共查询到20条相似文献,搜索用时 284 毫秒
1.
M. G. Pleshakov 《Journal of Approximation Theory》1999,99(2):519
Let 2s points yi=−πy2s<…<y1<π be given. Using these points, we define the points yi for all integer indices i by the equality yi=yi+2s+2π. We shall write fΔ(1)(Y) if f is a 2π-periodic continuous function and f does not decrease on [yi, yi−1], if i is odd; and f does not increase on [yi, yi−1], if i is even. In this article the following Theorem 1—the comonotone analogue of Jackson's inequality—is proved.
1. If fΔ(1)(Y), then for each nonnegative integer n there is a trigonometric polynomial τn(x) of order n such that τnΔ(1)(Y), and |f(x)−πn(x)|c(s) ω(f; 1/(n+1)), x
, where ω(f; t) is the modulus of continuity of f, c(s)=const. Depending only on s. 相似文献
2.
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
3.
Let F(s, t) = P(X > s, Y > t) be the bivariate survival function which is subject to random censoring. Let
be the bivariate product limit estimator (PL-estimator) by Campbell and Földes (1982, Proceedings International Colloquium on Non-parametric Statistical Inference, Budapest 1980, North-Holland, Amsterdam). In this paper, it was shown that
, where {ζi(s, t)} is i.i.d. mean zero process and Rn(s, t) is of the order O((n−1log n)3/4) a.s. uniformly on compact sets. Weak convergence of the process {n−1 Σi = 1n ζi(s, t)} to a two-dimensional-time Gaussian process is shown. The covariance structure of the limiting Gaussian process is also given. Corresponding results are also derived for the bootstrap estimators. The result can be extended to the multivariate cases and are extensions of the univariate case of Lo and Singh (1986, Probab. Theory Relat. Fields, 71, 455–465). The estimator
is also modified so that the modified estimator is closer to the true survival function than
in supnorm. 相似文献
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4.
Let B be a real separable Banach space with norm |ß|B, X, X1, X2, … be a sequence of centered independent identically distributed random variables taking values in B. Let sn = sn(t), 0 ≤ t ≤ 1 be the random broken line such that sn(0) = 0, sn(k/n) = n−1/2 Σi=1k Xi for n = 1, 2, … and k = 1, …, n. Denote |sn|B = sup0 ≤ t ≤ 1 |sn(t)|B and assume that w(t), 0 ≤ t ≤ 1 is the Wiener process such that covariances of w(1) and X are equal. We show that under appropriate conditions P(|sn|B > r) = P(|w|B > r)(1 + o(1)) and give estimates of the remainder term. The results are new already in the case of B having finite dimension. 相似文献
5.
Lofstrom J. 《Journal of Approximation Theory》1993,73(3)
We consider best approximation in Lp(
), 1 ≤ p ≤ ∞, by means of entire functions y of exponential type subject to additional constraints Γj(y) = 0, j = 1, ..., K. Here Γj are (unbounded) linear functionals of the form Γj(y) = Dny(sj) − ∑ akDky(sj) where sj are fixed points. 相似文献
6.
Let A = (aij) be an n × n Toeplitz matrix with bandwidth k + 1, K = r + s, that is, aij = aj−i, i, J = 1,… ,n, ai = 0 if i > s and if i < -r. We compute p(λ)= det(A - λI), as well as p(λ)/p′(λ), where p′(λ) is the first derivative of p(λ), by using O(k log k log n) arithmetic operations. Moreover, if ai are m × m matrices, so that A is a banded Toeplitz block matrix, then we compute p(λ), as well as p(λ)/p′(λ), by using O(m3k(log2 k + log n) + m2k log k log n) arithmetic operations. The algorithms can be extended to the computation of det(A − λB) and of its first derivative, where both A and B are banded Toeplitz matrices. The algorithms may be used as a basis for iterative solution of the eigenvalue problem for the matrix A and of the generalized eigenvalue problem for A and B. 相似文献
7.
Kazuo Nakajima S. Louis Hakimi 《Journal of Algorithms in Cognition, Informatics and Logic》1982,3(4):344-361
Suppose that n independent tasks are to be scheduled without preemption on a set of identical parallel processors. Each task Ti requires a given execution time τi and it may be started for execution on any processor at any of its prescribed starting times si1, si2, …, siki, with ki ≤ k for some fixed integer k. We first prove that the problem of finding a feasible schedule on a single processor is NP-complete in the strong sense even when τi ε {τ, τ′} and ki ≤ 3 for 1 ≤ i ≤ n. The same problem is, however, shown to be solvable in O(n log n) time, provided siki − si1 < τi for 1 ≤ i ≤ n. We then show that the problem of finding a feasible schedule on an arbitrary number of processors is strongly NP-complete even when τi ε {τ, τ′}, ki = 2 and si2 − si1 = δ < τi for 1 ≤ i ≤ n. Finally a special case with ki = 2 and si2 − si1 = 1, 1 ≤ i ≤ n, of the above multiprocessor scheduling problem is shown to be solvable in polynomial time. 相似文献
8.
Let X1,…, Xn be i.i.d. random variables symmetric about zero. Let Ri(t) be the rank of |Xi − tn−1/2| among |X1 − tn−1/2|,…, |Xn − tn−1/2| and Tn(t) = Σi = 1nφ((n + 1)−1Ri(t))sign(Xi − tn−1/2). We show that there exists a sequence of random variables Vn such that sup0 ≤ t ≤ 1 |Tn(t) − Tn(0) − tVn| → 0 in probability, as n → ∞. Vn is asymptotically normal. 相似文献
9.
In this paper a form of the Lindeberg condition appropriate for martingale differences is used to obtain asymptotic normality of statistics for regression and autoregression. The regression model is yt = Bzt + vt. The unobserved error sequence {vt} is a sequence of martingale differences with conditional covariance matrices {Σt} and satisfying supt=1,…, n
{v′tvtI(v′tvt>a) |zt, vt−1, zt−1, …}
0 as a → ∞. The sample covariance of the independent variables z1, …, zn, is assumed to have a probability limit M, constant and nonsingular; maxt=1,…,nz′tzt/n
0. If (1/n)Σt=1nΣt
Σ, constant, then √nvec(
n−B)
N(0,M−1Σ) and
n
Σ. The autoregression model is xt = Bxt − 1 + vt with the maximum absolute value of the characteristic roots of B less than one, the above conditions on {vt}, and (1/n)Σt=max(r,s)+1(Σtvt−1−rv′t−1−s)
δrs(ΣΣ), where δrs is the Kronecker delta. Then √nvec(
n−B)
N(0,Γ−1Σ), where Γ = Σs = 0∞BsΣ(B′)s. 相似文献
10.
Rong-Qing Jia 《Journal of Approximation Theory》1983,37(4):293-310
For an integer k 1 and a geometric mesh (qi)−∞∞ with q ε (0, ∞), let Mi,k(x): = k[qi + k](· − x)+k − 1, Ni,k(x): = (qi + k − qiMi,k(x)/k, and let Ak(q) be the Gram matrix (∝Mi,kNj,k)i,jεz. It is known that Ak(q)−1∞ is bounded independently of q. In this paper it is shown that Ak(q)−1∞ is strictly decreasing for q in [1, ∞). In particular, the sharp upper bound and lower bound for Ak (q)−1 are obtained: for all q ε (0, ∞). 相似文献
11.
A graph is k-linked if for every set of 2k distinct vertices {s1,…,sk,t1,…,tk} there exist disjoint paths P1,…,Pk such that the endpoints of Pi are si and ti. We prove every 6-connected graph on n vertices with 5n−14 edges is 3-linked. This is optimal, in that there exist 6-connected graphs on n vertices with 5n−15 edges that are not 3-linked for arbitrarily large values of n. 相似文献
12.
F. Mricz 《Journal of multivariate analysis》1989,30(2)
This is a systematic and unified treatment of a variety of seemingly different strong limit problems. The main emphasis is laid on the study of the a.s. behavior of the rectangular means ζmn = 1/(λ1(m) λ2(n)) Σi=1m Σk=1n Xik as either max{m, n} → ∞ or min{m, n} → ∞. Here {Xik: i, k ≥ 1} is an orthogonal or merely quasi-orthogonal random field, whereas {λ1(m): m ≥ 1} and {λ2(n): n ≥ 1} are nondecreasing sequences of positive numbers subject to certain growth conditions. The method applied provides the rate of convergence, as well. The sufficient conditions obtained are shown to be the best possible in general. Results on double subsequences and 1-parameter limit theorems are also included. 相似文献
13.
The existence of positive solutions of the Fredholm nonlinear equation y(t) = h(t) + ∫T0k(t, s)[f(y(s)) + g(y(s))] ds is discussed. It is assumed that f is a continuous, nondecreasing function and g is continuous, nonincreasing, and possibly singular. 相似文献
14.
H. A. Dzyubenko 《Ukrainian Mathematical Journal》2009,61(4):519-540
In the case where a 2π-periodic function f is twice continuously differentiable on the real axis ℝ and changes its monotonicity at different fixed points y
i
∈ [− π, π), i = 1,…, 2s, s ∈ ℕ (i.e., on ℝ, there exists a set Y := {y
i
}
i∈ℤ of points y
i
= y
i+2s
+ 2π such that the function f does not decrease on [y
i
, y
i−1] if i is odd and does not increase if i is even), for any natural k and n, n ≥ N(Y, k) = const, we construct a trigonometric polynomial T
n
of order ≤n that changes its monotonicity at the same points y
i
∈ Y as f and is such that
*20c || f - Tn || £ \fracc( k,s )n2\upomega k( f",1 \mathord\vphantom 1 n n ) ( || f - Tn || £ \fracc( r + k,s )nr\upomega k( f(r),1 \mathord | / |
\vphantom 1 n n ), f ? C(r), r 3 2 ), \begin{array}{*{20}{c}} {\left\| {f - {T_n}} \right\| \leq \frac{{c\left( {k,s} \right)}}{{{n^2}}}{{{\upomega }}_k}\left( {f',{1 \mathord{\left/{\vphantom {1 n}} \right.} n}} \right)} \\ {\left( {\left\| {f - {T_n}} \right\| \leq \frac{{c\left( {r + k,s} \right)}}{{{n^r}}}{{{\upomega }}_k}\left( {{f^{(r)}},{1 \mathord{\left/{\vphantom {1 n}} \right.} n}} \right),\quad f \in {C^{(r)}},\quad r \geq 2} \right),} \\ \end{array} 相似文献
15.
Let {Xn} be a strictly stationary φ-mixing process with Σj=1∞ φ1/2(j) < ∞. It is shown in the paper that if X1 is uniformly distributed on the unit interval, then, for any t [0, 1], |Fn−1(t) − t + Fn(t) − t| = O(n−3/4(log log n)3/4) a.s. and sup0≤t≤1 |Fn−1(t) − t + Fn(t) − t| = (O(n−3/4(log n)1/2(log log n)1/4) a.s., where Fn and Fn−1(t) denote the sample distribution function and tth sample quantile, respectively. In case {Xn} is strong mixing with exponentially decaying mixing coefficients, it is shown that, for any t [0, 1], |Fn−1(t) − t + Fn(t) − t| = O(n−3/4(log n)1/2(log log n)3/4) a.s. and sup0≤t≤1 |Fn−1(t) − t + Fn(t) − t| = O(n−3/4(log n)(log log n)1/4) a.s. The results are further extended to general distributions, including some nonregular cases, when the underlying distribution function is not differentiable. The results for φ-mixing processes give the sharpest possible orders in view of the corresponding results of Kiefer for independent random variables. 相似文献
16.
Miroslav Bartu
ek 《Journal of Mathematical Analysis and Applications》2003,280(2):232-240
In the paper sufficient conditions are given under which the differential equation y(n)=f(t,y,…,y(n−2))g(y(n−1)) has a singular solution y :[T,τ)→R, τ<∞ fulfilling
17.
Let F be a Banach space with a sufficiently smooth norm. Let (Xi)i≤n be a sequence in LF2, and T be a Gaussian random variable T which has the same covariance as X = Σi≤nXi. Assume that there exists a constant G such that for s, δ≥0, we have P(sTs+δ)Gδ. (*) We then give explicit bounds of Δ(X) = supi|P(|X|≤t)−P(|T|≤t)| in terms of truncated moments of the variables Xi. These bounds hold under rather mild weak dependence conditions of the variables. We also construct a Gaussian random variable that violates (*). 相似文献
18.
M. Deza 《Journal of Combinatorial Theory, Series A》1976,20(3):306-318
Le nombre maximal de lignes de matrices seront désignées par:
19.
Let ƒ be a continuous map of the compact unit interval I = [0, 1], such that ƒ2, the second iterate of ƒ, is topologically transitive in I. If for some x and y in I and any t in I there exists lim(1/n) # {i ≤ n; |ƒi(x) − ƒi(y)| < t} for n → ∞, denote it by φxy(t). In the paper we consider the class
(ƒ) if all φxy. The main results are that
(ƒ) is convex and pointwise closed. Using this we show that
(ƒ) is always bigger than the class
(ƒ) of probability distributions generated analogously by single trajectories (and corresponding to the class of probability invariant measures of ƒ), and prove that there are universal generators of probability distributions, i.e., maps ƒ such that
(ƒ) is the class
of all non-decreasing functions I I (contrary to this,
(ƒ)
for no ƒ). These results can be extended to more general continuous maps. One of the possible applications is to use the size of
(ƒ) as a measure of the degree of chaos of ƒ. 相似文献
20.
《Applied Mathematics Letters》2000,13(1):115-120
Existence results are presented for the singular Volterra integral equation y(t) = h(t) + ∫0t k(t, s) f(s, y(s)) ds, for t ∈ [0,T]. Here f may be singular at y = 0. As a consequence new results are presented for the nth order singular initial value problem. 相似文献
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