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1.
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
2.
Takafumi Isogai 《Annals of the Institute of Statistical Mathematics》1977,29(1):235-246
S
e
andS
n
are independent central and noncentral Wishart matrices having Wishart distributionsW
p
(n
e
, Σ) andW
p
(n
h
, Σ; Ω) respectively. Asymptotic expansions are given for the distributions of latent roots ofS
h
S
e
−1
and of certain test statistics in MANOVA under the assumption thatn=n
e
+n
h
becomes large with a fixed ration
e
∶n
h
=e∶h(e+h=1,e>0,h>0) andΩ=O(n). 相似文献
3.
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
4.
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
5.
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
6.
Sadanori Konishi Takakazu Sugiyama 《Annals of the Institute of Statistical Mathematics》1981,33(1):27-33
Summary Normalizing transformations of the largest and the smallest latent roots of a sample covariance matrix in a normal sample
are obtained, when the corresponding population roots are simple. Using our results, confidence intervals for population roots
may easily be constructed. Some numerical comparisons of the resulting approximations are made in a bivariate case, based
on exact values of the probability integral of latent roots. 相似文献
7.
In this paper, the authors consider the evaluation of the distribution functions of the ratios of the intermediate roots to the trace of the real Wishart matrix as well as the ratios of the individual roots to the trace of the complex Wishart matrix. In addition, the authors consider the evaluation of the distribution functions of the ratios of the extreme roots of the Wishart matrix in the real and complex cases. Some applications and tables of the above distributions are also given. 相似文献
8.
R. Michel 《Journal of multivariate analysis》1977,7(2):235-248
Given a suitable function Fn we define a class of estimators called asymptotic Fn-estimators (i.e., estimators which approximate the solution of Fn(θ) = 0). It is proved that this class is nonvoid if appropriate regularity conditions are fulfilled and if one has at hand a suitable initial estimator. Furthermore, it is shown that Fn-estimators admit a stochastic expansion (which enables to give results on asymptotic expansions for the distribution of these estimators). 相似文献