共查询到20条相似文献,搜索用时 31 毫秒
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.
The well known Shannon entroy −∑ pk log pk satisfies the inequality −∑ pk log pk − ∑ pk log qk. Extensive studies have been made on the inequality ∑ pkƒk(qk) ∑ pkƒk(pk) which contains the above inequality as a special case. In this paper, we consider the most general inequality ∑ gk(pk)ƒk(pk) ∑ gk(pk)ƒk(qk) above type and obtain its general solution on an open domain. 相似文献
3.
Weiyi Su 《Journal of Approximation Theory》1986,47(4)
At present there are only a few approximate identity kernels for the Walsh system, for example, the pN-truncated Dirichlet kernel DpN − 1(t) = ∑j = 0pN − 1 wj(t) [6]; the Abel-Poisson kernel λγ(t) = ∑k = 0∞ γkwk(t) [3], and so on. In [6], Zheng has introduced a new kind of approximate identity kernels for the Walsh system—the kernels of product type. In the present paper we discuss the approximation properties of such product type kernels. Estimates of their moments as well as a direct approximation theorem are obtained. Then, to establish an inverse approximation theorem, we need the p-adic derivative of product type kernels and we estimate this derivative in L1-norm. 相似文献
4.
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
5.
Summability of spherical h-harmonic expansions with respect to the weight function ∏j=1d |xj|2κj (κj0) on the unit sphere Sd−1 is studied. The main result characterizes the critical index of summability of the Cesàro (C,δ) means of the h-harmonic expansion; it is proved that the (C,δ) means of any continuous function converge uniformly in the norm of C(Sd−1) if and only if δ>(d−2)/2+∑j=1d κj−min1jd κj. Moreover, it is shown that for each point not on the great circles defined by the intersection of the coordinate planes and Sd−1, the (C,δ) means of the h-harmonic expansion of a continuous function f converges pointwisely to f if δ>(d−2)/2. Similar results are established for the orthogonal expansions with respect to the weight functions ∏j=1d |xj|2κj(1−|x|2)μ−1/2 on the unit ball Bd and ∏j=1d xjκj−1/2(1−|x|1)μ−1/2 on the simplex Td. As a related result, the Cesàro summability of the generalized Gegenbauer expansions associated to the weight function |t|2μ(1−t2)λ−1/2 on [−1,1] is studied, which is of interest in itself. 相似文献
6.
Let T = {T(t)}t ≥ 0 be a C0-semigroup on a Banach space X. In this paper, we study the relations between the abscissa ωLp(T) of weak p-integrability of T (1 ≤ p < ∞), the abscissa ωpR(A) of p-boundedness of the resolvent of the generator A of T (1 ≤ p ≤ ∞), and the growth bounds ωβ(T), β ≥ 0, of T. Our main results are as follows.
- 1. (i) Let T be a C0-semigroup on a B-convex Banach space such that the resolvent of its generator is uniformly bounded in the right half plane. Then ω1 − ε(T) < 0 for some ε > 0.
- 2. (ii) Let T be a C0-semigroup on Lp such that the resolvent of the generator is uniformly bounded in the right half plane. Then ωβ(T) < 0 for all β>¦1/p − 1/p′¦, 1/p + 1/p′ = 1.
- 3. (iii) Let 1 ≤ p ≤ 2 and let T be a weakly Lp-stable C0-semigroup on a Banach space X. Then for all β>1/p we have ωβ(T) ≤ 0.
7.
Certain path properties of a symmetric α-stable process X(t) = ∫Sh(t, s) dM(s), t T, are studied in terms of the kernel h. The existence of an appropriate modification of the kernel h enables one to use results from stable measures on Banach spaces in studying X. Bounds for the moments of the norm of sample paths of X are obtained. This yields definite bounds for the moments of a double α-stable integral. Also, necessary and sufficient conditions for the absolute continuity of sample paths of X are given. Along with the above stochastic integral representation of stable processes, the representation of stable random vectors due to[13], Ann. Probab.9, 624–632) is extensively used and the relationship between these two representations is discussed. 相似文献
8.
In this paper we study the asymptotic behaviors of the likelihood ratio criterion (TL(s)), Watson statistic (TW(s)) and Rao statistic (TR(s)) for testing H0s: μ
(a given subspace) against H1s: μ
, based on a sample of size n from a p-variate Langevin distribution Mp(μ, κ) when κ is large. For the case when κ is known, asymptotic expansions of the null and nonnull distributions of these statistics are obtained. It is shown that the powers of these statistics are coincident up to the order κ−1. For the case when κ is unknown, it is shown that TR(s) TL(s) TW(s) in their powers up to the order κ−1. 相似文献
9.
M. Deza 《Journal of Combinatorial Theory, Series A》1976,20(3):306-318
Le nombre maximal de lignes de matrices seront désignées par:
- 1. (a) R(k, λ) si chaque ligne est une permutation de nombres 1, 2,…, k et si chaque deux lignes différentes coïncide selon λ positions;
- 2. (b) S0(k, λ) si le nombre de colonnes est k et si chaque deux lignes différentes coïncide selon λ positions et si, en plus, il existe une colonne avec les éléments y1, y2, y3, ou y1 = y2 ≠ y3;
- 3. (c) T0(k, λ) si c'est une (0, 1)-matrice et si chaque ligne contient k unités et si chaque deux lignes différentes contient les unités selon λ positions et si, en plus, il existe une colonne avec les éléments 1, 1, 0.
10.
Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables 总被引:1,自引:0,他引:1
Consider Z+d (d2)—the positive d-dimensional lattice points with partial ordering , let {Xk,kZ+d} be i.i.d. random variables with mean 0, and set Sn=∑knXk, nZ+d. We establish precise asymptotics for ∑n|n|r/p−2P(|Sn||n|1/p), and for
, (0δ1) as 0, and for
as
. 相似文献
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11.
Let X ≡ (X1, …, Xt) have a multinomial distribution based on N trials with unknown vector of cell probabilities p ≡ (p1, …, pt). This paper derives admissibility and complete class results for the problem of simultaneously estimating p under entropy loss (EL) and squared error loss (SEL). Let
and f(x¦p) denote the (t − 1)-dimensional simplex, the support of X and the probability mass function of X, respectively. First it is shown that δ is Bayes w.r.t. EL for prior P if and only if δ is Bayes w.r.t. SEL for P. The admissible rules under EL are proved to be Bayes, a result known for the case of SEL. Let Q denote the class of subsets of
of the form T = j=1kFj where k ≥ 1 and each Fj is a facet of
which satisfies: F a facet of
such that F naFjF ncT. The minimal complete class of rules w.r.t. EL when N ≥ t − 1 is characterized as the class of Bayes rules with respect to priors P which satisfy P(
0) = 1, ξ(x) ≡ ∫ f(x¦p) P(dp) > 0 for all x in {x
: sup
0 f(x¦p) > 0} for some
0 in Q containing all the vertices of
. As an application, the maximum likelihood estimator is proved to be admissible w.r.t. EL when the estimation problem has parameter space Θ =
but it is shown to be inadmissible for the problem with parameter space Θ = (
minus its vertices). This is a severe form of “tyranny of boundary.” Finally it is shown that when N ≥ t − 1 any estimator δ which satisfies δ(x) > 0 x
is admissible under EL if and only if it is admissible under SEL. Examples are given of nonpositive estimators which are admissible under SEL but not under EL and vice versa. 相似文献
12.
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
13.
Krzysztof Przes
awski 《Journal of Approximation Theory》1996,85(3):288-296
It is shown that for each convex bodyARnthere exists a naturally defined family
AC(Sn−1) such that for everyg
A, and every convex functionf: R→Rthe mappingy∫Sn−1 f(g(x)−y, x) dσ(x) has a minimizer which belongs toA. As an application, approximation of convex bodies by balls with respect toLpmetrics is discussed. 相似文献
14.
Christian Elbert 《Journal of Approximation Theory》2001,109(2):708
For the horizontal generating functions Pn(z)=∑nk=1 S(n, k) zk of the Stirling numbers of the second kind, strong asymptotics are established, as n→∞. By using the saddle point method for Qn(z)=Pn(nz) there are two main results: an oscillating asymptotic for z(−e, 0) and a uniform asymptotic on every compact subset of
\[−e, 0]. Finally, an Airy asymptotic in the neighborhood of −e is deduced. 相似文献
15.
On a simplex SRd, the best polynomial approximation is En()Lp(S)=Inf{Pn−Lp(S): Pn of total degree n}. The Durrmeyer modification, Mn, of the Bernstein operator is a bounded operator on Lp(S) and has many “nice” properties, most notably commutativity and self-adjointness. In this paper, relations between Mn−z.dfnc;Lp(S) and E[√n]()Lp(S) will be given by weak inequalities will imply, for 0<α<1 and 1≤p≤∞, En()Lp(S)=O(n-2α)Mn−z.dfnc;Lp(S)=O(n-α). We also see how the fact that P(D)εLp(S) for the appropriate P(D) affects directional smoothness. 相似文献
16.
L∞ estimates are derived for the oscillatory integral ∫+0∞e−i(xλ + (1/m) tλm)a(λ) dλ, where 2 ≤ m
and (x, t)
×
+. The amplitude a(λ) can be oscillatory, e.g., a(λ) = eit
(λ) with
(λ) a polynomial of degree ≤ m − 1, or it can be of polynomial type, e.g., a(λ) = (1 + λ)k with 0 ≤ k ≤
(m − 2). The estimates are applied to the study of solutions of certain linear pseudodifferential equations, of the generalized Schrödinger or Airy type, and of associated semilinear equations. 相似文献
17.
Bergeron N. 《Advances in Mathematics》1995,110(2)
We construct Families {ρl, kn} of orthogonal idempotents of the hyperoctahedral group algebras
[Bn], which commute with the Hochschild boundary operators bn=∑ni=0 (−1)idi. We show that those idempotents are projections onto some hyperoactahedral symmetric powers of the free Lie algebra Lie(l, k)n(
). The commutations above then decompose the Hochshild homology Hn(C) obtained by any functor C:Δop → K-module that factor through Fin′B, the hyperoctahedral category. Moreover, we show that this decomposition is the finest possible for any such functor. In particular, the Hochschild homology of a commutative algebra equipped with an involutive automorphism splits into components indexed by (l, k) and the corresponding Harrison homology splits into two components indexed by (0, 1) and (1, 0). Generalizing the Harrison complex, we show that H(l,k)n(C)Hn(Sh(l,k)./Sh(l−1, k+1)., where Sh(r,s).are some shuffle complexes associated to C. We also give the characters of the representations related to Lie(l, k)n(
) as a direct sum of induced characters. 相似文献
18.
Pankaj K. Agarwal Sandeep Sen 《Journal of Algorithms in Cognition, Informatics and Logic》1996,20(3):581-601
Anm×nmatrix
=(ai, j), 1≤i≤mand 1≤j≤n, is called atotally monotonematrix if for alli1, i2, j1, j2, satisfying 1≤i1<i2≤m, 1≤j1<j2≤n.[formula]We present an[formula]time algorithm to select thekth smallest item from anm×ntotally monotone matrix for anyk≤mn. This is the first subquadratic algorithm for selecting an item from a totally monotone matrix. Our method also yields an algorithm of the same time complexity for ageneralized row-selection problemin monotone matrices. Given a setS={p1,…, pn} ofnpoints in convex position and a vectork={k1,…, kn}, we also present anO(n4/3logc n) algorithm to compute thekith nearest neighbor ofpifor everyi≤n; herecis an appropriate constant. This algorithm is considerably faster than the one based on a row-selection algorithm for monotone matrices. If the points ofSare arbitrary, then thekith nearest neighbor ofpi, for alli≤n, can be computed in timeO(n7/5 logc n), which also improves upon the previously best-known result. 相似文献
19.
Greedily Finding a Dense Subgraph 总被引:3,自引:0,他引:3
Yuichi Asahiro Kazuo Iwama Hisao Tamaki Takeshi Tokuyama 《Journal of Algorithms in Cognition, Informatics and Logic》2000,34(2):203
Given an n-vertex graph with nonnegative edge weights and a positive integer k ≤ n, our goal is to find a k-vertex subgraph with the maximum weight. We study the following greedy algorithm for this problem: repeatedly remove a vertex with the minimum weighted-degree in the currently remaining graph, until exactly k vertices are left. We derive tight bounds on the worst case approximation ratio R of this greedy algorithm: (1/2 + n/2k)2 − O(n − 1/3) ≤ R ≤ (1/2 + n/2k)2 + O(1/n) for k in the range n/3 ≤ k ≤ n and 2(n/k − 1) − O(1/k) ≤ R ≤ 2(n/k − 1) + O(n/k2) for k < n/3. For k = n/2, for example, these bounds are 9/4 ± O(1/n), improving on naive lower and upper bounds of 2 and 4, respectively. The upper bound for general k compares well with currently the best (and much more complicated) approximation algorithm based on semidefinite programming. 相似文献
20.
Let (X, X
;
d} be a field of independent identically distributed real random variables, 0 < p < 2, and {a
,
; (
,
)
d ×
d,
≤
} a triangular array of real numbers, where
d is the d-dimensional lattice. Under the minimal condition that sup
,
|a
,
| < ∞, we show that |
|− 1/p ∑
≤
a
,
X
→ 0 a.s. as |
| → ∞ if and only if E(|X|p(L|X|)d − 1) < ∞ provided d ≥ 2. In the above, if 1 ≤ p < 2, the random variables are needed to be centered at the mean. By establishing a certain law of the logarithm, we show that the Law of the Iterated Logarithm fails for the weighted sums ∑
≤
a
,
X
under the conditions that EX = 0, EX2 < ∞, and E(X2(L|X|)d − 1/L2|X|) < ∞ for almost all bounded families {a
,
; (
,
)
d ×
d,
≤
of numbers. 相似文献