text string | source string |
|---|---|
dimension of the output layer. A ( q+ 1)-layer DNN with layer-width his essentially a composite function ϕα:Rh0→Rhq+1recursively defined as ϕα(x) =ωqϕq(x) +vq, ϕq(x) =σ(ωq−1ϕq−1(x) +vq−1), ... ϕ2(x) =σ(ω1ϕ1(x) +v1), ϕ1(x) =σ(ω0x+v0),(3) where ω0, . . . ,ωqare the unknown weight matrices, and v0, . . . ,vqare the unknow... | https://arxiv.org/abs/2503.19763v1 |
that the unknown parameters in the DNN are bounded by one. In a DNN, the parameter sgoverns the network sparsity, and its selection typically leverages the dropout technique within each hidden layer, which randomly ignores partial neurons based on a specified dropout rate (Srivastava et al., 2014). This strategy enhanc... | https://arxiv.org/abs/2503.19763v1 |
we use the multi-index notation ∂ι=∂ι1. . . ∂ιdwithι= (ι1, . . . , ι d)⊤∈Nd +and |ι|=/summationtextd j=1ιj, and ⌊ξϕ⌋is the largest integer that is strictly smaller than ξϕ(Schmidt- Hieber, 2020). Let˜k∈N+andξϕ= (ξϕ1, . . . , ξϕ˜k)⊤∈R˜k +. Define ¯d= (¯d1, . . . , ¯d˜k+1)⊤∈N˜k+1 +and ˜d= (˜d1, . . . , ˜d˜k)⊤∈N˜k +with ˜... | https://arxiv.org/abs/2503.19763v1 |
For˜ξ≥1, the joint density f(t,x,w, δl, δi, δr) of ( T,X,W, δL, δI, δR) with respect totorwhas bounded ˜ξth partial derivatives. 15 (C8) Ifβ⊤X+f(t) = 0 for any t∈[a, b] with probability one, then β= 0 and f(t) = 0 fort∈[a, b]. Conditions (C1) and (C2) impose boundedness on the true parameters and covariates, which are ... | https://arxiv.org/abs/2503.19763v1 |
(Zeng et al., 2016) were also included to demonstrate the 17 advantages of the proposed method. In model (1), we let X= (X1, X2)⊤, where X1and X2were generated from the standard normal distribution and the Bernoulli distribution with a success probability of 0 .5, respectively. The length of Wwas set to d= 4 or 10, and... | https://arxiv.org/abs/2503.19763v1 |
in the SGD algorithm was set to 50. The neural network was trained for 20 epochs during each iteration of the proposed EM algorithm. We applied the dropout regu- larization with a rate of 0.1 to prevent overfitting and selected the optimal hyperparameter combination with the maximum likelihood principle on the validati... | https://arxiv.org/abs/2503.19763v1 |
0.176 0.965 -0.020 0.186 -0.038 0.186 0.069 0.206 -0.008 0.168 1000 β1 0.018 0.059 0.063 0.965 0.017 0.060 0.030 0.061 0.008 0.059 0.023 0.060 β2-0.015 0.125 0.121 0.965 -0.010 0.125 -0.028 0.134 0.011 0.129 -0.021 0.125 2 500 β1-0.015 0.087 0.097 0.985 -0.026 0.091 0.019 0.129 -0.087 0.087 -0.067 0.084 β2<0.001 0.171 ... | https://arxiv.org/abs/2503.19763v1 |
2.044 1.389 0.365 0.412 0.320 0.718 1000 0.313 0.285 0.421 1.039 1.947 0.545 0.263 0.206 0.216 0.337 5 500 0.358 1.606 0.545 3.212 1.574 0.900 0.430 1.803 0.426 2.489 1000 0.292 0.984 0.421 2.515 1.407 0.369 0.415 1.573 0.415 1.889 6 500 0.556 1.883 0.749 3.365 1.126 3.903 0.767 2.767 0.767 3.168 1000 0.488 1.262 0.630... | https://arxiv.org/abs/2503.19763v1 |
“Penalized PH” and “NPMLE-Trans”, they both yield conspicuously large REs and MSEs except for the cases of linear covariate effects (i.e., Case 1 and Case 4). The above comparison results all manifest the significant advantages of the proposed method in terms of prediction accuracy. In Section S.3 of the supplementary ... | https://arxiv.org/abs/2503.19763v1 |
That is, we assumed that the failure time of interest followed the 25 partially linear transformation model with the conditional hazard function: Λ(t|X,W) =G[Λ(t) exp{X1β1+X2β2+X3β3+ϕ(W)}], where G(x) = log(1 + rx)/rwith r≥0,X1=Age,X2=Gender ,X3=APOEϵ4andW includes all the aforementioned covariates except X1,X2andX3. N... | https://arxiv.org/abs/2503.19763v1 |
the four comparative methods, compared to the proposed method, the most notable difference is that the comparative methods all recognized Ageas insignificant. This conclusion may be erroneous since age has been widely acknowledged as one of the most significant influential factors for AD (e.g., James et al., 2019; Livi... | https://arxiv.org/abs/2503.19763v1 |
(Zhou et al., 2017; Li et al., 2024). Thus, generalizations of the proposed deep regression approach to account for the settings above warrant future research. Furthermore, multivariate interval-censored data arise frequently in many real-life studies, where each individual may undergo multiple events of interest that ... | https://arxiv.org/abs/2503.19763v1 |
Lee, B.L., Fine, J.P. (2004). Robust inference for univariate proportional hazards frailty regression models. The Annals of Statistics 32(10), 1448–1491. 31 Kvamme, H., Borgan, O., Scheel, I. (2019). Time-to-event prediction with neural networks and Cox regression. Journal of Machine Learning Research 20(129), 1–30. Le... | https://arxiv.org/abs/2503.19763v1 |
with Latent Mediators and Survival Outcome. Structural Equation Modeling: A Multidisciplinary Journal 28(5), 778–790. Sun, T., Ding, Y. (2023). Neural network on interval-censored data with application to the prediction of Alzheimer’s disease. Biometrics 79(3), 2677–2690. Sun, Y., Kang, J., Haridas, C., Mayne, N., Pott... | https://arxiv.org/abs/2503.19763v1 |
ti1=Ri(δL,i= 1) + Li(δL,i= 0). We introduce two independent Poisson random variables: Zi∼Pois(Λγ(ti1) exp{β⊤Xi+ϕα(Wi)}ηi) and Yi∼Pois({Λγ(ti2)−Λγ(ti1)}exp{β⊤Xi+ϕα(Wi)}ηi), where Pois(ν) indicates the Poisson distribution with mean ν. Moreover, by the spline represen- tation of Λ γ(t), we decompose ZiandYiasZi=/summatio... | https://arxiv.org/abs/2503.19763v1 |
. . . , L n, by setting ∂Q(β,γ,α(m+1);β(m),γ(m),α(m))/∂γ l= 0, we can derive a closed-form update for γlthat depends on the unknown β: γ(m+1) l (β;α(m+1)) =/summationtextn i=1E(Zil) + (δI,i+δR,i)E(Yil) /summationtextn i=1exp{β⊤Xi+ϕ(m+1) α (Wi)}E(ηi){(δL,i+δI,i)Ml(Ri) +δR,iMl(Li)}. Notably, given a nonnegative initial v... | https://arxiv.org/abs/2503.19763v1 |
q,h, B) for some B > 0. In what follows, Mdenotes a general positive constant whose value may vary from place to place. Before proving Theorem 1, we need the following lemma. Lemma 1. Under conditions (C2)–(C4), the class {ℓ(θn) :θn∈ D×M B×FB}is Glivenko- Cantelli . Proof. For any ε >0, we first know that there are O(ε... | https://arxiv.org/abs/2503.19763v1 |
=xlogx−x+ 1 and ( x−1)2/4≤m(x)≤(x−1)2ifx→1. Define h(z) = 1−exp{−Υ(X, z;θ)}, h0(z) = 1 −exp{−Υ0(X, z)}, and G(U,V;θ) = ( Υ(X,U;θ)− Υ0(X,U))2+ (Υ(X,V;θ)−Υ0(X,V))2. By Conditions (C1)–(C3), we utilize the Taylor series expansion and derive M(θ0)−M(θ) = E/braceleftbigg h(U)m/parenleftbiggh0(U) h(U)/parenrightbigg + (h(U)−... | https://arxiv.org/abs/2503.19763v1 |
0,n= arg min Λγ∈MB/2∥Λγ−Λ0∥L2andϕ0,n= arg min ϕα∈FB/2∥ϕα− ϕ0∥L2. LetLδ={ℓ(θn)−ℓ(θ0,n) :θn∈ D × M B× F B, d(θn,θ0,n)≤δ}. We have ∥Gn∥Lδ= sup θn∈D×M B×FB, d(θn,θ0,n)≤δ|Gnℓ(θn)−Gnℓ(θ0,n)|. By Condition (C1) and Theorem 9.23 of Kosorok (2008), for any δ >0 and 0 < ε < δ , theε-bracketing number of {β∈ D :∥β−β0∥≤δ}with radi... | https://arxiv.org/abs/2503.19763v1 |
. . . , ˙ℓϕ(θ0)[zp])⊤and˙ℓΛ(θ0)[q] = ( ˙ℓΛ(θ0)[q1], . . . , ˙ℓΛ(θ0)[qp])⊤, where z= (z1, . . . , z p)⊤ ∈Ωp ϕ0,q= (q1, . . . , q p)⊤∈Ωp Λ0. Referring to Theorem 1 in Bickel et al. (1993)(pp. 70), under Conditions (C1)–(C3) and (C7), the efficient score vector for βis defined as ℓ∗ β(θ0) =˙ℓβ(θ0)−˙ℓΛ(θ0)[q∗]−˙ℓϕ(θ0)[z∗],... | https://arxiv.org/abs/2503.19763v1 |
S1 and S2 suggest similar conclusions as Section 4 of the main manuscript regarding the comparisons of the proposed method, “Spline-Trans” and “NPMLE-Trans”. In addition, it is worth pointing out that the performances of “deep PH” and “penalized PH” deteriorate remarkably here due to using a misspecified PH model. Sect... | https://arxiv.org/abs/2503.19763v1 |
0.174 0.194 0.139 0.009 0.155 5 500 β1-0.001 0.128 0.129 0.945 -0.163 0.145 0.115 0.167 -0.184 0.109 -0.024 0.122 β2 0.015 0.231 0.233 0.955 0.131 0.250 -0.099 0.309 0.281 0.223 0.039 0.227 1000 β1-0.022 0.089 0.086 0.940 -0.194 0.100 0.038 0.098 -0.189 0.066 -0.048 0.083 β2 0.035 0.165 0.155 0.930 0.169 0.178 -0.029 0... | https://arxiv.org/abs/2503.19763v1 |
and 30-minute delayed recognition, respec- tively. Higher values for TrailA and TrailB reflect longer processing times to complete tasks in Test Part A and Part B, respectively, suggesting poorer executive function. The continuous variables were all normalized to have a mean of 0 and a variance of 1. To measure the pre... | https://arxiv.org/abs/2503.19763v1 |
Asymptotic Statistics . Cambridge: Cambridge University Press. van der Vaart, A.W., Wellner, J.A. (1996). Weak Convergence and Empirical Processes: with Applications to Statistics . New York, NY: Springer. Yuan, C., Zhao, S., Li, S., Song, X. (2024). Sieve Maximum Likelihood Estimation of Partially Linear Transformatio... | https://arxiv.org/abs/2503.19763v1 |
ТЕОРЕТИЧНІ ТА ПРИКЛАДНІ ПРОБЛЕМИ МАТЕМАТИКИ 17 UDC 519.21 DOI: 10.20535/1810-05 46.2017.4.105428 O.M. Mokliachuk* Igor Sikorsky Kyiv Polytechni c Institute, Kyiv, Ukraine ESTIMATION OF ACCURACY AND RELIABILITY OF MODELS OF -SUB-GAUSSIAN STOCHASTIC PROCESSES IN С (T ) SPACES Background. At present, in the theory of sto... | https://arxiv.org/abs/2503.19789v1 |
Needless to say, errors of these approximations will have the influence on the reliability and accuracy of the developed model. This problem has been previously considered in [5—8]. Research objective In this paper, we consider models of stochastic processes in C (T ) space when decomposition ele- ments ()kat cannot be... | https://arxiv.org/abs/2503.19789v1 |
process X, r {r (u ): u 1} is the continuous function such that ()ru 0 for 1,u and function () ( e x p {} ) , 0 ,st r t t is con- vex. Then, if (1 ) 0( ( ( ))) ,BrN u d u stochastic process X (t ) is bounded with probability 1, and for all (0,1)p and 0x next inequalities are true: (0, ) (0, ) (0, ) 0 (... | https://arxiv.org/abs/2503.19789v1 |
(4) If : ( ) ( ( )), sup supuB uBuX u formula (4) reaches its minimum when 1 1 (1 ), (1 )xp pp and minimal value of fo rmula (4) is equal to 1 1 1(1 )( (1 )). (( 1 ) )xp pp ... | https://arxiv.org/abs/2503.19789v1 |
space, let (,)B be the com- pact and let X be separable on (,) .B Let (t) ,1 2 ,t and let the process X satisfy the conditions of the theorem 2. Then, the model XN (t ) approximates the process X (t ) with given reliability 1 and accuracy in C (B ) space, if 1 (1 ) 1 1 1 (1 ) 10( 1)( (1 ))2exp (( 1 ) ) 1... | https://arxiv.org/abs/2503.19789v1 |
() ,u we obtain ТЕОРЕТИЧНІ ТА ПРИКЛАДНІ ПРОБЛЕМИ МАТЕМАТИКИ 21 æ æ ææ æ ææ æ1/1// (1 ) / 00 1/ 1/1 / 1/ 1/11 () 1( ).1/ ()( 1 / )ppTT Cdu dupp uu TC p T C p p When 0 we will have ææ æ æææ1/ 1/ 1/ 1/ 1/ 1/. ()( 1 / ) ()TC TCe pp B... | https://arxiv.org/abs/2503.19789v1 |
are needed. □ Models of stochastic processes that allow rep- resentation in series with independent elements Assume that stochastic process XN (t ) {XN (t ), t T } can be represented as 1() () ,kk kXt a t (6) where Sub ( ).k Let () | () () | , ()kk k kta t a t a t be approximations of () ,kat and ... | https://arxiv.org/abs/2503.19789v1 |
Proof. 11 11(( ) )sup () () sup ( ) () ( ) () .sup supN N uB N kk kk uB kk N N kk kk uB uB kk Nu ua u ua u The last inequality follows from the properties of function and theorem 1. □ Let the process X be def... | https://arxiv.org/abs/2503.19789v1 |
in series can- not be found explicitly and requires application of se- ries’ elements approximations. Influence of the error of such approximations is studied for reliability and accuracy of a model of stochastic process in C (T ) space. Theorems that allow developing a model that approximates -sub-Gaussian stochastic... | https://arxiv.org/abs/2503.19789v1 |
help of Karhunen —Loeve de- composition”, Theory of Stochastic Processes , iss. 13 (29), no. 4, pp. 90—94, 2007. [6] Yu.V. Kozachenko and O.M. Moklyachuk, “Sample continuity and modeling of stochastic processes from the spaces DV,W”, Theor. Probab. Math. Statist ., no 83, pp. 95—110, 2011. [7] O.M. Moklyachuk, “Models ... | https://arxiv.org/abs/2503.19789v1 |
arXiv:2503.19892v1 [math.PR] 25 Mar 2025A MARTINGALE APPROACH TO LARGE– θEWENS–PITMAN MODEL RODRIGO RIBEIRO1 1 Abstract. We investigate the asymptotic behavior of the number of part s Knin the Ewens–Pitman partition model under the regime where t he diversity parameter is scaled linearly with the sample size, that is, ... | https://arxiv.org/abs/2503.19892v1 |
(Strong Law of Large Numbers) .Fixλ>0, then forα∈[0,1) lim n→∞Kα,λn,n n=mλ,α, almost surely. We also show a Central Limit Theorem with Berry-Esseen-type boun ds Theorem 2 (CLT and Berry-Esseen-type bounds) .Fixλ>0andα∈[0,1), and letFnbe the CDF of√n sλ,α/parenleftbiggKα,λn,n n−mλ,α/parenrightbigg , Then, for any ǫ∈(0,1... | https://arxiv.org/abs/2503.19892v1 |
extra advantage that ther e is no predictable component. However, identifying such a transformation may be ch allenging. In our work, we adopt the latter approachby constructing a suita ble transforma- tionTofKinto a martingale. We apply Azuma’s inequality and a Borel-Cantelli argument to establish a Law of Large Numbe... | https://arxiv.org/abs/2503.19892v1 |
andφθ,1= 1. When α= 0,φθ,j= 1 for alljandθ. We also define ψθ,jandZθ,jas (2.2) ψθ,j:= (θ+α)φθ,jandZθ,j:=θ+αKθ,j. Finally, given any process {Xj}j, we let ∆Xjbe (2.3) ∆ Xj:=Xj−Xj−1. We are now ready for the main result of this section. Proposition 1 (A useful martingale) .The process {Zθ,j/ψθ,j}j∈Nis a martingale with re... | https://arxiv.org/abs/2503.19892v1 |
P/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleZλn,n ψλn,n−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥ε/parenrightbigg ≤2exp/braceleftbigg −ε2n 2C/bracerightbigg , which, by the first Borel-Cantelli lemma, implies thatZλn,n ψλn,nconvergesto 1 a.s. as n goes to infinity. To show the ... | https://arxiv.org/abs/2503.19892v1 |
3.7) and write n/summationdisplay i=1Γ(λn+i+1) (λn+i)Γ(λn+i+1+α)=aλ,α nα+O(n−1−α), where we bounded the other sums that appear in ( 3.7) by their integral counter- parts to obtain terms of order at most n−1−α. Finally, getting back to ( 3.6), and expanding the factors in Γ again using ( 3.2), yields φλn,n nn/summationd... | https://arxiv.org/abs/2503.19892v1 |
−1/bracketrightbigg , which combined imply √n sλ,α/parenleftbiggψλn,n αn−mλ,α−λ α/parenrightbigg =√n sλ,α/bracketleftbigg(λn+α)φλn,n αn−λ α/parenleftbigg 1+1 λ/parenrightbiggα/bracketrightbigg =√n sλ,α/bracketleftbiggλ α/parenleftbigg φλn,n−/parenleftbigg 1+1 λ/parenrightbiggα/parenrightbigg +φλn,n n/bracketrightbigg T... | https://arxiv.org/abs/2503.19892v1 |
Lemma 4. LetYn,jbe as in (4.2). Then, there exists a positive constant Cde- pending on λandαonly, such that P(|Yn,j−1|>ε,for somej≤n)≤2ne−Cε2n Proof.We already have all we need for the proof. Firstly, recall from Prop osition1 that{Yn,j}jis a bounded-increment martingale of mean 1. Secondly, by ( 3.4), for anyj≤n j/sum... | https://arxiv.org/abs/2503.19892v1 |
fact that by technical Lemma6,α2n σ2 λ,α/summationtextn j=11 (λn+j+α)ψλn,j+1converges as ngoes to infinity. As for the second summation of ( 4.17), we handle it in a similar manner. First. notice that Yn,jis bounded. In fact, by ( 3.3), we have that |Yn,j| ≤j/summationdisplay i=1|∆Yn,i| ≤j/summationdisplay i=1C (λn+α)1−... | https://arxiv.org/abs/2503.19892v1 |
In P. L. Hennequin, editor, ´Ecole d’ ´Et´ e de Probabilit´ es de Saint-Flour XIII — 1983 , pages 1–198, Berlin, Heidelberg, 1985. Springer Berlin Heidelberg. [2] Bernard Bercu and Stefano Favaro. Laws of iterated logar ithm for the number of species in the ewens–pitman model. arXiv preprint arXiv:2401.00000 , 2024. [3... | https://arxiv.org/abs/2503.19892v1 |
Submitted to the Annals of Statistics FUNCTIONAL STRUCTURAL EQUATION MODELS WITH OUT-OF-SAMPLE GUARANTEES BYPHILIP KENNERBERG1,a, ERNST C. W IT1,b 1Università della Svizzera italiana, Switzerland,aphilip.kennerberg@usi.ch;bwite@usi.ch Statistical learning methods typically assume that the training and test data origina... | https://arxiv.org/abs/2503.20072v1 |
developed invariant risk minimization , which seeks to identify data representations yielding consistent classifiers across multiple environments. However, as shown by Rosenfeld, Ravikumar and Risteski (2020) and Kamath et al. (2021), these strict causal methods often struggle when the training and test data differ sig... | https://arxiv.org/abs/2503.20072v1 |
— as opposed toT— this will allow for a large class of unbounded operators to be considered. In Section 3 we provide the central result of this paper, the functional worst-risk decomposition. This re- sult considers the worst risk among all the shifted environments corresponding to shifts in the out-of-sample shift set... | https://arxiv.org/abs/2503.20072v1 |
well. Let V=( UisF −B L2([T1,T2])p+1 measurable:p+1X i=1Z [T1,T2]E Ut(i)2 dt <∞) . The expected value of an element in Vexists in the sense of a Bochner integral. LetT:D(T)→L2([T1,T2])p+1, where D(T)⊆L2([T1,T2])p+1, be any operator such that •Range (I− T) =L2([T1,T2])p+1; note that this is trivially true if Range (... | https://arxiv.org/abs/2503.20072v1 |
with a bounded, linear map Sas illus- trated in the next example, which is related to functional regression. FUNCTIONAL STRUCTURAL EQUATION MODELS WITH OUT-OF-SAMPLE GUARANTEES 5 Xt(1) Yt Xt(2)βx1y βyx2 FIG1. Observational environment: a functional system that serves as an illustration of a structural system throughout... | https://arxiv.org/abs/2503.20072v1 |
(ϕ1,...,ϕ 10) Y=ζtϕ, X (1) = ξt 1ϕ, X (2) = ξt 2ϕ, whereby the random scores (ζ,ξ1,ξ2)∈R30in the observational environment Oare related according to (5) ζO ξO 1 ξO 2 =B ζO ξO 1 ξO 2 +ϵO, where ϵO∼N(0,Σ)withΣ =I30×30. In our case, we assume homogeneous effects across all the basis functions and choose, B= 0bx1... | https://arxiv.org/abs/2503.20072v1 |
Since we have a multivariate functional model, the natural gen- eralization is therefore the following definition. DEFINITION 3.2. Collection of out-of-sample environments. ForA∈ V,A ⊆ V and γ∈R+let the collection of future out-of-sample environments be defined as, Cγ A(A)=n A′∈A:R [T1,T2]2g(s)KA′(s,t)g(t)Tdsdt≤γR [T1,... | https://arxiv.org/abs/2503.20072v1 |
Bühlmann and Mein- shausen, 2019). Also, the above decomposition is an exact analogue of the non-functional case (Kania and Wit, 2022). We wish to stress the case A=˜Vwhich restricts us to shifts for which the SEMs (3), in case the range of I− T is only dense (in this case Tcannot be closed) and (3) is fulfilled, with ... | https://arxiv.org/abs/2503.20072v1 |
precise we have the SEM, YA′ t=S XA′(1),...,XA′(p) t+A′ t(1) + ϵA′ t(1), FUNCTIONAL STRUCTURAL EQUATION MODELS WITH OUT-OF-SAMPLE GUARANTEES 11 withXA′ t(i) =A′ t(i+ 1) + ϵA′ t(i+ 1), for1≤i≤pandA′∈ V, so that XA′−A′∼XO. In particular with YA′ t=Z [T1,T2]pX i=1(β(t,τ))(i)XA′ τ(i)dτ+A′ t(1) + ϵA′ t(1), we have classic... | https://arxiv.org/abs/2503.20072v1 |
ξξ Ze=Me ξζ, whereby the subscripts indicate the submatrices of the second moment matrix. For each γ∈ [1/2,∞)we can now define the regularized covariance operator for each of the covariates X(1)andX(2), through the two 10×10submatrices Cγ 1andCγ 2, Cγ 1 Cγ 2 = γGA+ (1−γ)GO−1 γZA+ (1−γ)ZO . With these matrices, w... | https://arxiv.org/abs/2503.20072v1 |
as in the population case will not affect our estimator. In order to avoid imposing any extra moment conditions for our esti- mator we will do separate (independent) estimation for the denominator coefficients. To that end, let Fk=σ XA,1,XO,1,YA,1,YO,1,...,XA,k,XO,k,YA,k,YO,k and assume X′A,m,X′O,m m∈Nis such that... | https://arxiv.org/abs/2503.20072v1 |
for all m∈N. In order to avoid imposing any extra moment conditions for our estimator we will do separate (independent) estimation for the denominator coefficients. To that end, let Fk=σ XA,m(1),...,XA,m(p),XO,m(1),...,XO,m(p),YA,m,YO,m:m≤k and assume X′A,m,X′O,m m∈Nis such that X′A,m,X′O,m is independent of Fma... | https://arxiv.org/abs/2503.20072v1 |
case corresponds to the causal solution. 6. Conclusion. In this paper, we introduced a novel framework for functional structural equation models (SEMs) that extends worst-risk minimization to the functional domain. By leveraging linear, potentially unbounded operators, we provided a formulation that circum- vents limit... | https://arxiv.org/abs/2503.20072v1 |
Method- ology) 78947-1012. https://doi.org/10.1111/rssb.12167 ROSENFELD , E., R AVIKUMAR , P. and R ISTESKI , A. (2020). The risks of invariant risk minimization. arXiv preprint arXiv:2010.05761 . ROTHENHÄUSLER , D., M EINSHAUSEN , N., B ÜHLMANN , P. and P ETERS , J. (2021). Anchor regression: Het- erogeneous data meet... | https://arxiv.org/abs/2503.20072v1 |
arXiv:2503.20193v1 [math.ST] 26 Mar 2025Nonparametric MLE for Gaussian Location Mixtures: Certifie d Computation and Generic Behavior Yury Polyanskiy* Mark Sellke† Abstract We study the nonparametric maximum likelihood estimator /hatwideπfor Gaussian location mixtures in one dimension. It has been known since [ Lin83a ]... | https://arxiv.org/abs/2503.20193v1 |
in how new locations (atoms) are added at each iteration, see e.g. [ Der86 , B¨ oh86 ,LK92 ,BSL92 ] and [ Lin83b , Chapter 6] for a detailed survey. However, a decade ago [ KM14 ] discovered that due to the progress in convex optimization, the (empirically) fastest and most accurate way to maximize ( 1.2) is to fix the ... | https://arxiv.org/abs/2503.20193v1 |
succeeds almost surely for generic X, onceεis sufficiently small. As preparation, we recall the classical stationarity condi tions characterizing /hatwideπ. Define the function 1Our genericity results also hold if x1,...,x nare drawn independently from different probability densit ies. In fact this is an immediate coroll... | https://arxiv.org/abs/2503.20193v1 |
,(III) hold, then: (A) There is an open neighborhood of /hatwideπinΠkon which ℓXis locally c-strongly concave for some c >0. (B) There is an open neighborhood of XinRn, such that/hatwideπ(ˆX)∈Πkfor allˆXin this neighborhood. Moreover,/hatwideπis a smooth function on this neighborhood. (C) The expectation-maximization (... | https://arxiv.org/abs/2503.20193v1 |
that one obtains a q uantitative rate of convergence of πεto/hatwideπ. Worse, this proof does not give any stopping rule at which one can guarantee that πεis withinεdistance from/hatwideπ. From the point of view of computational complexity theory, this means one does not yet have an actual algorithm to compute /hatwide... | https://arxiv.org/abs/2503.20193v1 |
exactly for a Newton–Raphson iteration to make sense. Indeed, it underlines the point that exact computation of /hatwideπinherently requires exact computation of the support size. 1.3 Na ¨ıve Brute-Force Approximation of /hatwideπ To further motivate and illustrate our algorithmic results , we discuss two na¨ ıve algor... | https://arxiv.org/abs/2503.20193v1 |
kcan be exactly computed. We equip Zε(recall ( 1.11)) with the adjacent-neighbors graph structure, making it isomorp hic to a path. Below we write OX(·)to indicate an implicit constant factor which is random and depends on X, but not on e.g. ε. 3For example let S1,...,S K⊆Zε= [−L,L]∩εZbe IID uniformly random subsets of... | https://arxiv.org/abs/2503.20193v1 |
we employ Theorem 1.2(II) and(III) to show the necessary conditions hold once /tildewideπis a sufficiently accurate approximation for /hatwideπ. While Newton–Raphson iteration is appealing due to its quad ratic local convergence rate, other ap- proaches also suffice for asymptotic convergence from an app roximate solutio... | https://arxiv.org/abs/2503.20193v1 |
Other Related Work Gaussian mixture models have been studied since the pioneer ing work of Pearson [ Pea94 ], which proposed that the ratio of forehead width to body length of crabs might follow such a distribution. Much work has focused on statistical recovery of such mixtures. In the 1-dimensional Gaussian location m... | https://arxiv.org/abs/2503.20193v1 |
sure local linear convergence for generic datasets, without even requiring the existence of an underl ying mixture model generating the data. On the other hand, it is currently limited to dimension 1and does not give quantitative bounds on η. Among the vast literature in this direction, we also mention a few rece nt wo... | https://arxiv.org/abs/2503.20193v1 |
D: |D(j) π,X(z)| ≤CjLje2L2, (1.19) |D(j) π,X(z)−D(j) π′,X(z)| ≤CjLje4L2δ. (1.20) Lemma 1.19. Fixπ=/summationtextk i=1pjδyiandπ′=/summationtextk i=1qiδyi, both inΠL k. Consider ℓ(t)/definesℓX(πt) =ℓX((1−t)π+tπ′) Ifmaxi|pi−qi| ≤τ, then sup 0≤t≤1/vextendsingle/vextendsingle/vextendsingle/vextendsingled dtℓ(t)/vextendsingl... | https://arxiv.org/abs/2503.20193v1 |
=f(x). The next key estimate follows from [ SM93 , Page 334], see also [ Mar88 ,BGG+07]. 13 Proposition 2.2. Letγbe a non-degenerate smooth loop, and faC1function on its range. Then the Fourier transform ˆµofµsatisfies: |ˆµ(ω)| ≤C(γ)(1+∝⌊a∇⌈⌊lω∝⌊a∇⌈⌊l)−1/d·∝⌊a∇⌈⌊lf∝⌊a∇⌈⌊lC1,∀ω∈Rd. We next deduce that for non-degenerate ... | https://arxiv.org/abs/2503.20193v1 |
NPMLE Proposition 2.1implies that for any π0∈Πk, having/hatwideπ∈Bδ(π0)requires the vector Vπ0(X)/defines/parenleftbig Dπ0,X(y1),...,D π0,X(yk), D′ π0,X(y1),...,D′ π0,X(yk)/parenrightbig to be within distance C(n,L)δof the half-ones vector (1,...,1,0,...,0). Note that the pj-weighted aver- age of the first kcoordinates ... | https://arxiv.org/abs/2503.20193v1 |
e.g. Proposition 2.1) and show that the probability for L-bounded generic data to violate the claims is 0, which suffices by countable additivity. Weclaim that for/hatwideπ∈Πk,ε, and any y∗∈[−L,L]withmin1≤j≤k|y∗−yj| ≥2ε, there isC(n,L,ε,k) such that for δ≤δ∗(n,L,ε,k)small enough, the probability that both ( 2.1) and ( 2... | https://arxiv.org/abs/2503.20193v1 |
W1(/tildewideπε,/hatwideπ)≤O/parenleftBig LeL2/radicalbig ηk/λ/parenrightBig . The intuition is that conditions 1 and 2 imply /hatwideπcan be approximated by a distribution (denoted /tildewideπ∗ below) supported on supp(/tildewideπε)and achieving a similar value of likelihood, while conditio n (3.1) implies any distrib... | https://arxiv.org/abs/2503.20193v1 |
that D/hatwideπ,X(y) = 1 . Outside of c-neighborhoods of each support element of /hatwideπεthere can be no atoms of /hatwideπbecause by ( 1.20) withj= 0 we should have that D/hatwideπ,X(y)<1for alld(y,supp(/tildewideπ))≥c. 3.2 Approximability of /hatwideπ Here we synthesize the preceding results to obtain an approx ima... | https://arxiv.org/abs/2503.20193v1 |
Together these convergence statements yield ( 3.5). Proposition 1.8ensures that uniformly in the choice of Zε, we have limε→0W1(/hatwideπ,/hatwideπε) = 0 . By Lemma 1.19, this implies convergence of D/hatwideπε,XtoD/hatwideπ,Xin the space C2([−L,L]). The other two state- ments of the proposition now follow from the con... | https://arxiv.org/abs/2503.20193v1 |
3.6that, as claimed, D/hatwideπε,X(y)≤D/hatwideπε,X(/hatwidey)−AX(c1−2ε)2 4≤1−/parenleftbiggAX(c1−2ε)2 4−δ/parenrightbigg = 1−c2. 22 In particular for εsmall enough, we can set c1=3/radicalbig 10Lδ/AX, c2=3/radicalbig L2δ2AX=⇒η=O(3/radicalbig Lδ/AX). Further, Proposition 3.2ensures that ( 3.2) holds for εsmall enough, ... | https://arxiv.org/abs/2503.20193v1 |
that the function (X,π)∝ma√sto→ℓX(π)defined in ( 1.2) is jointly continuous for X,π supported inside the radius Rball. Therefore any subsequential limit of NPMLEs /hatwideπ(n)forXnmust be an NPMLE for µsp d,R. By Lemma 4.1this means /hatwideπ(n)d→µsp d,r. In particular their support sizes must grow to infinity, whi ch co... | https://arxiv.org/abs/2503.20193v1 |
E. Price. Tight bounds for learning a mixt ure of two Gaussians. In Proceedings of the forty-seventh annual ACM symposium on Theory of computing , pages 753–760, 2015. [Jag13] M. Jaggi. Revisiting Frank-Wolfe: Projection-fre e sparse convex optimization. In International confer- ence on machine learning , pages 427–435... | https://arxiv.org/abs/2503.20193v1 |
rithm. Linear Algebra and its Applications , 199:413–425, 1994. [MSS23] S. Mukherjee, B. Sen, and S. Sen. A Mean Field Approac h to Empirical Bayes Estimation in High- dimensional Linear Regression. arXiv preprint arXiv:2309.16843 , 2023. [MV10] A. Moitra and G. Valiant. Settling the polynomial lea rnability of mixture... | https://arxiv.org/abs/2503.20193v1 |
Statistics and Learning , 4(3):143–220, 2022. [XJ96] L. Xu and M. I. Jordan. On convergence properties of th e EM algorithm for Gaussian mixtures. Neural Computation , 8(1):129–151, 1996. [Zha09] C.-H. Zhang. Generalized maximum likelihood estim ation of normal mixture densities. Statistica Sinica , pages 1297–1318, 20... | https://arxiv.org/abs/2503.20193v1 |
estimate cannot hold with Tcon the left-hand side. Therefore we have an upper bound for cwhich yields: W1(/hatwideπ,/tildewideπε)≤W1(/hatwideπ,πt)+W1(πt,/tildewideπε)≤Lc+O(η)≤O/parenleftBig LeL2/radicalbigg ηk λ+η/parenrightBig . The former term dominates the latter when ( 3.2) holds, finishing the proof of the main bou... | https://arxiv.org/abs/2503.20193v1 |
proof below shows it is the only NPMLE. Spherical Symmetry of /hatwideπGiven the preceding discussion, we know that any NPMLE /hatwideπis supported on Sd,rfor some unique r. Recalling ( 4.1), note that without the logarithm, the quantity/integraltext Pπ(x) dµ(x)is constant over all such /hatwideπ. Therefore by concavit... | https://arxiv.org/abs/2503.20193v1 |
on Land fort≥ε−8, the probability measure ˘π(t)satisfies: D˘π(t),X(y)−1≥ −O/parenleftBigg e4L2 ε1/24√ t/parenrightBigg ,∀y∈supp(˘π(t)), D˘π(t),X(y)−1≤O/parenleftBigg e4L2 ε1/24√ t/parenrightBigg ,∀y∈Zε. 31 Proof. We apply [ Jag13 , Theorem 2] to the algorithm ( B.1). The conclusion is a bound g(π(t))≤10diam(P(Zε))2Lip(∇... | https://arxiv.org/abs/2503.20193v1 |
Forε≤ε0(X), the conditions of Proposition 3.1apply to˚πεwith: /tildewideδ=O(L2e4L2δ/AX) =O(ε2L4e8L2/AX), c1=4√ CLδ, c2=AX√ δ, η= 2c1, λε/definesλ(˚πε) =λ(/hatwideπ)±oε(1). 33 In particular, the estimator ˚πεobeys certifiable bounds of the form W1(/hatwideπ,˚πε)≤OX(ε1/4)as well as |supp(/hatwideπ)| ≥ |supp(˚πε)|. Proof. ... | https://arxiv.org/abs/2503.20193v1 |
to certify that all approximate local maxima of ℓXsupported on supp(˘πS,ε)are within O(λ)of˘πS,ε(using now the fact that ℓ has3bounded derivatives). Since we could also certify above tha t/hatwideπhas at most εmass outside supp(˘πS,ε), 35 these certificates thus combine to certify that W1(/hatwideπS,˘πS,ε)≤C(S,X)εalmost... | https://arxiv.org/abs/2503.20193v1 |
we say such Xis k-good; this condition implies /hatwideπ∈Πk. We claim that conditioned on (xm+1,...,x n), the conditional law of/hatwideπdoes not admit a density on Πk. The proof is motivated by Remark 2.5. The main remaining step is the smooth dependence of /hatwideπon(x1,...,x k). To show it, we rely on the following... | https://arxiv.org/abs/2503.20193v1 |
any non-zero entry interacting with M′sinceMis already positive semi-definite. But then we are reduced to the strict positive definiteness of −Jkshown above! Lemma C.6. Suppose/hatwideπis the NPMLE for Xand satisfies min y∈supp(/hatwideπ)D′′ /hatwideπ,X(y)<0. Suppose further thatx1=···=xm. Forε >0andZ= (z1,...,zm)∈[0,1]m,... | https://arxiv.org/abs/2503.20193v1 |
Πk-neighborhood of ˚πεwhich certifiably contains /hatwideπ, forεsmall enough. We will apply Proposition D.1to suchπ= ˚πεand conclude the desired result. Indeed we can take C=C(L)as mentioned just above, and β= 2/c. Neither depends on ε, and so we can take α≤ε1/5 for small enough ε. Thenh≤1/10<1andr≤2α≤2ε1/5is smaller th... | https://arxiv.org/abs/2503.20193v1 |
matrix/bracketleftbiggpi/n p ipr/n 1/n p r/n/bracketrightbigg is similar to/bracketleftBigg pi/n√pipr n √pipr npr/n/bracketrightBigg via conjugation by M= diag(√p1,...,√pk,1,...,1); in particular their eigenvalues are equivalent. This conjuga tion extends to the preceding display, and shows I2k−JF+diag(0,...,0,D′′(y1),... | https://arxiv.org/abs/2503.20193v1 |
Estimation and variable selection in nonlinear mixed-effects models. Antoine Caillebotte1,2& Estelle Kuhn2& Sarah Lemler3 1Universit´ e Paris-Saclay, INRAE, UMR GQE-Moulon, France, caillebotte.antoine@inrae.fr, 2Universit´ e Paris-Saclay, INRAE, UR MaIAGE, France, estelle.kuhn@inrae.fr, 3Universit´ e Paris-Saclay, Cent... | https://arxiv.org/abs/2503.20401v1 |
variables of the environment as covariates acting directly on the plant development process. The parameters of these models are often physical quantities such as leaf appear- ance speeds or light interception capacities. These parameters may vary when considering a population of plants from different varieties each cha... | https://arxiv.org/abs/2503.20401v1 |
opening new possibilities for maximum likelihood inference. In order to achieve the objective of selecting variables from a set of high-dimensional variables in mixed effects models, several regularization approaches have been developed. J¨ urg Schelldorfer and B¨ uhlmann [2014] Schelldorfer et all proposed a maximum l... | https://arxiv.org/abs/2503.20401v1 |
a matrix of size q×pandβis a vector of sizep. The noise term εi,jis usually assumed centered Gaussian with unknown covariance matrix Σ. The unknown parameters of the nonlinear mixed-effects model are therefore θ= (α, β, Ω,Σ)∈Ra×Rp×S++ q×S++ dwhere S++ dstands for the set of symmetric positive definite matrices of size ... | https://arxiv.org/abs/2503.20401v1 |
the constrained model. 3.1 Estimation in Latent Variable Model We consider the maximum likelihood estimator to infer the Non-Linear Mixed-Effects model’s parameters. In the context of latent variable models, the marginal likelihood, denoted by g, is obtained by integrating the complete likelihood over the latent variab... | https://arxiv.org/abs/2503.20401v1 |
important to choose a well-balanced 5 value for the regularization parameter. We choose the best parameter by minimizing the extended Bayesian Information Criterion (eBIC) (see Chen and Chen [2008]). The eBIC adds a penalty to the high-dimensional model, whereas the BIC (see Schwarz [1978]) will tend to under-penalize ... | https://arxiv.org/abs/2503.20401v1 |
a computational point of view. Therefore we consider rather an adaptive vectorial learning rate. We also add a step to take into account the non dif- ferentiable penalization term. The main idea is to combine the PSG with a Proximal Forward-Backward algorithm (Chen and Rockafellar [1997]; Tseng [2000]). The algo- rithm... | https://arxiv.org/abs/2503.20401v1 |
use an adaptative preconditioning diagonal matrix and LASSO penalization. From now on will be denoted s=diag(A) for the sake of clarity. In this configuration the proximal operator has an explicit form : (Prox s,penλ(β))i= 0 if |βi|< λs i βi−λsiifβi≥λsi βi+λsiifβi≤ −λsi;∀i∈ {1, ..., p}. (9) Algorithm 1 provides in ... | https://arxiv.org/abs/2503.20401v1 |
the selection capacity of the method. Sensitivity (Se) measures the proportion of true positives (TP) correctly identified, while Specificity (Sp) quantifies the proportion of true negatives (TN) correctly identified. Accuracy (Ac) represents the overall proportion of correctly classified instances, including both TP a... | https://arxiv.org/abs/2503.20401v1 |
3.93 17.32 0.88 78.11 4.03 12.72 0.88 77.92 σ21.00 0.99 4.99 0.98 5.00 1.01 3.91 0.98 3.97 β18.00 8.14 7.33 8.13 7.45 8.08 5.35 7.98 5.45 β2-10.00 -9.98 6.05 -10.00 6.63 -9.99 3.91 -9.96 4.22 β320.00 19.98 3.30 19.97 3.42 19.98 1.91 19.97 2.14 10 Table 2: Average of sensitivity (Se), specificity (Sp), accuracy (Ac), me... | https://arxiv.org/abs/2503.20401v1 |
the convergence of the algorithm to the true parameter values used in the simulation. 11 Table 3: Average Relative Root Mean Square Errors (RRMSE) and parameter estimates over 100 repetitions for the Non Linear Mixed Effects Model (NLMEM) with N= 100. N = 100 P = 200 P = 500 P = 1000 θ∗ ˆθ RRMSE ˆθ RRMSE ˆθ RRMSE µ1200... | https://arxiv.org/abs/2503.20401v1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.