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1.
R Michel 《Journal of multivariate analysis》1979,9(3):393-400
It is shown that—under appropriate regularity conditions—the conditional distribution of the first p components of a normalized sum of i.i.d. m-dimensional random vectors, given the complementary subvector, admits a Chebyshev-Cramér asymptotic expansion of order , uniformly over all Borelsets in p and uniformly in a region of the conditioning subvector that includes moderate deviations. 相似文献
2.
Asymptotic expansions are given for the distributions of latent roots of matrices in three multivariate situations. The distribution of the roots of the matrix S1(S1 + S2)?1, where S1 is and S2 is Wm(n2, Σ), is studied in detail and asymptotic series for the distribution are obtained which are valid for some or all of the roots of the noncentrality matrix Ω large. These expansions are obtained using partial-differential equations satisfied by the distribution. Asymptotic series are also obtained for the distributions of the roots of n?1S, where S in Wm(n, Σ), for large n, and S1S2?1, where S1 is Wm(n1, Σ) and S2 is Wm(n2, Σ), for large n1 + n2. 相似文献
3.
Y. Fujikoshi 《Journal of multivariate analysis》1978,8(1):63-72
In this paper we derive asymptotic expansions for the distributions of some functions of the latent roots of the matrices in three situations in multivariate normal theory, i.e., (i) principal component analysis, (ii) MANOVA model and (iii) canonical correlation analysis. These expansions are obtained by using a perturbation method. Confidence intervals for the functions of the corresponding population roots are also obtained. 相似文献
4.
Y. Fujikoshi 《Journal of multivariate analysis》1977,7(3):386-396
Asymptotic expansions are given for the density function of the normalized latent roots of S1S2?1 for large n under the assumption of , where S1 and S2 are independent noncentral and central Wishart matrices having the and Wp(n, Σ) distributions, respectively. The expansions are obtained by using a perturbation method. Asymptotic expansions are also obtained for the density function of the normalized canonical correlations when some of the population canonical correlations are zero. 相似文献
5.
In this paper, the authors obtained asymptotic expressions for the joint distributions of certain functions of the eigenvalues of the Wishart matrix, correlation matrix, MANOVA matrix and canonical correlation matrix when the population roots have multiplicity. 相似文献
6.
Asymptotic distributions of the latent roots of the covariance matrix with multiple population roots
Yasuko Chikuse 《Journal of multivariate analysis》1976,6(2):237-249
An asymptotic expansion for large sample size n is derived by a partial differential equation method, up to and including the term of order n?2, for the 0F0 function with two argument matrices which arise in the joint density function of the latent roots of the covariance matrix, when some of the population latent roots are multiple. Then we derive asymptotic expansions for the joint and marginal distributions of the sample roots in the case of one multiple root. 相似文献
7.
D.J. de Waal 《Journal of multivariate analysis》1977,7(1):223-228
The asymptotic distributions of the elementary symmetric functions (esf's) of the characteristic roots of a noncentral multivariate beta matrix and of the generalized correlation matrix (noncentral under the assumption of linearity) are derived. 相似文献
8.
Haruhiko Ogasawara 《Journal of multivariate analysis》2007,98(9):1726-1750
Asymptotic expansions of the distributions of typical estimators in canonical correlation analysis under nonnormality are obtained. The expansions include the Edgeworth expansions up to order O(1/n) for the parameter estimators standardized by the population standard errors, and the corresponding expansion by Hall's method with variable transformation. The expansions for the Studentized estimators are also given using the Cornish-Fisher expansion and Hall's method. The parameter estimators are dealt with in the context of estimation for the covariance structure in canonical correlation analysis. The distributions of the associated statistics (the structure of the canonical variables, the scaled log likelihood ratio and Rozeboom's between-set correlation) are also expanded. The robustness of the normal-theory asymptotic variances of the sample canonical correlations and associated statistics are shown when a latent variable model holds. Simulations are performed to see the accuracy of the asymptotic results in finite samples. 相似文献
9.
In this paper asymptotic nonnull distributions are derived for two statistics used in testing for the reality of the covariance matrix in a complex Gaussian distribution. 相似文献
10.
The authors investigated the asymptotic joint distributions of certain functions of the eigenvalues of the sample covariance matrix, correlation matrix, and canonical correlation matrix in nonnull situations when the population eigenvalues have multiplicities. These results are derived without assuming that the underlying distribution is multivariate normal. In obtaining these expressions, Edgeworth type expansions were used. 相似文献
11.
This paper examines asymptotic distributions of the canonical correlations between and with q≤p, based on a sample of size of N=n+1. The asymptotic distributions of the canonical correlations have been studied extensively when the dimensions q and p are fixed and the sample size N tends toward infinity. However, these approximations worsen when q or p is large in comparison to N. To overcome this weakness, this paper first derives asymptotic distributions of the canonical correlations under a high-dimensional framework such that q is fixed, m=n−p→∞ and c=p/n→c0∈[0,1), assuming that and have a joint (q+p)-variate normal distribution. An extended Fisher’s z-transformation is proposed. Then, the asymptotic distributions are improved further by deriving their asymptotic expansions. Numerical simulations revealed that our approximations are more accurate than the classical approximations for a large range of p,q, and n and the population canonical correlations. 相似文献
12.
Haruhiko Ogasawara 《Journal of multivariate analysis》2010,101(9):2149-2167
Asymptotic expansions of the distributions of the pivotal statistics involving log-likelihood derivatives under possible model misspecification are derived using the asymptotic cumulants up to the fourth-order and the higher-order asymptotic variance. The pivots dealt with are the studentized ones by the estimated expected information, the negative Hessian matrix, the sum of products of gradient vectors, and the so-called sandwich estimator. It is shown that the first three asymptotic cumulants are the same over the pivots under correct model specification with a general condition of the equalities. An application is given in item response theory, where the observed information is usually used rather than the estimated expected one. 相似文献
13.
14.
Asymptotic expansions of posterior expectations,distributions and densities for stochastic processes
Martin Crowder 《Annals of the Institute of Statistical Mathematics》1988,40(2):297-309
Asymptotic expansions are derived for Bayesian posterior expectations, distribution functions and density functions. The observations constitute a general stochastic process in discrete or continuous time. 相似文献
15.
《Indagationes Mathematicae》2022,33(6):1221-1235
In a recent paper (Temme, 2021) new asymptotic expansions are given for the Kummer functions and for large positive values of and , with fixed and special attention for the case . In this paper we extend the approach and also accept large values of . The new expansions are valid when at least one of the parameters , , or is large. We provide numerical tables to show the performance of the expansions. 相似文献
16.
This paper deals with the asymptotic distribution of Wishart matrix and its application to the estimation of the population matrix parameter when the population eigenvalues are block-wise infinitely dispersed. We show that the appropriately normalized eigenvectors and eigenvalues asymptotically generate two Wishart matrices and one normally distributed random matrix, which are mutually independent. For a family of orthogonally equivariant estimators, we calculate the asymptotic risks with respect to the entropy or the quadratic loss function and derive the asymptotically best estimator among the family. We numerically show (1) the convergence in both the distributions and the risks are quick enough for a practical use, (2) the asymptotically best estimator is robust against the deviation of the population eigenvalues from the block-wise infinite dispersion. 相似文献
17.
Nico M. Temme 《Journal of Computational and Applied Mathematics》2009,232(2):201-215
Riesz fractional derivatives of a function, (also called Riesz potentials), are defined as fractional powers of the Laplacian. Asymptotic expansions for large x are computed for the Riesz fractional derivatives of the Airy function of the first kind, Ai(x), and the Scorer function, Gi(x). Reduction formulas are provided that allow one to express Riesz potentials of products of Airy functions, and , via and . Here Bi(x) is the Airy function of the second type. Integral representations are presented for the function A2(a,b;x)=Ai(x−a)Ai(x−b) with a,b∈R and its Hilbert transform. Combined with the above asymptotic expansions they can be used for computing asymptotics of the Hankel transform of . These results are used for obtaining the weak rotation approximation for the Ostrovsky equation (asymptotics of the fundamental solution of the linearized Cauchy problem as the rotation parameter tends to zero). 相似文献
18.
For a normally distributed random matrix Y with mean zero and general covariance matrix ΣY and for a symmetric matrix W, necessary and sufficient conditions are derived for the Wishartness of Y′WY. 相似文献
19.
Chelo Ferreira 《Journal of Mathematical Analysis and Applications》2004,298(1):210-224
The Hurwitz-Lerch zeta function Φ(z,s,a) is considered for large and small values of a∈C, and for large values of z∈C, with |Arg(a)|<π, z∉[1,∞) and s∈C. This function is originally defined as a power series in z, convergent for |z|<1, s∈C and 1−a∉N. An integral representation is obtained for Φ(z,s,a) which define the analytical continuation of the Hurwitz-Lerch zeta function to the cut complex z-plane C?[1,∞). From this integral we derive three complete asymptotic expansions for either large or small a and large z. These expansions are accompanied by error bounds at any order of the approximation. Numerical experiments show that these bounds are very accurate for real values of the asymptotic variables. 相似文献
20.
We consider asymptotic distributions of maximum deviations of sample covariance matrices, a fundamental problem in high-dimensional inference of covariances. Under mild dependence conditions on the entries of the data matrices, we establish the Gumbel convergence of the maximum deviations. Our result substantially generalizes earlier ones where the entries are assumed to be independent and identically distributed, and it provides a theoretical foundation for high-dimensional simultaneous inference of covariances. 相似文献