共查询到20条相似文献,搜索用时 31 毫秒
1.
Jorge L.C Sanz Thomas S Huang 《Journal of Mathematical Analysis and Applications》1984,104(1):302-308
In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z1…,zn), zi ? , i=1, …, n, is uniquely determined from the magnitude of f(x1…,xn): | f(x1…,xn)|, xi ? , i=1,…, n, except for (1) linear shifts: i(α1z1+…+αn2n+β), β, αi?, i=1,…, n; and (2) conjugation: . 相似文献
2.
R.S. Singh 《Journal of multivariate analysis》1976,6(2):338-342
Let Xj = (X1j ,…, Xpj), j = 1,…, n be n independent random vectors. For x = (x1 ,…, xp) in Rp and for α in [0, 1], let Fj(x) = αI(X1j < x1 ,…, Xpj < xp) + (1 ? α) I(X1j ≤ x1 ,…, Xpj ≤ xp), where I(A) is the indicator random variable of the event A. Let Fj(x) = E(Fj(x)) and Dn = supx, α max1 ≤ N ≤ n |Σ0n(Fj(x) ? Fj(x))|. It is shown that P[Dn ≥ L] < 4pL exp{?2(L2n?1 ? 1)} for each positive integer n and for all L2 ≥ n; and, as n → ∞, with probability one. 相似文献
3.
K.B. Athreya 《Statistics & probability letters》1983,1(3):147-150
Let X1, X2, X3, … be i.i.d. r.v. with E|X1| < ∞, E X1 = μ. Given a realization X = (X1,X2,…) and integers n and m, construct Yn,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution for 1 ? j ? n. ( denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X1 are given to ensure the almost sure convergence of toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981). 相似文献
4.
The probability measure of X = (x0,…, xr), where x0,…, xr are independent isotropic random points in n (1 ≤ r ≤ n ? 1) with absolutely continuous distributions is, for a certain class of distributions of X, expressed as a product measure involving as factors the joint probability measure of (ω, ?), the probability measure of p, and the probability measure of . Here ω is the r-subspace parallel to the r-flat η determined by X, ? is a unit vector in ω⊥ with ‘initial’ point at the origin [ω⊥ is the (n ? r)-subspace orthocomplementary to ω], p is the norm of the vector z from the origin to the orthogonal projection of the origin on η, and , where α is a scale factor determined by p. The probability measure for ω is the unique probability measure on the Grassmann manifold of r-subspaces in n invariant under the group of rotations in n, while the conditional probability measure of ? given ω is uniform on the boundary of the unit (n ? r)-ball in ω⊥ with centre at the origin. The decomposition allows the evaluation of the moments, for a suitable class of distributions of X, of the r-volume of the simplicial convex hull of {x0,…, xr} for 1 ≤ r ≤ n. 相似文献
5.
Stanley J Benkoski 《Journal of Number Theory》1976,8(2):218-223
If r, k are positive integers, then denotes the number of k-tuples of positive integers (x1, x2, …, xk) with 1 ≤ xi ≤ n and (x1, x2, …, xk)r = 1. An explicit formula for is derived and it is shown that .If S = {p1, p2, …, pa} is a finite set of primes, then 〈S〉 = {p1a1p2a2…psas; pi ∈ S and ai ≥ 0 for all i} and denotes the number of k-tuples (x1, x3, …, xk) with 1 ≤ xi ≤ n and (x1, x2, …, xk)r ∈ 〈S〉. Asymptotic formulas for are derived and it is shown that . 相似文献
6.
The following estimate of the pth derivative of a probability density function is examined: , where hk is the kth Hermite function and Σi = 1nhk(p)(Xi) is calculated from a sequence X1,…, Xn of independent random variables having the common unknown density. If the density has r derivatives the integrated square error converges to zero in the mean and almost completely as rapidly as O(n?α) and O(n?α log n), respectively, where . Rates for the uniform convergence both in the mean square and almost complete are also given. For any finite interval they are O(n?β) and , respectively, where . 相似文献
7.
Let Fn(x) be the empirical distribution function based on n independent random variables X1,…,Xn from a common distribution function F(x), and let be the sample mean. We derive the rate of convergence of to normality (for the regular as well as nonregular cases), a law of iterated logarithm, and an invariance principle for . 相似文献
8.
J.E Nymann 《Journal of Number Theory》1975,7(4):406-412
Given a set S of positive integers let denote the number of k-tuples 〈m1, …, mk〉 for which and (m1, …, mk) = 1. Also let denote the probability that k integers, chosen at random from , are relatively prime. It is shown that if P = {p1, …, pr} is a finite set of primes and S = {m : (m, p1 … pr) = 1}, then if k ≥ 3 and where d(S) denotes the natural density of S. From this result it follows immediately that as n → ∞. This result generalizes an earlier result of the author's where and S is then the whole set of positive integers. It is also shown that if S = {p1x1 … prxr : xi = 0, 1, 2,…}, then as n → ∞. 相似文献
9.
The Fréchet distance between two multivariate normal distributions having means μX, μY and covariance matrices ΣX, ΣY is shown to be given by . The quantity d0 given by is a natural metric on the space of real covariance matrices of given order. 相似文献
10.
D.C Doss 《Journal of multivariate analysis》1979,9(3):460-464
The probability generating function (pgf) of an n-variate negative binomial distribution is defined to be [β(s1,…,sn)]?k where β is a polynomial of degree n being linear in each si and k > 0. This definition gives rise to two characterizations of negative binomial distributions. An n-variate linear exponential distribution with the probability function h(x1,…,xn) is negative binomial if and only if its univariate marginals are negative binomial. Let St, t = 1,…, m, be subsets of {s1,…, sn} with empty ∩t=1mSt. Then an n-variate pgf is of a negative binomial if and only if for all s in St being fixed the function is of the form of the pgf of a negative binomial in other s's and this is true for all t. 相似文献
11.
Let be a polynomial in the variables x1,…,xp with nonnegative real coefficients which sum to one, let A1,…,Ap be stochastic matrices, and let be the stochastic matrix which is obtained from ? by substituting the Kronecker product of An11,…,Anppfor each term Xn11·?·Xnpp. In this paper, we present necessary and sufficient conditions for the Cesàro limit of the sequence of the powers of to be equal to the Kronecker product of the Cesàro limits associated with each of A1,…,Ap. These conditions show that the equality of these two matrices depends only on the number of ergodic sets under and?or the cyclic structure of the ergodic sets under A1,…,Ap, respectively. As a special case of these results, we obtain necessary and sufficient conditions for the interchangeability of the Kronecker product and the Cesàro limit operator. 相似文献
12.
Given a polynomial , we calculate a subspace Gp of the linear space 〈X〉 generated by the indeterminates which is minimal with respect to the property (the algebra generated by Gp, and prove its uniqueness. Furthermore, we use this result to characterize the pairs (P,Q) of polynomials P(X1,…,Xn) and Q(X1,…,Xn) for which there exists an isomorphism T:〈X〉 →〈X〉 that “separates P from Q,” i.e., such that for some k(1<k<n) we can write P and Q as and respectively, where . 相似文献
13.
Let ρ21,…,ρ2p be the squares of the population canonical correlation coefficients from a normal distribution. This paper is concerned with the estimation of the parameters δ1,…,δp, where , i = 1,…,p, in a decision theoretic way. The approach taken is to estimate a parameter matrix Δ whose eigenvalues are δ1,…,δp, given a random matrix F whose eigenvalues have the same distribution as , i = 1,…,p, where r1,…,rp are the sample canonical correlation coefficients. 相似文献
14.
R.N. Buttsworth 《Journal of Number Theory》1980,12(4):487-498
The polynomial functions f1, f2,…, fm are found to have highest common factor h for a set of values of the variables x1, x2,…,xm whose asymptotic density is For the special case f1(x) = f2(x) = … = fm(x) = x and h = 1 the above formula reduces to , the density if m-tuples with highest common factor 1. Necessary and sufficient conditions on the polynomials f1, f2,…, fm for the asymptotic density to be zero are found. In particular it is shown that either the polynomials may never have highest common factor h or else h is the highest common factor infinitely often and in fact with positive density. 相似文献
15.
C.G Khatri 《Journal of multivariate analysis》1978,8(3):453-467
Optimization problems are connected with maximization of three functions, namely, geometric mean, arithmetic mean and harmonic mean of the eigenvalues of (X′ΣX)?1ΣY(Y′ΣY)?1Y′ΣX, where Σ is positive definite, X and Y are p × r and p × s matrices of ranks r and s (≥r), respectively, and X′Y = 0. Some interpretations of these functions are given. It is shown that the maximum values of these functions are obtained at the same point given by X = (h1 + ?1hp, …, hr + ?rhp?r+1) and , where h1, …, hp are the eigenvectors of Σ corresponding to the eigenvalues λ1 ≥ λ2 ≥ … ≥ λp > 0, ?j = +1 or ?1 for j = 1,2,…, r and , are linear functions of hr+1,…, hp?r. These results are extended to intermediate stationary values. They are utilized in obtaining the inequalities for canonical correlations θ1,…,θr and they are given by expressions (3.8)–(3.10). Further, some new union-intersection test procedures for testing the sphericity hypothesis are given through test statistics (3.11)–(3.13). 相似文献
16.
This paper presents a demonstrably convergent method of feasible directions for solving the problem min{φ(ξ)| gi(ξ)?0i=1,2,…,m}, which approximates, adaptively, both φ(x) and ▽φ(x). These approximations are necessitated by the fact that in certain problems, such as when , a precise evaluation of φ(x) and ▽φ(x) is extremely costly. The adaptive procedure progressively refines the precision of the approximations as an optimum is approached and as a result should be much more efficient than fixed precision algorithms.It is outlined how this new algorithm can be used for solving problems of the form under the assumption that Ωmξ={x|gi(x)?0, j=1,…,s} ∩n, Ωy={y|ζi(y)?0, i-1,…,t} ∩ m, with f, gj, ζi continuously differentiable, f(x, ·) concave, ζi convex for compact. 相似文献
17.
L.R. Haff 《Journal of multivariate analysis》1977,7(3):374-385
Let Sp×p ~ Wishart (Σ, k), Σ unknown, k > p + 1. Minimax estimators of Σ?1 are given for L1, an Empirical Bayes loss function; and L2, a standard loss function (Ri ≡ E(Li ∣ Σ), i = 1, 2). The estimators are , a, b ≥ 0, r(·) a functional on . Stein, Efron, and Morris studied the special cases and , for certain, a, b. From their work , a = k ? p ? 1, b = p2 + p ? 2; whereas, we prove . The reversal is surprising because a.e. (for a particular L2). Assume (compact) ? , the set of p × p p.s.d. matrices. A “divergence theorem” on functions Fp×p : → implies identities for Ri, i = 1, 2. Then, conditions are given for , i = 1, 2. Most of our results concern estimators with r(S) = t(U)/tr(S), U = p ∣S∣1/p/tr(S). 相似文献
18.
Dudley Paul Johnson 《Stochastic Processes and their Applications》1985,19(1):183-187
We show that under mild conditions the joint densities Px1,…,xn) of the general discrete time stochastic process Xn on can be computed via where ? is in a Hilbert space , and T (x), x ? are linear operators on . We then show how the Central Limit Theorem can easily be derived from such representations. 相似文献
19.
William E Strawderman 《Journal of multivariate analysis》1974,4(3):255-264
Families of minimax estimators are found for the location parameters of a p-variate distribution of the form , where G(·) is a known c.d.f. on (0, ∞), p ≥ 3 and the loss is sum of squared errors. The estimators are of the form where 0 ≤ a ≤ 2, r(X′X) is nondecreasing, and is nonincreasing. Generalized Bayes minimax estimators are found for certain G(·)'s. 相似文献
20.
Robert Chen 《Journal of multivariate analysis》1978,8(2):328-333
Let {Xn}n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let Sn = Σj=1nXj and . In this paper, we prove that (1) lim?→0+?α(r?1)E{N∞(r, t, ?)} = K(r, t) if E(X1) = 0, Var(X1) = 1, and E(| X1 |t) < ∞, where 2 ≤ t < 2r ≤ 2t, , and ; (2) if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(|X1|t) < ∞, where G(t, ?) = E{N∞(t, t, ?)} = Σn=1∞nt?2P{| Sn | > ?n} → ∞ as ? → 0+ and , i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(| X1 |t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution. 相似文献