Let E be a Moran set on R1 associated with a bounded closed interval J and two sequences(nk)k =1 ^∞ and(Ck =(ck,j)j=1)^nk)k≥1. Let μ be the Moran measure on E associated with a sequence(Pk)k≥1 of positive probabil...Let E be a Moran set on R1 associated with a bounded closed interval J and two sequences(nk)k =1 ^∞ and(Ck =(ck,j)j=1)^nk)k≥1. Let μ be the Moran measure on E associated with a sequence(Pk)k≥1 of positive probability vectors with Pk =(pk,j)j =1,^nk, k ≥ 1. We assume that k≥1 1≤j≤nk^inf min Ck,j>0,k≥1 1≤j≤nk^inf minCk,j>0,k≥1 1≤j≤nk ^inf min pk,j>0. For every n ≥ 1, let αn be αn n optimal set in the quantization for μ of order r ∈(0,∞) and{Pa(αn)}a∈α∈an an arbitrary Voronoi partition with respect to αn. We write Iα(α,μ):=∫Pα(αn)^d(x,αn)^τ dμ(x),α∈αn;J(αn,μ):=α∈αn^minⅠα(α,μ),J(αn,μ):=α∈αn^max Ⅰα(α,μ).We show that J(αn,μ),J(αn,μ) and en^r,r(μ)-en^r +1,r(μ) are of the same order as 1/n en^r ,r(μ), where en^r ,r(μ):=∫d(x,αn)^r dμ(x) is the nth quantization error for μ of order r. In particular, for the class of Moran measures on R1, our result shows that a weaker version of Gersho’s conjecture holds.展开更多
Consider the standard non-linear regression model y_i=g(x_i,θ_o)+ε_i,i=1,...,n whereg(x,θ)is a continuous function on a bounded closed region X×Θ,θ_o is the unknown parametervector in θ■R_p,{x_1,x_2,...,x_...Consider the standard non-linear regression model y_i=g(x_i,θ_o)+ε_i,i=1,...,n whereg(x,θ)is a continuous function on a bounded closed region X×Θ,θ_o is the unknown parametervector in θ■R_p,{x_1,x_2,...,x_n}is a deterministic design of experiment and{ε_1,ε_2,...,ε_n}is asequence of independent random variables.This paper establishes the existences of M-estimates andthe asymptotic uniform linearity of M-scores in a family of non-linear regression models when theerrors are independent and identically distributed.This result is then used to obtain the asymptoticdistribution of a class of M-estimators for a large class of non-linear regression models.At the sametime,we point out that Theorem 2 of Wang(1995)(J.of Multivariate Analysis,vol.54,pp.227-238,Corrigenda.vol.55,p.350)is not correct.展开更多
For a subset K of a metric space （X,d） and x ∈ X,Px（x）={y ∈ K ： d（x,y） = d（x,K）≡ inf{d（x,k） ： k ∈ K}}is called the set of best K-approximant to x. An element go E K is said to be a best simulta- neous ...For a subset K of a metric space （X,d） and x ∈ X,Px（x）={y ∈ K ： d（x,y） = d（x,K）≡ inf{d（x,k） ： k ∈ K}}is called the set of best K-approximant to x. An element go E K is said to be a best simulta- neous approximation of the pair y1,y2 E ∈ if max{d（y1,go）,d（y2,go）}=inf g∈K max {d（y1,g）,d（y2,g）}.In this paper, some results on the existence of common fixed points for Banach operator pairs in the framework of convex metric spaces have been proved. For self mappings T and S on K, results are proved on both T- and S- invariant points for a set of best simultaneous approximation. Some results on best K-approximant are also deduced. The results proved generalize and extend some results of I. Beg and M. Abbas^[1], S. Chandok and T.D. Narang^[2], T.D. Narang and S. Chandok^[11], S.A. Sahab, M.S. Khan and S. Sessa^[14], P. Vijayaraju^[20] and P. Vijayaraju and M. Marudai^[21].展开更多
基金supported by National Natural Science Foundation of China(Grant No.11571144)
文摘Let E be a Moran set on R1 associated with a bounded closed interval J and two sequences(nk)k =1 ^∞ and(Ck =(ck,j)j=1)^nk)k≥1. Let μ be the Moran measure on E associated with a sequence(Pk)k≥1 of positive probability vectors with Pk =(pk,j)j =1,^nk, k ≥ 1. We assume that k≥1 1≤j≤nk^inf min Ck,j>0,k≥1 1≤j≤nk^inf minCk,j>0,k≥1 1≤j≤nk ^inf min pk,j>0. For every n ≥ 1, let αn be αn n optimal set in the quantization for μ of order r ∈(0,∞) and{Pa(αn)}a∈α∈an an arbitrary Voronoi partition with respect to αn. We write Iα(α,μ):=∫Pα(αn)^d(x,αn)^τ dμ(x),α∈αn;J(αn,μ):=α∈αn^minⅠα(α,μ),J(αn,μ):=α∈αn^max Ⅰα(α,μ).We show that J(αn,μ),J(αn,μ) and en^r,r(μ)-en^r +1,r(μ) are of the same order as 1/n en^r ,r(μ), where en^r ,r(μ):=∫d(x,αn)^r dμ(x) is the nth quantization error for μ of order r. In particular, for the class of Moran measures on R1, our result shows that a weaker version of Gersho’s conjecture holds.
基金This research was supported by the Natural science Foundation of china(Grant No.19831010 and grant No.39930160)and the Doctoral Foundation of China
文摘Consider the standard non-linear regression model y_i=g(x_i,θ_o)+ε_i,i=1,...,n whereg(x,θ)is a continuous function on a bounded closed region X×Θ,θ_o is the unknown parametervector in θ■R_p,{x_1,x_2,...,x_n}is a deterministic design of experiment and{ε_1,ε_2,...,ε_n}is asequence of independent random variables.This paper establishes the existences of M-estimates andthe asymptotic uniform linearity of M-scores in a family of non-linear regression models when theerrors are independent and identically distributed.This result is then used to obtain the asymptoticdistribution of a class of M-estimators for a large class of non-linear regression models.At the sametime,we point out that Theorem 2 of Wang(1995)(J.of Multivariate Analysis,vol.54,pp.227-238,Corrigenda.vol.55,p.350)is not correct.
文摘For a subset K of a metric space （X,d） and x ∈ X,Px（x）={y ∈ K ： d（x,y） = d（x,K）≡ inf{d（x,k） ： k ∈ K}}is called the set of best K-approximant to x. An element go E K is said to be a best simulta- neous approximation of the pair y1,y2 E ∈ if max{d（y1,go）,d（y2,go）}=inf g∈K max {d（y1,g）,d（y2,g）}.In this paper, some results on the existence of common fixed points for Banach operator pairs in the framework of convex metric spaces have been proved. For self mappings T and S on K, results are proved on both T- and S- invariant points for a set of best simultaneous approximation. Some results on best K-approximant are also deduced. The results proved generalize and extend some results of I. Beg and M. Abbas^[1], S. Chandok and T.D. Narang^[2], T.D. Narang and S. Chandok^[11], S.A. Sahab, M.S. Khan and S. Sessa^[14], P. Vijayaraju^[20] and P. Vijayaraju and M. Marudai^[21].
