1671–1691, 1995. Simon Duane, Anthony D Kennedy, Brian J Pendleton, and Dunca n Roweth. Hybrid monte carlo. Physics letters B , 195(2):216–222, 1987. A. Durmus, A. Guillin, and P. Monmarch´ e. Piecewise determi nistic Markov proce... | https://arxiv.org/abs/2504.13322v1 |
Gradient-Based MCMC. Journal of the Royal Statistical Society Series B: Statisti cal Methodology , 84(2):496–523, 01 2022. ISSN 1369-7412. doi: 10.1111/rss b.12482. I. Lytras and P. Mertikopoulos. Tamed Langevin sampling und er weaker conditions, 2024. Jonathan C Mattingly, Andrew M Stuart, and Desmond J Higham. Ergodi... | https://arxiv.org/abs/2504.13322v1 |
Neural Information Processing Systems , 35:23867–23880, 2022. Haoran Sun, Bo Dai, Charles Sutton, Dale Schuurmans, and Han jun Dai. Any-scale Balanced Sam- plers for Discrete Space. In The Eleventh International Conference on Learning Representa tions, FOUNDATIONS OF LOCALLY-BALANCED MARKOV PROCESSES 23 2023. Achille T... | https://arxiv.org/abs/2504.13322v1 |
inft≥0{Yt=k}for the hitting time of k. Assume that the current state of the process is n∈N. We denote the probabilities associated with the process moving to the right or to the left p(n) :=P(X1=n+1|X0=n) =g/parenleftBig π(n+1) π(n)/parenrightBig g/parenleftBig π(n−1) π(n)/parenrightBig +g/parenleftBig π(n+1) π(n)/pare... | https://arxiv.org/abs/2504.13322v1 |
the same uniform distribut ions to generate their jumps. A simple comparison between the two coupled processes shows that for anyn≥k En+1[hn]≤En+1/bracketleftBig ˜hn/bracketrightBig , and therefore En+1[hn]≤En+1/bracketleftBig ˜hn/bracketrightBig ≤1 λEn+1/bracketleftbigg inf m∈N/braceleftBig ˜Xm=n/bracerightBig/bracket... | https://arxiv.org/abs/2504.13322v1 |
8exp{ξ}/bracketleftBig b′′(e(x,σ(1) n))/bracketrightBig2 e(x,σn)4. At the same time, from a Taylor expansion up to second degree, ∃σ(2) n∈(0,σn) such that for any Z∈Rd f(x+σnZ)−f(x) =σn∇f(x)·Z+1 2σ2 nZT∇2f(x)Z+1 6σ3 nd/summationdisplay i,j,k=1∂i∂j∂kf(x+σ(2) nZ)ZiZjZk. Overall this gives FOUNDATIONS OF LOCALLY-BALANCED ... | https://arxiv.org/abs/2504.13322v1 |
Continuous-time filtering in Lie groups : estimation via the Fréchet mean of solutions to stochastic differential equations Marc Arnaudon1, Magalie Bénéfice2, and Audrey Giremus3 1Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400 Talence, France 2Université de Lorraine, CNRS, IECL, F-54000 Nancy, France 3Univ.... | https://arxiv.org/abs/2504.13502v1 |
variable XonGwith support supppXq, we say thatEpXq PGis the exponential barycenter of Xif there exists a G-valued integrable and centered random variable νpXqsuch that X“EpXqexppνpXqq, (1) the segmenttEpXqexpptνpXqq, tP r0,1suis included in the convex hull of supppXqand the couple pEpXq, νpXqqis unique. It is well know... | https://arxiv.org/abs/2504.13502v1 |
achieves the proof of Theorem 1. [ \ With Theorem 1 at hand, we propose to look for computable approximations of theFréchetmeans Etofthesolution Xttothestochasticdifferentialequation(3), together with the error term νt. Theorem 2. We have the expansions dEt“Ethtdtwith ht“bpEtq `1 2˜ Hess EtbpEtp¨q,Etp¨qq´r TEtbpEtp¨q,¨... | https://arxiv.org/abs/2504.13502v1 |
side term. As Xτ t“Etexppντ tq, we get: bpXτ tq“bpEtq`TEtbpEtντ tq`1 2Hess EtbpEtντ tbEtντ tq`Op|ντ t|3q.(22) We now turn to the right-hand side term. We first have: exppadp´ντ tqqphtq“ht`rht, ντ ts`1 2rrht, ντ ts, ντ ts`` Op|ντ t|3q.(23) ForuPG, the differential of the exponential map at point uisduexp“ exppuqř kě0adp... | https://arxiv.org/abs/2504.13502v1 |
ht`bpEtqqdντ tdt`´ TEtbpEt¨qdId´1 6mÿ i“1rrστ i,¨s, στ isdId ´1 2rht`bpEtq,¨sdId¯ pντ tbντ tqdt`dRt,3. (37) To obtain (10), we then just have to look at the means of (31) and (37) and use again that ht“bpEtq`Optq,Erντ ts“0andtÞÑErντ tbντ tsis smooth (with the same arguments as for the smoothness of tÞÑEt). 5 Simulation... | https://arxiv.org/abs/2504.13502v1 |
Bayesian Model Averaging in Causal Instrumental Variable Models Gregor Steiner∗Mark Steel† Department of Statistics, University of Warwick May 14, 2025 Abstract Instrumental variables are a popular tool to infer causal effects under unobserved confounding, but choosing suitable instruments is challenging in practice. W... | https://arxiv.org/abs/2504.13520v3 |
adding to the computational cost. 4. We prove that our model selection procedure is consistent in the sense that the conditional Bayes factors used in the model updates tend to infinity in favor of the true model as the sample size grows. 5. We allow for the BMA framework to freely assign variables as either instrument... | https://arxiv.org/abs/2504.13520v3 |
error criterion (e.g. Donald and Newey, 2001; DiTraglia, 2016), or model-averaging to obtain model-averaged first-stage predictions of the endogenous variable (e.g. Kuersteiner and Okui, 2010). An important strand of the literature focuses on identification and estimation when some instrumental variables are invalid (K... | https://arxiv.org/abs/2504.13520v3 |
of τ. We assume the residuals are jointly normal [ϵ:H]∼MN(0,In,Σ), where the structural covariance matrix Σcan be partitioned into Σ=σyyΣyx Σ⊺ yxΣxx = var(ϵi) cov( ϵi,Hi) cov(Hi, ϵi) var( Hi) . IfΣyx̸= 0, the outcome and the treatment residuals are correlated, signaling the presence of unobserved confounding or end... | https://arxiv.org/abs/2504.13520v3 |
the treatment model to infer the covariance “ratio” Σ−1 xxΣ⊺ yxand subsequently identify τ. Our approach does not do this explicitly, but instead, in the outcome model, we condition on HΣ−1 xxΣ⊺ yx, which is known conditionally on the treatment and covariance parameters. The treatment residual H contains all the variat... | https://arxiv.org/abs/2504.13520v3 |
σy|x ,Q∼MN(ιΓ + [Z: W]∆,In,Σxx), F(y)andF(xj)are the distributions of yandxj, and hyandhxjare invertible univariate link functions that map the latent Gaussian to the appropriate parameter of F(y)andF(xj), j= 1. . . , l. If required, ryandrxjgroup any additional parameters of F(y)andF(xj), which are assumed to be the ... | https://arxiv.org/abs/2504.13520v3 |
conjugate for our sampling model, so we obtain closed-form expressions for conditional posteriors and marginal likelihoods (see Subsection 4.2). In addition, we only have to elicit two scalar hyperparameters, gLandgM. The choice of gL, gM>0 controls the prior variance and the complexity penalty of the Bayes factors. We... | https://arxiv.org/abs/2504.13520v3 |
by l+1 as our default choice throughout the remainder of the paper. This prior is quite uninformative on νitself while implying a prior on the covariance ratio Σ−1 xxΣ⊺ yxthat is (almost) identical to fixing ν= 3 (as done by Karl and Lenkoski, 2012). Figure S.1 in the supplementary material illustrates the implied prio... | https://arxiv.org/abs/2504.13520v3 |
We can obtain closed-form conditional posteriors and marginal likelihoods based on the prior specification described above. A detailed derivation is provided in Section B of the supplementary material. In the 7 Algorithm 1: The gIVBMA Gibbs Sampling Algorithm Input: DataD= (y,X,Z,W), number of posterior samples S, prio... | https://arxiv.org/abs/2504.13520v3 |
step to the Gibbs sampler updating ν. Given Σ,νis independent of everything else, so the full conditional is proportional to the prior on Σgiven νtimes the prior on ν. We use an adaptive MH step, targeting an acceptance rate of 0 .234. 4.3 The non-Gaussian case Here, we discuss the computational strategy to deal with t... | https://arxiv.org/abs/2504.13520v3 |
two-component prior as there is no a priori distinction between instruments and covariates. As before, our method works well when at least lrelevant and valid instruments are available without re- quiring the analyst to identify these instruments in advance. The main benefit of this data-driven instrument selection is ... | https://arxiv.org/abs/2504.13520v3 |
. , (p+k)l} lim n→∞Z ℜ+gc/2 Mp(gM)dgM=∞ 1This is what is often referred to as an M-closed setting. In most situations, model selection consistency naturally extends to the M-open framework in an intuitive manner (Mukhopadhyay et al., 2015). 10 For the two-component prior explained in Section 3 with l= 1, we have consis... | https://arxiv.org/abs/2504.13520v3 |
standard normal distributions and two different sample sizes n∈ {50,500}. Then, we multiply all instruments and covariates with even indices by 100 to have more variation in the variables’ scale. We expect IVBMA to struggle as its prior variance does not account for the scale, while the g-prior used in gIVBMA is scale-... | https://arxiv.org/abs/2504.13520v3 |
1.37 O-TSLS 0.33 0.33 0.66 1.33 0.08 0.06 0.92 1.4 JIVE 0.96 0.79 0.82 1.72 0.18 -0.15 0.98 1.57 RJIVE 0.97 0.74 0.82 1.75 0.18 -0.16 0.98 1.57 MATSLS 0.34 0.02 0.87 1.62 0.1 0.01 0.96 1.45 Post-LASSO 1.13 0.26 1.0 - 0.11 0.02 0.99 - Table 1: Many Weak Instruments: MAE, bias, coverage, and mean LPS on 100 simulated dat... | https://arxiv.org/abs/2504.13520v3 |
be beneficial. All methods except for sisVIVE are largely unaffected by the plurality rule. 6.3 Multiple endogenous variables with correlated instruments We consider an example with two endogenous variables, a Gaussian and a Beta, and correlated instruments. We generate 15 valid instruments (five of which are also rele... | https://arxiv.org/abs/2504.13520v3 |
a substantial negative impact on income. DiTraglia (2016) reanalyzes these data to illustrate the usefulness of their proposed Focused Moment Selection Criterion. They consider the regression model log gdpci=β1+β2rulei+β3malfal i+ϵi, where gdpc is real GDP per capita in 1995 prices, rule is an average governance indica... | https://arxiv.org/abs/2504.13520v3 |
performance on this dataset, we compute the LPS using leave-one-out cross- validation. For each observation, the model is trained excluding that observation, and the LPS is then 14 gIVBMA (BRIC) gIVBMA (hyper- g/n) BMA (hyper- g/n) Mean 95% CI Mean 95% CI Mean 95% CI rule 0.85 [0.59, 1.12] 0.87 [0.61, 1.12] 0.79 [0.45,... | https://arxiv.org/abs/2504.13520v3 |
set of potential exogenous covariates and instruments includes age and age squared, college proximity (distance to a 2 or 4-year college), variables on family background, marital status, race, and regional indicators. The dataset also includes information on parents’ educational attainment. However, due to a substantia... | https://arxiv.org/abs/2504.13520v3 |
to those from the full dataset (see Supplementary Section F.1). These findings suggest that the differences, at least for gIVBMA, are primarily driven by the additional instruments rather than the missing observations, highlighting how posterior inference can be highly sensitive to the choice of the instrument set. We ... | https://arxiv.org/abs/2504.13520v3 |
the return to schooling. In Christofides, L. N., Grant, E. K., and Swidinsky, R., editors, Aspects of Labour Market Behaviour: Essays in Honour of John Vanderkamp , pages 201–222. University of Toronto Press, Toronto. Carrasco, M. (2012). A regularization approach to the many instruments problem. Journal of Econometric... | https://arxiv.org/abs/2504.13520v3 |
Society Series B: Statistical Methodology , 84(2):496–523. Mukhopadhyay, M., Samanta, T., and Chakrabarti, A. (2015). On consistency and optimality of Bayesian variable selection based on g-prior in normal linear regression models. Annals of the Institute of Statistical Mathematics , 67:963–997. Mullahy, J. (1997). Ins... | https://arxiv.org/abs/2504.13520v3 |
= exp −1 2σy|x ˜y⊺˜y+ρ⊺gL+ 1 gLU⊺U ρ−2˜y⊺Uρ +gL gL+ 1˜y⊺PU˜y−gL gL+ 1˜y⊺PU˜y ∝N ρ|gL gL+ 1(U⊺U)−1U⊺˜y,gL gL+ 1σy|x(U⊺U)−1 ·exp −1 2σy|x˜y⊺ In−gL gL+ 1PU ˜y ∝N ρ|gL gL+ 1(U⊺U)−1U⊺˜y,gL gL+ 1σy|x(U⊺U)−1 . The (conditional) marginal likelihood is the normalising constant of this distribution (adjusted by the... | https://arxiv.org/abs/2504.13520v3 |
now depend on both the actual treatment Xthrough the projection on U= [ι:X:W] and its latent Gaussian representation Qthrough the latent residual H=Q−VΛ. For the treatment model, define ˜Q=Q−1 σy|x(q−Uρ)Σyx B−1 Σ⊺ such that we have Λ|q,Q,X,ρ,Σ∼MN (V⊺V)V⊺˜Q (Il+g−1 MB−1 Σ)−1⊺ ,(V⊺V)−1,(BΣ+g−1 MIl)−1Σxx and p(q,Q|X... | https://arxiv.org/abs/2504.13520v3 |
we distinguish between the following two prior options: Fixed gL:Here a necessary and sufficient condition for CBF( Li, Lj)→ ∞ with nis that gLincreases without bound in n. This is the case in e.g.the BRIC prior we use, and would also hold for the unit information prior, and many more. Random gL:if we assume a hyperpri... | https://arxiv.org/abs/2504.13520v3 |
true model Li. Summarising, for any model Lj, we have shown that CBF( Li, Lj) in favour of the true model Liincreases without bounds for the choice of the outcome model. Outcome model selection on the basis of gIVBMA is thus consistent, if and only if •limn→∞gL=∞for fixed gL •we have that lim n→∞Z ℜ+(1 +gL)c/2p(gL)dgL=... | https://arxiv.org/abs/2504.13520v3 |
26 The assumption for the non-nested case in Lemma 1 is satisfied if Vcorresponding to the full model has full column rank, which was assumed in Subsection 3.1. Applying Lemma 1, we have that the exponential term behaves (as n→ ∞ ) as exp (n·tr (AΣDj)). Note that AΣcan be written as the quadratic form AΣ= Il+g−1 MB−1... | https://arxiv.org/abs/2504.13520v3 |
that the prior on any additional parameters in ryandrx= (rx1, . . . , r xl) is proper and independent of q,Q, Li, Mj. The conditional Bayes factor for the outcome model will now be based on the marginal likelihood for y|X,Q, Li, Mj,Λ,Σ, which can be written as p(y|X,Q, Li, Mj,Λ,Σ) =Z p(y|q, ry)p(q|X,Q, Li, Mj,Λ,Σ)p(ry)... | https://arxiv.org/abs/2504.13520v3 |
E Additional details on the simulation experiments E.1 Performance measures This Section defines the performance measures used in our simulation experiments. Define the median absolute error (MAE) as the median ℓ1difference between the point estimator and the true parameter, MAE( τ,ˆτ) = Median i=1,...,M τ−ˆτ(i) 1, whe... | https://arxiv.org/abs/2504.13520v3 |
variables on the instru- ments and uses the resulting fitted values in the outcome model. This is equivalent to using the linear projection PVXinstead of X. Accordingly, the TSLS estimator is given by ˆρTSLS = (U⊺PVU)−1U⊺PVy and its variance is estimated as ˆ varTSLS = ˆσ2(U⊺PVU)−1, where ˆ σ2=∥y−UˆρTSLS∥2 2/(n−kU). Th... | https://arxiv.org/abs/2504.13520v3 |
to Julia. •sisVIVE: Kang et al. (2016) propose the sisVIVE estimator for settings with potentially invalid instruments. Instruments can be included in the outcome model but are subject to an ℓ1penalty. The estimation strategy is to minimise the moment condition subject to that ℓ1penalty term. We use the sisVIVE R packa... | https://arxiv.org/abs/2504.13520v3 |
To investigate the performance in the presence of invalid instruments, we consider a simulation setup similar to the ones in Kang et al. (2016) and Windmeijer et al. (2019). Again, we consider two different sample sizes n∈ {50,500}andp= 10 potential instruments simulated from independent standard Normals. Their coeffic... | https://arxiv.org/abs/2504.13520v3 |
of Kang et al. (2016). We also add what we call oracle gIVBMA (O-gIVBMA), which correctly separates the valid and invalid instruments (or covariates) a priori. For this oracle, the invalid instruments can be included in both models, but the valid instruments can only be included in the treatment model. This corresponds... | https://arxiv.org/abs/2504.13520v3 |
latter does affect sisVIVE, however, which performs very poorly when s= 6, as expected. E.5 Multiple endogenous variables with correlated instruments We consider an example with two endogenous variables, one following a Gaussian distribution and one following a Beta distribution, and correlated instruments. We take ins... | https://arxiv.org/abs/2504.13520v3 |
Table S.5 includes definitions of all the variables used. Tables S.6 and S.7 present the posterior inclusion probabilities obtained with the Card (1995) dataset, including and excluding parental education, respectively, by gIVBMA, IVBMA, and naive BMA (only the outcome model). Figure S.2 and Table S.8 present the poste... | https://arxiv.org/abs/2504.13520v3 |
- 6 expersq Experience squared nearc2 =1 if near 2-year college in 1966 nearc4 =1 if near 4-year college in 1966 fatheduc Father’s schooling in years motheduc Mother’s schooling in years momdad14 =1 if live with mom and dad at 14 sinmom14 =1 if with single mom at 14 step14 =1 if with step parent at 14 black =1 if black... | https://arxiv.org/abs/2504.13520v3 |
0.108 1.0 smsa 1.0 0.944 1.0 0.958 1.0 0.982 1.0 married 1.0 0.879 1.0 0.983 1.0 0.993 1.0 reg662 0.001 0.002 0.007 0.011 0.025 0.135 0.106 reg663 0.042 0.007 0.163 0.017 0.25 0.126 0.555 reg664 0.007 0.016 0.04 0.044 0.144 0.464 0.169 reg665 0.003 0.022 0.008 0.032 0.029 0.257 0.094 reg666 0.0 0.004 0.011 0.015 0.032 ... | https://arxiv.org/abs/2504.13520v3 |
in ounces, and the endogenous regressor is the typical number of cigarettes smoked daily during the pregnancy. As instruments, we use the father’s years of education, the mother’s years of education, the cigarette price in their home state, and the excise tax on cigarettes in their home state. These have a plausible ef... | https://arxiv.org/abs/2504.13520v3 |
arXiv:2504.13620v1 [math.PR] 18 Apr 2025Set-valued conditional functionals of random sets Tobias Fissler∗Ilya Molchanov† April 21, 2025 Abstract Many key quantities in statistics and probability theory su ch as the expectation, quantiles, expectiles and many risk measures are law-deter mined maps from a space of random... | https://arxiv.org/abs/2504.13620v1 |
depth defined by Koshevoy and Mosler (1997) andMosler(2002).Molchanov and Turin (2021) showed that many such constructions arise from an application of a sublinear expectation to the projections of random vectors and interpreting the obtain ed function as the support function of the depth-trimmed region. Motivated by fin... | https://arxiv.org/abs/2504.13620v1 |
riskm anagement, orthegauges based on moments as defined by Fischer(2003). Third, we construct a conditional version of the set-valued gauge of the random set X. That means, for a sub- σ-algebra Aof theσ- algebra of the underlying probability space, a gauge function g, and a random closed convex setX, we construct anoth... | https://arxiv.org/abs/2504.13620v1 |
k ind of a depth-trimmed region for the random set. This construction is principally different f rom the one developed byCascos et al. (2021), where a depth-trimmed region for a random set is a collection of sets. Followingtheworksby Hamel and Heyde (2010),Hamel et al. (2011),Hamel et al. (2013), andMolchanov and Cascos... | https://arxiv.org/abs/2504.13620v1 |
if the right-hand side is well defined; (g8) superadditivity: g(X+Y)≥g(X)+g(Y) if the right-hand side is well defined; (g9) sensitivity with respect to infinity: P{X=∞}>0 implies that g(X) =∞. 5 Note that in (g7) and (g8) the right-hand sides are well defined unle ss the expressions (∞−∞) or (−∞+∞) appear. Remark 2.2. It s... | https://arxiv.org/abs/2504.13620v1 |
assume that Tg:Mp→¯RisB(Mp)/B(¯R)-measurable. This assump- tion holds for all practically relevant examples discussed in Section 2.3due to the results of Fissler and Holzmann (2022). Thismeasurabilityassumption directlyyields thefollowingre- sult, establishing A-measurabilityof g(X|A). ItfollowsfromLemma2of Fissler and... | https://arxiv.org/abs/2504.13620v1 |
by Fissler and Holzmann (2022), quantilefunctionalsare B(Mp)/B(¯R)-measurable. Theysatisfyproperties(g1)–(g5), butnot (g9). For α∈(0,1) they are neither sub- nor superadditive. We refer to de Castro et al. (2023) for an exhaustive overview of the properties of conditional quan tiles. Even though conditional quantiles a... | https://arxiv.org/abs/2504.13620v1 |
defined on Lp(R) as /bardblX/bardblp:=/parenleftbig E|X|p/parenrightbig1/p. This function is not translation equivariant and not necessarily mono tone. It is possible to turn it into a gauge on Lp(R) by letting ep,a(X) :=/braceleftBigg EX+a/parenleftbig E(X−EX)p +/parenrightbig1/p,ifX∈Lp(R), ∞, ifX=∞with positive probab... | https://arxiv.org/abs/2504.13620v1 |
convex cone byCand a random one 11 byC. IfX=C, thenBX=Coand soBXis a random closed convex set. Let X=X+C, whereXis a random vector and Cis a random closed convex cone in Rdwhich contains (−∞,0]d. In the financial setting, such a cone is called the set of portfolios a vailable at price zero, see Kabanov and Safarian (200... | https://arxiv.org/abs/2504.13620v1 |
almost surely nonempty random closed convex set in Rd. Then, for each Castaing representation {Wn,n≥1}ofBX∩Sd−1, X=/intersectiondisplay n≥1/braceleftBig x∈Rd:/a\}bracketle{tWn,x/a\}bracketri}ht ≤h(X,Wn)/bracerightBig . (3.4) IfBXis regular closed, then (3.4)holds with Wn=wn, where{wn,n≥1}is a deterministic dense set in... | https://arxiv.org/abs/2504.13620v1 |
order) A-measurable random closed convex set Ysuch that h(Y,W)≤g(h(X,W)|A)a.s. (4.1) for allW∈L∞(Rd;A). Proof.Consider Y′:=/intersectiondisplay W∈L∞(Rd;A)HW/parenleftbig g(h(X,W)|A)/parenrightbig . (4.2) By construction, Y′has closed realisations, so it is a random closed set in a σ-algebra chosen to be suitably rich. ... | https://arxiv.org/abs/2504.13620v1 |
The conditional gauge G(·|A)is a map from the family of almost surely nonempty ( p-integrable) random closed convex sets to the family of A-measurable random closed convex sets which is (G1) law-determined, that is, G(X|A) =G(X′|A)ifXandX′have the same conditional distribution given A; (G2) constant preserving: G(X|A) ... | https://arxiv.org/abs/2504.13620v1 |
to be A-measurable, g(h(X,W)|A) = 0, W 1≤0,W2≤0, W1g(V1|V2)+W2V2, W1>0,W2>0, W1g(V1) W1>0,W2≤0, W2V2 W1≤0,W2>0. Thus,G(X|V2) is the rectangle/bracketleftbig 0,g(V1|V2)/bracketrightbig ×/bracketleftbig 0,V2/bracketrightbig . 5 Special cases of set-valued gauges While we write G(X|A) for a generic conditional s... | https://arxiv.org/abs/2504.13620v1 |
follows from L´ epinette and Molchanov (2019, Theorem 5.2) that the support function of M(X|A) equals esssup h(X,W) for allW∈L0(Sd−1;A). Ifgis the essential supremum, then the right-hand side of ( 4.2) equalsM(X|A), meaning that M(X|A) = esssup( X|A). (5.2) This equality is due to the fact that the essential supremum i... | https://arxiv.org/abs/2504.13620v1 |
Xis uniformly distributed in a convex set K, the set G({X}) is called the floating body of K, seeNagy et al. (2019). It is known from Bobkov(2010) that, if the distribution of Xis log-concave and α∈(1/2,1), then h(G({X}),u) =q− α(/a\}bracketle{tX,u/a\}bracketri}ht), u∈Sd−1. Ifα= 1, then gis theessential supremum, andess... | https://arxiv.org/abs/2504.13620v1 |
Letg=ep,abe defined at ( 2.4). Assume that E(X|A) = 0 a.s. for X∈L1(Rd). Then the corresponding conditional depth-trimmed region is the largest convex set such that h(G({X}|A),W) =/parenleftBig E/bracketleftbig |/a\}bracketle{tX,W/a\}bracketri}ht|p +/vextendsingle/vextendsingleA]/parenrightBig1/p for allW∈L0(Sd−1;A). Th... | https://arxiv.org/abs/2504.13620v1 |
Molchanov (2019, Proposition 5.5), ( m(Co|A))oequals the conditional convex hull M(X|A) ofC. Since the essential supremum also satisfies (g9), we obtain as a corollary of P roposition 7.3that esssup(C|A) equals the conditional convex hull of C, which also follows from ( 5.2). Ifgis the essential infimum, then the situati... | https://arxiv.org/abs/2504.13620v1 |
(1,q− α(κ1)) and (q− α(κ2),1) on its boundary. Assume that X=X+CforX∈Lp(Rd). Then h(X,w) =/a\}bracketle{tX,w/a\}bracketri}htifw∈Coand otherwise h(X,w) =∞. If the gauge function satisfies (g9), then G(X) is the sum of G({X}(which is the corresponding depth-trimmed region of X) andG(C) as described above. If the gauge is ... | https://arxiv.org/abs/2504.13620v1 |
KW (1990) Symmetric Multivariate and Related Dis tributions. Cjap- man and Hall Fischer T (2003) Risk capital allocation by coherent risk measures b ased on one-sided mo- ments. Insurance Math Econom 32: 135–146 Fissler T, Holzmann H (2022) Measurability of functionals and of ideal point forecasts. Electronic Journal o... | https://arxiv.org/abs/2504.13620v1 |
Testing Random Effects for Binomial Data Lucas Kania, Larry Wasserman and Sivaraman Balakrishnan Department of Statistics and Data Science Carnegie Mellon University In modern scientific research, small-scale studies with limited participants are in- creasingly common. However, interpreting individual outcomes can be c... | https://arxiv.org/abs/2504.13977v1 |
better account for known individual differences in task performance. Binomial mixtures also play an important role in other fields. They are used to account for variation in mouse mortality rates [Brooks et al., 1997], word frequencies [Lowe, 1999], welfare program participation [Melkersson and Saarela, 2004], genetic ... | https://arxiv.org/abs/2504.13977v1 |
has been observed in binomial, Multinomial, Poisson distributions and smooth densities in the context of fixed effects testing [Valiant and Valiant, 2014, Balakrishnan and Wasserman, 2019, Chhor and Carpentier, 2022]. A key application of locality arises in statistical meta-analysis of treatment effectiveness. Suppose ... | https://arxiv.org/abs/2504.13977v1 |
denote its l-th moment, and V(π) =m2(π)−m2 1(π) to denote its variance. Let δA(a) equal 1 if a∈A, and 0 otherwise. Alternatively, we use the following notation I(condition) to denote the indicator function that evaluates to 1 whenever the condition is true and returns 0 otherwise. Finally, let a∧b= min( a, b) and a∨b= ... | https://arxiv.org/abs/2504.13977v1 |
the type I error by α∈(0,1) for any null distribution, called valid tests, Ψ =∩π0∈DΨ(π0) where Ψ( π0) = ψ:Pn π0(ψ(X) = 1) ≤α . where X= (X1, . . . , X n) is a vector containing nobservations. The risk of a valid test is given by its maximum type II error, i.e., the probability of choosing the null hypothesis when the ... | https://arxiv.org/abs/2504.13977v1 |
null distributions cW1(X) =W1(bπ, π 0) where bπ=1 nnX i=1δXi/t. We reject the null hypothesis when cW1(X) exceeds its 1 −αquantile under the null distri- bution, denoted by qα(Pπ0,cW1), ensuring type I error control: ψα cW1(X) =I cW1(X)≥qα(Pπ0,cW1) . (9) Intuitively, the test works whenever W1(bπ, π 0) is a good appr... | https://arxiv.org/abs/2504.13977v1 |
to be suboptimal. If we consider all polynomials of degree t, then the best uniform approximation of a Lipschitz function has an error of at most π/2t[Plaskota, 2021]. Therefore, the plug-in test can be improved by modifying the polynomial approximation of the witness function in (5). In the following section, we show ... | https://arxiv.org/abs/2504.13977v1 |
n) max 1 t+1, µbj,t(π0). We call the corresponding test the debiased Pearson’s chi-squared test ψα bℓ2=I bℓ2(X)> qα(Pπ0,bℓ2) , (18) since expanding the definition of eK2, it is clear that is a centered chi-squared statistic eK2(nj, bj,t(π0), n) =nj n−bj,t(π0)2 −µnj/n n−1. Unlike the usual chi-squared statistic, th... | https://arxiv.org/abs/2504.13977v1 |
and mixing distribution π0andπ1inD, it holds that V(Pπ0, Pπ1)≤1 2·q Mp(π0, π1)where Mp(π0, π1) =tX m=1t m ·∆2 m(π1, π0) µm p and∆m(π1, π0) =Eu∼π1[u−p]m−Eu∼π0[u−p]mis the moment difference centered at p. 11 The proof, found in appendix C.1, relies on the orthogonality of the Kravchuk polynomials. It follows a structur... | https://arxiv.org/abs/2504.13977v1 |
} Test H0:π=π0v.s. H1:W1(π, π 0)≥ϵ, for which the separation rate is known to be ϵ≍n−1/2[Ba et al., 2011]. In the intermediate regime, where log n≲t≲logn/n, the lower bound follows from con- structing mixing distributions that match tlognmoments, which is the optimal number of moments that one needs to estimate to appr... | https://arxiv.org/abs/2504.13977v1 |
better in the worst case. Conversely, when tis much larger, the plug-in test should be able to obtain higher power. Both phenomena can be observed in the figure. Finally, figure 4 illustrates the empirical critical separation, which represents the minimum distance between the null and alternative mixing distributions r... | https://arxiv.org/abs/2504.13977v1 |
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