Asymptotic expansions of the distributions of estimators in canonical correlation analysis under nonnormality

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.     

Approximations to the distribution of the sample correlation matrix

In this article, multivariate density expansions for the sample correlation matrix R are derived. The density of R is expressed through multivariate normal and through Wishart distributions. Also, an asymptotic expansion of the characteristic function of R is derived and the main terms of the first three cumulants of R are obtained in matrix form. These results make it possible to obtain asymptotic density expansions of multivariate functions of R in a direct way.     

Asymptotic distributions of functions of the eigenvalues of some random matrices for nonnormal populations

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.     

Asymptotic distributions of the latent roots of the covariance matrix with multiple population roots

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.     

Further asymptotic formulas for the non-null distributions of three statistics for multivariate linear hypothesis

Summary New asymptotic expansions of the non-null distributions of the likelihood ratio, Hotelling's and Pillai's statistics for multivariate linear hypothesis are given in terms of normal distribution function and its derivatives, assuming the matrix of noncentrality parameters is of the same order as the sample size.     

Accurate distribution and its asymptotic expansion for the tetrachoric correlation coefficient

Accurate distributions of the estimator of the tetrachoric correlation coefficient and, more generally, functions of sample proportions for the 2 by 2 contingency table are derived. The results are obtained given the definitions of the estimators even when some marginal cell(s) are empty. Then, asymptotic expansions of the distributions of the parameter estimators standardized by the population asymptotic standard errors up to order O(1/n) and those of the studentized ones up to the order next beyond the conventional normal approximation are derived. The asymptotic results can be obtained in a much shorter computation time than the accurate ones. Numerical examples were used to illustrate advantages of the studentized estimator of Fisher’s z transformation of the tetrachoric correlation coefficient.     

Merging asymptotic expansions for cooperative gamblers in generalized St. Petersburg games

Merging asymptotic expansions are established for the distribution functions of suitably centered and normed linear combinations of winnings in a full sequence of generalized St. Petersburg games, where a linear combination is viewed as the share of any one of n cooperative gamblers who play with a pooling strategy. The expansions are given in terms of Fourier-Stieltjes transforms and are constructed from suitably chosen members of the classes of subsequential semistable infinitely divisible asymptotic distributions for the total winnings of the n players and from their pooling strategy, where the classes themselves are determined by the two parameters of the game. For all values of the tail parameter, the expansions yield best possible rates of uniform merge. Surprisingly, it turns out that for a subclass of strategies, not containing the averaging uniform strategy, our merging approximations reduce to asymptotic expansions of the usual type, derived from a proper limiting distribution. The Fourier-Stieltjes transforms are shown to be numerically invertible in general and it is also demonstrated that the merging expansions provide excellent approximations even for very small n.     

Tail Calculus with Remainder,Applications to Tail Expansions for Infinite Order Moving Averages,Randomly Stopped Sums,and Related Topics

We derive asymptotic expansions for tails of infinite weighted convolutions of some heavy-tailed distributions. Applications are given to tail expansion of the marginal distribution of ARMA processes, randomly stopped sums, as well as limiting waiting time distribution. AMS 2000 Subject Classifications. Primary—62E99, Secondary—41A60, 44A35, 60G50, 60K25     

Central limit asymptotics for shifts of finite type

We study the rate of convergence and asymptotic expansions in the central limit theorem for the class of Hölder continuous functions on a shift of finite type endowed with a stationary equilibrium state. It is shown that the rate of convergence in the theorem isO(n ^?1/2) and when the function defines a non-lattice distribution an asymptotic expansion to the order ofo(n ^?1/2) is given. Higher-order expansions can be obtained for a subclass of functions. We also make a remark on the central limit theorem for (closed) orbital measures.     

A systematization of the saddle point method. Application to the Airy and Hankel functions

The standard saddle point method of asymptotic expansions of integrals requires to show the existence of the steepest descent paths of the phase function and the computation of the coefficients of the expansion from a function implicitly defined by solving an inversion problem. This means that the method is not systematic because the steepest descent paths depend on the phase function on hand and there is not a general and explicit formula for the coefficients of the expansion (like in Watson's Lemma for example). We propose a more systematic variant of the method in which the computation of the steepest descent paths is trivial and almost universal: it only depends on the location and the order of the saddle points of the phase function. Moreover, this variant of the method generates an asymptotic expansion given in terms of a generalized (and universal) asymptotic sequence that avoids the computation of the standard coefficients, giving an explicit and systematic formula for the expansion that may be easily implemented on a symbolic manipulation program. As an illustrative example, the well-known asymptotic expansion of the Airy function is rederived almost trivially using this method. New asymptotic expansions of the Hankel function Hn(z) for large n and z are given as non-trivial examples.     

Asymptotic expansions for the distribution of the crossing number of a strip by a Markov-modulated random walk

We obtain complete asymptotic expansions for the distribution of the crossing number of a strip in n steps by sample paths of a random walk defined on a finite Markov chain. We assume that the Cramér condition holds for the distribution of jumps and the width of the strip grows with n. The method consists in finding factorization representations of the moment generating functions of the distributions under study, isolating the main terms in the asymptotics of the representations, and inverting those main terms by the modified saddle-point method.     

Asymptotic expansion of the null distribution of test statistic for linear hypothesis in nonnormal linear model

This paper is concerned with the null distribution of test statistic T for testing a linear hypothesis in a linear model without assuming normal errors. The test statistic includes typical ANOVA test statistics. It is known that the null distribution of T converges to χ² when the sample size n is large under an adequate condition of the design matrix. We extend this result by obtaining an asymptotic expansion under general condition. Next, asymptotic expansions of one- and two-way test statistics are obtained by using this general one. Numerical accuracies are studied for some approximations of percent points and actual test sizes of T for two-way ANOVA test case based on the limiting distribution and an asymptotic expansion.     

Asymptotic expansions in mean and covariance structure analysis

Asymptotic expansions of the distributions of parameter estimators in mean and covariance structures are derived. The parameters may be common to, or specific in means and covariances of observable variables. The means are possibly structured by the common/specific parameters. First, the distributions of the parameter estimators standardized by the population asymptotic standard errors are expanded using the single- and the two-term Edgeworth expansions. In practice, the pivotal statistic or the Studentized estimator with the asymptotically distribution-free standard error is of interest. An asymptotic distribution of the pivotal statistic is also derived by the Cornish-Fisher expansion. Simulations are performed for a factor analysis model with nonzero factor means to see the accuracy of the asymptotic expansions in finite samples.     

An asymptotic expansion of the distribution of Rao's U-statistic under a general condition

In this paper we consider the problem of testing the hypothesis about the sub-mean vector. For this propose, the asymptotic expansion of the null distribution of Rao's U-statistic under a general condition is obtained up to order of n^-1. The same problem in the k-sample case is also investigated. We find that the asymptotic distribution of generalized U-statistic in the k-sample case is identical to that of the generalized Hotelling's T² distribution up to n^-1. A simulation experiment is carried out and its results are presented. It shows that the asymptotic distributions have significant improvement when comparing with the limiting distributions both in the small sample case and the large sample case. It also demonstrates the equivalence of two testing statistics mentioned above.     

On a test of equality of two-parameter exponential distributions

In this paper, an asymptotic expansion of the distribution of the statistic for testing the equality of p two-parameter exponential distributions is obtained upto the order n^?4 with the second term of the order n^?3 where n is the size of the sample drawn from the i th exponential population. The asymptotic expansion can therefore be used to obtain accurate approximations to the critical values of the test statistic even for comparatively small values of n. Also we have shown that F-tables can be used to test the hypothesis when the sample size is moderately large.     

ASYMPTOTIC DISTRIBUTION OF LIKELIHOOD RATIO STATISTIC FOR TESTING SPHERICTY IN GROWTH CURVE MODELS

1IntroductionTheideaofgeneralizedgrowthcurvemodelsmeanswhereY=(yi,y2,''tAN)isapxNobservationmains;bothBandAarematrixeswithknownconstantelementsoforderpxqandmxNrespectively:(isaqxinparametermatriX;e=(el,ez,'',eN)isarandomerrorrnatrisoforderNxN;Zisanonnegativedefitiveparametermatrixoforderpxp,W=(wij)isaNxNnon-negativedefitivernatriswithknownelements.LetYdenoteamatriXwhosecoluxnnvectorsarethecorrelatedsaxnpleswithsizeNfroma"dimensionalnormalpopulation.Thecovariancematrixofthepopulationis…     

The Accuracy of Merging Approximation in Generalized St. Petersburg Games

Merging asymptotic expansions of arbitrary length are established for the distribution functions and for the probabilities of suitably centered and normalized cumulative winnings in a full sequence of generalized St. Petersburg games, extending the short expansions due to Csörg? (Acta Sci. Math. (Szeged) 73:297–331, 2007). These expansions are given in terms of suitably chosen members from the classes of subsequential semistable infinitely divisible asymptotic distribution functions and certain derivatives of these functions. The length of the expansion depends upon the tail parameter. Both uniform and nonuniform bounds are presented.     

A two-stage rank test using density estimation

For the one-sample problem, a two-stage rank test is derived which realizes a required power against a given local alternative, for all sufficiently smooth underlying distributions. This is achieved using asymptotic expansions resulting in a precision of orderm ^–1, wherem is the size of the first sample. The size of the second sample is derived through a number of estimators of e. g. integrated squared densities and density derivatives, all based on the first sample. The resulting procedure can be viewed as a nonparametric analogue of the classical Stein's two-stage procedure, which uses at-test and assumes normality for the underlying distribution. The present approach also generalizes earlier work which extended the classical method to parametric families of distributions.     

Large degree asymptotics of generalized Bessel polynomials

Asymptotic expansions are given for large values of n of the generalized Bessel polynomials . The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the origin in the z-plane. New forms of expansions in terms of elementary functions valid in sectors not containing the turning points z=±i/n are derived, and a new expansion in terms of modified Bessel functions is given. Earlier asymptotic expansions of the generalized Bessel polynomials by Wong and Zhang (1997) and Dunster (2001) are discussed.