Hypothesis testing for the generalized multivariate modified Bessel model

In this paper, we consider hypothesis testing problems in which the involved samples are drawn from generalized multivariate modified Bessel populations. This is a much more general distribution that includes both the multivariate normal and multivariate-t distributions as special cases. We derive the distribution of the Hotelling's T²-statistic for both the one- and two-sample problems, as well as the distribution of the Scheffe's T²-statistic for the Behrens–Fisher problem. In all cases, the non-null distribution of the corresponding F-statistic follows a new distribution which we introduce as the non-central F-Bessel distribution. Some statistical properties of this distribution are studied. Furthermore, this distribution was utilized to perform some power calculations for tests of means for different models which are special cases of the generalized multivariate modified Bessel distribution, and the results compared with those obtained under the multivariate normal case. Under the null hypothesis, however, the non-central F-Bessel distribution reduces to the central F-distribution obtained under the classical normal model.     

Onm-dependence and Edgeworth expansions

This paper contains two results. The first establishes, under mild assumptions, the validity of an Edgeworth expansion with remaindero(N ^–1/2) for aU-statistic with a kernel of degree two using observations from anm-dependent shift. The second result gives a necessary and sufficient condition for the distribution of a sum ofm-dependent random variables to possess an Edgeworth expansion. This generalizes a result of Bickel and Robinson from the i.i.d. case to them-dependent case.This research was supported in part by National Science Foundation, Grant DMS 89-23071.     

Approximations for multivariate U-statistics

Edgeworth approximations for multivariate U-statistics hold up to the order o(n^−1/2) under moment conditions and the assumption that the projection of the U-statistic to sums of i.i.d. random vectors is strongly nonlattice.     

Weighted bootstrap for U-statistics

In this paper we investigate the weighted bootstrap for U-statistics and its properties. Under very general choices of random weights and certain regularity conditions, we show that the weighted bootstrap method with U-statistics provides second-order accurate approximations to the distribution of U-statistics. We shall prove this via one-term Edgeworth expansions of weighted U-statistics.     

Edgeworth expansion in censored linear regression model

For the censored simple linear regression model, we establish a oneterm Edgeworth expansion for the Koul, Susarla and Van Ryzin type estimator of the regression coefficient. Our approach is to represent the estimator of the regression coefficient as an asymptoticU-statistic plus some ignorable terms and hence apply the known results on the Edgeworth expansions for asymptoticU-statistic. The counting process and martingale techniques are used to provide the proof of the main results.     

On the k-sample Behrens-Fisher problem for high-dimensional data

For several decades, much attention has been paid to the two-sample Behrens-Fisher (BF) problem which tests the equality of the means or mean vectors of two normal populations with unequal variance/covariance structures. Little work, however, has been done for the k-sample BF problem for high dimensional data which tests the equality of the mean vectors of several high-dimensional normal populations with unequal covariance structures. In this paper we study this challenging problem via extending the famous Scheffe’s transformation method, which reduces the k-sample BF problem to a one-sample problem. The induced one-sample problem can be easily tested by the classical Hotelling’s T ² test when the size of the resulting sample is very large relative to its dimensionality. For high dimensional data, however, the dimensionality of the resulting sample is often very large, and even much larger than its sample size, which makes the classical Hotelling’s T ² test not powerful or not even well defined. To overcome this difficulty, we propose and study an L ²-norm based test. The asymptotic powers of the proposed L ²-norm based test and Hotelling’s T ² test are derived and theoretically compared. Methods for implementing the L ²-norm based test are described. Simulation studies are conducted to compare the L ²-norm based test and Hotelling’s T ² test when the latter can be well defined, and to compare the proposed implementation methods for the L ²-norm based test otherwise. The methodologies are motivated and illustrated by a real data example. The work was supported by the National University of Singapore Academic Research Grant (Grant No. R-155-000-085-112)     

TheK-theory of AF embeddings of the rational rotation algebras

Our main result is the construction of an embedding ofC(T²) into an approximately finite-dimensionalC ^*-algebra which induces an injection onK ₀(C(T²)). The existence of this embedding implies that Cech cohomology cannot be extended to a stable, continuous homology theory forC ^*-algebras which admits a well-behaved Chern character. Homotopy properties ofC ^*-algebras are also considered. For example, we show that the second homotopy functor forC ^*-algebras is discontinuous. Similar embeddings are constructed for all the rational rotation algebras, with the consequence that none of the rational rotation algebras satisfies the homotopy property called semiprojectivity.     

Moments of randomly stopped sums-revisited

Conditions are obtained for (*)E|S _T |^γ<∞, γ>2 whereT is a stopping time and {S _n=∑ ₁ ⁿ ,X _j ℱ _n ,n⩾1} is a martingale and these ensure when (**)X _n ,n≥1 are independent, mean zero random variables that (*) holds wheneverET ^γ/2<∞, sup _n≥1 E|X _n |^γ<∞. This, in turn, is applied to obtain conditions for the validity ofE|S _k,T |^γ<∞ and of the second moment equationES _k,T ² =σ ² EΣ _j=k ^T S _k−1,j−1 ² where and {X _n , n≥1} satisfies (**) and ,n≥1. The latter is utilized to elicit information about a moment of a stopping rule. It is also shown for i.i.d. {X _n , n≥1} withEX=0,EX ²=1 that the a.s. limit set of {(n log logn)^−k/2 S _k,n ,n≥k} is [0,2 ^k/2/k!] or [−2 ^k/2/k!] according ask is even or odd and this can readily be reformulated in terms of the corresponding (degenerate kernel)U-statistic .     

An Edgeworth Expansion for Studentized Finite Population Statistics

We show the validity of the one-term Edgeworth expansion for Studentized asymptotically linear statistics based on samples drawn without replacement from finite populations. Replacing the moments defining the expansion by their estimators we obtain an empirical Edgeworth expansion. We show the validity of the empirical Edgeworth expansion in probability.     

Edgeworth Expansions for Studentized Versions of Symmetric Finite Population Statistics

We show the validity of the one-term Edgeworth expansion for Studentized asymptotically linear statistics based on samples drawn without replacement from finite populations. Replacing the moments defining the expansion by their estimators, we obtain an empirical Edgeworth expansion. We show the validity of the empirical Edgeworth expansion in probability.     

E *-Stable Theories

S. Shelah proved that stability of a theory is equivalent to definability of every complete type of that theory. T. Mustafin introduced the concept of being T ^*-stable, generalizing the notion of being stable. However, T ^*-stability does not necessitate definability of types. The key result of the present article is proving the definability of types for E ^*-stable theories. This concept differs from that of being T ^*-stable by adding the condition of being continuous. As a consequence we arrive at the definability of types over any P-sets in P-stable theories, which previously was established by T. Nurmagambetov and B. Poizat for types over P-models.     

On Positive Operators on Some Ordered Banach Spaces

In this paper we study to what extent some classical results concerning operators T, from a -space to a Banach space, or from a Banach space to a L ¹-space, can be precised, when the Banach spaces involved are ordered (by a normal cone in the first case, by a closed generating proper convex cone in the second case) and when the operators T are positive.     

Finite-sample inference with monotone incomplete multivariate normal data, I

We consider problems in finite-sample inference with two-step, monotone incomplete data drawn from , a multivariate normal population with mean and covariance matrix . We derive a stochastic representation for the exact distribution of , the maximum likelihood estimator of . We obtain ellipsoidal confidence regions for through T², a generalization of Hotelling’s statistic. We derive the asymptotic distribution of, and probability inequalities for, T² under various assumptions on the sizes of the complete and incomplete samples. Further, we establish an upper bound for the supremum distance between the probability density functions of and , a normal approximation to .     

Vector measure Maurey–Rosenthal-type factorizations and -sums of -spaces

Consider a vector measure of bounded variation m with values in a Banach space and an operator T:XL¹(m), where L¹(m) is the space of integrable functions with respect to m. We characterize when T can be factorized through the space L²(m) by means of a multiplication operator given by a function of L²(|m|), where |m| is the variation of m, extending in this way the Maurey–Rosenthal Theorem. We use this result to obtain information about the structure of the space L¹(m) when m is a sequential vector measure. In this case the space L¹(m) is an ℓ-sum of L¹-spaces.     

A Note on Edgeworth Expansions for <Emphasis Type="Italic">U</Emphasis>-Statistics under Minimal Conditions

A one-term Edgeworth expansion for U-statistics with kernel h(x, y) was derived by Jing and Wang [3] under optimal moment conditions. In this note, we show that one of the optimal moment conditions E| h(X ₁, X ₂|^5/3 < ∞ can be weakened to lim _t→∞ t ^5/3 P(|h(X ₁, X ₂)| > t) → 0. Printed in Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 453–440, July–September, 2005.     

An everywhere convergent series representation of the distribution of Hotelling's generalized T0

A new series representation of the exact distribution of Hotelling's generalized T₀² statistic is obtained. Unlike earlier work, the series representation given here is everywhere convergent. Explicit formulae are given for both the null and the non-central distributions. Earlier results by [1], 215–225), which are convergent on the interval [0, 1), are also derived quite simply from our formulae. The paper therefore provides a solution to the long standing problem of the exact distribution of the T₀² statistic in the general case.     

Partial pivoting strategies for symmetric gaussian elimination

In an earlier paper [6] it is shown how the use of symmetric additions of rows and columns enables a stableLDL ^T factorization of symmetric indefinite matrices. In this paper we describe partial pivoting strategies. These strategies are faster than the complete pivoting strategies that were introduced in the first paper. Numerical experiments are included. The results show that some of the new strategies share the stable behaviour of complete pivoting while running almost as fast as the well-known Cholesky decomposition.     

A Cramér Type Large Deviation Result for Student's t-Statistic

Let X, X ₁, X ₂,... be independent and identically distributed random variables with a finite third moment, and let T _n be the Student's t-statistic. This paper shows that lim _n P(T _n>x)/P(t _n>x)=1 holds uniformly in 0xo(n ^1/6), where t _n has a t-distribution with n–1 degrees of freedom. An example is also given to show that a finite third moment is necessary for this result.     

Two Commuting Involutions Fixing F n ∪ F n-1

Consider G=Z ₂² as the group generated by two commuting involutions, and let be a smooth G-action on a smooth and closed manifold M. Suppose that the fixed point set of Φ consists of two connected components, F ⁿ and F ^n-1, with dimensions n and n−1, respectively. In this paper we prove that, if in the fixed data of Φ at least two eigenbundles over F ⁿ have dimension greater than n, and at least one eigenbundle over F ^n-1 has dimension greater than n−1, then the action (M,Φ) bounds equivariantly.It is well known that, if is a smooth involution on a smooth and closed m-dimensional manifold M ^m such that the fixed point set of T has constant dimension n, and if m > 2n, then (M ^m ,T) bounds equivariantly; this fact was proved by R. E. Stong and C. Kosniowski 27 years ago. As a consequence of our result, we will see that the same fact is true when, besides n-dimensional components, the fixed point set contains additional (n−1)-dimensional components.     

Test for a specified signal when the noise covariance matrix is unknown

In the univariate case it is well known that the one sided t test is uniformly most powerful for the null hypothesis against all one sided alternatives. Such a property does not easily extend to the multivariate case. In this paper, a test derived for the hypothesis that the mean of a vector random variable is zero against specified alternatives, when the covariance matrix is unknown. This test depends on the given alternatives and is more powerful than Hotelling's T². The results are derived both for real and complex vector observations and under normal and spherical distributions. The properties of the proposed tests are investigated in detail when a single alternative is specified.