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and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6 Discussion 21 ∗Department of Mathematics and Statistics, Washington State University, Pullman, WA 99164, USA; E-mail: xiongzhi.chen@wsu.edu 1arXiv:2503.16590v1 [math.ST] 20 Mar 2025 A Type I location-shift families and closed or half-closed null... | https://arxiv.org/abs/2503.16590v1 |
. . . . . . . . . . . 47 C.2.3 Specialization to Gaussian family . . . . . . . . . . . . . . . . . . . . . . . 50 D Proofs Related to Construction II 51 D.1 Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 D.2 Proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2503.16590v1 |
the null parameter belongs. The motivations behind using such composite nulls, the important roles of these proportions in statistical modeling and inference, the current status of works on estimating these proportions, and the advantages of using Lebesgue-Stieltjes integral equations over existing methods to estimate ... | https://arxiv.org/abs/2503.16590v1 |
( ψ, K) converts proportion estimation into solving a specific Lebesgue-Stieltjes integral equation. When the difference em(t) = ˆφm(t,z)−φm(t,µ) (5) is small for large t, ˆφm(t,z) will accurately estimate π1,m. Since φm(t,µ) =π1,mor ˆφm(t,z) = π1,mrarely happens, ˆ φm(t,z) usually employs an increasing sequence {tm}m≥... | https://arxiv.org/abs/2503.16590v1 |
concentration inequalities for three complicated empirical processes that are induced by unbounded or non-Lipschitz functions, and these inequalities have independent interests; see Section B.3 in the supplementary material. As stated in Chen (2025), we have found similar things: (i) for estimating the proportion of fa... | https://arxiv.org/abs/2503.16590v1 |
such estimator to estimate “proportions” induced by a continuous function of a bounded null that is of bounded variation. We provide in Section 5 a simulation study and end the article with a discussion in Section 6. Some auxiliary results are provided in the appendix, and technical proofs are given in the supplementar... | https://arxiv.org/abs/2503.16590v1 |
e.g., Giraud and Peschanski (2014); Tuck (2006)), we will only provide in Section A.1 two methods to construct these families. As revealed by Jin (2008); Chen (2019) and will be seen later, the key appeal of a Type I location-shift family in constructing proportion estimators for both a one-sided null and a bounded nul... | https://arxiv.org/abs/2503.16590v1 |
ydyandD2(t, µ;b) =1 πZt 0sin{(µ−b)y} ydy and D1,∞(µ;a, b) = lim t→∞D1(t, µ;a, b) = 1 if a < µ < b 2−1ifµ=aorµ=b 0 if µ < a orµ > b, (7) and D2,∞(µ;b) = lim t→∞D2(t, µ;b) = 2−1ifµ > b 0 if µ=b −2−1ifµ < b. (8) The identities (7) and (8) are a consequence of Lemma 3 in the supplementary material. Letfµbe the dens... | https://arxiv.org/abs/2503.16590v1 |
bounded null, and requires additional innovation. Instead, finding K1for a one-sided null employs a different method, and K1is the real part of K† 1(t, x) =1 2πZ1 0dyZ1 −11 ιyd dsexp (ιtysx ) r0(tys) ds; see details in the proof of Theorem 4. If we set ψ(t, µ) = 2−1−ψ1(t, µ)−2−1ψ1,0(t, µ; 0), then limt→∞ψ(t, µ) = 1 (... | https://arxiv.org/abs/2503.16590v1 |
the estimator induced by K1,0(t, x;a) and K1,0(t, x;b) in (15) to consistently estimate the proportion of µj’s that are equal to aorb; see Theorem 3 below and Theorems 2 and 3 of Chen (2019). For the concept of uniform consistency, we adopt the following definition and interpretation from Chen (2025): Definition 3.1. G... | https://arxiv.org/abs/2503.16590v1 |
imposed on the average reciprocal modulus gvia Υ ( λ,˜τ) in (22). An example of (25) is Um= µ∈ B1,m(ρm) :ϑ= 3/4;q= 7/4;τm=√ 2−1σ−2lnm 0≤ϑ′<1/4;R1(ρm)≤˜Cmϑ′ π−1 1,m≤˜C√ lnm;um≥(τm)−1ln lnm (26) for a constant ˜C > 0, for which supµ∈Um π−1 1,msupt∈{τm}ˆφm(t,z)−1 ⇝0 asm→ ∞ . We remark that for the Gaussian fami... | https://arxiv.org/abs/2503.16590v1 |
speed of convergence of the corresponding φm(t,µ) toπ1,mdoes not depend on π1,mbut depends only on um,0viaτmum,0→ ∞ , and thus the sparsest π1,mcontained in ˜Qcan be of order arbitrarily close to (even though not equal to) m−0.5when C √2γlnm−1mγ−0.5π−1 1,m=o(1). For two Type I location-shift families F1andF2, their co... | https://arxiv.org/abs/2503.16590v1 |
ˆ φm(t,z) for the Gaussian family has a much simpler formulation. Theorem 5. Set˜um= min {j:µj̸=0}|µj|. Under the hypotheses of Theorem 4 and assumeR |x|2dFµ(x)<∞for each µ∈U. Suppose that {zi}m i=1are independent. Then V{em(t)} ≤4 π2mh ¯r2 0(t) +t2ˇr2 0(t)˜Dmi +∥ω∥2 ∞ 2mg2(t,0), (32) where ˜Dm=m−1Pm i=1 σ2 i+µ2 i ,σ... | https://arxiv.org/abs/2503.16590v1 |
to deal with three empirical processes simultaneously, one of which is induced by unbounded and non-Lipschitz functions (as can been from the function K1(t, x) in Theorem 4 with a ziin place ofxand the proof of Theorem 5). Firstly, the first five rows of conditions in Q(F) ensure that the concentration inequality (34) ... | https://arxiv.org/abs/2503.16590v1 |
of a one-sided null using the same techniques in Section 3.2, respectively. However, we will not pursue this here. Example 2. Laplace family Laplace µ,2σ2 with mean µand standard deviation√ 2σ >0for whichdFµ dν(x) =fµ(x) =1 2σexp −σ−1|x−µ| and the CF of fµisˆFµ(t) = 1 +σ2t2−1exp (ιtµ). So, r−1 µ(t) = 1+ σ2t2andˆF... | https://arxiv.org/abs/2503.16590v1 |
16 On the other hand, if ϕis continuous and of bounded variation on [a, b], then Dϕ,∞(µ;a, b) := lim t→∞Dϕ(t, µ;a, b) = ϕ(µ) ifa < µ < b 2−1ϕ(µ)ifµ=aorµ=b 0 ifµ < a orµ > b. (44) If in addition both W+ µandW− µare well-defined and of bounded variation for each fixed µ∈U, then |Dϕ(t, µ;a, b)− D ϕ,∞(µ;a, b)| ≤4Cµ(ϕ) ... | https://arxiv.org/abs/2503.16590v1 |
τmm−1/2, τm . If in addition condition C3) holds, then a uniform consistency class for either ˆφm(t,z)of ˇπ0,morˆφ1,m(t,z)of˜π0,mis the Q(F)in (23), with ˇπ0,mor˜π0,min place of π1,m, for which Q(F)in (23) takes for the form of (25) with an example given by (26), when Fis the Gaussian family with variance σ2>0, again ... | https://arxiv.org/abs/2503.16590v1 |
or explore here which density function ω(s) on [−1,1] should be used to give the best performances to the proposed estimators among all continuous densities on [ −1,1] that are of bounded variation. By default, we will choose the triangular density ω(s) = (1 − |s|) 1[−1,1](s), since numerical evidence in Jin (2008); Ch... | https://arxiv.org/abs/2503.16590v1 |
µi’s independently from U(b−4, b−um). In this setting, C−1π1,m≤˜π1,m≤Cπ1,mholds for some constant 19 C > 0 and t−1 m=o(˜π1,m) holds, ensuring the consistency of the proposed estimator ˆ φ1,m as per Theorem 7. Scenario 1 models the setting that when testing a bounded null in practice, it is unlikely that there is always... | https://arxiv.org/abs/2503.16590v1 |
discovery rate (FDR) control in nonasymptotic settings, an adaptive FDR procedure that uses the new estimators ˆψm(t,µ) may fail to maintain a prespecified nominal FDR, even though such a procedure may have larger power compared to its non-adaptive coun- terparts. We remark that the accuracy and speed of convergence of... | https://arxiv.org/abs/2503.16590v1 |
the corresponding proportion of false null hypotheses via solutions to Lebesgue-Stieltjes integral equations, for which consistency, speeds of convergence, and uniform consistency classes have been obtained under independence between these random variables. The strategy proposed in the Discussion section of Chen (2019)... | https://arxiv.org/abs/2503.16590v1 |
to nandp; see, e.g., Zhao and Yu (2006); Lv and Fan (2009); Zhang (2010); Su and Cand´ es (2016); Javanmard and Montanari (2018). However, there does not seem to exist a consistent estimator of or test on ϖwhen Xis not a diagonal matrix. If we set µ=Xβ= (µ1, . . . , µ n)⊺and assume Σ=σ2Ifor some σ2>0 where Iis the iden... | https://arxiv.org/abs/2503.16590v1 |
a real CF ˆF0must have zero expectation. So, Lemma 2 tells us that, if we pick a P´ olya-type CF h0that is nowhere 0 on Rand is Lebesgue integrable on R, then its Fourier inverse ˆh0generates a Type I location-shift family with members CDFs as Fµ(x) =Rx −∞ˆh0(y−µ)dy, µ∈R. A second method to construct a Type I location-... | https://arxiv.org/abs/2503.16590v1 |
of the closed null Θ 0= [a, b], (52) implies that (54) becomes em(t) = ˆφm(t,z)−φm(t,µ) =−e1,m(t)−2−1e1,0,m(t, a)−2−1e1,0,m(t, b) (57) However, (55) and (56) remain valid for em(t) in (57). When Θ 0= (a, b), φm(t,µ) = 1−φ1,m(t,µ) + 2−1φ1,0,m(t,µ;a) + 2−1φ1,0,m(t,µ;b) =5X i=1ed1,m (58) and π−1 1,mφm(t,µ)−1 =π−1 1,med1,m... | https://arxiv.org/abs/2503.16590v1 |
specifically ¯d2,m=m−1P {i:µi=0}1. Further, when Θ 0= (−∞,0), to upper bound π−1 1,mφm(t,µ)−1 , we have replaced each ¯dj,m ,3≤j≤4 by its upper bound and replaced π−1 1,m ¯d1,m+¯d2,m −1 by its upper bound in the inequality π−1 1,mφm(t,µ)−1 ≤ π−1 1,m ¯d1,m+¯d2,m −1 +π−1 1,m ¯d3,m +π−1 1,m ¯d4,m , where π−1 1,m ¯d1,m+¯... | https://arxiv.org/abs/2503.16590v1 |
[a, b], we have φm(t,µ) =m−1mX i=1h ψ1(t, µi) + 2−1n ϕ(a)ψ1,0(t, µ;a) +ϕ(b)˜ψ1,0(t, µ;b)oi and φm(t,µ) =dϕ,1(t,µ) +d∗ ϕ,2(t,µ) +d∗ ϕ,3(t,µ)−dϕ,4(t,µ) +dϕ,5(t,µ) 27 where d∗ ϕ,2(t,µ) =m−1P {i:µi=a} ψ1(t, µi) + 2−1ϕ(a)˜ψ1,0(t, µi;a) d∗ ϕ,3(t,µ) =m−1P {i:µi=b} ψ1(t, µi) + 2−1ϕ(b)˜ψ1,0(t, µi;b). Then ˇπ−1 0,mφm(t,... | https://arxiv.org/abs/2503.16590v1 |
0(1−s) cos ( tsx) exp 2−1t2s2 ds. Further, the estimator of π1,m=m−1Pm i=11U\(−∞,0)(µi) is ˆφm(t,z) =m−1mX i=1[1−K(t, zi)], where K(t, x) = 2−1−K1(t, x)−2−1K1,0(t, x; 0). Namely, ˆφm(t,z) = 2−1+1 πm−1mX i=1Z1 0dyZ1 0syt2sin (ytsz i) exp 2−1y2t2s2 ds +1 πm−1mX i=1Z1 0dyZ1 0txcos (tysz i) exp 2−1t2y2s2 ds +m−1mX i=... | https://arxiv.org/abs/2503.16590v1 |
normal maxima, Journal of Mathematical Analysis and Applications 422(1): 376–396. Giraud, B. G. and Peschanski, R. (2014). On the positivity of Fourier transforms, arXiv:1405.3155 . Hoang, A.-T. and Dickhaus, T. (2022a). On the usage of randomized p-values in the schweder- spjøtvoll estimator, Annals of the Institute o... | https://arxiv.org/abs/2503.16590v1 |
Section 5.1. When π1,m= 0.2, our proposed estimators “New” show a clear trend of convergence to 0 as mincreases. For π1,m= 1/ln (ln m) though, ˜δmfor our “New” estimators does not show a clear trend of convergence to 0 asmincreases. However, this is an artifact of the numerical error when implementing our “New” estimat... | https://arxiv.org/abs/2503.16590v1 |
100000 MR -0.4539 0.0021 One-sided null π1,m= 0.2 100000 New 0.2705 0.0884 One-sided null π1,m= 0.2 500000 HD -0.3362 0.0066 One-sided null π1,m= 0.2 500000 MR -0.4528 0.0009 One-sided null π1,m= 0.2 500000 New 0.2385 0.0785 One-sided null π1,m= 1/ln (ln m) 1000 HD -0.2543 0.0469 One-sided null π1,m= 1/ln (ln m) 1000 M... | https://arxiv.org/abs/2503.16590v1 |
exp{ιys(µ−b)}ds. Third, we quote the speed of convergence of the Riemann-Lebesgue lemma: Lemma 5. Let−∞< a1< b1<∞. Iff: [a1, b1]→Ris of bounded variation, then Z [a1,b1]f(s) cos ( ts)ds ≤4 (∥f∥TV+∥f∥∞)|t|−11{t̸=0}(t) and Z [a1,b1]f(s) sin ( ts)ds ≤4 (∥f∥TV+∥f∥∞)|t|−11{t̸=0}(t). B.2 Speeds of convergence of discriminant... | https://arxiv.org/abs/2503.16590v1 |
1κ2−R1(ρm)−2V X(0) . (76) 3. Case 3: Assume a0= 1, b0= 0 and[c2, c3] ={0}. Ifµ∈ B 2,m(ρm)and2−1∆−1 1κ2− 2−1R2(ρm)>0, then Prn supv∈[0,c1]Gm(v,0)≥κ1o ≤ˆp3,m(Ξ3), where Ξ3= (κ1, κ2, c1,∆1, F0, ρm, κ),R2(ρm) = 4Eh X2 (0)i + 4ρm, ˆp3,m(Ξ3) = 2 c1∆−1 1+ 1 exp −2−1mκ−2(κ1−κ2)2 +m−1 2−1∆−1 1κ2−2−1R2(ρm)−2Vh X2 (0)i ... | https://arxiv.org/abs/2503.16590v1 |
b0= 0,[c2, c3] ={0}and supv∈[0,c1]|Gm(v,0)|. Part II, Case 1: a0= 0, b0= 1, c2< c3and sup(y,v)∈D|Gm(v, y)|. In this case, fbecomes f(x, v, y, 0,1, c0) = cos ( v(x−y)−c0), and (81) and (82) respectively reduce to sup(v,y)∈D|∂vGm(v, y)| ≤2Cc2,c3+H1,m(z) sup(v,y)∈D|∂yGm(v, y)| ≤2c1, (83) where Hj,m(z) =1 mmX i=1h |zi|j+E... | https://arxiv.org/abs/2503.16590v1 |
<∞. •(90) and (93) reduce to B1,m= Pr sup1≤i≤l∗|Gm(vi, c2)| ≥κ1−κ2 ≤2l∗exp −2−1m(κ1−κ2)2 . 42 With these in mind, we see that sup v∈[0,c1]|Gm(v, c2)| ≤ sup 1≤i≤l∗|Gm(vi, c2)|+ ∆ 1[R1(ρm) +T1,m(x, F0)]. On the other hand, by Chebyshev’s inequality, ˜B2,m= Pr T1,m(x, F0)≥∆−1 1κ2−R1(ρm) ≤m−1V X(0) ∆−1 1κ2−R1(ρm... | https://arxiv.org/abs/2503.16590v1 |
m = 1− 1−2 Pr X(0)> κ− ∥µ∥∞ m= 1−Cm,µ,F0, where Cm,µ,F0= 1−2 Pr X(0)> κ− ∥µ∥∞ m. Thus, when µ∈ B2,m(ρm) and Eh X2 (0)i < ∞, we have B0,m= Prn supv∈[0,c1]Gm(v,0)≥κ1o ≤B1,m+ˆB2,m+ 1−Prn ˜Bκo ≤ˆp3,m(Ξ3), where Ξ 3= (κ1, κ2, c1,∆1, F0, ρm, κ) and ˆp3,m(Ξ3) = 2 c1∆−1 1+ 1 exp −2−1mκ−2(κ1−κ2)2 44 +m−1 2−1∆−1 1κ2−... | https://arxiv.org/abs/2503.16590v1 |
C.2.2 Uniform bound on error and consistency Second, let us show the uniform bound on em(t) = ˆφm(t,z)−φm(t,µ), for which we will use Lemma 9. Let {τm}m≥1,{γm}m≥1and{ρm}m≥1be three positive sequences to be determined later. In the rest of the proof, q, γandϑare positive constants and ϑ′is a non-negative constant whose ... | https://arxiv.org/abs/2503.16590v1 |
τm (125) forτ∈ {a, b}. Therefore, in view of ˆφm(t,z) = ˆφ1,m(t,z)−2−1X τ∈{a,b}ˆφ1,0,m(t,z;τ), (123) and (125), a union bound for probability implies that sup µ∈B1,m(ρm)sup t∈[0,τm]|em(t)| ≤n (b−a)τm 2π+∥ω∥∞o Υ γmm−1/2, τm (126) with probability at least 1 −2ˆp∗ 2,m−ˆp∗ 1,m. 48 Third, we derive conditions that ensur... | https://arxiv.org/abs/2503.16590v1 |
t∈[0,˜τ]exp 2−1t2sσ2 ds= 2Z1 0exp 2−1s˜τ2σ2 ds =4 exp 2−1˜τ2 −1 ˜τ2σ2≤4 exp 2−1˜τ2σ2 ˜τ2σ2. (135) This implies Υ (λ,˜τ)≤λ4 exp 2−1˜τ2σ2 ˜τ2σ2and Υ γmm−1/2, τm ≤4√2qlnm 2γlnm×σ2mσ2γ−1/2, (136) andτmπ−1 1,mΥ γmm−1/2, τm =o(1) if π−1 1,mmσ2γ−1/2=o(1). Namely, (126) becomes sup µ∈B1,m(ρm)sup t∈[0,√2γlnm]|em... | https://arxiv.org/abs/2503.16590v1 |
0dyZ1 −11 ιyd dsexp (ιysx) r0(ys)ds =ℜ1 2πZt 0dyZ1 −1exp (ιysx) ιy˜r′ 0(ys)ds +ℜ1 2πZt 0dyZ1 −1xexp (ιysx) r0(ys)ds =ℜ1 2πZt 0dyZ1 −1exp (ιysx) ιy˜r′ 0(ys)ds +1 2πZt 0dyZ1 −1xcos (ysx) r0(ys)ds =1 2πZt 0dyZ1 −1sin (ysx) y˜r′ 0(ys)ds+1 2πZt 0dyZ1 −1xcos (ysx) r0(ys)ds =1 2πZ1 0dyZ1 −1sin (ytsx) yd ds1 r0(tys) ... | https://arxiv.org/abs/2503.16590v1 |
1andϑ2are positive constants and ϑ′andϑ′′ are non-negative constants whose ranges will be determined later. Set D= [0, τm]×[0,1]×[0,1]. The continuity of ˜∆1,m,0(·,·) and that of ˜∆1,m,1(·,·) respectively imply ( sup(t,s,y)∈D˜∆1,m,0(ts, y) = supt∈[0,τm]˜∆1,m,0(t,1) sup(t,s,y)∈D˜∆1,m,1(ts, y) = supt∈[0,τm]˜∆1,m,1(t,1). ... | https://arxiv.org/abs/2503.16590v1 |
µ∈˜Bm(˜ρm)sup t∈[0,τm]Z1 0dyZ1 0|˜r0(tys)|sup (t,s,y)∈D ˜∆1,m,0(ts, y) ds +2 2πsup µ∈˜Bm(˜ρm)sup t∈[0,τm]Z1 0dyZ1 0t r0(tys)sup (t,s,y)∈D ˜∆1,m,1(ts, y) ds ≤Υ1 τmm−1/2, τm +τmΥ2 τmm−ϑ1, τm where we recall ˜ r0(tys) =y−1∂s{1/r0(tys)}and that r0(·) is a positive function, and have set Υ1(λ,˜τ) =λ πsupt∈[0,˜τ... | https://arxiv.org/abs/2503.16590v1 |
1,m=o(π1,m) Υ γmm−1/2, τm + Υ 1 γmm−1/2, τm supµ∈Ωm(˜ρm)π−1 1,m=o(1) τmΥ2 γmm−ϑ1, τm supµ∈Ωm(˜ρm)π−1 1,m=o(1) . D.2.3 Specialization to Gaussian family Finally, we specialize the above results to Gaussian family with scale parameter σ >0. There are three key steps to achieve this. Firs... | https://arxiv.org/abs/2503.16590v1 |
58 ϑ2≥1. Then p0(˜κ)→0, ˜κ→ ∞ and mln (1−2p0(˜κ)) =m(−2p0(˜κ) +o(p0(˜κ))) =−2×σ+o(1)√2σ2ϑ2lnmm1−ϑ2+o(1) = o(1). Namely, κ− ∥µ∥∞=√2σ2ϑ2lnmforϑ2≥1 forces Cm,µ,F0= 1 + o(1). In addition, by the remark right after (152), R2(˜ρm) = O mϑ′′ implies R1 √˜ρm = O mϑ′′/2 . So, ( (γmR2(˜ρm))−1mϑ−ϑ1lnτm→ ∞ when R2(˜ρm) =O mϑ′... | https://arxiv.org/abs/2503.16590v1 |
because the only difference between the construction in Theorem 3 and the construction in Theorem 7 (this Theorem) is that the integrand of the latter construction has a factor ϕ(y) compared to the former, so that the variance bounds (for em(t)) for the latter will differ by at most a factor of ∥ϕ∥2 ∞from those for the... | https://arxiv.org/abs/2503.16590v1 |
[0,1]ds r0(ts)forλ,˜τ >0. Third, we show the consistency of ˆ φm(t,z). Again the results of in the proof of Theorem 3 can be directly used here. In particular, since (127) there holds here, i.e., (2τm+R1(ρm))−1mϑ−1/2γ−1 mlnτm→ ∞ τ2 mm2ϑexp −2−1γ2 m +m−2ϑγ2 mln−2τm=o(1)=⇒ˆp∗ 2,m=o(1) = ˆ p∗ 1,m, 62 and (180) holds, ... | https://arxiv.org/abs/2503.16590v1 |
7/4;τm=√ 2−1σ−2lnm 0≤ϑ′<1/4;R1(ρm)≤˜Cmϑ′ ˇπ−1 0,m≤˜C√ lnm;um≥(τm)−1ln lnm , (187) then supµ∈Um ˇπ−1 0,msupt∈{τm}ˆφm(t,z)−1 ⇝0 as m→ ∞ .Note that Q(F) and Umare the same as those for the Gaussian family provided by Theorem 3. Finally, we show the claim about the estimator ˆ φ1,m(t,z) =m−1Pm i=1K1(t, zi) with K1in... | https://arxiv.org/abs/2503.16590v1 |
here Notations in proof Theorem 3 Chen (2019) ˆφ0,m(t,z) ˆφm(t,z) φ0,m(t,µ) φm(t,µ) g(t;µ0) a(t;µ0) Bm(ρm) Bm(ρ) X(µ0) X1 um,0 um Q(F) Qm(µ, t;F) Notations with the different definitions A0,m= Pr max y∈Gm|Sm(y)| ≥√2qγm√m B0= Pr sup µ∈Bm(ρ)max y∈Gm|Sm(y)| ≥√2qγm√m! A1,m= Pr max 1≤i≤l∗ d(0) m(yi) ≥˜p0,m,1 B1= Pr sup ... | https://arxiv.org/abs/2503.16590v1 |
and continuous in tonRfor each µ∈U. SinceR |x|2dFµ(x)<∞for each µ∈U, we know that for each fixed µ∈U, the CF ˆFµ(t), t∈Ris differentiable in t∈R, and hence a branch of hµ(t) that is differentiable in t∈Rcan be chosen. Therefore,d dyhµ0(y) is well-defined. Further, ∂ywi(y) =−(zi−∂yhµ0(y)) sin ( yzi−hµ0(y)), and ∂yE[wi(y... | https://arxiv.org/abs/2503.16590v1 |
µi;µ0) =Z [−1,1]ω(s) cos ( ts(µi−µ0))ds. We obtain supt∈[0,τm]ˆφ0,m(t,z) π1,m−1 = sup t∈[0,τm]ˆφ0,m(t,z)−φ0,m(t,µ) π1,m+φ0,m(t,µ) π1,m−1 ≤ sup t∈[0,τm]ˆφ0,m(t,z)−φ0,m(t,µ) π1,m + sup t∈[0,τm]φ0,m(t,µ) π1,m−1 and π−1 1,msupt∈[0,τm]ˆφ0,m(t,z)−1 ≤π−1 1,msup t∈[0,τm]|ˆφ0,m(t,z)−φ0,m(t,µ)| | {z } =ϵ0,m,1(τm,µ) + π−1... | https://arxiv.org/abs/2503.16590v1 |
Revenue Maximization Under Sequential Price Competition Via The Estimation Of s-Concave Demand Functions Daniele Bracale dbracale@umich.edu Department of Statistics University of Michigan Moulinath Banerjee moulib@umich.edu Department of Statistics University of Michigan Cong Shi congshi@bus.miami.edu University of Mia... | https://arxiv.org/abs/2503.16737v2 |
regret with respect to a dynamic benchmark. In our work, we extend the conventional multi-period pricing framework by incorporating a nonlinear demand model. Here, Nsellers simultaneously set prices over Tperiods, and each seller’s observed demand results from a complex interplay of their own pricing and that of their ... | https://arxiv.org/abs/2503.16737v2 |
contrast to prior studies (Li et al., 2024; Kachani et al., 2007; Gallego et al., 2006), we analyze the more general unknown monotone single index expected demand model λi(p) =ψi −βipi+/summationdisplay j∈N\{i}γijpj =ψi(−βipi+γ⊤ ip−i) =ψi(θ⊤ ip), fori∈Nwhere p−i≜(pj)j∈N\{i}andθiis a vector of dimension Nwithi-th en... | https://arxiv.org/abs/2503.16737v2 |
phase, it typically is not sufficient to have a convergence in L2 ofx∝⇕⊣√∫⊔≀→/hatwideψi(/hatwideθ⊤ ix)to derive an upper bound on the total regret, while a supremum norm converge is sufficient for this scope. For this reason, we propose to split the exploration phase into two sub-phases for which we separately learn θi... | https://arxiv.org/abs/2503.16737v2 |
ψ∈Fd,sif ψ((1−θ)u0+θu1)≥Ms(ψ(u0),ψ(u1) ;θ), for allu0,u1∈Rdandθ∈(0,1), where Ms(y0,y1;θ)≜ ((1−θ)ys 0+θys 1)1/s, s̸= 0,y0,y1>0 0, s< 0,y0=y1= 0 y1−θ 0yθ 1, s = 0. This notion generalizes log-concavity which holds for s= 0, in the sense that lims→0Ms(y0,y1;θ) = M0(y0,y1;θ). The class of log-concave densities has be... | https://arxiv.org/abs/2503.16737v2 |
reformulation allows the development of a fully data-driven, tuning-parameter-free algorithm using shape constraints. Regret upper bound and convergence to equilibrium. We establish an upper bound on the total expected regret and analyze the convergence to the Nash equilibrium (NE) for a general exploration length of τ... | https://arxiv.org/abs/2503.16737v2 |
ordered. The parameter vector γimeasures how selleri’s demand is affected by competitor prices. We assume that the parameter space is such that the average demand λiis non-negative and ∂piλi<0among all values of {θi,ψi}i∈N; a similar assumption is found in Birge et al. (2024); Li et al. (2024). The above conditions hol... | https://arxiv.org/abs/2503.16737v2 |
Proposition 4.3. For everyi∈N,φ′ i≥ciiffψiis(ci−1)-concave. For the uniqueness of the Nash equilibrium (i.e., the fixed-point equation in Equation (7)), a sufficient condition is to ensure that the Lipschitz constant of LΓis strictly less than 1inP. The following condition guarantees the uniqueness of the Nash equilibr... | https://arxiv.org/abs/2503.16737v2 |
made public . At the end of the exploration phase, firm iestimates (θi,ψi)using data{(p(t),y(t) i)}t≤τ. More precisely, each firm i chooses a proportion κiof initial data points in the exploration phase T(1) i={1,2,...,κiτ}to estimate θi, and subsequent time points T(2) i={κiτ+ 1,...,τ}to estimate ψi(a scheme in presen... | https://arxiv.org/abs/2503.16737v2 |
that adhere to the full exploration phase. Remark 5.2 (Need for two different phases for model estimation). In principle, one could estimate (θi,ψi)jointly using the full exploration phase. For instance, Balabdaoui et al. (2019) employ a profile least squares approach to achieve L2convergence for the joint LSE of (θi,ψ... | https://arxiv.org/abs/2503.16737v2 |
the midpoint of this interval with variance σ2 itaken to be an adequately small fraction of the length of Pi. If the sellers function independently, as is generally the case, the joint distribution is certainly elliptically symmetric. More generally, view the parameters Λ,m, andgas concentrating the mass of DinPwith ov... | https://arxiv.org/abs/2503.16737v2 |
setU, then{w(t) i}t∈[n]are “asymptotically dense” within each interval contained inU, which condition is sufficient to produce a uniform convergence result of our estimator of ψi,θon any interval contained in U. In our case, fwis supported on all R, and hence fwis bounded away from 0on any bounded set K, therefore the ... | https://arxiv.org/abs/2503.16737v2 |
=O(Tξ+T1−2ξ/5N1/2(logT)2/5). (2)Convergence to NE :E/bracketleftbig ∥p(T)−p∗∥2 2/bracketrightbig =O(N1/2T−2ξ/5(logT)2/5). More precisely, we have E/bracketleftig ∥p(t)−p∗∥2 2/bracketrightig ≲N1/2T−2ξ/5(logT)2/5+L2(t−τ−1) Γ, t≥τ+ 1. In Corollary 6.10, we provide the exploration-exploitation trade off for our Algorithm... | https://arxiv.org/abs/2503.16737v2 |
research direction is achieving uniform convergence of the joint estimator (θi,ψi). While this does not affect the theoretical convergence rate of the regret or the Nash 13 equilibrium – since it influences only a constant factor (given that θiis estimated using κiτdata points and ψiusing (1−κi)τ, whereτis the length o... | https://arxiv.org/abs/2503.16737v2 |
to interfere: Information revelation and price-setting incentives in a multiagent learning environment. Operations Research , 2024. SergeyBobkovandMokshayMadiman. Concentrationoftheinformationindatawithlog-concavedistributions. The Annals of Probability , 39(4):1528–1543, 2011. 14 GR Mohtashami Borzadaran and HA Mohtas... | https://arxiv.org/abs/2503.16737v2 |
Kang, and Robert Phillips. Price competition with the attraction demand model: Existence of unique equilibrium and its stability. Manufacturing & Service Operations Management , 8(4):359–375, 2006. Negin Golrezaei, Adel Javanmard, and Vahab Mirrokni. Dynamic incentive-aware learning: Robust pricing in contextual auctio... | https://arxiv.org/abs/2503.16737v2 |
Proposition A.1. Letψ: (a,b)→(0,∞), a twice differentiable function. Then ψiss-concave iff ψ·ψ′′+ (s−1)(ψ′)2≤0in(a,b). A.1 Proof of Proposition A.1 As fors= 0is a known result, we prove it for s̸= 0. A function ψiss-concave if and only if ds◦ψis concave, wheredsis defined in Equation (3). Then ψiss-concave if and only ... | https://arxiv.org/abs/2503.16737v2 |
that with probability at least 1−2e−cς2 minn/16we have (ςmin/2)I≼Σn, where Σn≜1 nn/summationdisplay t=1x(t)x(t)⊤. Proof.To get a lower bound of the minimum eigenvalue of Σnwe use the following remark 5.40 of Vershynin (2010): let Abe an×Nmatrix whose rows Aiare independent sub-gaussian random vectors in RNwith 20 secon... | https://arxiv.org/abs/2503.16737v2 |
λiθ⊤ ix(t)−y(t) i/parenrightig x(t) i/vextendsingle/vextendsingle/vextendsingle≥u/parenrightig ≤2 exp/parenleftbigg −1 2min/braceleftbiggu2 ν2 i,u νi/bracerightbigg/parenrightbigg . Now, using that Xt= 2(λiθ⊤ ix(t)−y(t) i)x(t) i∼SE(ν2 i,νi)are independent for all t, with E(Xt) = 0we get P/parenleftig/vextenddouble/v... | https://arxiv.org/abs/2503.16737v2 |
θ⊤Λθ/vextenddouble/vextenddouble/vextenddouble/vextenddouble 2 ≲∥θi−θ∥2∥α∥2+∥θi−θ∥2cmax∥θ∥2 cmin∥θ∥2 ≲∥θi−θ∥2(c+∥α∥2), for somec>0, where we used that ∥θi∥2=∥θ∥2= 1and thatu∈U, which is a bounded set. We then have |ψi,θ(u)−ψi(u)|≤/integraldisplay RN−1|θ⊤ imu+θ⊤ iAα−u|˜g(α)dα ≲∥θi−θ∥2/parenleftbigg c+/integraldisplay RN... | https://arxiv.org/abs/2503.16737v2 |
Γ ≲N1/2(logTξ Tξ)2/5+L2(t−Tξ−1) Γ =N1/2T−2ξ/5(logT)2/5+L2(t−Tξ−1) Γ, and fort=T E[∥p(T)−p∗∥2 2]≲N1/2T−2ξ/5(logT)2/5+L2(T−Tξ−1) Γ. E.1 Proof of Lemma E.1 Letτbe the minimum exploration phase among the Nfirms andt∈{τ+ 1,τ+ 2,...,T}. Letnbe the length of the exploration phase to estimate θiand/tildewidenthe remaining leng... | https://arxiv.org/abs/2503.16737v2 |
whole R. Proposition F.1, which is followed by Lemma F.7, shows the constraints on the codomain of ϕ0whenψ0iss-concave for s̸= 1. Proposition F.1 (Constraints for s-concave transformations) .Suppose that ψ0iss-concave, that is ψ0= hs◦ψ0for somes̸= 1, wherehs=d−1 sas defined in Equation (3). Then ϕ0∈[a′,b′]≜ds([lψ,uψ]) ... | https://arxiv.org/abs/2503.16737v2 |
{2}, s = 2 (d′′ s(uψ),d′′ s(lψ)), s∈(1,2)c, and specifically 0<C′ s(lψ)≤d′ s≤C′ s(uψ)andC′′ s(lψ)≤d′′ s≤C′′ s(uψ),uniformly on [lψ,uψ], where C′ s(lψ) = d′ s(lψ), s> 1, 1, s = 1, d′ s(uψ), s< 1andC′′ s(lψ) = d′′ s(lψ),1<s< 2 {0}, s = 1 {2}, s = 2 d′′ s(uψ), s∈(1,2)c C′′ s<0fors<1andC′′ s>0fors>1. F.2 Te... | https://arxiv.org/abs/2503.16737v2 |
Thus ˇϕnis one particular LS estimator for ϕ0. The following technical Lemma F.12 can be found in Dümbgen et al. (2004, Lemma 3). 33 Lemma F.12. For0<u≤b−alet /tildewiderMn(u) := min c∈[a,b−u]Mn[c,c+u]. Suppose that|/hatwideϕn−ϕ0|≥ξon some interval [c,c+δ]⊂[a,b]with length δ >0. Then there is a function ∆∈D 6such that ... | https://arxiv.org/abs/2503.16737v2 |
1{u=Uk}andζ(2) jk(u) := 0. This defines a collection Φof at most n2different nonzero functions ζ(e) jk. Then for any fixed γo>2, Equation (32) implies P/braceleftig Sn≤γoσ(logn)1/2/bracerightig ≥1−2 nγ2o/2, where Sn:= max ζ∈Φ/vextendsingle/vextendsingle/vextendsingle/summationtextn i=1ζ(Ui)h′(/hatwideϕn(Ui))Ei/vexten... | https://arxiv.org/abs/2503.16737v2 |
the assumptions of Theorem 6.6 imply those of Theorem F.5. (1)ψi,θissi-concave for some si>−1. According to Proposition 6.5, this condition is satisfied provided ψi issi-concave for some si>−1, which holds by Proposition 4.3 and Assumption 4.2. (2)Assumption F.2 holds uniformly in θ∈SN−1because the design points are dr... | https://arxiv.org/abs/2503.16737v2 |
Nonparametric Factor Analysis and Beyond Yujia Zheng1Yang Liu2Jiaxiong Yao2Yingyao Hu†,3Kun Zhang†,1,4 1Carnegie Mellon University 2International Monetary Fund 3Johns Hopkins University 4Mohamed bin Zayed University of Artificial Intelligence Abstract Nearly all identifiability results in unsuper- vised representation ... | https://arxiv.org/abs/2503.16865v1 |
models (Reiersøl, 1950; Lawley and Maxwell, 1962; Bekker and ten Berge, 1997) can incorporate noise but remain subject to certain limitations. First, like sev- eral approaches in noisy ICA and other models (Ikeda and Toyama, 2000; Beckmann and Smith, 2004; Bon-arXiv:2503.16865v1 [cs.LG] 21 Mar 2025 Nonparametric Factor... | https://arxiv.org/abs/2503.16865v1 |
density of VonU. The capital letter Pdenotes the distribution. Throughout this work, for any matrix S, we use Si,:to indicate its i-th row and S:,jto indicate its j-th col- umn. For any index set I ⊂ { 1, . . . , a } × { 1, . . . , b}, we define Ii,::={j|(i, j)∈ I} andI:,j:={i| (i, j)∈ I} . Based on this, we define the... | https://arxiv.org/abs/2503.16865v1 |
is possible to transform an unbounded density over a bounded support to a bounded density over an unbounded support, so it can be extended to some cases where densities are unbounded. Before introducing more assumptions, we define an in- tegral operator corresponding to pXA|Z, which maps pZover support ZtopXAover suppo... | https://arxiv.org/abs/2503.16865v1 |
more precise inference. Therefore, it becomes necessary to go beyond distributional identi- fiability and focus on recovering the individual latent components. We first propose the following theorem for identifying the submanifold of the latent variables. Theorem 2. Consider two models θ= (f, pZ, pϵ)and ˆθ= (ˆf, p ˆZ, ... | https://arxiv.org/abs/2503.16865v1 |
has been introduced in (Zheng et al., 2022). However, the identifiability results in (Zheng et al., 2022) are limited to deterministic trans- formations, thus requiring the generative process to be a diffeomorphism without any noise. In Theorem 3, we prove that, even in the presence of non-negligible noise, we can stil... | https://arxiv.org/abs/2503.16865v1 |
Theorem 2, suppose the following assumptions hold: i. The density pZis positive and smooth. ii. (Distributional Variability ) There exist 2n+1val- ues of U, i.e., U(i)with i∈ {0,1, . . . , 2n}, s.t. the2nvectors w(Z, U(i))−w(Z, U(0))with i∈ {1, . . . , 2n}are linearly independent, where vector w(Z, U)is defined as foll... | https://arxiv.org/abs/2503.16865v1 |
distribution functions based on sample ( ⃗X1,⃗X2, ...,⃗Xm,⃗ˆZ). Notice that Genters the loss function through⃗ˆZ= (ˆZ(1),ˆZ(2), ...,ˆZ(N))Tin density estimators. To be specific, we can have a kernel density estimator ˆp(x1, . . . , x k,ˆz) =1 mmX i=1Kh∗ ˆz−ˆZ(i)kY j=1Khj xj−X(i) j , where Kh(u) =1 hKu h , and the... | https://arxiv.org/abs/2503.16865v1 |
ϵ1∼ N(0,4), ϵ2∼Beta (2,4)−1 3, ϵ3∼Laplace (0,2). Table 1 demonstrates the min, median and max cor- relations of ⃗Zand⃗ˆZin the test sample for the three Yujia Zheng1, Yang Liu2, Jiaxiong Yao2, Yingyao Hu†,3, Kun Zhang†,1,4 Table 1: Basis Validation for Continuous Data Simulation Name corr( ⃗Z,⃗ˆZ) corr( ⃗Z,⃗X1) min med... | https://arxiv.org/abs/2503.16865v1 |
of multiple latent variables. We begin by conducting experiments with a noise vari- ance of one, i.e., ϵi∼ N (0,1), while varying the number of latent variables from 2 to 10. The re- sults, presented in Figure 1, show that our estimator consistently achieves MCC values close to one across datasets of different dimensio... | https://arxiv.org/abs/2503.16865v1 |
data reveal important patterns in official GDP data and are useful in a number of aspects. First, most countries’ official GDP growth data align well with our refined estimates. For example, both Chile and South Africa have differences within 0.15 percentage points despite volatile economic growth. It suggests that GEE... | https://arxiv.org/abs/2503.16865v1 |
M Smith. Prob- abilistic independent component analysis for func- tional magnetic resonance imaging. IEEE transac- tions on medical imaging , 23(2):137–152, 2004. Paul A Bekker and Jos MF ten Berge. Generic global indentification in factor analysis. Linear Algebra and its Applications , 264:255–263, 1997. Robert Beyer,... | https://arxiv.org/abs/2503.16865v1 |
Statistical Society. Series D (The Statistician) , 12 (3):209–229, 1962. Francesco Locatello, Stefan Bauer, Mario Lucic, Gun- nar Raetsch, Sylvain Gelly, Bernhard Sch¨ olkopf, and Olivier Bachem. Challenging common assumptions in the unsupervised learning of disentangled repre- sentations. In international conference o... | https://arxiv.org/abs/2503.16865v1 |
Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 C Supplementary Experiments 21 C.1 Supplementary details of the settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 C.2 Supplementary experimental results . . . . . . . . . . . . . . . .... | https://arxiv.org/abs/2503.16865v1 |
. . . ,∂logp(zn|U(i)) ∂zn , v′(Z, U(i)) =∂2logp(z1|U(i)) ∂z2 1, . . . ,∂2logp(zn|U(i)) ∂z2n . Then, the vector w(Z, U(i)) is given by: w(Z, U(i)) = v(Z, U(i)),v′(Z, U(i)) . B Proofs B.1 Proof of Theorem 2 Theorem 2. Consider two models θ= (f, pZ, pϵ)andˆθ= (ˆf, p ˆZ, pˆϵ)following the process in Section 2, under A... | https://arxiv.org/abs/2503.16865v1 |
of the observed variables. From Equation (24), we derive the operator equivalence: KXC;XA|XB=KXA|ZΛXC;ZKZ|XB. (25) This equivalence holds over the space of functions G(Z), given the factorization properties of the conditional densities established earlier. Now, integrating over XC, we use the fact that: Z KXC;XA|XBf′(x... | https://arxiv.org/abs/2503.16865v1 |
Λ XC;Zhas eigenvalues governed by pXC|Z. Clearly, without additional constraints, we could have distinct Zvalues leading to the same eigenvalue. However, Assumption 4 avoids this by ensuring the set {x:p(xC|z)̸=p(xC|z′)}has positive probability for all z, z′∈ Zwith z̸=z′. Even if the eigenvalues are distinct, one can p... | https://arxiv.org/abs/2503.16865v1 |
is equivalent to proving that Jtis a generalized permutation matrix. Let us denote JtbyT. According to our assumption, for each index i, the set of vectors {Jf(z(ℓ))i,:}|Fi,:| ℓ=1spans the space Rn Fi,:. This means any vector in Rn Fi,:can be expressed as a linear combination of these vectors. Specifically, for any sta... | https://arxiv.org/abs/2503.16865v1 |
. . , 2n}are linearly independent, where vector w(Z, U)is defined as follows: w(Z, U(i)) = v(Z, U(i)),v′(Z, U(i)) , where v(Z, U(i)) =∂logp(z1|U(i)) ∂z1,···,∂logp(zn|U(i)) ∂zn , v′(Z, U(i)) =∂2logp(z1|U(i)) (∂z1)2,···,∂2logp(zn|U(i)) (∂zn)2 . Then there exists a component-wise invertible function hand a permutati... | https://arxiv.org/abs/2503.16865v1 |
2 , X4= Φ( Z/3)·(−1)I(ϵ4>0.5), Z∼Binomial (10,0.5). For the linear error case, we use: ϵ1=N(0,1 4Z2), ϵ3=Laplace (0,1 2|Z|). Nonparametric Factor Analysis and Beyond Figure 4: Results w.r.t. different numbers of latent variables for model satisfying distributional variability. For the double error case, we use: ϵ1=N(0... | https://arxiv.org/abs/2503.16865v1 |
arXiv:2503.17179v1 [stat.ME] 21 Mar 2025An improved nonparametric test and sample size procedures for the randomized complete block designs Show-Li Jan Department of Applied Mathematics Chung Yuan Christian University 200 Zhongbei Road Taoyuan 320314, Taiwan Email: sljan@cycu.edu.tw Gwowen Shieh * Department of Managem... | https://arxiv.org/abs/2503.17179v1 |
prescribed F-transformation with intrablock ranks to an Fdistribution with two smaller degrees of freedom than those of the usual ANOV A Fdis- tribution. Thus, reduction of numerator and denominator de grees of freedom can help improve the liberal Fapproximation of Iman and Davenport (1980) to some extent. U nfortunate... | https://arxiv.org/abs/2503.17179v1 |
the con trol of Type I error rates and accuracy of power computations. 2 Test procedures The usual model for randomized complete block designs is defi ned as follows: Xi j=µ+θi+γj+εi j, (1) where Xi jis the response of the ith treatment in the jth block, µis the overall mean, θiis the ith treatment effect, γjis the jth ... | https://arxiv.org/abs/2503.17179v1 |
deserves critical recognition and 5 further investigations. Following the moment-matching pr inciples of Box and Andersen (1955), Pitman (1938), and Welch (1937), improved procedures are de scribed next to give better Type I error control than the liberal FRtest. 2.2 A general class of rank procedures To provide altern... | https://arxiv.org/abs/2503.17179v1 |
freedom f1asl1=K−1 or, equivalently, d=1. Then, the quantity Scan be derived as L=B(K+1)−2 and the denominator degrees of freedom f2has the specific form l2=(B−1)(K+1). The suggested test statistic and approximate distribution are formulated as FL=(K+1)(B−1)T (K−1)(L−T)˙∼F(l1,l2). (7) Accordingly, the test rejects the n... | https://arxiv.org/abs/2503.17179v1 |
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