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1.
Bimal Kumar Sinha 《Journal of multivariate analysis》1976,6(4):617-625
Treated in this paper is the problem of estimating with squared error loss the generalized variance | Σ | from a Wishart random matrix S: p × p Wp(n, Σ) and an independent normal random matrix X: p × k N(ξ, Σ Ik) with ξ(p × k) unknown. Denote the columns of X by X(1) ,…, X(k) and set ψ(0)(S, X) = {(n − p + 2)!/(n + 2)!} | S |, ψ(i)(X, X) = min[ψ(i−1)(S, X), {(n − p + i + 2)!/(n + i + 2)!} | S + X(1) X′(1) + + X(i) X′(i) |] and Ψ(i)(S, X) = min[ψ(0)(S, X), {(n − p + i + 2)!/(n + i + 2)!}| S + X(1) X′(1) + + X(i) X′(i) |], i = 1,…,k. Our result is that the minimax, best affine equivariant estimator ψ(0)(S, X) is dominated by each of Ψ(i)(S, X), i = 1,…,k and for every i, ψ(i)(S, X) is better than ψ(i−1)(S, X). In particular, ψ(k)(S, X) = min[{(n − p + 2)!/(n + 2)!} | S |, {(n − p + 2)!/(n + 2)!} | S + X(1)X′(1)|,…,| {(n − p + k + 2)!/(n + k + 2)!} | S + X(1)X′(1) + + X(k)X′(k)|] dominates all other ψ's. It is obtained by considering a multivariate extension of Stein's result (Ann. Inst. Statist. Math. 16, 155–160 (1964)) on the estimation of the normal variance. 相似文献
2.
Birkholl quadrature formulae (q.f.), which have algebraic degree of precision (ADP) greater than the number of values used, are studied. In particular, we construct a class of quadrature rules of ADP = 2n + 2r + 1 which are based on the information {ƒ(j)(−1), ƒ(j)(−1), j = 0, ..., r − 1 ; ƒ(xi), ƒ(2m)(xi), i = 1, ..., n}, where m is a positive integer and r = m, or r = m − 1. It is shown that the corresponding Birkhoff interpolation problems of the same type are not regular at the quadrature nodes. This means that the constructed quadrature formulae are not of interpolatory type. Finally, for each In, we prove the existence of a quadrature formula based on the information {ƒ(xi), ƒ(2m)(xi), i = 1, ..., 2m}, which has algebraic degree of precision 4m + 1. 相似文献
3.
Let Xi, i ≥ 1, be a sequence of φ-mixing random variables with values in a sample space (X, A). Let L(Xi) = P(i) for all i ≥ 1 and let
n, n ≥ 1, be classes of real-valued measurable functions on (X, A). Given any function g on (X, A), let Sn(g) = Σi = 1n {g(Xi) − Eg(Xi)}. Under weak metric entropy conditions on
n and under growth conditions on both the mixing coefficients and the maximal variance V V(n) maxi ≤ n supg
n ∫ g2 dP(i), we show that there is a numerical constant U < ∞ such that
a.s.
*, where
i = 1xP(i) and H H(n) is the square root of the entropy of the class
n. Additionally, the rate of convergence H−1(n/V)1/2 cannot, in general, be improved upon. Applications of this result are considered. 相似文献
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4.
Given a set function, that is, a map ƒ:
(E) →
{−∞} from the set
(E) of subsets of a finite set E into the reals including −∞, the standard greedy algorithm (GA) for optimizing ƒ starts with the empty set and then proceeds by enlarging this set greedily, element by element. A set function ƒ is said to be tractable if in this way a sequence x0 , x1, . . ., xN E (N #E) of subsets with max(ƒ) {ƒ(x0), ƒ(x1), . . ., ƒ(xN)} will always be found. In this note, we will reinterpret and transcend the traditions of classical GA-theory (cf., e.g., [KLS]) by establishing necessary and sufficient conditions for a set function ƒ not just to be tractable as it stands, but to give rise to a whole family of tractable set functions ƒ(η) :
(E) →
: x ƒ(x) + Σe xη(e), where η runs through all real valued weighting schemes η : E →
, in which case ƒ will be called rewarding. In addition, we will characterize two important subclasses of rewarding maps, viz. truncatably rewarding (or well-layered) maps, that is, set functions ƒ such that [formula] is rewarding for every i = 1, . . ., N, and matroidal maps, that is, set functions ƒ such that for every η : E →
and every ƒeta-greedy sequence x0, x1, . . ., xN as above, one has max(ƒη) = ƒη(xi) for the unique i {0, . . ., N} with ƒη(x0) < ƒη(x1) < ··· < ƒη(xi) ≥ ƒη(xi + 1). 相似文献
5.
Let ga(t) and gb(t) be two positive, strictly convex and continuously differentiable functions on an interval (a, b) (−∞ a < b ∞), and let {Ln} be a sequence of linear positive operators, each with domain containing 1, t, ga(t), and gb(t). If Ln(ƒ; x) converges to ƒ(x) uniformly on a compact subset of (a, b) for the test functions ƒ(t) = 1, t, ga(t), gb(t), then so does every ƒ ε C(a, b) satisfying ƒ(t) = O(ga(t)) (t → a+) and ƒ(t) = O(gb(t)) (t → b−). We estimate the convergence rate of Lnƒ in terms of the rates for the test functions and the moduli of continuity of ƒ and ƒ′. 相似文献
6.
We study the bootstrap distribution for U-statistics with special emphasis on the degenerate case. For the Efron bootstrap we give a short proof of the consistency using Mallows′ metrics. We also study the i.i.d. weighted bootstrap [formula] where (Xi) and (ξi) are two i.i.d. sequences, independent of each other and where Eξi = 0, Var(ξi) = 1. It turns out that, conditionally given (Xi), this random quadratic form converges weakly to a Wiener-Ito double stochastic integral ∫10 ∫10h(F−1(x), F−1(y)) dW(x) dW(y). As a by-product we get an a.s. limit theorem for the eigenvalues of the matrix Hn=((1/n)h(Xi, Xj))1 ≤ i, j ≤ n. 相似文献
7.
Simsa J. 《Journal of Approximation Theory》1994,76(3)
It is known that if a smooth function h in two real variables x and y belongs to the class Σn of all sums of the form Σnk=1ƒk(x) gk(y), then its (n + 1)th order "Wronskian" det[hxiyj]ni,j=0 is identically equal to zero. The present paper deals with the approximation problem h(x, y) Σnk=1ƒk(x) gk(y) with a prescribed n, for general smooth functions h not lying in Σn. Two natural approximation sums T=T(h) Σn, S=S(h) Σn are introduced and the errors |h-T|, |h-S| are estimated by means of the above mentioned Wronskian of the function h. The proofs utilize the technique of ordinary linear differential equations. 相似文献
8.
We consider boolean circuits C over the basis Ω={,} with inputs x1, x2,…,xn for which arrival times are given. For 1in we define the delay of xi in C as the sum of ti and the number of gates on a longest directed path in C starting at xi. The delay of C is defined as the maximum delay of an input.Given a function of the form
f(x1,x2,…,xn)=gn−1(gn−2(…g3(g2(g1(x1,x2),x3),x4)…,xn−1),xn)