共查询到20条相似文献,搜索用时 31 毫秒
1.
Rasul A. Khan 《Journal of multivariate analysis》1978,8(4):550-558
Let X1, X2,… be idd random vectors with a multivariate normal distribution N(μ, Σ). A sequence of subsets {Rn(a1, a2,…, an), n ≥ m} of the space of μ is said to be a (1 − α)-level sequence of confidence sets for μ if P(μ Rn(X1, X2,…, Xn) for every n ≥ m) ≥ 1 − α. In this note we use the ideas of Robbins Ann. Math. Statist. 41 (1970) to construct confidence sequences for the mean vector μ when Σ is either known or unknown. The constructed sequence Rn(X1, X2, …, Xn) depends on Mahalanobis'
or Hotelling's
according as Σ is known or unknown. Confidence sequences for the vector-valued parameter in the general linear model are also given. 相似文献
2.
Dale Umbach 《Journal of multivariate analysis》1978,8(4):518-531
The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters.Let X = (X1, X2, …, Xn) be an observation from a multivariate exponential distribution with that natural parameter Θ = (Θ1, Θ2, …, Θn). Let θx* be the posterior mode. Sufficient conditions are presented for the distribution of Θ − θx* given X = x to converge to a multivariate normal with mean vector 0 as |x| tends to infinity. These same conditions imply that E(Θ | X = x) − θx* converges to the zero vector as |x| tends to infinity.The posterior for an observation X = (X1, X2, …, Xn is considered for a location vector Θ = (Θ1, Θ2, …, Θn) as x gets large along a path, γ, in Rn. Sufficient conditions are given for the distribution of γ(t) − Θ given X = γ(t) to converge in law as t → ∞. Slightly stronger conditions ensure that γ(t) − E(Θ | X = γ(t)) converges to the mean of the limiting distribution.These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let δ be a Bayes estimator for a loss function of this type. Generally, if the distribution of Θ − E(Θ | X = γ(t)) given X = γ(t) converges in law to a symmetric distribution as t → ∞, it is shown that δ(γ(t)) − E(Θ | X = γ(t)) → 0 as t → ∞. 相似文献
3.
George A. Anastassiou 《Journal of Approximation Theory》1985,45(4):383-388
In this paper we give a sufficient condition for the pointwise Korovkin property on B(X), the space of bounded real valued functions on an arbitrary countable set X = {xl,…, xj,…}. Our theorem follows from its Lp(X, μ) analogue (and conversely); here 1 p < ∞ and μ is a positive finite measure on X such that μ({xj}) > 0 for all j. 相似文献
4.
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
5.
Let Ω be a plane bounded region. Let U = {Uμ(P):μ(P)εL∞(Ω), uμ ε H22, 0(Ω) and a(P, μ(P))uμ,xx + 2b(P, μ(P))uμ,xy + c(P, μ(P))uμ,vv = ƒ(P) for P ε Ω; here we are given a(P, X), b(P, X), c(P, X) ε L∞(Ω × E1), ƒ(P) ε Lp(Ω) with p > 2, and our partial differential equation is uniformly elliptic. The functions μ(P) are called profiles. We establish sufficient conditions—which when they apply are constructive—that there exist a μ0 ε L∞(Ω) such that uμ0 (P) uμ(P) for all P ε Ω and for each μ ε L∞(Ω). Similar results are obtained for a difference equation and convergence is proved. 相似文献
6.
Let (X, Y) be an
d ×
-valued random vector and let (X1, Y1),…,(XN, YN) be a random sample drawn from its distribution. Divide the data sequence into disjoint blocks of length l1, …, ln, find the nearest neighbor to X in each block and call the corresponding couple (Xi*, Yi*). It is shown that the estimate mn(X) = Σi = 1n wniYi*/Σi = 1n wni of m(X) = E{Y|X} satisfies E{|mn(X) − m(X)|p}
0 (p ≥ 1) whenever E{|Y|p} < ∞, ln
∞, and the triangular array of positive weights {wni} satisfies supi ≤ nwni/Σi = 1n wni
0. No other restrictions are put on the distribution of (X, Y). Also, some distribution-free results for the strong convergence of E{|mn(X) − m(X)|p|X1, Y1,…, XN, YN} to zero are included. Finally, an application to the discrimination problem is considered, and a discrimination rule is exhibited and shown to be strongly Bayes risk consistent for all distributions. 相似文献
7.
We give a direct formulation of the invariant polynomials μGq(n)(, Δi,;, xi,i + 1,) characterizing U(n) tensor operators p, q, …, q, 0, …, 0 in terms of the symmetric functions Sλ known as Schur functions. To this end, we show after the change of variables Δi = γi − δi and xi, i + 1 = δi − δi + 1 thatμGq(n)(,Δi;, xi, i + 1,) becomes an integral linear combination of products of Schur functions Sα(, γi,) · Sβ(, δi,) in the variables {γ1,…, γn} and {δ1,…, δn}, respectively. That is, we give a direct proof that μGq(n)(,Δi,;, xi, i + 1,) is a bisymmetric polynomial with integer coefficients in the variables {γ1,…, γn} and {δ1,…, δn}. By making further use of basic properties of Schur functions such as the Littlewood-Richardson rule, we prove several remarkable new symmetries for the yet more general bisymmetric polynomials μmGq(n)(γ1,…, γn; δ1,…, δm). These new symmetries enable us to give an explicit formula for both μmG1(n)(γ; δ) and 1G2(n)(γ; δ). In addition, we describe both algebraic and numerical integration methods for deriving general polynomial formulas for μmGq(n)(γ; δ). 相似文献
8.
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
9.
Let (X, Y) be a random vector such that X is d-dimensional, Y is real valued, and θ(X) is the conditional αth quantile of Y given X, where α is a fixed number such that 0 < α < 1. Assume that θ is a smooth function with order of smoothness p > 0, and set r = (p − m)/(2p + d), where m is a nonnegative integer smaller than p. Let T(θ) denote a derivative of θ of order m. It is proved that there exists estimate
of T(θ), based on a set of i.i.d. observations (X1, Y1), …, (Xn, Yn), that achieves the optimal nonparametric rate of convergence n−r in Lq-norms (1 ≤ q < ∞) restricted to compacts under appropriate regularity conditions. Further, it has been shown that there exists estimate
of T(θ) that achieves the optimal rate (n/log n)−r in L∞-norm restricted to compacts. 相似文献
10.
Anders Grimvall 《Stochastic Processes and their Applications》1973,1(4):335-368
Starting from a real-valued Markov chain X0,X1,…,Xn with stationary transition probabilities, a random element {Y(t);t[0, 1]} of the function space D[0, 1] is constructed by letting Y(k/n)=Xk, k= 0,1,…,n, and assuming Y (t) constant in between. Sample tightness criteria for sequences {Y(t);t[0,1]};n of such random elements in D[0, 1] are then given in terms of the one-step transition probabilities of the underlying Markov chains. Applications are made to Galton-Watson branching processes. 相似文献
11.
Let {Xn}∞n=1be a sequence of free, identically distributed random variables with common distributionμ. Then there exist sequences {Bn}∞n=1and {An}∞n=1of positive and real numbers, respectively, such that sequence of random variables[formula]converges in distribution to the semicircle law if and only if the function[formula]is slowly varying in Karamata's sense. In other words, the free domain of attraction of the semicircle law coincides with the classical domain of attraction of the Gaussian. We prove an analogous result for normal domains of attraction in the sense of Linnik. 相似文献
12.
Akio Arimoto 《Journal of Approximation Theory》2001,112(2)
Necessary and sufficient conditions are given which ensure the completeness of the trigonometric systems with integer indices; {einx; x
}∞n=−∞ or {einx; x
}∞n=1 in Lα(μ,
), α1. If there exists a support Λ of the measure μ which is a wandering set, that is, Λ+2kπ, k=0, ±1, ±2, … are mutually disjoint for different k's, then the linear span of our trigonometric system {einx; x
}∞n=−∞ is dense in Lα(μ,
) α1. The converse statement is also true. 相似文献
13.
Let {α1,α2,…} be a sequence of real numbers outside the interval [−1,1] and μ a positive bounded Borel measure on this interval satisfying the Erd
s–Turán condition μ′>0 a.e., where μ′ is the Radon–Nikodym derivative of the measure μ with respect to the Lebesgue measure. We introduce rational functions n(x) with poles {α1,…,αn} orthogonal on [−1,1] and establish some ratio asymptotics for these orthogonal rational functions, i.e. we discuss the convergence of n+1(x)/n(x) as n tends to infinity under certain assumptions on the location of the poles. From this we derive asymptotic formulas for the recurrence coefficients in the three-term recurrence relation satisfied by the orthonormal functions. 相似文献
14.
We consider a conditional empirical distribution of the form Fn(C x)=∑nt=1 ωn(Xt−x) I{YtC} indexed by C
, where {(Xt, Yt), t=1, …, n} are observations from a strictly stationary and strong mixing stochastic process, {ωn(Xt−x)} are kernel weights, and
is a class of sets. Under the assumption on the richness of the index class
in terms of metric entropy with bracketing, we have established uniform convergence and asymptotic normality for Fn(· x). The key result specifies rates of convergences for the modulus of continuity of the conditional empirical process. The results are then applied to derive Bahadur–Kiefer type approximations for a generalized conditional quantile process which, in the case with independent observations, generalizes and improves earlier results. Potential applications in the areas of estimating level sets and testing for unimodality (or multimodality) of conditional distributions are discussed. 相似文献
15.
Vladimir Koltchinskii 《Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques》2003,39(6):1143-978
Let
be a probability space and let Pn be the empirical measure based on i.i.d. sample (X1,…,Xn) from P. Let
be a class of measurable real valued functions on
For
define Ff(t):=P{ft} and Fn,f(t):=Pn{ft}. Given γ(0,1], define n,γ(δ):=1/(n1−γ/2δγ). We show that if the L2(Pn)-entropy of the class
grows as −α for some α(0,2), then, for all
and all δ(0,Δn), Δn=O(n1/2), and where
and c(σ)↓1 as σ↓0 (the above inequalities hold for any fixed σ(0,1] with a high probability). Also, define Then for all
uniformly in
and with probability 1 (for
the above ratio is bounded away from 0 and from ∞). The results are motivated by recent developments in machine learning, where they are used to bound the generalization error of learning algorithms. We also prove some more general results of similar nature, show the sharpness of the conditions and discuss the applications in learning theory. 相似文献
16.
For all integers m3 and all natural numbers a1,a2,…,am−1, let n=R(a1,a2,…,am−1) represent the least integer such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to
a1x1+a2x2++am−1xm−1=xm.