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can replace the condition in Assumption 2, by N2= XTX− D+ X −p D+ XD− X −p D+ XD− X D− X! ≤(C− x)2 √nC− y C+ y(86) with D+ Xthediagonalmatrixwithdiagonalterms ||xi||2, and D− Xthediagonalmatrixwith diagonal terms ||x−i||2. Thus, adding antipodal vectors will result in the same equations: d dt⟨wj∗ i|xi⟩ ≥|aj∗ i| nh ri||... | https://arxiv.org/abs/2502.16977v1 |
QUASI-LIKELIHOOD RATIO TEST FOR JUMP-DIFFUSION PROCESSES BASED ON ADAPTIVE MAXIMUM LIKELIHOOD INFERENCE HIROMASA NISHIKAWA1, TETSUYA KAWAI2, AND MASAYUKI UCHIDA1,3 Abstract. In this paper, we consider parameter estimation and quasi-likelihood ratio tests for multidi- mensional jump-diffusion processes defined by stocha... | https://arxiv.org/abs/2502.17058v1 |
Japan 3Center for Mathematical Modeling and Data Science (MMDS), Osaka University and JST CREST Key words and phrases. Adaptive estimation, Asymptotic theory; Discrete time observation; Quasi-log likelihood estimation; Stochastic differential equation; Jump-diffusion model, Consistent test; Quasi-likelihood ratio test.... | https://arxiv.org/abs/2502.17058v1 |
two, along with the proposed adaptive quasi-maximum likelihood estimator. We also discuss its asymptotic properties. This approach improves the computational efficiency and stabilizes numerical calculations. In particular, for the test statistic, we introduce five thresholds in the construction of the adaptive estimato... | https://arxiv.org/abs/2502.17058v1 |
measure π(dx): For any π-integrable function f, it holds that 1 TZT 0f(Xt)dtP−→Z f(x)π(dx) asT→ ∞ . Moreover, we assume the stationarity of Xfor simplicity. [A3] For any p≥1, sup t≥0Eθ[|Xt|p]<∞. [A4] For each αandβ, the derivatives ∂k xa(x, α) and ∂k xb(x, β) (k= 0,1,2,3,4) exist on Rdand they are continuous in x. More... | https://arxiv.org/abs/2502.17058v1 |
us introduce a truncation function φnto ensure the integrability of quasi-log likelihood functions in the next section. [A13] At least one of the following two conditions holds true. (i) For k= 0,1,2,3 and l, m= 0,1, there exists a constant C >0 such that ∂m x∂l y∂k βlog Ψ β(y, x) ≤C(1 +|y|)C(1 +|x|)C((x, y, β )∈Rd×E×Θ... | https://arxiv.org/abs/2502.17058v1 |
ρ3<1/2, l(1) n(α) :=−1 2nX i=1 h−1 n(∆Xn i)⊤S−1 i−1(α)∆Xn i+ log det Si−1(α) 1{|∆Xn i|≤D1hρ1n}, (3.4) l(2) n(β|¯α) :=¯l(2) n(β|¯α) +˜l(2) n(β), ¯l(2) n(β|¯α) :=−1 2hnnX i=1(¯Xi,n(β))⊤S−1 i−1(¯α)¯Xi,n(β)1{|∆Xn i|≤D3hρ3n}, (3.5) ˜l(2) n(β) :=nX i=1 log Ψ β(∆Xn i, Xtn i−1) φn(Xtn i−1,∆Xn i)1{|∆Xn i|>D2hρ2n} −hnnX i=1Z ... | https://arxiv.org/abs/2502.17058v1 |
2)∩[1+δ 6,1 2),ρ2∈[δ 2,1 2) and [A13] -(i). [D21] Fulfill ρ1∈B3(δ),ρ2∈B2and[A13] -(ii). Then, (√n(ˆαn−α0),√nhn(ˆβn−β0))d→N(0, I(θ0;θ0)−1). Remark 3.2 Under the conditions [B2],[B3],B2̸=∅, B3(δ)̸=∅. Remark 3.3 Under the conditions [B2],[B3],B2⊂B1(3). Remark 3.4 The proposed adaptive estimator ˆθnhas asymptotic efficienc... | https://arxiv.org/abs/2502.17058v1 |
S(x, α0) for a.s. all x=⇒α=α0. b(x, β) =b(x, β0) for a.s. all ( x, y) =⇒β=β0. Ψγ(y, x) = Ψ γ0(y, x) for a.s. all ( x, y) =⇒γ=γ0. [A12’] J(θ0;θ0) is non-singular for θ0∈Int(Θ). Under these fixed assumptions, the following consistency and asymptotic normality are hold. Theorem 3.3 Assume [A1]-[A8],[A9’] ,[A10] ,[A11] ,[B... | https://arxiv.org/abs/2502.17058v1 |
that we can choose up to five thresholds to compose the test statistics Λn; in five thresholds, three thresholds are for adaptive estimators, and two thresholds are for joint quasi-log likelihood function. Corollary 4.1 Assume [A1]-[A12] ,[B2],[B3], and either “[ C21] of Theorem 3.2 and [ D21] of Corollary 3.2” or “[ C... | https://arxiv.org/abs/2502.17058v1 |
. , h l(β) = 0 and i1(γ) = 0 , . . . , i m(γ) = 0 with the some smooth real valued functions g1, . . . , g k,h1, . . . , h landi1, . . . , i m, can be put into H0by a reparametrization. Let˜θnand ˜θ∗ nbe estimators on Θ and Θ 0, respectively. Then, we define the quasi-likelihood ratio test statistics Λ nwith the joint ... | https://arxiv.org/abs/2502.17058v1 |
(ˇαn,ˇβn,ˇλn,ˇµn,ˇσ2 n) can be calculated as ˇαn=vuut1 n1hnnX i=1(∆Xn i)21{|∆Xn i|≤D1hρ1n},ˇβn=−Pn i=1Xtn i−1∆Xn i1{|∆Xn i|≤D3hρ3n} hnPn i=1X2 tn i−11{|∆Xn i|≤D3hρ3n}, ˇλn=n2 nhn,ˇµn=1 n2nX i=1∆Xn i1{|∆Xn i|>D2hρ2n},ˇσ2 n=1 n2nX i=1(∆Xn i−ˇµn)21{|∆Xn i|>D2hρ2n}. In our simulation, we set θ0= (2,2.5,6,0,4.5) and for sim... | https://arxiv.org/abs/2502.17058v1 |
σ2= 20.25, H1: not H0. First, we simulate the asymptotic behavior of the adaptive test statistic under H0. Joint quasi-log likelihood function, used for composing the adaptive test statistic, is as follows: ln(θ) =−1 2nX i=1n h−1 nα−2(∆Xn i+βhnXtn i−1)2+ 2 log αo 1{|∆Xn i|≤¯D1h¯ρ1n} +nX i=1logfγ(∆Xn i)1{|∆Xn i|>¯D2h¯ρ2... | https://arxiv.org/abs/2502.17058v1 |
threshold for the continuous component in the joint quasi-log likelihood function, which is used for the construction of quasi-likelihood ratio, should be determined separately. However, the remaining thresholds can be aligned. Figure 4 shows QQ-plot of the simulated adaptive test statistic with ρ1= 0.285, and with ¯ ρ... | https://arxiv.org/abs/2502.17058v1 |
6.1 (Shimizu and Yoshida (2003, 2006) ) Suppose [A1],[A3] and[A5]. For k≥2, k∈N,tn i−1≤t≤tn i, Eh |Xt−Xtn i−1|k|Fn i−1i ≤Ck|t−tn i−1|(1 +|Xtn i−1|)Ck. (6.1) Ifgis a function defined on Rd×Θ and is polynomial growth in xuniformly in θ, then, E |g(Xt, θ)| |Fn i−1 ≤C(1 +|Xtn i−1|)C. (6.2) Remark 6.1 Assumptions in Propo... | https://arxiv.org/abs/2502.17058v1 |
|∆Zτn i| ≤3Dhρ n c0, Jn i= 1|Fn i−1 , where c0is the constant in condition [A7] and ∆ Zτn ihas density Fβ0under Fn i−1. Under [A7], (Xtn i−Xτn i) + (Xτn i−−Xtn i−1) + ∆ Xτn i ≤Dhρ n,|∆Zτn i|>3Dhρ n c0, and|∆Xτn i|is small enough, then it holds that Xtn i−Xτn i + Xτn i−−Xtn i−1 ≥ |∆Xτn i| −Dhρ n = c(Xτn i,∆Zτn i, β0) −... | https://arxiv.org/abs/2502.17058v1 |
i=1g(n) i−1(θ)1{|∆Xn i|≤Dhρ n}−Z g(x, θ)π(dx) P→0 (n→ ∞ ), (iii) sup θ∈Θ 1 nhnnX i=1g(n) i−1(θ)1{|∆Xn i|>Dhρ n}−λ0Z g(x, θ)π(dx) P→0 (n→ ∞ ). Remark 6.3 The statements of Proposition 6.4 is similar to those of Proposition 3.3 of Shimizu and Yoshida (2003, 2006). However, our balance conditions for εnare milder than the... | https://arxiv.org/abs/2502.17058v1 |
i=11 nhng(n) i−1(θ)P(|∆Xn i|> Dhρ n|Fn i−1)−λ0 nhnZnhn 0g(n)(Xs, θ)ds # ≤3 nhnεnX i=1Ztn i tn i−1Eh g(n) i−1(θ)h−1 nP(|∆Xn i|> Dhρ n|Fn i−1)−λ0g(n)(Xs, θ) i ds ≤3 nhnεnX i=1Ztn i tn i−1n Eh g(n) i−1(θ)h−1 nP(|∆Xn i|> Dhρ n|Fn i−1)−λ0g(n) i−1(θ) i +λ0Eh g(n) i−1(θ)−g(n)(Xs, θ) io ds ≤3 nεnX i=1E g(n) i−1(θ)21 2 Eh ... | https://arxiv.org/abs/2502.17058v1 |
balance conditions for εnare milder than theirs. If the statements in this proposition hold for k= 3, then it is easy to show that the statements for k= 1,2 also hold. Thus, it is sufficient to prove the case of k= 3. However, we show this proof for k∈ {1,2,3}since we utilize the proof of this proposition in the case o... | https://arxiv.org/abs/2502.17058v1 |
solution of the following stochastic differential equation under the set {Jn i= 1}: ˜Xt−˜Xtn i−1=Ht+Zt tn i−1b(˜Xs)ds+Zt tn i−1a(˜Xs)dWs, ADAPTIVE INFERENCE FOR JUMP DIFFUSION PROCESSES 27 where ˜Xtn i−1=Xtn i−1,Ht=c(Xu−, z)1[u,tn i](t),uis [tn i−1, tn i]-valued uniform random variable which is independent of ( Wt)t≥0a... | https://arxiv.org/abs/2502.17058v1 |
tn i−1Z gn(θ, ci−1(z, β0), Xtn i−1)(p−qβ0)(ds, dz ) ADAPTIVE INFERENCE FOR JUMP DIFFUSION PROCESSES 29 ×E"Ztn j tn j−1Z gn(θ, cj−1(z, β0), Xtn j−1)(p−qβ0)(ds, dz )|Fn j−1## =25 n2h2nε2nX i=1E"Ztn i tn i−1Z g2 n(θ, ci−1(z, β0), Xtn i−1)qβ0(ds, dz )# = O 1 nhn (under [P1]), O 1 nhnε2kn (under [P2],k= 1,2)(6.12) →... | https://arxiv.org/abs/2502.17058v1 |
−1 2nnX i=1log det Si−1(α)1{|∆Xn i|≤D1hρ1n} =−1 2nhnnX i=1¯Xi,n(β0)⊤S−1 i−1(α)¯Xi,n(β0)1{|∆Xn i|≤D1hρ1n} −hn·1 nhnnX i=1b⊤ i−1(β0)S−1 i−1(α)¯Xi,n(β0)1{|∆Xn i|≤D1hρ1n} −hn·1 2nnX i=1b⊤ i−1(β0)S−1 i−1(α)bi−1(β0)1{|∆Xn i|≤D1hρ1n} −1 2nnX i=1log det Si−1(α)1{|∆Xn i|≤D1hρ1n}. ADAPTIVE INFERENCE FOR JUMP DIFFUSION PROCESSES ... | https://arxiv.org/abs/2502.17058v1 |
β) ≤sup θ∈Θ 1 nhnnX i=1(bi−1(β)−bi−1(β0))⊤S−1 i−1(¯α)¯Xi,n(β0)1{|∆Xn i|≤D3hρ3n} + sup θ∈Θ −1 2nnX i=1(bi−1(β)−bi−1(β0))⊤S−1 i−1(¯α)(bi−1(β)−bi−1(β0))1{|∆Xn i|≤D3hρ3n}−¯U(2) β0(α, β) P→0. (6.20) By the assumption [A8], it follows that ¯U(2) β0(α0, β) =−1 2Z (b(x, β)−b(x, β0))⊤S−1(x, α0)(b(x, β)−b(x, β0))π(dx)≤0 (6.21) w... | https://arxiv.org/abs/2502.17058v1 |
Si−1(α)o 1{|∆Xn i|≤D1hρ1n}, that for 1 ≤m2, m′ 2≤q, ∂2 βm2βm′ 2l(2) n(β|¯α) =nX i=1n (∂2 βm2βm′ 2bi−1(β))⊤S−1 i−1(¯α)¯Xi,n(β)−hn(∂βm2bi−1(β))⊤S−1 i−1(¯α)∂βm′ 2bi−1(β)o 1{|∆Xn i|≤D3hρ3n} +nX i=1n ∂2 βm2βm′ 2log Ψ β(∆Xn i, Xtn i−1)o φn(Xtn i−1,∆Xn i)1{|∆Xn i|>D2hρ2n} −hnnX i=1Z B∂2 βm2βm′ 2Ψβ(y, X tn i−1)φn(Xtn i−1, y)dy... | https://arxiv.org/abs/2502.17058v1 |
−1 nnX i=1Z B∂2 βm2βm′ 2Ψβ(y, X tn i−1)φn(Xtn i−1, y)dy+I(m2,m′ 2) b,c(¯α, β) ≤sup (¯α,β)∈Θ 1 nnX i=1(∂2 βm2βm′ 2bi−1(β))⊤S−1 i−1(¯α)(bi−1(β0)−bi−1(β))1{|∆Xn i|≤D3hρ3n} +Z (∂2 βm2βm′ 2b(x, β))⊤S−1(x,¯α)(b(x, β)−b(x, β0))π(dx) + sup (¯α,β)∈Θ 1 nhnnX i=1(∂2 βm2βm′ 2bi−1(β))⊤S−1 i−1(¯α)¯Xi,n(β0)1{|∆Xn i|≤D3hρ3n} + sup (¯α... | https://arxiv.org/abs/2502.17058v1 |
i−1(α0)¯Xi,n(β0) +∂αm1log det Si−1(α0) 1{|∆Xn i|≤D1hρ1n}|Fn i−1i +1√nnX i=1 Eh bi−1(β0)⊤∂αm1S−1 i−1(α0)¯Xi,n(β0) 1{|∆Xn i|≤D1hρ1n}|Fn i−1i +hn 2√nnX i=1 bi−1(β0)⊤∂αm1S−1 i−1(α0)bi−1(β0) P |∆Xn i| ≤D1hρ1 n|Fn i−1 40 N NISHIKAWA, T KAWAI, AND M UCHIDA ≤1 2√nnX i=1 dX k1,k2=1h−1 n∂αm1S−1(k1,k2) i−1 (α0)Eh ¯X(k1) i,n(β0... | https://arxiv.org/abs/2502.17058v1 |
ci−1(z, β0), Xtn i−1)qβ0(ds, dz )−hnZ B∂βm2Ψβ0(y, Xtn i−1)φn(Xtn i−1, y)dy|Fn i−1# =:5X i=1Hi n. It is obvious from martingale property that H4 n= 0. Since it follows from change of variables that Ztn i tn i−1Z E∂βm2gn(β0, ci−1(z, β0), Xtn i−1)qβ0(ds, dz ) =Ztn i tn i−1Z E∂βm2log Ψ β0(ci−1(z, β0), Xtn i−1)φn(Xtn i−1, c... | https://arxiv.org/abs/2502.17058v1 |
i−1Eh ¯X(k1) i,n¯X(k2) i,n¯X(k3) i,n1{|∆Xn i|≤D1hρ1n}|Fn i−1i +1 4nhnnX i=1∂αm′ 1log det Si−1dX k1,k2=1∂αm1S−1(k1,k2) i−1Eh ¯X(k1) i,n¯X(k2) i,n1{|∆Xn i|≤D1hρ1n}|Fn i−1i +1 4nhnnX i=1∂αm1log det Si−1dX k1,k2=1∂αm′ 1S−1(k1,k2) i−1Eh ¯X(k1) i,n¯X(k2) i,n1{|∆Xn i|≤D1hρ1n}|Fn i−1i +1 nnX i=1R(θ,1, Xtn i−1)dX k1,k2=1Eh ¯X(k... | https://arxiv.org/abs/2502.17058v1 |
β0(∆Xn i, Xtn i−1) φn(Xtn i−1,∆Xn i)1{|∆Xn i|>D2hρ2n} −hnZ B∂βm2Ψβ0(y, X tn i−1)φn(Xtn i−1, y)dy ×n (∂βm′ 2bi−1(β0))⊤S−1 i−1(α0)¯Xi,n(β0)1{|∆Xn i|≤D3hρ3n} + ∂βm′ 2log Ψ β0(∆Xn i, Xtn i−1) φn(Xtn i−1,∆Xn i)1{|∆Xn i|>D2hρ2n} −hnZ B∂βm′ 2Ψβ0(y, X tn i−1)φn(Xtn i−1, y)dy |Fn i−1 =1 nhnnX i=1dX k1,k2=1dX k3,k4=1∂βm2b(... | https://arxiv.org/abs/2502.17058v1 |
i−1Z Egn(β0, ci−1(z, β0), Xtn i−1)p(ds, dz ) −1 nhnnX i=1Ztn i tn i−1Z Egn(β0, ci−1(z, β0), Xtn i−1)qβ0(ds, dz )|Fn i−1# + E" 1 nhnnX i=1Ztn i tn i−1Z Egn(β0, ci−1(z, β0), Xtn i−1)qβ0(ds, dz )−1 nnX i=1Z A∂βm2Ψβ0∂βm′ 2Ψβ0 Ψβ0(y, Xtn i−1)φ2 n(Xtn i−1, y)dy|Fn i−1# ≤1 nhnnX i=1Eh |gn(β0,∆Xn i, Xtn i−1)−gn(β0,∆Xτn i, Xtn ... | https://arxiv.org/abs/2502.17058v1 |
Xtn i−1) × P {|∆Xn i| ≤D3hρ3 n} ∩ {| ∆Xn i|> D 2hρ2 n} |Fn i−1 +E |∆Xn i|C|Fn i−1 (under [ C21]) 1 nhnnX i=1dX k1,k2=1R(θ, hρ3 nε−1 n, Xtn i−1)P {|∆Xn i| ≤D3hρ3 n} ∩ {| ∆Xn i|> D 2hρ2 n} |Fn i−1 (under [ C22]) ≤ 1 nhnnX i=1dX k1,k2=1R(θ, h1+ρ3 n, Xtn i−1) (under [ C21]) 1 nhnnX i=1dX k1,k2=1R(θ, h1... | https://arxiv.org/abs/2502.17058v1 |
i|≤D1hρ1n}|Fn i−1i +√hn 2nnX i=1dX k1,k2=1dX k3,k4=1 b(k1) i−1 ∂αm1S−1 i−1(k1,k2) b(k2) i−1 ∂βm2b(k3) i−1 S(k3,k4) i−1 Eh ¯X(k4) i,n1{|∆Xn i|≤D1hρ1n}|Fn i−1i +1 2n√hnnX i=1 ∂αm1log det Si−1 dX k1,k2=1 ∂βm2b(k1) i−1 S(k1,k2) i−1 Eh ¯X(k2) i,n1{|∆Xn i|≤D1hρ1n}|Fn i−1i +1 2nhn√hnnX i=1dX k1,k2=1 ∂αm1S−1 i−1(k1,k2) ×Eh ∆Xn... | https://arxiv.org/abs/2502.17058v1 |
of (6.34), (6.35), it is easy to show that nX i=1E1√nξm1 i(α0)|Fn i−1 E1√nhn ηm2 i,1(β0|α0) +ηm2 i,2(β0) |Fn i−1 =nX i=1 R θ,hn√n, Xtn i−1 +R θ,h3ρ1n√n, Xtn i−1 × R θ,hnε−2 n√n, Xtn i−1 +R θ,s h1+4ρ1n n, Xtn i−1 +oP(1) 52 N NISHIKAWA, T KAWAI, AND M UCHIDA P→0. Proof of (6.42) .Note that sinc... | https://arxiv.org/abs/2502.17058v1 |
y)dy ) ≤1 (nhn)ν2 2( 1 nhnnX i=1R(θ, h(2+ν2)ρ3 n , Xtn i−1) +1 nnX i=1R(θ, ε−(2+ν2) n , Xtn i−1) +1 nnX i=1R(θ, h1+ν2 n, Xtn i−1)) ≤Op (nhn)−ν2 2h(2+ν2)ρ3−1 n +Op (nhn)−ν2 2ε−(2+ν2) n +Op (nhn)−ν2 2h1+ν2 n . It is obvious from ν2> 4−2δ δ∨2 that the third term on the right-hand side converges to 0 in probability.... | https://arxiv.org/abs/2502.17058v1 |
1 nhn¯ln(α, β)−1 nhn¯ln(α, β0)−¯U(2) β0(α, β) P→0. (6.62) Moreover, by the definition of ˆβn, it follows that for all ε >0, P1 nhnl(2) n(ˆβn|ˆαn) +ε <1 nhnl(2) n(β0|ˆαn) = 0. (6.63) Furthermore, by using (6.58), in an analogous manner to (6.25), one has |¯U(2) β0(ˆαn,ˆβn)−¯U(2) β0(α0,ˆβn)|P→0. (6.64) Therefore, in a ... | https://arxiv.org/abs/2502.17058v1 |
+1 nnX i=1R(θ, hρ1 n, Xtn i−1) +1 nnX i=1R(θ, hn, Xtn i−1) P→0. Therefore, sup θ∈Θ 1 n∂2 αln(θ) +Ia(α) P→0. (6.70) 58 N NISHIKAWA, T KAWAI, AND M UCHIDA Since for all u∈[0,1],α0+u(ˆαn−α0)∈ {α∈Θα| |α−α0|< εn}on the set ˆAn, it holds from (6.70), continuity of Ia(α) and consistency of ˆ αnthat for all ε >0, P sup u∈[0,1]... | https://arxiv.org/abs/2502.17058v1 |
i=1E1√nψm1 i(θ0)|Fn i−1 P→0. 60 N NISHIKAWA, T KAWAI, AND M UCHIDA By using (6.49), we have nX i=1E1√nψm1 i(θ0)|Fn i−1 = 1√nnX i=1Eh (bi−1)⊤∂αm1S−1 i−1¯Xi,n1{|∆Xn i|≤D1hρ1n}|Fn i−1i +hn 2√nnX i=1Eh (bi−1)⊤∂αm1S−1 i−1bi−11{|∆Xn i|≤D1hρ1n}|Fn i−1i ≤1√nnX i=1dX k1,k2=1|b(k1) i−1||∂αm1S−1 i−1(k1,k2)| Eh ¯X(k2) i,n1{|∆X... | https://arxiv.org/abs/2502.17058v1 |
+1 nnX i=1R(θ, h2 n, Xtn i−1) 62 N NISHIKAWA, T KAWAI, AND M UCHIDA P→0. Proof of (6.79) .By simple computation, nX i=1E1 n√hn(ξm1 i(α0) +ψm1 i(θ0)) ηm2 i,1(β0|α0) +ηm2 i,2(β0) |Fn i−1 =nX i=1E1 n√hnξm1 i(α0) ηm2 i,1(β0|α0) +ηm2 i,2(β0) |Fn i−1 +nX i=1E1 n√hnψm1 i(θ0) ηm2 i,1(β0|α0) +ηm2 i,2(β0) |Fn i−1 , a... | https://arxiv.org/abs/2502.17058v1 |
∩ {| ∆Xn i|> D 2hρ2n} |Fn i−1 (under [ C21]) 1 n√hnnX i=1dX k1,k2=1R(θ, hρ1 nε−1 n, Xtn i−1) ×P {|∆Xn i| ≤D1hρ1n} ∩ {| ∆Xn i|> D 2hρ2n} |Fn i−1 (under [ C22]) ≤ 1 n√hnnX i=1dX k1,k2=1R θ, h1+2ρ1 n , Xtn i−1 (under [ C21]) 1 n√hnnX i=1dX k1,k2=1R θ, h1+2ρ1 n ε−1 n, Xtn i−1 (under [ C22]) ≤ Op h... | https://arxiv.org/abs/2502.17058v1 |
probability if5ρ1−1 1−2ρ1≥ν1>0. Therefore, it is sufficient to show that for5ρ1−1 1−2ρ1≥ν′ 1>0, nX i=1E" 1√nψm1 i(θ0) 2+ν′ 1 |Fn i−1# P→0. 66 N NISHIKAWA, T KAWAI, AND M UCHIDA Since|¯Xi,n(β0)|2+ν′ 11{|∆Xn i|≤D1hρ1n}=R(θ, h(2+ν′ 1)ρ1 n , Xtn i−1)1{|∆Xn i|≤D1hρ1n}, it follows that nX i=1E" 1√nψm1 i(θ0) 2+ν′ 1 |Fn i−1# ≤... | https://arxiv.org/abs/2502.17058v1 |
¯I(¯θ;θ0) =¯Ia,3(¯α;α0) O O ¯Ib,c,3(¯θ;θ0) , where ¯Ia,3(¯α;α0) :=Ia,3((0,···,0, α(k+1),···, α(p))⊤;α0), ¯Ib,c,3(¯θ;θ0) :=Ib,c,3((0,···,0, α(k+1),···, α(p),0,···,0, β(l+1),···, β(q))⊤;θ0). Since, from [A12] ,I(θ0;θ0) is non-singular, ¯I(¯θ0;θ0) is non-singular, too. Then it follows from Corollary 3.2 that√n(¯ˆα∗ n−¯... | https://arxiv.org/abs/2502.17058v1 |
I(θ0;θ0)−1−H D−1 2n∂θln(θ0) +I(θ0;θ0)−1ˆTn−HˆT∗ n. For the right-hand side, it holds, by using (6.93), (6.99), Slutsky’s theorem and the continuous mapping theorem, that I(θ0;θ0)−1−H D−1 2n∂θln(θ0) +I(θ0;θ0)−1ˆTn−HˆT∗ nd→ I(θ0;θ0)−1−H Y (6.100) since it follows from (6.69) that D−1 2n∂θln(θ0)d→Y. 70 N NISHIKAWA, ... | https://arxiv.org/abs/2502.17058v1 |
(6.104) We can express (6.102) as follows: Λn(ˆθn,ˆθ∗ n) =−2 D1 2n(ˆθ∗ n−ˆθn)⊤ I(2) n(ˆθ∗ n,ˆθn) D1 2n(ˆθ∗ n−ˆθn) (ω∈ˆAn∩ˆA∗ n). (6.105) Under H0, it holds from Lemma 6.2 and (6.104) that D1 2n(ˆθn−ˆθ∗ n), I(2) n(ˆθ∗ n,ˆθn)d→ I(θ0;θ0)−1−H Y,−1 2I(θ0;θ0) , 72 N NISHIKAWA, T KAWAI, AND M UCHIDA where Y∼Np+q(0,... | https://arxiv.org/abs/2502.17058v1 |
(6.104), we obtain I(2) n(˜θ∗ n,ˆθ∗ n)P→ −1 2I(θ0;θ0) under H0. Since, by [T1],D1 2n(˜θ∗ n−ˆθ∗ n) =op(1) under H0, it follows for the right-hand side of (6.112) that under H0, ¯C(1) n(ˆθ∗ n)⊤D1 2n(˜θ∗ n−ˆθ∗ n) + D1 2n(˜θ∗ n−ˆθ∗ n) I(2) n(˜θ∗ n,ˆθ∗ n) D1 2n(˜θ∗ n−ˆθ∗ n) =op(1). Hence, for all ε >0, we see from (6.11... | https://arxiv.org/abs/2502.17058v1 |
term on the right-hand side converges to 0 in probability. Thus, 1√nhn ∂βl(2) n(β0|α0)−∂βl(2) n(β0|ˇαn)P→0, and since we see from (6.121) that ( ˇY(2) n)−1=Op(1), we obtain ˇS(2) nP→0. (6.124) By using (6.123) and (6.124), we have ˇSnP→0. (6.125) Similarly, it follows from Taylor’s theorem that −1√n∂αln(θ0) =Z1 01 n... | https://arxiv.org/abs/2502.17058v1 |
have 1{|∆Xn i|≤h¯ρ1n}−1{|∆Xn i|≤hρ1n}=−1{|∆Xn i|≤hρ1n}1{|∆Xn i|>h¯ρ1n}. (6.136) 78 N NISHIKAWA, T KAWAI, AND M UCHIDA Hence, since |¯Xi,n(β0)|21{|∆Xn i|≤hρ1n}=R(θ, h2ρ1n, Xtn i−1), it follows from (6.136), Markov’s inequality, and Proposition 6.2 that, for the first term, for all ε >0, P 1 2√nhnnX i=1 ¯Xi,n⊤∂αm1S−1 i−... | https://arxiv.org/abs/2502.17058v1 |
0. Hence, we evaluate the case where ¯ ρ1̸=ρ3. First, we discuss the case where ¯ρ1> ρ3. From (6.136), it holds that 1{|∆Xn i|≤h¯ρ1n}−1{|∆Xn i|≤hρ3n}=−1{|∆Xn i|≤hρ3n}1{|∆Xn i|>h¯ρ1n}. (6.144) Since|¯Xi,n(β0)|1{|∆Xn i|≤hρ3n}=R(θ, hρ3n, Xtn i−1), by using (6.144), Markov’s inequality, and Proposi- tion 6.2, it holds that... | https://arxiv.org/abs/2502.17058v1 |
n ,···,ˇβ∗(q) n)⊤, where ˇα∗ n= (0,···,0,ˇα∗(k+1) n ,···,ˇα∗(p) n,0)⊤, ˇβ∗ n= (0,···,0,ˇβ∗(l+1) n ,···,ˇβ∗(q) n)⊤, and let ¯Θα0:={¯α∈Rp−k| ∃α∈Θα0,¯α= (α(k+1),···, α(p))⊤}and ¯Θβ0:={¯β∈Rq−l| ∃β∈ Θβ0,¯β= (β(l+1),···, β(q))⊤}. We define U(1) n(¯α) and U(2) n(¯β|¯α) as follows with l(1) n(α) and l(2) n(β|α): U(1) n (α(k+1... | https://arxiv.org/abs/2502.17058v1 |
n)−ln(˜α∗ n, β1))−V∗ β1(˜α∗ n,˜β∗ n) + 2 V∗ β1(˜α∗ n,˜β∗ n)−V∗ β1(α∗, β∗) ≤2 sup θ∈Θ 1 nhn(ln(θ)−ln(α, β1))−V∗ β1(α, β) + 2 V∗ β1(˜α∗ n,˜β∗ n)−V∗ β1(α∗, β∗) ≤2 sup θ∈Θ 1 nhn(¯ln(θ)−¯ln(α, β1))−¯U(2)∗ β1(α, β) + 4 sup θ∈Θ 1 nhn˜ln(θ)−˜U(2)∗ β1(β) + 2 V∗ β1(˜α∗ n,˜β∗ n)−V∗ β1(α∗, β∗) P→0. Hence, we obtain 1 nhn¯Λn(˜θn,˜θ... | https://arxiv.org/abs/2502.17058v1 |
arXiv:2502.17142v1 [math.ST] 24 Feb 2025The feasibility of multi-graph alignment: a Bayesian appro ach Louis Vassaux LOUIS .VASSAUX @ENS.PSL.EU INRIA, DMA/ENS, PSL Research University, Paris, France Laurent Massouli ´e LAURENT .MASSOULIE @INRIA .FR INRIA, DI/ENS, PSL Research University, Paris, France Abstract We estab... | https://arxiv.org/abs/2502.17142v1 |
(2018 ),Fan et al. (2019a ),Fan et al. (2019b ),Ganassali et al. (2019 ), Mao et al. (2022 ),Ganassali et al. (2024 ),Even et al. (2024 )). These algorithms, in turn, can be used in various applications: for instance, de-anonymisat ion problems ( Narayanan and Shmatikov (2009 )), natural language processing ( Bayati et... | https://arxiv.org/abs/2502.17142v1 |
os–R´ enyi graph H1=G1∩(G2∪...∪Gp), and so our theorem-conjecture pair states that the feasibility of partial alignment should depend only on whether the graph H1has a giant component. This idea is supported by the results f rom the recent preprint Ameen and Hajek (2024 ), where it is proven that the cutoff for exact a... | https://arxiv.org/abs/2502.17142v1 |
have observed a p-uple of random graphs (G′ 1,...,G′ p), sampled from a distribution Pσfor some unknown σ∈ Sp−1 n, whose prior distribution is assumed to be the uniform distributio nU(Sp−1 n); again, we wish to estimate σ∗. This reformulation of the problem places it squarely within the zoo of Bayesian inference proble... | https://arxiv.org/abs/2502.17142v1 |
Sn. Definition 2.1 Apartial estimator ˆXnofXnis aYn-measurable random variable such that, for some0≤r <1, E[lr(ˆXn,Xn)]−→ n→+∞0i.e.P(d(ˆXn,Xn)> r)−→ n→+∞0 (2.5) On the other hand, we will say that partial estimation of Xnis intractable if, for any 0≤r <1, and for any sequence (ˆXn)of estimators, E[lr(ˆXn,Xn)]−→ n→+∞1 (2... | https://arxiv.org/abs/2502.17142v1 |
d≤dw≤(p−1)d. Naturally, a converse proposition also holds for alignment intractability. For technical reasons, we will also want to work with a ”squar e correlated overlap”, defined by ovc(σ,σ′)2=1 p(p−1)/summationdisplay 1≤i/\e}atio\slash=j≤pov(σ−1 iσ′ i,(σj)−1σ′ j)2(2.13) Note that, setting dc=/radicalbig 1−ov2c,1 pdw... | https://arxiv.org/abs/2502.17142v1 |
(π∗)−1(G′)whereG′∼ N(n,ρ′)for anyρ′≤ρ and solve the alignment problem for G′. In what follows, for σ∈ Sp−1 nand1≤i/\e}atio\slash=j≤p, we will set σij=σj(π∗ j)−1π∗ iσ−1 i, which depends upon the true alignment π∗. It may be helpful to imagine that π∗= Id , since this is true up to a relabelling of nodes; in this case, σ... | https://arxiv.org/abs/2502.17142v1 |
/BDC=ρ/summationdisplay 1≤i/\e}atio\slash=j≤p/parenleftbigg/parenleftbiggn 2/parenrightbigg −/parenleftbiggdij(σ) 2/parenrightbigg/parenrightbigg +Voff(σ)+O(n(logn)3 4) (3.9) where the error term is bounded uniformly in σ. We now turn to Voff, whose fluctuations are harder to control. Proposition 3.4 Letσ,σ′∈ Sp−1 n. Then... | https://arxiv.org/abs/2502.17142v1 |
Section 2.4, starting from a plausible permutation π, our proof method is to build many other plausible-looking permutations (σkπ)1≤k≤N. If the map π→(σkπ)is close enough to being injective then this will imply the diffusene ss ofPpost. Ifp= 2, this was done in Ganassali et al. (2021 ): if the graph G1∩π(G2)has automor... | https://arxiv.org/abs/2502.17142v1 |
- versity Press, 2 edition, 2001. St´ ephane Boucheron, Gabor Lugosi, and Pascal Massart. Concentration inequalities : a non asymp- totic theory of independence . Oxford University Press, 2013. Section 2.8. Daniel Cullina and Negar Kiyavash. Exact alignment recover y for cor- related erdos renyi graphs. ArXiv , abs/171... | https://arxiv.org/abs/2502.17142v1 |
17, 12 2016. doi: 10.1186/s12859-016-1395-9. Chung-Shou Liao, Kanghao Lu, Michael Baym, Rohit Singh, and Bonnie Berger. Isorankn: Spectral methods for global alignment of multiple protein networks. Bioinformatics (Oxford, England) , 25:i253–8, 07 2009. doi: 10.1093/bioinformatics/btp203. Cheng Mao, Yihong Wu, Jiaming X... | https://arxiv.org/abs/2502.17142v1 |
subset d efined in ( B.3). Fix(dij)i/\e}atio\slash=j. We are going to bound the number of possible choices for σ1, then bound the number of possible choices for σ2given a choice of σ1and so on; finally bounding the number of choices for σpgiven a choice of σ1,...,σ p−1. There are n!choices for σ1. In order to fix σ2givenσ... | https://arxiv.org/abs/2502.17142v1 |
anye,e′∈E,1≤i,j≤p; and, as described in ( 1.2), the covariance matrix of Gis Σ = (1−ρ)IN+ρ(I(n 2)⊗Jp) = def(1−ρ)IN+ρ˜J (C.6) whereJpis thep×pall-ones matrix. For the sake of convenience, we will set Mσ= ΣMσfor the rest of this proof. We will also note that, for symmetric positive U,V , TrUV≤ /ba∇dblV/ba∇dblopTrU. (C.7)... | https://arxiv.org/abs/2502.17142v1 |
SN) =/summationtext iλi. We will apply the following theorem from Fan et al. (2015 ): Proposition D.2 Letξ1,...,ξNbe independent centred random variables such that, for some A > 0, E[|ξi|k]≤Akk! (D.4) Then, setting Φ(x) =1√ 2π/integraltextx −∞e−u2 2du, ands2=/summationtext iE[ξ2 i], if1≪x3≪s, P/parenleftigg 1 sN/summa... | https://arxiv.org/abs/2502.17142v1 |
σ∈Sp−1 n σij(e)=e′e−βV(σ) Z = def2β/summationdisplay e′∈E,j/\e}atio\slash=iG(j) e′α(e,i),(e,j)(F.3) (We setα(e,i),(e′,j)= 0wheni=j.) Defined as such, we may notice that 0≤α(e,i),(e,j)≤1, and that for any fixed e′,j(resp. any fixed e,i), /summationdisplay e∈E,1≤i≤pα(e,i),(e′,j)= 2resp./summationdisplay e′∈Eα(e,i),(e′... | https://arxiv.org/abs/2502.17142v1 |
0≤(dij)1≤i/ne}ationslash=j≤p≤nF(dij)exp/parenleftigg −2(p−1)nlogn 1+1 p(p−1)n2/summationtext 1≤i/\e}atio\slash=j≤pd2 ij(1+O(ρ))/parenrightigg (F.19) However, instead of using Theorem 3.2to boundF(dij), we will need to use the more powerful Theorem B.1. For any τ∈ Sp, E[N2]≤/summationdisplay 0≤(dij)1≤i/ne}ationslash=j... | https://arxiv.org/abs/2502.17142v1 |
n/parenrightbigge∅(π)/productdisplay X⊆/llbracket1,p/rrbracket\∅/parenleftigg λs|X|(1−s)p−|X| n/parenrightiggeX(π)(H.3) 30 THE FEASIBILITY OF MULTI -GRAPH ALIGNMENT :ABAYESIAN APPROACH Since/summationtext X⊆/llbracket1,p/rrbracketeX(π) =/parenleftbign 2/parenrightbig is independent from π, we may therefore rewrite: P... | https://arxiv.org/abs/2502.17142v1 |
of S|Z. This allows us to use the following lemma to conclude. Lemma K.1 LetHbe a graph over some set V, with degree distribution (dv)v∈V. Then,Hhas at most/productdisplay v∈V(dv!)automorphisms which send each connected component of Hto itself. Indeed, by construction of X(π) =Z, any connected component of Sintersects ... | https://arxiv.org/abs/2502.17142v1 |
(since all of the edges of the spanning tree are ≥x) but the quantity2/summationtext 1≤i/ne}ationslash=j≤px2 ij 1+/summationtext 1≤i/ne}ationslash=j≤px2 ijwill have increased. Noting|B1|=k1and|B2|=k2=p−k1, this means that (p−1)S∗= (k1−1)S∗((xij)i,j∈B1)+(k2−1)S∗((xij)i,j∈B2)+x. (L.4) However, by induction, we have (k1−1... | https://arxiv.org/abs/2502.17142v1 |
Multivariate R´ enyi inaccuracy measures based on copulas: properties and application Shital Saha∗and Suchandan Kayal† Department of Mathematics, National Institute of Technology Rourkela, Rourkela-769008, Odisha, India Abstract We propose R´ enyi inaccuracy measure based on multivariate copula and multivariate sur- vi... | https://arxiv.org/abs/2502.17215v1 |
to investigate the yield loss probability to various drought conditions in south-eastern Australia. It is of recent interest to study copula-based information and divergence measures. For example, Ma and Sun (2011) combined the concepts of copula and entropy, and then in- troduced copula entropy. They established that ... | https://arxiv.org/abs/2502.17215v1 |
entropy is the most eminent measure of uncertainty (see R´ enyi (1961)), given by Hγ(X) =ψ(γ) logZ∞ 0fγ X(x)dx, 0< γ(̸= 1), (1.3) where ψ(γ) =1 1−γ.Note that Hγ(X) is a parametric generalization of H(X), which can be deduced taking γtending to 1 .For some applications of the Renyi entropy, the interested readers may re... | https://arxiv.org/abs/2502.17215v1 |
be from the same class) can be com- bined by copula. Further, it gives way more options when explaining relationship between different variables, since they do not restrict the dependence structure to be linear. In the main results, we have used multivariate survival copula in the place of the joint SF in (1.7) and the... | https://arxiv.org/abs/2502.17215v1 |
and preliminary results We recall and discuss various basic definitions and properties of copula functions. We note that although in the following sections most of the results are based on the multivariate 5 random vectors with more than 2 components, here we have presented the preliminary results for the bivariate ran... | https://arxiv.org/abs/2502.17215v1 |
isolate and analyse the dependency structure enhances traditional methods of studying information transfer, entropy, and mutual information in multivariate contexts. They are crucial for advancing applications in data science, signal processing, and statistical learning, where understanding complex dependencies is esse... | https://arxiv.org/abs/2502.17215v1 |
we conclude that the proposed copula-based inaccuracy measure is capable to capture more discrepancy between two multivariate copula functions than the CCI measure. 8 -1.0 -0.5 0.0 0.5 1.00123 θCCI(X,Y)CCRI(X,Y)(a) -1.0 -0.5 0.0 0.51.00.00.51.01.52.02.53.03.5 αCCI(X,Y)CCRI(X,Y) (b) 0 1 2 3 4 50.00.51.01.52.0 λ1CCI(X,Y)... | https://arxiv.org/abs/2502.17215v1 |
given by (3.11) and (3.12), respectively. Thus, the inequality in (3.8) follows. This completes the proof. Next, we discuss comparison study between two MCCRI measures. The comparison of two multi-dimensional inaccuracy measures are needed to understand the insights of the complex interactions and dependencies in multi... | https://arxiv.org/abs/2502.17215v1 |
logZ1 0···Z1 0CX(u1,···, un)n CZ G1(F−1 1(u1)),···, Gn(F−1 n(un))oγ−1 du1···dun ⇒CCRI (X,Y)≤CCRI (X,Z). Thus, the proof of Part ( A) is completed. Part ( B) can be established similarly. The lower orthant order in multivariate information theory provides a principle and rig- orous way to compare joint distributions, ... | https://arxiv.org/abs/2502.17215v1 |
Gn(F−1 n(un))oγ−1 du1···dun ≤ψ(γ) logZ1 0···Z1 0CX(u1,···, un)n CY G1(F−1 1(u1)),···, Gn(F−1 n(un))oγ−1 du1···dun ⇒CCRI (Z,Y)≤CCRI (X,Y). (3.31) Further, Z≤LOYandγ >1 together imply that n CZ(H1(F−1 1(u1)),···, Hn(F−1 n(un)))oγ−1 ≥n CY(G1(F−1 1(u1)),···, Gn(F−1 n(un)))oγ−1 ⇒ψ(γ) logZ1 0···Z1 0CX(u1,···, un)n CZ(H1(F... | https://arxiv.org/abs/2502.17215v1 |
is difficult to obtain the explicit forms of the MSCRI measure in (4.4) and SCI measure in (4.5). Thus, we have presented the graphs of these measures in order to study their behaviours with respect to θ, α, λ 1andλ2(see Figures 2 (a-d)). It is observed from these figures that the areas captured by the curves of the MS... | https://arxiv.org/abs/2502.17215v1 |
yn) × {G(y1,···, yn)}γ−1dy1···dyn =SCRI (X,Y). Therefore, the proof is finished. The comparison of two multivariate statistical inaccuracy measures are very important to select a better model. In the following we discuss the comparison study for two MSCRI measures. Proposition 4.3. Suppose X,YandZhave survival copula f... | https://arxiv.org/abs/2502.17215v1 |
(B)IfX≤UOZ≤UOYand the random variables Ziare i.d. with Yifori= 1,···, n∈N, then (i)forγ >1,SCRI (Z,X)≥SCRI (Y,X)≥SCRI (Y,Z); (ii)for0< γ < 1,SCRI (Y,X)≥maxn SCRI (Y,Z), SCRI (Z,X)o . (C)IfX≤UOZ≤UOYand the random variables Ziare i.d. with Xifori= 1,···, n∈N, then (i)forγ >1,SCRI (X,Z)≥SCRI (X,Y)≥SCRI (Z,Y); (ii)for0< γ ... | https://arxiv.org/abs/2502.17215v1 |
for the purpose of studying their behaviours with respect to θ, α, λ 1andλ2(see Figures 3 (a-d)). From Figure 3, we observe that MCoCRI and MDCRI are monotone functions. Remark 5.1. Properties similar to Propositions 3.2 and 3.3 can be obtained for the case of MCoCRI after replacing multivariate copula by multivariate ... | https://arxiv.org/abs/2502.17215v1 |
(2010) and Keziou and Regnault (2016). Using the semiparametric copula estimator in (3.2), we propose a semiparametric MCCRI estimator, given below. Definition 6.1. Suppose CX(·,·,·)andCY(·,·,·)are two trivariate copula functions of X andY, respectively. Then, the semiparametric estimator of MCCRI measure for 0< γ̸= 1 ... | https://arxiv.org/abs/2502.17215v1 |
W) copulas. For illustration purposes, we have chosen γ= 3. The values of MCCRI measures are reported in Table 3. From Table 3, we observe that MCCRI measure between Frank and Gumbel-Hougaarad copulas is lesser than the MCCRI measure between Frank and Joe copulas and Frank and Product copulas, as expected. Thus, we con... | https://arxiv.org/abs/2502.17215v1 |
and MDCRI measures and studied their various properties. A semiparametric es- timator has been proposed of MCCRI measure. In this regard, a Monte Carlo simulation study has been performed for illustration purposes. Using simulation, we have obtained the values of SD, AB and MSE of the proposed estimator in (6.1). Final... | https://arxiv.org/abs/2502.17215v1 |
Raton, FL, USA. Karci, A. (2016). Fractional order entropy: New perspectives, Optik .127(20), 9172–9177. Kayal, S., Madhavan, S. S. and Ganapathy, R. (2017). On dynamic generalized measures of inaccuracy, Statistica .77(2), 133–148. Kayal, S. and Sunoj, S. (2017). Generalized kerridge’s inaccuracy measure for condition... | https://arxiv.org/abs/2502.17215v1 |
M. and Shanthikumar, J. G. (2007). Stochastic orders , Springer. Shannon, C. E. (1948). A mathematical theory of communication, The Bell System Technical Journal . 27(3), 379–423. Sunoj, S. and Nair, N. U. (2023). Survival copula entropy and dependence in bivariate distributions: Accepted-february 2023, REVSTAT-Statist... | https://arxiv.org/abs/2502.17215v1 |
On the admissibility of bounds on the mean of discrete, scalar probability distributions from an iid sample Erik Learned-Miller February 25, 2025 Abstract We address the problem of producing a lower bound for the mean of a discrete probability distribution, with known support over a finite set of real numbers, from an ... | https://arxiv.org/abs/2502.17223v1 |
consistent with specific sample orderings . . . . . . . . . . . . . . . . . . . 10 2.3 How conditioning on a sample space ordering makes bound specification easy . . . 11 2.3.1 Error sets and upper sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.2 The central optimization problem . . . . . . . . . .... | https://arxiv.org/abs/2502.17223v1 |
the sample space of counts for the categories in S. In this work, we consider the problem of establishing a lower bound on the mean of such a cate- gorical distribution defined over a subset of real numbers from a multinomial sample. We consider the specific setting in which the support Sof the distribution is known, b... | https://arxiv.org/abs/2502.17223v1 |
validity, we investigate issues of admissibility and optimality of lower bounds on the mean. Admissible bounds are valid bounds that are not uniformly dominated by any other valid bound. That is, given an admissible bound Aand another valid bound Bover a sample space Ω, there is at least one sample in Ω for which Agive... | https://arxiv.org/abs/2502.17223v1 |
die would be S={1,2, ...,6}. Suppose one rolls such a die 5 times and obtains the values 3,4,2,6,2. We represent this as the sorted multinomial sample x= (2,2,3,4,6). Example 1.2. Consider a sample space S={0,1,3}and a sample size of n= 4. Then the induced sample space Ω(S, n)is {(0,0,0,0),(0,0,0,1),(0,0,0,3),(0,0,1,1)... | https://arxiv.org/abs/2502.17223v1 |
of the full sample space. Bottom. The multinomial likelihoods of the subsets on the top. The black lines illustrate a particular iso- contour of the probability function, which is relevant to the central optimization problem discussed below. Notice that each subset likelihood is a sum of some subset of the sample likel... | https://arxiv.org/abs/2502.17223v1 |
set is to analyze its performance with respect to each distribution in such a simplex. We now provide some more precise definitions. Definition 1.8 (open probability simplex and closed probability simplex) .LetSbe a finite sup- port set. Let G(S)represent the open set of probability distributions that assign strictly p... | https://arxiv.org/abs/2502.17223v1 |
with the stipulation that no two components of Tare equivalent. That is, Tis a permutation of the lexicographic ordering. Definition 1.13 (order consistent bound) .LetTbe a sample ordering for a sample space Ω. A bound BonΩisconsistent with the order T(ororder-consistent ) if and only if it satisfies B(xt1)≤B(xt2)≤...≤... | https://arxiv.org/abs/2502.17223v1 |
particular it is U+ 1, where Uis the number of unique values in the range of the bound. 1.3 Methods for comparing bounding functions Let Ω be a sample space over a support set Sand a sample size n. Let AandBbe two bounding functions which produce lower bounds for each sample xin the sample space. Of course, among multi... | https://arxiv.org/abs/2502.17223v1 |
B(at the same confidence level) such that EF[B(X)]> EF[A(X)] is a stronger bound with respect to the distribution F. Of course, for some other distribution Gover the same sample space this relationship could be reversed, with Astronger than B. One method for comparing two different bound functions across the full set o... | https://arxiv.org/abs/2502.17223v1 |
set of bound functions B. A bound is optimal if it is the only admissible bound inB. Equivalently, a bound Bis optimal with respect to the set Bif it is valid and it dominates every other valid bound in B. Note that a bound must be valid to be optimal. Also, an optimal bound need only dominate valid bounds, not invalid... | https://arxiv.org/abs/2502.17223v1 |
these sample-specific bounds specify an optimal bound function over the sample space, with respect to other bounds that obey the same total order. We proceed as follows. Given a total order T, suppose we wish to specify a lower bound value B(xtk) for the kth element in the order. That is, the total order specifies that... | https://arxiv.org/abs/2502.17223v1 |
Thus, in general, the error sets of a sample are supersets of the upper sets of the sample. We shall develop bounds based on an optimization over the upper sets of each sample. We will first argue that such a procedure produces order-consistent bounds when the resulting optimization results are all distinct, i.e., that... | https://arxiv.org/abs/2502.17223v1 |
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