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in (1). Under the null hy- pothesis, all random effects are equal to p0. Under the alternative, the distance between the unobserved empirical distribution of random effects eπ(10) and the null distribution is a good proxy for W1(π, π 0), provided ϵis not too small. Lemma 5. Letδ∈(0,1), ifW1(π, δp0)≥ϵand there exists a ... | https://arxiv.org/abs/2504.13977v1 |
the mean (30) is easier than testing variance (31). This is reflected by equation (33), where the fast testing rates near the boundaries, i.e. µp0≲t−1n−1/2, are achieved by testing for deviations in the mean (30), and the slow testing rates away from the boundaries are achieved by testing for deviations in the variance... | https://arxiv.org/abs/2504.13977v1 |
p0values, while (37) captures the√p0dependence for null hypotheses that are close to 0 .5. Figure 7 shows the empirical local critical separation when considering all families of dis- tributions, approximating the local critical separation in (33). When considering all three kinds of mixing distributions, all tests hav... | https://arxiv.org/abs/2504.13977v1 |
we develop a test by debiasing the plug-in estimator of V(π). Although minimax optimal, this test is overly conservative, leaving room to improve type I error control. Section 5.2 introduces a debiased Cochran’s chi-squared test that achieves better type I control while preserving optimality. 5.1 A conservative test Th... | https://arxiv.org/abs/2504.13977v1 |
Furthermore, the proof suggests that unbiased estimation of µm1(π)is not crucial; instead, estimating max ( m1(π),1−m1(π)) is sufficient since one of the terms must behave like a constant. Consequently, bµ(Y) can be replaced by µbm1(Y)in (48). Examining the variance of the statistic under a point in Sversus the maximum... | https://arxiv.org/abs/2504.13977v1 |
Threshold Cochran's asymptotic threshold Max. quantile debiased Cochran's 2 (I) Max. quantile debiased Cochran's 2 (II) Max. quantile mod. Cochran's 2 Figure 9: 0.95% quantile as a function of the null hypothesis distribution for the studied statistics. Additionally, the asymptotic threshold used by Cochran’s χ2test (5... | https://arxiv.org/abs/2504.13977v1 |
the proportion of patients that suffered a myocardial infarction and died from cardiovascular causes due to the application of the rosiglitazone treatment for the treatment and control groups. Following Park [2019], we test the homogeneity of myocardial infarction and death propor- tions across studies in both treatmen... | https://arxiv.org/abs/2504.13977v1 |
of moments (MOM) to estimate the mixing distribution, while Vinayak et al. [2019] applied maximum likelihood estimation (MLE). Vinayak et al. [2019] noted that the MLE and MOM estimators look qualitatively different, although they match their first 8 moments. Here, we extend their analysis by applying the goodness-of-f... | https://arxiv.org/abs/2504.13977v1 |
demonstrate that combining a test based on the plug-in and debiased Pearson’s χ2tests [Balakrishnan and Wasserman, 2019] is minimax optimal. The critical sep- aration rates match previously established estimation rates [Tian et al., 2017, Vinayak et al., 2019], indicating that testing is as hard as estimation, a common... | https://arxiv.org/abs/2504.13977v1 |
the critical separation . . . . . . . . . . . . . . . . . . . . . 52 H Local minimax critical separation with a reference effect 56 H.1 Upper bounds on the critical separation . . . . . . . . . . . . . . . . . . . . . 59 H.2 Lower bound for random effects . . . . . . . . . . . . . . . . . . . . . . . . . 62 H.3 Lower b... | https://arxiv.org/abs/2504.13977v1 |
k. Letpkbe the polynomial that achieves the infimum; it can be expressed in the shifted Chebyshev basis pk(x) =kX m=0am(f)·Tm(x) where ∥a∥2 2≤1 . (53) Furthermore, since k≤t,pkcan be expressed in the Bernstein basis of degree t. Vinayak et al. [2019] Showed that the Chebyshev basis can be rewrriten as Tm(x) =tX j=0Ct,m... | https://arxiv.org/abs/2504.13977v1 |
p·(l)! for 0 < p < 1 and l∈ {0, . . . , t}. C.1 Bounding the total variation distance by moment differences Lemma 12 (Extension of lemma 3) .Letπ0andπ1be two mixing distributions supported on the [0,1]interval. For p∈(0,1), it follows that p pt∧(1−p)t·Mp(π1, π0) 2≤V(Pπ1, Pπ0)≤Mp(π1, π0) 2. Proof of lemma 12. We prove t... | https://arxiv.org/abs/2504.13977v1 |
≤δalmost surely for u∼π0where δ≤log(2−C)·p/tfor some C∈(0,1]. Then, for any mixing distribution π1, it holds that χ2(Pπ0, Pπ1)≤C−1·Mp(π1, π0). Proof of corollary 3. By proposition 3, it holds that Bj,t(u) Bj,t(p)≥2− 1 +|u−p| pt ≥2−exptδ p ≥C The statement follows by lemma 13. C.3 Expectations and variances of 1st a... | https://arxiv.org/abs/2504.13977v1 |
. Finally, J(π)satisfies J(π)≤J(π0) +√ 3·p W1(π, π 0). Proof of lemma 15. By theorem 3.2 of Bobkov and Ledoux [2019],we have that EπW1(eπ, π)≤J(π)√n Due to the triangle inequality in the space L2 VπW1(eπ, π)≤Eπ[W1(eπ, π)]2≤ EπW2 1(eπ, π) ≤J(π)√n2 43 Alternatively, the generalized Minkowski inequality can be used to... | https://arxiv.org/abs/2504.13977v1 |
+ Eπ0∼Γ0Pn π0(ψ(X) = 1) −α. By theorem 2.2 of Tsybakov [2009], it follows that R∗(ϵ)≥1−V Eπ0∼Γ0Pn π0, Pn π1 −α. Finally, by the assumption V Eπ0∼Γ0Pn π0, Pn π1 < C α, it follows that R∗(ϵ)≥1−Cα−α > β . Alternatively, the same claim follows whenever χ2 Eπ0∼Γ0Pn π0, Pn π1 < C2 αsince V Eπ0∼Γ0Pn π0, Pn π1 ≤q χ2 E... | https://arxiv.org/abs/2504.13977v1 |
the type II error by βwhenever ϵ(n, t, π 0)≥C·" J(π0)√n+r Ep∼π0[p(1−p)] t+1 n+1 t# . Proof of theorem 1. LetT(X) =W1(bπ, π 0). By lemma 16, the test control the type II error byβif EPπ[T]−EPπ0[T]≥s VPπ[T] β+r VPπ0[T] α. (66) Since the standard deviation of the statistic is dominated by its expectation p VPπ[T]≤q EPπ[W2... | https://arxiv.org/abs/2504.13977v1 |
separation (6)is lower-bounded by ϵ∗(n, t)≳ 1 tfort≲logn 1√tlognforlogn≲t≲n logn 1√nfort≳n logn(25) Proof of theorem 3. The statement is equivalent to proving that there exists mixing distri- butions π1andπ0inDthat satisfy W1(π1, π0)≳ 1 tfort≲logn 1√tlognfor log n≲t≲n logn 1√nfor... | https://arxiv.org/abs/2504.13977v1 |
lemma 19. By Chebyshev inequality, it follows that |W1(eπ, π 0)−W1(π, π 0)| ≤W1(eπ, π) ≤EπW1(eπ, π) +|W1(eπ, π)−EW 1(eπ, π)| ≤EπW1(eπ, π) + (1 + δ−1/2)p VπW1(eπ, π) with prob. 1 −δ By theorem 3.2 of Bobkov and Ledoux [2019],we have that EπW1(eπ, π)≤J(π)√n and VπW1(eπ, π)≤Eπ[W1(eπ, π)]2≤ EπW2 1(eπ, π) ≤J(π)√n2 Conse... | https://arxiv.org/abs/2504.13977v1 |
Note that s VPπ[T1] β+r VPπ0[T1] α≤C n1/2· 2p0 t+1 td1+ (1−1 t)·d2−d2 11/2 , where C=2√ 2h 1 β∨1 αi1/2 , is implied by d1≥C1/2·" 21/2·p1/2 0 n1/2t1/2∨1 nt∨d1/2 2 n1/2# . (73) Recalling that d1≥ϵ≥2p0, we obtain the following simple upper-bound d2≤Z |p−p0|dπ(p)≤m1(π) +p0=d1+ 2p0≤2d1. Combining it with (73), we get that... | https://arxiv.org/abs/2504.13977v1 |
mixing distributions Pπ0|A=Pπ0andPπ|A=Pπ0. Consequently, by lemma 22, the local minimax risk is lower-bounded by a constant R∗(ϵ, π0)≥πn(A)−πn 0(A)·V(Pπ0, Pπ0)−α= (1−Cα)n−α > β . Since the distance between πandπ0satisfies W1(π1, π0)≥Cα 2n, the statement of the lemma follows. H.3 Lower bounds for fixed effects H.3.1 Sma... | https://arxiv.org/abs/2504.13977v1 |
that ϵ∗(n, t, π 0)≳p1/2 0 t1/2n1/4for1 n1/2t≲p0. I Minimax critical separation for homogeneity testing without a reference effect I.1 Upper bounds on the critical separation I.1.1 Conservative test Lemma 9. For testing problem (42), the test (45) controls type I error by α. Furthermore, there exists a universal positiv... | https://arxiv.org/abs/2504.13977v1 |
1 t22t−1 t−1+ 4d2 (t−1)t+ 8d3 tt−2 t−1(1−2m1) + 4m4 1 t22t−1 t−1+ 8m1d2 (t−1)t2(1 + 2( t−3)t) −4d4 (t−1)t(2t−3)−4d2 2 t2(2t−1) −8m2 1d2 (t−1)t2(1 + 2( t−3)t)−8m3 1 t2(t−1)(2t−1) ≲m2 1 t2+d2 t2+d3 t+m4 1 t2+m1d2 tsince m1≤1 2andt≥3 ≲m2 1 t2+d2 t2+d3 t+m1d2 tsince m1≤1 (84) 69 Regarding B, we have that B=Ep,qiid∼π(p−q)4−... | https://arxiv.org/abs/2504.13977v1 |
Therefore, analogously to (83), it follows that VPπ[bµ]≤1 2n·VX,Yiid∼Pπh eh(X, Y)i We split the analysis of the variance in two terms VPπ[h(X1, X2)] =A+B where A=Ep,qiid∼πVX1|p∼Bin(t,p),X2|q∼Bin(t,p)h eh(X1, X2)i B=Vp,qiid∼πEX1|p∼Bin(t,p),X2|q∼Bin(t,p)h eh(X1, X2)i By direct computation, the first term can be bounded b... | https://arxiv.org/abs/2504.13977v1 |
if we choose γsuch that γ=r n+ 1 2n2·log 2 +logC2 α 2n2 we control the χ2distance between the distributions by an arbitrary constant χ2 Ep∼Γ0h Pn δpi , Pn π1 ≤C2 αfor Consequently, by lemma 17, it follows that R∗(ϵ)> β forϵ≥W1(π1, S) Note that W1(π1, S) = min p0∈[0,1]W1(π, δp0) = min p0∈[0,1]1 2+γ−2γp0=1 2−γ≥γ 2forγ ... | https://arxiv.org/abs/2504.13977v1 |
testing with simple null In the following, let π0=δp0. 79 ℓ2test The following test statistic is used T=1 nnX i=1Xi ti−p02 The corresponding test is ψ=I(T≥qα(Pπ0, T)) Modified Pearson’s χ2test The following test statistic is usually employed T=1 nnX i=1ti· Xi ti−p02 µp0 To avoid numerical instabilities in the appli... | https://arxiv.org/abs/2504.13977v1 |
des acad´ emies royales, 1912. Sergey Bobkov and Michel Ledoux. One-dimensional empirical measures, order statistics, and Kantorovich transport distances. Memoirs of the American Mathematical Society , 261 (1259):0–0, 2019. S. P. Brooks, B. J. T. Morgan, M. S. Ridout, and S. E. Pack. Finite Mixture Models for Proportio... | https://arxiv.org/abs/2504.13977v1 |
for nonparametric alternatives. I.Math. Methods Statist , 2(2):85–114, 1993a. Yuri I. Ingster. Asymptotically minimax hypothesis testing for nonparametric alternatives. II.Math. Methods Statist , 2(2):85–114, 1993b. 84 Yuri I. Ingster. Asymptotically minimax hypothesis testing for nonparametric alternatives. III.Math. ... | https://arxiv.org/abs/2504.13977v1 |
in Applied Mathematics , 3(4):608–622, 2003. Filippo Santambrogio. Optimal Transport for Applied Mathematicians: Calculus of Varia- tions, PDEs, and Modeling , volume 87. Springer International Publishing, 2015. Lars Snipen, Trygve Almøy, and David W Ussery. Microbial comparative pan-genomics using binomial mixture mod... | https://arxiv.org/abs/2504.13977v1 |
NONPARAMETRIC ESTIMATION IN UNIFORM DECONVOLUTION AND INTERV AL CENSORING BYPIETGROENEBOOMaAND GEURT JONGBLOEDb Delft Institute of Applied Mathematics, Mekelweg 4, 2628 CD Delft, The Netherlands. aP .Groeneboom@tudelft.nl;bG.Jongbloed@tudelft.nl In the uniform deconvolution problem one is interested in estimating the d... | https://arxiv.org/abs/2504.14555v2 |
general model is studied in [5] and [6]. There the length of the support of the uni- form random variable V, sayE, is actually random (but observable). This model is used for estimating the distribution of the incubation time of a disease. In this interpretation there is a positive random variable E, the duration of th... | https://arxiv.org/abs/2504.14555v2 |
this a random vari- ableYwith a distribution function G. Instead of observing X, the pair (Y,∆)is observed, where ∆indicates whether Xis smaller than or equal to Y. So, ∆ = 1 {X≤Y}. In survival UNIFORM DECONVOLUTION 3 analysis terminology, Xis the event time, Yis the inspection time and we observe the “sta- tus” of the... | https://arxiv.org/abs/2504.14555v2 |
regression problem’, the MLE ˆFnis known to coincide with the solution of the isotonic regression problem of minimizing the sum of squares nX i=1(∆i−F(Yi))2, over all distribution functions F. The one step algorithm for computing the solution, using the so-called cusum diagram of the ∆i’s is described, e.g., on p. 30 o... | https://arxiv.org/abs/2504.14555v2 |
the IC- 1model. Let us now turn to the nonparametric MLE. Define mn= max j:Sj>1(Sj−1). (2.6) Denoting by Qnthe empirical distribution of (S1,...,S n), the nonparametric MLE is defined as maximizer of the log likelihood function ℓn(F) =Z log{F(s)−F(s−1)}dQn(s) =Z s≤1logF(s)dQn(s) +Z 1<s≤mnlog{F(s)−F(s−1)}dQn(s) +Z s>1∨m... | https://arxiv.org/abs/2504.14555v2 |
deconvolution problem. UNIFORM DECONVOLUTION 7 3. The mixed model. The mixed uniform deconvolution model as described in the in- troduction (so with the random length Eof the support of the uniform noise V) is clearly a generalization of the what we from now will call the fixed uniform deconvolution model . The first r... | https://arxiv.org/abs/2504.14555v2 |
deconvolution function is intrinsically more complicated than based on a direct sample from the unknown distribution, reflected in a slower rate of estimation than the ‘parametric rate’√n, there are functionals of F0that can be estimated at rate√nin the uniform deconvolution model. For the functionals discussed in this... | https://arxiv.org/abs/2504.14555v2 |
0(x+i−1)−K′ h(t−(x+i−1)),i= 1...,m andm=⌈M⌉, where Mis the upper bound of the support of f0. We show that σ2 t, given by (4.4) can be simplified to σ2 t=F0(t){1−F0(t)}Z K′(u)2du. (4.5) 10 PIET GROENEBOOM AND GEURT JONGBLOED We have, for t∈(0,1):Z θ2 h,t,F 0(x){F0(x)−F0(x−1)}dx =Zt+h x=t−hK′ h(t−x)2{1−F0(x)}2F0(x)dx +Zt... | https://arxiv.org/abs/2504.14555v2 |
and asymptotic values is shown in Figure 2, where we take n= 10,000, andF0the truncated standard exponential and the uniform distribution function on [0,2], respectively. IfM > 1, as in this example where M= 2, the values of the asymptotic variances are no longer of the form (5.1) in the fixed model. We shall now expla... | https://arxiv.org/abs/2504.14555v2 |
of F0. Then, for t0∈(a,b): n1/3{ˆFn(t0)−F0(t0)}/(4f0(t0)F0(t0)(1−F0(t0))})1/3d−→argmint∈R W(t) +t2 ,(5.6) where Wis two-sided Brownian motion on R, originating from zero. So in this case we expect the MLE to have exactly the same limit behavior as in the case that the support of the distribution is contained in [0,1](... | https://arxiv.org/abs/2504.14555v2 |
The asymptotics in the mixed uniform deconvolution model (where the length of the sup- port of the uniform variable is random) was studied in [5] under a smoothness condition on the distribution of the interval length. In that setting there is no distinction depending on the support of the distribution corresponding to... | https://arxiv.org/abs/2504.14555v2 |
Neerl. 75 161–179. https://doi.org/10.1111/stan.12231 MR4245907 [5] G ROENEBOOM , P. (2024a). Nonparametric estimation of the incubation time distribution. Electron. J. Stat. 181917–1969. https://doi.org/10.1214/24-ejs2243 MR4736274 [6] G ROENEBOOM , P. (2024b). Estimation of the incubation time distribution in the sin... | https://arxiv.org/abs/2504.14555v2 |
arXiv:2504.14659v2 [eess.SP] 29 Apr 20251 Markovian Continuity of the MMSE Elad Domanovitz and Anatoly Khina Abstract —Minimum mean square error (MMSE) estimation is widely used in signal processing and related fields. While it is known to be non-continuous with respect to all standard notions of stochastic convergence,... | https://arxiv.org/abs/2504.14659v2 |
an empirical distribution resulting from a finite sample of length ndrawn from the distribution of (X,Y), or a finite-percision variant of (X,Y)with the machine percision increasing with n. Then, MMSE(Xn∣Yn) /⟶MMSE(X∣Y) in general, where MMSE(X∣Y)denotes the MMSE in esti- mating the random parameter Xfrom the measurement... | https://arxiv.org/abs/2504.14659v2 |
the MMSE is u.s.c. in distri- bution as long as the second moment of Xnconverges to that ofX(requirement 1 above). This requirement is weaker than the uniform boundness requirement of [22]. For the case of a common parameter X=Xnwith a finite second moment, and a sequence /b{aceleft.big1PYn∣X/b{ace{ight.big1∞ n=1of chan... | https://arxiv.org/abs/2504.14659v2 |
assume degrade dness between XiandXjfori≠j. where the infimum is over all RVs ˆXwith finite second moment that satisfy X⊸− −Y⊸− −ˆX. The following is a known characterization of the MMSE [3, Chapter 4], [30, Chapter 9.1.5], [7, Appendix for Chapter 3] which is often used as its definition. Theorem II.1. The MMSE estimate ... | https://arxiv.org/abs/2504.14659v2 |
is u.s.c. in distribution: lim n→∞MMSE(Xn∣Yn)≤MMSE(X∣Y). (3) When restricting the possible statistical relations, the f ol- lowing continuity results have been proved. Theorem II.3 ([22, Theorem 4]) .LetXandNbe a pair of RVs of the same length {(Xn,Yn)}∞ n=1be a sequence of pairs of RVs, such that •Y=X+N, andYn=Xn+Nfor... | https://arxiv.org/abs/2504.14659v2 |
meaning that lim n→∞MMSE(X∣Yn)=0=MMSE(X∣Y) by the squeeze theorem: 0≤lim n→∞MMSE(X∣Yn)≤lim n→∞∥X−Yn∥RV=0. However, since requirement 5 in Theorem II.4 does not hold for anyn∈N, this theorem cannot be applied for this case. The conditions of Theorem II.3 (recall Remark II.1) do not hold either since the uniform distribu... | https://arxiv.org/abs/2504.14659v2 |
of pairs of RVs such that (Xn,Yn)M.p.− −−−→ n→∞(X,Y), and lim n→∞E/b{acketleft.big2X2 n/b{acket{ight.big2=E/b{acketleft.big2X2/b{acket{ight.big2. (6) Then,E[Xn∣Yn]m.s.− −−−→ n→∞E[X∣Y]and the MMSE is Markov continuous in probability: lim n→∞MMSE(Xn∣Yn)=MMSE(X∣Y). Remark III.4.Condition (6) can be replaced by Xnm.s.− −−−... | https://arxiv.org/abs/2504.14659v2 |
which implies that requirement 5 of Theorem II.4, Xd⊸− −Yn+1d⊸− −Yn, is not necessary for the contiuity of the MMSE as long as requirement 5, Xd⊸− −Yd⊸− −Yn, continues to hold. Namely, the nested garbling requirement of Figure 1 ma y be replaced by the individual garbling requirement depicte d in Figure 2. Since we foc... | https://arxiv.org/abs/2504.14659v2 |
convergenc e of the second moment of {Yn}∞ n=1to that of Yin requirement 2 of Theorem V .2. Example V .2. LetXbe some random variable with zero mean and unit variance. Set Y=X,Xn=Xfor alln∈N, and Yn=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩/{adical.te}n,w.p.1 2n −/{adical.te}n,w.p.1 2n X,w.p.1−1 n. Clearly, requirements 1 and 3 of Theorem V .2 hold... | https://arxiv.org/abs/2504.14659v2 |
continuous at x. We denote this convergence by Xnd− −−−→ n→∞X. 2)Convergence in probability. {Xn}∞ n=1converges in prob- ability to Xif, for all ǫ>0,2 lim n→∞P(∥Xn−X∥>ǫ)=0. 2Other metrics between XnandXcan be used as well.We denote this convergence by Xnp− −−−→ n→∞X. 3)Almost-sure convergence. {Xn}∞ n=1converges almost... | https://arxiv.org/abs/2504.14659v2 |
n→∞E/b{acketleft.big2X2 n/b{acket{ight.big2=E/b{acketleft.big2X2/b{acket{ight.big2; •Xnp− −−−→ n→∞Xand/b{aceleft.big2X2 n/b{ace{ight.big2∞ n=1is u.i. Furthermore, if one of the above statements holds, then lim n→∞E[Xn]=E[X]. B. Stochastic Degradedness / Garbling The following notion of stochastic degradedness orgarblin... | https://arxiv.org/abs/2504.14659v2 |
space of compactly-supported continuous functions (an d hence also bounded) on Rk. Clearly g[i]∈L2/pa{enleft.big2Rk/pa{en{ight.big2for all i∈{1,2,...,k}since ∥g(Y)∥RV(a)=∥E[X∣Y]∥RV(b)≤∥X∥RV(c)<∞, where (a) holds by the definition of g(13a), (b) follows from (2b) in Theorem II.1 and the non-negativity of the MMSE, and (c... | https://arxiv.org/abs/2504.14659v2 |
follows from limit-superior arithmetics, and (26c) follow s from (25).APPENDIX E PROOFS OF COROLLARIES III.1 AND III.2 Proof of Corollary III.1: Let{γ∈R}∞ n=1and{λ∈R}∞ n=1 be some sequences that converge to zero. Denote Xn=X+ γnNandYn=Y+λnMforn∈N. Since∥X∥RV,∥N∥RV<∞, ∥Xn∥RV≤∥X∥RV+∣γn∣∥N∥RV<∞ ∀n∈N, where the inequality ... | https://arxiv.org/abs/2504.14659v2 |
new approach to linear filtering and predi ction problems,” Transactions of the ASME–Journal of Basic Engineering , vol. 82, no. 1, pp. 35–45, 1960. [5] N. Wiener, Extrapolation, interpolation, and smoothing of stationar y time series . The MIT press, 1964. [6] A. N. Kolmogorov, “Stationary sequences in Hilbert spac e,”... | https://arxiv.org/abs/2504.14659v2 |
ori and G. J. Sz´ ekely, “Four simple axioms of dep endence measures,” Metrika , vol. 82, no. 1, pp. 1–16, 2019. [25] S. Y¨ uksel and T. Bas ¸ar, Stochastic Teams, Games, and Control under Information Constraints . Springer, 2024. [26] I. Hogeboom-Burr and S. Y¨ uksel, “Continuity properti es of value functions in info... | https://arxiv.org/abs/2504.14659v2 |
arXiv:2504.15150v1 [stat.ME] 18 Apr 2025PREVALENCE ESTIMATION IN INFECTIOUS DISEASES WITH IMPERFECT TESTS : A C OMPARISON OF FREQUENTIST AND BAYESIAN LOGISTIC REGRESSION METHODS WITH MISCLASSIFICATION CORRECTION Jorge Mario Estrada Alvarez Caja de Compensacion Familiar de Risaralda Salud - Comfamiliar Pereira, PA 66003... | https://arxiv.org/abs/2504.15150v1 |
information on the misclassification rates (sensitivity and specificity) is available or can be reasona bly assumed. In that case, methods that explicitly incorpor ate this information should be employed to correct the estimate d prevalence and avoid bias toward the null value in the resulting estimates [1]. Logistic reg... | https://arxiv.org/abs/2504.15150v1 |
estimate the prevalence of low-prevalenc e sexually transmitted infections (STIs) while correcting for diagnostic misclassification. We implemented and compa red four logistic regression models that differ in their handling of misclassification and statistical inference fr ameworks. 2.1 Crude Prevalence Estimation Crude ... | https://arxiv.org/abs/2504.15150v1 |
22, 2025 Table 2: Descriptive characteristics of the study populati on (n = 11,452) Characteristic Age (mean [SD]) 34 (14) Male sex 5759 (50%) HIV reactivity 155 (1.4%) Syphilis reactivity 905 (7.9%) Hepatitis B reactivity 8 (<0.1%) Key population type MSM 3248 (28%) LGTBIQ 224 (2.0%) Other populations 193 (1.7%) Gener... | https://arxiv.org/abs/2504.15150v1 |
misclassification correction. Table 4: Comparison of regression coefficients for HIV accor ding to the implemented model Coefficient STD Liu Liu Change (%) BC BEC BEC Change (%) β0 -5.864 -4.648 -20.73 -5.418 -4.717 -12.93 β1 -0.001 0.002 -325.76 -0.001 -0.015 2798.64 β2 0.991 0.356 -64.09 0.749 0.512 -31.68 β3 1.946 1.49... | https://arxiv.org/abs/2504.15150v1 |
the BC model. Another relevant finding is observed in the coefficient for ag e group (β1), which changed from -0.001 in the standard model to -0.015 in BEC, representing a relative increase of a pproximately 3 times compared to its value in the BC model. Although its magnitude remains small, this change re flects the sens... | https://arxiv.org/abs/2504.15150v1 |
β6 0.193 0.285 -0.544 β7 0.191 0.230 -0.311 β8 0.168 0.164 0.047 The Liu model showed a systematic reduction in most coefficie nts, with the most notable decreases observed for male sex (β2), with a change of -74.15%, and the sex worker category ( β8), with a decrease of 51.43%. These results may reflect an overestimatio... | https://arxiv.org/abs/2504.15150v1 |
Fine [14] point out that sensitivity—traditionally c onsidered an intrinsic property of the test—is a significant factor influencing prevalence estimates due to factors such as changes in the severity spectrum of cases or variations in the clinical context of diagnosis. Consequently, adjust ing for sensitivity and specifi... | https://arxiv.org/abs/2504.15150v1 |
22, 2025 to misclassification error, which is especially relevant in population-based studies and epidemiological surveillan ce systems. The results revealed substantial differences in adjusted p revalence estimates and regression coefficients depending on the approach used. In particular, the Bayesian model with mi scla... | https://arxiv.org/abs/2504.15150v1 |
contexts where cla ssical estimation yields unstable coefficients and extensi ve standard errors. However, the use of non-informative priors also holds pract ical applicability, as demonstrated by Gordóvil-Merino et al. [17] through a simulation study. They employed weakly informative priors, providing excellent stabili... | https://arxiv.org/abs/2504.15150v1 |
R interface to Stan, 2 025. R package version 2.32.7. [13] Ben Goodrich, Jonah Gabry, Imad Ali, and Sam Brilleman. rstanarm: Bayesian applied regression modeling via Stan., 2024. R package version 2.32.1. [14] Jialiang Li and Jason P. Fine. Assessing the dependence of sensitivity and specificity on prevalence in meta- a... | https://arxiv.org/abs/2504.15150v1 |
arXiv:2504.15186v1 [math.ST] 21 Apr 2025Sum of Independent XGamma Distributions Therrar Kadria,b, Rahil Omairib, Khaled Smailicand Seifedine Kadryd aDepartment of Science, Northwestern Polytechnic, Grande Prair e, Canada; bDepartment of Art and Science, Lebanese International Univers ity, Lebanon cDepartment of Applied... | https://arxiv.org/abs/2504.15186v1 |
called the X Gamma distribution. They list some of its statistical parameters and clarify an a pplication that this model is provides an adequate fit for the data set more than the Expon ential distribution. For this reason, they consider this new finite mixture and fou nd that the XGamma model provides better fit to the ... | https://arxiv.org/abs/2504.15186v1 |
XGamma distributions denoted asSn∼HypoXG(− →θ). Then the PDF of Snis given as fSn(t) =n/summationtext i=13/summationtext k=1RikfYik(t) (6) with Rik=Aik θ4−k in/producttext l=1θ2 l (1+θl), (7) andYik∼Erl(4−k,θi) 3 Furthermore, Ai1=θin/producttext j=1,j/negationslash=i/parenleftbigg(θj−θi)2+θj (θj−θi)3/parenrightbigg Ai2... | https://arxiv.org/abs/2504.15186v1 |
lim t→+∞FSn(t) = lim t→+∞n/summationtext i=13/summationtext k=1RikFYik(t) =n/summationtext i=13/summationtext k=1Riklim t→+∞FYik(t).Hence,n/summationtext i=13/summationtext k=1Rik= 1. Theorem 2.4. LetSn∼HypoXG(− →θ). Then we have ΦSn(t) =n/summationtext i=13/summationtext k=1RikΦYik(t) (18) Proof.ReferringtoTheorem2 .1... | https://arxiv.org/abs/2504.15186v1 |
∂ ∂θp/parenleftbiggN/summationtext u=1logK/parenrightbigg =N(2+θp) θp(1+θp). Moreover, ∂ ∂θp/parenleftbiggN/summationtext u=1log/bracketleftbiggn/summationtext i=1e−θitu/parenleftbiggAi,1t2 u 2+Ai,2tu+Ai,3/parenrightbigg/bracketrightbigg/parenrightbigg =N/summationtext u=1n/summationtext i=1∂ ∂θp/parenleftBig e−θitu/pa... | https://arxiv.org/abs/2504.15186v1 |
on the histogram of the data sets and shown in the following Figure: Figure 1. Fitted pdfs and cdfs to the Survival Times for BallBearings d ata set. References Abdelkader, Y. H. (2003). Erlang distributed activity times in stoch astic activity networks. Kybernetika ,39(3), 347–358. Amari, S. V. & Misra, R. B. (1997). ... | https://arxiv.org/abs/2504.15186v1 |
ASTOCHASTIC METHOD TO ESTIMATE A ZERO -INFLATED TWO -PART MIXED MODEL FOR HUMAN MICROBIOME DATA John Barrera∗ Instituto de Ingeniería Matemática Facultad de Ingeniería Universidad de Valparaíso Valparaíso, Chile Cristian Meza CIMFA V Universidad de Valparaíso Valparaíso, Chile Ana Arribas-Gil Departamento de Estadístic... | https://arxiv.org/abs/2504.15411v1 |
2025 Stochastic estimation of a zero-inflated mixed model for microbiome data Regarding data analysis, Kodikara et al. (2022) compiled some recent models developed to study longitudinal micro- biome data from sample sequencing, considering those that model count data and those that model relative abundances. One of the... | https://arxiv.org/abs/2504.15411v1 |
data models for which the EM algorithm (Dempster et al., 1977) is not directly applicable because the complexity of the likelihood function does not allow for exact calculation of its conditional expectation. This would be the case of the ZIBR model. The SAEM algorithm not only preserves the good behaviour of the EM al... | https://arxiv.org/abs/2504.15411v1 |
and 2, it can be seen that the ZIBR model explicitly includes a component that is responsible for the presence of zeros in the data. It is also clear that conveniently defined covariates XitandZitcan influence both the probability of presence or absence of a bacterial taxon (through the logistic regression that defines... | https://arxiv.org/abs/2504.15411v1 |
|y, θ(q−1)/bracketrightbig : sq(θ) =sq−1(θ) +γq/parenleftig logp/parenleftig y,φ(q);θ/parenrightig −sq−1(θ)/parenrightig where {γq}q∈Nis a decreasing sequence of stepsizes with γ1= 1. •Maximization (M) step: Update θ(q)according to θ(q)= arg maxθsq(θ). There are some important remarks on the the working details of ... | https://arxiv.org/abs/2504.15411v1 |
ZIBR model cannot be considered part of the exponential family (Eggers, 2015). However, the decomposition presented in Equation 5 allows us to propose a simplified structure for the SAEM algorithm (Equation 4). For the multivariate normal part corresponding to the random effects, the actualization in the SA step is don... | https://arxiv.org/abs/2504.15411v1 |
of population parameter estimates, that is LLy(ˆθ) = log p(y;ˆθ) where p(y;ˆθ) =L(ˆθ;y)is the joint probability distribution function of the observed data given ˆθ. Notice that LLy(ˆθ) = log p(y;ˆθ) =/summationtextN i=1logp(yi;ˆθ)and, for some proposal distribution ˜pφiabsolutely continuous with respect topφi, we have ... | https://arxiv.org/abs/2504.15411v1 |
SAEM on the unbalanced datasets is compared with the use of the GHQ algorithm on balanced datasets obtained from imputation, and with gamlss without imputation. Covariates significance analysis based on the LRT and the Wald test are also presented in Appendix B, Supplementary Materials. 3.1 Setup We use two different s... | https://arxiv.org/abs/2504.15411v1 |
the parameter controlling the overdispersion of the data, i.e., ϕ, is particularly challenging (as well as the estimation of σ2in Setting 2). According to Stasinopoulos et al. (2017), by defining ZIBR as a model with several random effects, GAMLSS estimates the variance components with a method prone to generating bias... | https://arxiv.org/abs/2504.15411v1 |
Ti= 3 -0.1720 0.2979 0.2429 -0.3206 0.6612 0.4347 -0.0142 0.2478 0.1957 Ti= 5 -0.1639 0.2779 0.2270 -0.4970 0.8065 0.5605 -0.0899 0.2425 0.1932 Ti= 10 -0.1699 0.2635 0.2162 -0.8565 1.0651 0.8674 -0.1457 0.2480 0.2018 ϕ 6.4 Ti= 3 0.1301 1.2251 0.9465 -0.3696 2.0104 1.4482 4.6834 5.1779 4.6834 Ti= 5 0.0895 0.8059 0.6283 ... | https://arxiv.org/abs/2504.15411v1 |
the previous section, given that the ZIBR model has more than one random effect, GAMLSS uses an estimation method that can cause a bias in the estimation of the variance parameters, and also does not allow the evaluation of the tests based on the calculation of the log-likelihood that we will use in the following secti... | https://arxiv.org/abs/2504.15411v1 |
0.03) compared to those of the Beta component of the abundance uit(FDR p-value 0.80), and by the Wald test (Table 9, Appendix D). These results show that at the 5% significance level the treatment is significant in the logistic part but not in the Beta part, proving that the definition of the ZIBR model and the combina... | https://arxiv.org/abs/2504.15411v1 |
abandoning the follow-up. This could be one of the reasons contributing to the high non-publication rate in many medical studies, which according to certain sources could be close to 50% (Chan et al., 2014). Therefore, developing analysis methods that can deal accurately with unbalanced data is of great interest. The d... | https://arxiv.org/abs/2504.15411v1 |
zero-inflated mixed model for microbiome data References Arribas-Gil, A., Bertin, K., Meza, C., and Rivoirard, V . (2014). LASSO-type estimators for semiparametric nonlinear mixed-effects models estimation. Statistics and Computing , 24(3):443–460. Baldelli, V ., Scaldaferri, F., Putignani, L., and Del Chierico, F. (20... | https://arxiv.org/abs/2504.15411v1 |
J., Wang, C., Blaser, M. J., and Li, H. (2022). Joint modeling of zero-inflated longitudinal proportions and time-to-event data with application to a gut microbiome study. Biometrics , 78(4):1686–1698. Jeyakumar, T., Beauchemin, N., and Gros, P. (2019). Impact of the microbiome on the human genome. Trends in Parasitolo... | https://arxiv.org/abs/2504.15411v1 |
, 6:57–65. Rigby, R. A. and Stasinopoulos, D. (2005). Generalized additive models for location, scale and shape, (with discussion). Applied Statistics , 54:507–554. Samson, A., Lavielle, M., and Mentré, F. (2007). The SAEM algorithm for group comparison tests in longitudinal data analysis based on non-linear mixed effe... | https://arxiv.org/abs/2504.15411v1 |
the three specifications is 8 (IQR: 7 to 9). Given the drop-out method we chose to simulate an unbalanced data situation, we can assume that we are in a case of MCAR (Missing Completely At Random) (Rubin, 1976). Finally, we will compare the performance of SAEM on unbalanced data with GHQ on interpolated data. For each ... | https://arxiv.org/abs/2504.15411v1 |
0.0531 β 0.5 N= 50 -0.0118 0.2283 0.1828 -0.0163 0.2171 0.1766 -0.0096 0.2131 0.1714 N= 100 0.0073 0.1578 0.1257 -0.0015 0.1497 0.1203 0.0070 0.1484 0.1177 N= 200 0.0042 0.1084 0.0864 -0.0054 0.1064 0.0842 0.0022 0.1012 0.0801 σ1 0.7 N= 50 -0.1083 0.2935 0.2164 0.1867 0.2706 0.2205 -0.0787 0.1859 0.1443 N= 100 -0.0288 ... | https://arxiv.org/abs/2504.15411v1 |
is performed to test the null hypothesis H0:α=β= 0. We will now analyze its type I error with the SAEM estimation method. As for parameter estimation, we are interested in the performance on both balanced and unbalanced data, and we will compare it with the results of the LRT based on the GHQ procedure, only in the bal... | https://arxiv.org/abs/2504.15411v1 |
ar(ˆβ) where V ar(ˆα) and V ar(ˆβ) are estimated by the procedure described in Section 2.3 of the main document. Under the null hypothesis, these variables follow an asymptotic χ2distribution with one degree of freedom. We keep the simulation settings used for the LRT. To improve the convergence properties of the SAEM ... | https://arxiv.org/abs/2504.15411v1 |
analysis is performed on count data, we will analyse the data as proportions using the ZIBR model with SAEM estimation. Furthermore, a comparison of the results with the GHQ method can not be established since the number of time points is different between individuals; that is, the data is unbalanced. Table 5: Characte... | https://arxiv.org/abs/2504.15411v1 |
sanguinegens . The information in Table 6 also confirms these findings, showing that the coefficients associated with pregnancy for these taxa in the abundance part have different sign. In a previous work (Romero et al., 2014) it is found that bacteria of the genus Sneathia , potentially pathogenic, reduce their presen... | https://arxiv.org/abs/2504.15411v1 |
-3.1960 -4.3473 0.0634 -0.3158 0.1018 Coriobacteriaceae 0.7464 -3.6256 -5.1390 -0.6120 -0.2043 -0.2116 Veillonellaceae 1.3996 -3.8063 -6.2298 -0.2779 0.1731 -0.2848 Eggerthella 3.9614 -3.8869 -4.4172 0.2629 -1.0546 -1.0317 Lachnospiraceae 0.4803 -3.3161 -4.0073 -1.3028 -0.0050 -0.7322 Bacteroidales 0.8925 -2.6188 -5.75... | https://arxiv.org/abs/2504.15411v1 |
0.0999 Peptoniphilus asaccharolyticus -0.0138 -5.6375 1.3374 -6.6523 -0.5590 -0.1947 0.3948 0.0481 Peptoniphilus harei 0.7404 1.5115 1.1492 -6.6702 -0.3795 -0.0607 0.5226 -0.0205 Actinomyces 1.3715 -7.1737 0.7501 -6.2155 0.3464 -0.4793 0.1216 -0.0661 Sneathia 1.1231 0.1323 0.1920 -5.1472 -0.6467 -0.5812 -0.4221 0.4894 ... | https://arxiv.org/abs/2504.15411v1 |
based on the SAEM algorithm for the bacterial taxa of the IBD patients data. The p-values were corrected using the Benjamini-Hochberg process to decrease the false discovery rate. Species Baseline Time Treat Bacteroides 0.0000 0.1172 0.2133 Ruminococcus 0.0008 0.1203 0.0033 Faecalibacterium 0.0000 0.2645 0.0009 Bifidob... | https://arxiv.org/abs/2504.15411v1 |
arXiv:2504.15515v2 [math.ST] 23 Apr 2025TRANSPORT f-DIVERGENCES WUCHEN LI Abstract. We define a class of divergences to measure differences betwee n probability density functions in one-dimensional sample space. The con struction is based on the convex function with the Jacobi operator of mapping functio n that pushforwa... | https://arxiv.org/abs/2504.15515v2 |
there are joint studies between information divergences and optimal trans- port distances [7, 12, 19, 22]. On the one hand, [22] applies t he second-order derivatives of information divergences in Wasserstein-2 space to prove the first-order entropy power inequalities and their generalizations. On the other hand, the an... | https://arxiv.org/abs/2504.15515v2 |
ns. We call them transport f-divergences . We demonstrate several properties of transport f-divergences, including invariances, dualities, local behaviors and Taylor expans ions in Wasserstein-2 spaces. 3.1.Review of Wasserstein- 2distances. We first review the definition of optimal transport mapping functions in one-dim... | https://arxiv.org/abs/2504.15515v2 |
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