Error bounds for asymptotic expansions of Wilks’ lambda distribution

Let Λ=|S_e|/|S_e+S_h|, where S_h and S_e are independently distributed as Wishart distributions W_p(q,Σ) and W_p(n,Σ), respectively. Then Λ has Wilks’ lambda distribution Λ_p,q,n which appears as the distributions of various multivariate likelihood ratio tests. This paper is concerned with theoretical accuracy for asymptotic expansions of the distribution of T=-nlogΛ. We derive error bounds for the approximations. It is necessary to underline that our error bounds are given in explicit and computable forms.     

High-dimensional Edgeworth expansion of a test statistic on independence and its error bound

In this paper, we calculate Edgeworth expansion of a test statistic on independence when some of the parameters are large, and simulate the goodness of fit of its approximation. We also calculate an error bound for Edgeworth expansion. Some tables of the error bound are given, which show that the derived bound is sufficiently small for practical use.     

Cornish-Fisher expansions using sample cumulants and monotonic transformations

General formulas of the asymptotic cumulants of a studentized parameter estimator are given up to the fourth order with the added higher-order asymptotic variance. Using the sample counterparts of the asymptotic cumulants, formulas for the Cornish-Fisher expansions with third-order accuracy are obtained. Some new methods of monotonic transformations of the studentized estimator are presented. In addition, similar transformations of a fixed normal deviate are proposed up to the same order with some asymptotic comparisons to the transformations of the studentized estimator. Applications to a mean and a binomial proportion are shown with simulations for estimation of the proportion.     

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.     

Asymptotic expansions of test statistics for dimensionality and additional information in canonical correlation analysis when the dimension is large

This paper examines asymptotic expansions of test statistics for dimensionality and additional information in canonical correlation analysis based on a sample of size N=n+1 on two sets of variables, i.e.,  and . These problems are related to dimension reduction. The asymptotic approximations of the statistics have been studied extensively when dimensions p₁ and p₂ are fixed and the sample size N tends to infinity. However, the approximations worsen as p₁ and p₂ increase. This paper derives asymptotic expansions of the test statistics when both the sample size and dimension are large, assuming that and have a joint (p₁+p₂)-variate normal distribution. Numerical simulations revealed that this approximation is more accurate than the classical approximation as the dimension increases.     

A comparison of higher-order local powers of a class of one-way MANOVA tests under general distributions

The purpose of this paper is to investigate the effect of nonnormality upon the nonnull distributions of some MANOVA test statistics under normality. It is shown that whatever the underlying distributions, the difference of the local powers up to order N^-1 (N is the total number of observations) after either Bartlett's type adjustment or Cornish-Fisher's type adjustment under nonnormality coincides with that in Anderson [An Introduction to Multivariate Statistical Analysis, second ed., 1984 and third ed., 2003, Wiley, New York] under normality. The performance of higher-order results in finite samples is examined using simulation studies.     

Third-order power comparisons for a class of tests for multivariate linear hypothesis under general distributions

The purpose of this paper is, in multivariate linear regression model (Part I) and GMANOVA model (Part II), to investigate the effect of nonnormality upon the nonnull distributions of some multivariate test statistics under normality. It is shown that whatever the underlying distributions, the difference of local powers up to order N⁻¹ after either Bartlett’s type adjustment or Cornish-Fisher’s type size adjustment under nonnormality coincides with that in Anderson [An Introduction to Multivariate Statistical Analysis, 2nd ed. and 3rd ed., Wiley, New York, 1984, 2003] under normality. The derivation of asymptotic expansions is based on the differential operator associated with the multivariate linear regression model under general distributions. The performance of higher-order results in finite samples, including monotone Bartlett’s type adjustment and monotone Cornish-Fisher’s type size adjustment, is examined using simulation studies.     

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.     

Asymptotic expansions of the distributions of the latent roots in MANOVA and the canonical correlations

Asymptotic expansions are given for the density function of the normalized latent roots of S₁S₂^?1 for large n under the assumption of $\text{Ω = O(n)}$, where S₁ and S₂ are independent noncentral and central Wishart matrices having the $\text{W}{}_{\mathrm{p}}\text{(b, Σ; Ω)}$ and W_p(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.     

Some properties of canonical correlations and variates in infinite dimensions

In this paper the notion of functional canonical correlation as a maximum of correlations of linear functionals is explored. It is shown that the population functional canonical correlation is in general well defined, but that it is a supremum rather than a maximum, so that a pair of canonical variates may not exist in the spaces considered. Also the relation with the maximum eigenvalue of an associated pair of operators and the corresponding eigenvectors is not in general valid. When the inverses of the operators involved are regularized, however, all of the above properties are restored. Relations between the actual population quantities and their regularized versions are also established. The sample functional canonical correlations can be regularized in a similar way, and consistency is shown at a fixed level of the regularization parameter.     

Asymptotic theory for maximum deviations of sample covariance matrix estimates

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.     

The generalized near-integer Gamma distribution: a basis for ‘near-exact’ approximations to the distribution of statistics which are the product of an odd number of independent Beta random variables

In this paper the concept of near-exact approximation to a distribution is introduced. Based on this concept it is shown how a random variable whose exponential has a Beta distribution may be closely approximated by a sum of independent Gamma random variables, giving rise to the generalized near-integer (GNI) Gamma distribution. A particular near-exact approximation to the distribution of the logarithm of the product of an odd number of independent Beta random variables is shown to be a GNI Gamma distribution. As an application, a near-exact approximation to the distribution of the generalized Wilks Λ statistic is obtained for cases where two or more sets of variables have an odd number of variables. This near-exact approximation gives the exact distribution when there is at most one set with an odd number of variables. In the other cases a near-exact approximation to the distribution of the logarithm of the Wilks Lambda statistic is found to be either a particular generalized integer Gamma distribution or a particular GNI Gamma distribution.     

Canonical representation of conditionally specified multivariate discrete distributions

Most work on conditionally specified distributions has focused on approaches that operate on the probability space, and the constraints on the probability space often make the study of their properties challenging. We propose decomposing both the joint and conditional discrete distributions into characterizing sets of canonical interactions, and we prove that certain interactions of a joint distribution are shared with its conditional distributions. This invariance opens the door for checking the compatibility between conditional distributions involving the same set of variables. We formulate necessary and sufficient conditions for the existence and uniqueness of discrete conditional models, and we show how a joint distribution can be easily computed from the pool of interactions collected from the conditional distributions. Hence, the methods can be used to calculate the exact distribution of a Gibbs sampler. Furthermore, issues such as how near compatibility can be reconciled are also discussed. Using mixed parametrization, we show that the proposed approach is based on the canonical parameters, while the conventional approaches are based on the mean parameters. Our advantage is partly due to the invariance that holds only for the canonical parameters.     

Flexible bivariate beta distributions

Bivariate beta distributions which can be used to model data sets exhibiting positive or negative correlation are introduced. Properties of these bivariate beta distributions and their applications in Bayesian analysis are discussed. Three methods for parameter estimation are presented. The performance of these estimators is evaluated based on Monte Carlo simulations. Examples are provided to illustrate how additional parameters can be introduced to gain even more modeling flexibility. A possible extension of the proposed bivariate beta model and a multivariate generalization are also discussed.     

The asymptotic behavior of linear placement statistics

Orban and Wolfe (1982) and Kim (1999) provided the limiting distribution for linear placement statistics under null hypotheses only when one of the sample sizes goes to infinity. In this paper we prove the asymptotic normality and the weak convergence of the linear placement statistics of Orban and Wolfe (1982) and Kim (1999) when the sample sizes of each group go to infinity simultaneously.     

Local asymptotic normality in a stationary model for spatial extremes

De Haan and Pereira (2006) [6] provided models for spatial extremes in the case of stationarity, which depend on just one parameter β>0 measuring tail dependence, and they proposed different estimators for this parameter. We supplement this framework by establishing local asymptotic normality (LAN) of a corresponding point process of exceedances above a high multivariate threshold. Standard arguments from LAN theory then provide the asymptotic minimum variance within the class of regular estimators of β. It turns out that the relative frequency of exceedances is a regular estimator sequence with asymptotic minimum variance, if the underlying observations follow a multivariate extreme value distribution or a multivariate generalized Pareto distribution.     

A weighted multivariate signed-rank test for cluster-correlated data

A weighted multivariate signed-rank test is introduced for an analysis of multivariate clustered data. Observations in different clusters may then get different weights. The test provides a robust and efficient alternative to normal theory based methods. Asymptotic theory is developed to find the approximate p-value as well as to calculate the limiting Pitman efficiency of the test. A conditionally distribution-free version of the test is also discussed. The finite-sample behavior of different versions of the test statistic is explored by simulations and the new test is compared to the unweighted and weighted versions of Hotelling’s T² test and the multivariate spatial sign test introduced in [D. Larocque, J. Nevalainen, H. Oja, A weighted multivariate sign test for cluster-correlated data, Biometrika 94 (2007) 267-283]. Finally, a real data example is used to illustrate the theory.     

On the covariance of the asymptotic empirical copula process

Conditions are given under which the empirical copula process associated with a random sample from a bivariate continuous distribution has a smaller asymptotic covariance function than the standard empirical process based on observations from the copula. Illustrations are provided and consequences for inference are outlined.     

Asymptotic expansions for the pivots using log-likelihood derivatives with an application in item response theory

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.     

Edgeworth expansion of Wilks’ lambda statistic

An asymptotic expansion of the null distribution of the Wilks’ lambda statistic is derived when some of the parameters are large. Cornish-Fisher expansions of the upper percent points are also obtained. A monotone transformation which reduces the third and the fourth order cumulants is also derived. In order to study the accuracy of the approximation formulas, some numerical experiments are done, with comparing to the classical expansions when only the sample size tends to infinity.