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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
3.
It is shown that the conditional probability density function of Y1 given (1/n) Σi=1n Yi=1Yit = Σ, where Y1, Y2,…, Yn are i.i.d, p-variate uniform random vectors with mean 0 equals to that of Y1 given (1/n) Σi=1n YiYit,…, Yn are i.i.d, p-variate normal random vectors with mean 0 and covariance matrix Σ. 相似文献
4.
We consider the problem of discriminating between two independent multivariate normal populations, Np(μ1, Σ1) and Np(μ2, Σ2), having distinct mean vectors μ1 and μ2 and distinct covariance matrices Σ1 and Σ2. The parameters μ1, μ2, Σ1, and Σ2 are unknown and are estimated by means of independent random training samples from each population. We derive a stochastic representation for the exact distribution of the “plug-in” quadratic discriminant function for classifying a new observation between the two populations. The stochastic representation involves only the classical standard normal, chi-square, and F distributions and is easily implemented for simulation purposes. Using Monte Carlo simulation of the stochastic representation we provide applications to the estimation of misclassification probabilities for the well-known iris data studied by Fisher (Ann. Eugen.7 (1936), 179–188); a data set on corporate financial ratios provided by Johnson and Wichern (Applied Multivariate Statistical Analysis, 4th ed., Prentice–Hall, Englewood Cliffs, NJ, 1998); and a data set analyzed by Reaven and Miller (Diabetologia16 (1979), 17–24) in a classification of diabetic status. 相似文献
5.
Marianna Pensky 《Journal of multivariate analysis》1999,69(2):242
We consider independent pairs (X1, Σ1), (X2, Σ2), …, (Xn, Σn), where eachΣiis distributed according to some unknown density functiong(Σ) and, givenΣi=Σ,Xihas conditional density functionq(xΣ) of the Wishart type. In each pair the first component is observable but the second is not. After the (n+1)th observationXn+1is obtained, the objective is to estimateΣn+1corresponding toXn+1. This estimator is called the empirical Bayes (EB) estimator ofΣ. An EB estimator ofΣis constructed without any parametric assumptions ong(Σ). Its posterior mean square risk is examined, and the estimator is demonstrated to be pointwise asymptotically optimal. 相似文献
6.
Let μ(· ; Σ, G1) and μ(· ; Ω, G2) be elliptically contoured measures on
k centered at 0, having scale parameters (Σ, Ω) and radial cdf′s (G1, G2). Elliptical measures vm(·) and vM(·), depending on (Σ, Ω, G1, G2), are constructed such that Vm(C) ≤ {μ(C; Σ, G1), μ(C; Ω, G2)} for every symmetric convex set C
k with equality for certain sets. These in turn rely on the construction of spectral lower and upper matrix bounds for (Σ, Ω). Extensions include bounds for certain ensembles and mixtures, including versions having star-shaped contours. The lindings specialize to give envelopes for some nonstandard distributions of quadratic forms, with applications to stochastic characteristics of ballistic systems. 相似文献
7.
The predictive ratio is considered as a measure of spread for the predictive distribution. It is shown that, in the exponential families, ordering according to the predictive ratio is equivalent to ordering according to the posterior covariance matrix of the parameters. This result generalizes an inequality due to Chaloner and Duncan who consider the predictive ratio for a beta-binomial distribution and compare it with a predictive ratio for the binomial distribution with a degenerate prior. The predictive ratio at x1 and x2 is defined to be pg(x1)pg(x2)/[pg(
)]2 = hg(x1, x2), where pg(x1) = ∫ ƒ(x1θ) g(θ) dθ is the predictive distribution of x1 with respect to the prior g. We prove that hg(x1, x2) ≥ hg*(x1, x2) for all x1 and x2 if ƒ(xθ) is in the natural exponential family and Covgx(θ) ≥ Covg*x(θ) in the Loewner sense, for all x on a straight line from x1 to x2. We then restrict the class of prior distributions to the conjugate class and ask whether the posterior covariance inequality obtains if g and g* differ in that the “sample size” 相似文献
8.
Let (x, Xβ, V) be a linear model and let A′ = (A′1, A′2) be a p × p nonsingular matrix such that A2X = 0, Rank A2 = p − Rank X. We represent the BLUE and its covariance matrix in alternative forms under the conditions that the number of unit canonical correlations between y1 ( = A1x) and y2 ( = A2x) is zero. For the second problem, let x′ = (x′1, x′2) and let a g-inverse V− of V be written as (V−)′ = (A′1, A′2). We investigate the reations (if any) between the nonzero canonical correlations {1 1 … 1 > 0} due to y1 ( = A1x) and y2 ( = A2x), and the nonzero canonical correlations {1 λ1 … λv+r > 0} due to x1 and x2. We answer some of the questions raised by Latour et al. (1987, in Proceedings, 2nd Int. Tampere Conf. Statist. (T. Pukkila and S. Puntanen, Eds.), Univ. of Tampere, Finland) in the case of the Moore-Penrose inverse V+ = (A′1, A′2) of V. 相似文献
9.
A function f(x) defined on
=
1 ×
2 × … ×
n where each
i is totally ordered satisfying f(x y) f(x y) ≥ f(x) f(y), where the lattice operations and refer to the usual ordering on
, is said to be multivariate totally positive of order 2 (MTP2). A random vector Z = (Z1, Z2,…, Zn) of n-real components is MTP2 if its density is MTP2. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies −DΣ−1D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP2 properties. The MTP2 property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP2 kernels are presented and amplified by a wide variety of applications. 相似文献
10.
Let the kp-variate random vector X be partitioned into k subvectors Xi of dimension p each, and let the covariance matrix Ψ of X be partitioned analogously into submatrices Ψij. The common principal component (CPC) model for dependent random vectors assumes the existence of an orthogonal p by p matrix β such that βtΨijβ is diagonal for all (i, j). After a formal definition of the model, normal theory maximum likelihood estimators are obtained. The asymptotic theory for the estimated orthogonal matrix is derived by a new technique of choosing proper subsets of functionally independent parameters. 相似文献
11.
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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12.
Sequential procedures are proposed to estimate the unknown mean vector of a multivariate linear process of the form Xt − μ = ∑∞j = 0AjZt − j, where the Zt are i.i.d. (0, Σ) with unknown covariance matrix Σ. The proposed point estimation is asymptotically risk efficient in the sense of Starr (1966, Ann. Math. Statist.37 1173-1185). The fixed accuracy confidence set procedure is asymptotically efficient with prescribed coverage probability in the sense of Chow and Robbins (1965, Ann. Math. Statist.36 457-462). A random central limit theorem for this process, under a mild summability condition on the coefficient matrices Aj, is also obtained. 相似文献
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Rudi Brits 《Quaestiones Mathematicae》2016,39(1):103-114
Let A be a complex unital Banach algebra. Using a connection between the spectral distance and the growth characteristics of a certain entire map into A, we derive a generalization of Gelfand’s famous power boundedness theorem. Elaborating on these ideas, with the help of a Phragm´en-Lindel¨of device for subharmonic functions, it is then shown, as the main result, that two normal elements of a C?-algebra are equal if and only if they are quasinilpotent equivalent. 相似文献
14.
Let x=(x1,…,xn) be a sequence of positive integers. An x-parking function is a sequence (a1,…,an) of positive integers whose non-decreasing rearrangement b1bn satisfies bix1++xi. In this paper we give a combinatorial approach to the enumeration of (a,b,…,b)-parking functions by their leading terms, which covers the special cases x=(1,…,1), (a,1,…,1), and (b,…,b). The approach relies on bijections between the x-parking functions and labeled rooted forests. To serve this purpose, we present a simple method for establishing the required bijections. Some bijective results between certain sets of x-parking functions of distinct leading terms are also given. 相似文献
15.
For a scale mixture of normal vector, X=A1/2G, where X, G
n and A is a positive variable, independent of the normal vector G, we obtain that the conditional variance covariance, Cov(X2 X1), is always finite a.s. for m2, where X1
n and m<n, and remains a.s. finite even for m=1, if and only if the square root moment of the scale factor is finite. It is shown that the variance is not degenerate as in the Gaussian case, but depends upon a function SA, m(·) for which various properties are derived. Application to a uniform and stable scale of normal distributions are also given. 相似文献
16.
R 《Journal of Approximation Theory》1993,74(3)
In this paper, we determine the exact value of average n − K width
n(Wrpq(R), Lq(R)) of Sobolev-Wiener class Wrpq(R) in the metric Lq(R) for 1 > q ≥ p > ∞ and get the value of
n(Wrp(R), Lqp(R)) for the dual case. We also solve the optimal interpolation problems of Wrpq(R) in the metric Lq(R) and Wrp(R) in the metric Lqp(R) for 1 < q ≤ p < ∞. 相似文献
17.
In this paper, we determine the X-inner automorphisms of the smash product R # U(L) of a prime ring R by the universal enveloping algebra U(L) of a characteristic 0 Lie algebra L. Specifically, we show that any such automorphism σ stabilizing R can be written as a product σ = σ1σ2, where σ1 is induced by conjugation by a unit of Q3(R), the symmetric Martindale ring of quotients of R, and σ2 is induced by conjugation by a unit of Q3(T). Here S = Ql(R) is the left Martindale ring of quotients of R and T is the centralizer of S in S # U(L) - R # U(L). One of the subtleties of the proof is that we must work in several unrelated overrings of R # U(L). 相似文献
18.
Let A=c1A1+c2A2, wherec1, c2 are nonzero complex numbers and (A1,A2) is a pair of two n×n nonzero matrices. We consider the problem of characterizing all situations where a linear combination of the form A=c1A1+c2A2 is (i) a tripotent or an involutive matrix when are commuting involutive or commuting tripotent matrices, respectively, (ii) an idempotent matrix when are involutive matrices, and (iii) an involutive matrix when are involutive or idempotent matrices. 相似文献
19.
C. G. Khatri 《Journal of multivariate analysis》1979,9(4):589-598
It is established that a vector (X′1, X′2, …, X′k) has a multivariate normal distribution if (i) for each Xi the regression on the rest is linear, (ii) the conditional distribution of X1 about the regression does not depend on the rest of the variables, and (iii) the conditional distribution of X2 about the regression does not depend on the rest of the variables, provided that the regression coefficients satisfy some more conditions that those given by [4]J. Multivar. Anal. 6 81–94]. 相似文献
20.
A graph H is called a supersubdivison of a graph G if H is obtained from G by replacing every edge uv of G by a complete bipartite graph K2,m (m may vary for each edge) by identifying u and v with the two vertices in K2,m that form one of the two partite sets. We denote the set of all such supersubdivision graphs by SS(G). Then, we prove the following results.
- 1. Each non-trivial connected graph G and each supersubdivision graph HSS(G) admits an α-valuation. Consequently, due to the results of Rosa (in: Theory of Graphs, International Symposium, Rome, July 1966, Gordon and Breach, New York, Dunod, Paris, 1967, p. 349) and El-Zanati and Vanden Eynden (J. Combin. Designs 4 (1996) 51), it follows that complete graphs K2cq+1 and complete bipartite graphs Kmq,nq can be decomposed into edge disjoined copies of HSS(G), for all positive integers m,n and c, where q=|E(H)|.
- 2. Each connected graph G and each supersubdivision graph in SS(G) is strongly n-elegant, where n=|V(G)| and felicitous.
- 3. Each supersubdivision graph in EASS(G), the set of all even arbitrary supersubdivision graphs of any graph G, is cordial.