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.     

The Whitney problem of existence of a linear extension operator

We prove that a linear bounded extension operator exists for the trace of C ^1·ω (R ⁿ )to an arbitrary closed subset of R ⁿ .The similar result is obtained for some other spaces of multivariate smooth functions. We also show that unlike the one-dimensional case treated by Whitney, for some trace spaces of multivariate smooth functions a linear bounded extension operator does not exist. The proofs are based on a relation between the problem under consideration and a similar problem for Lipschitz spaces defined on hyperbolic Riemannian manifolds.     

Testing for Spherical Symmetry of a Multivariate Distribution

We consider a test for spherical symmetry of a distribution in ^dwith an unknown center. It is a multivariate version of the tests suggested by Schuster and Barker and by Arcones and Giné. The test statistic is based on the multivariate extension of the distribution and quantile functions, recently introduced by Koltchinskii and Dudley and by Chaudhuri. We study the asymptotic behavior of the sequence of test statistics for large samples and for a fixed spherically asymmetric alternative as well as for a sequence of local alternatives converging to a spherically symmetric distribution. We also study numerically the performance of the test for moderate sample sizes and justify a symmetrized version of bootstrap approximation of the distribution of test statistics.     

Robust and accurate inference for generalized linear models

In the framework of generalized linear models, the nonrobustness of classical estimators and tests for the parameters is a well known problem, and alternative methods have been proposed in the literature. These methods are robust and can cope with deviations from the assumed distribution. However, they are based on first order asymptotic theory, and their accuracy in moderate to small samples is still an open question. In this paper, we propose a test statistic which combines robustness and good accuracy for moderate to small sample sizes. We combine results from Cantoni and Ronchetti [E. Cantoni, E. Ronchetti, Robust inference for generalized linear models, Journal of the American Statistical Association 96 (2001) 1022–1030] and Robinson, Ronchetti and Young [J. Robinson, E. Ronchetti, G.A. Young, Saddlepoint approximations and tests based on multivariate M-estimators, The Annals of Statistics 31 (2003) 1154–1169] to obtain a robust test statistic for hypothesis testing and variable selection, which is asymptotically χ²-distributed as the three classical tests but with a relative error of order O(n⁻¹). This leads to reliable inference in the presence of small deviations from the assumed model distribution, and to accurate testing and variable selection, even in moderate to small samples.     

The Distribution of Condition Numbers of Rational Data of Bounded Bit Length

We prove that rational data of bounded input length are uniformly distributed (in the classical sense of H. Weyl, in [42]) with respect to the probability distribution of condition numbers of numerical analysis. We deal both with condition numbers of linear algebra and with condition numbers for systems of multivariate polynomial equations. For instance, we prove that for a randomly chosen n\times n rational matrix M of bit length O(n ⁴ log n) + log w , the condition number k(M) satisfies k(M) ≤ w n ^5/2 with probability at least 1-2w ^-1 . Similar estimates are established for the condition number μ_ norm of M. Shub and S. Smale when applied to systems of multivariate homogeneous polynomial equations of bounded input length. Finally, we apply these techniques to estimate the probability distribution of the precision (number of bits of the denominator) required to write approximate zeros of systems of multivariate polynomial equations of bounded input length. March 7, 2001. Final version received: June 7, 2001.     

Reduction Algorithm for the NPMLE for the Distribution Function of Bivariate Interval-Censored Data

This article considers computational aspects of the nonparametric maximum likelihood estimator (NPMLE) for the distribution function of bivariate interval-censored data. The computation of the NPMLE consists of a parameter reduction step and an optimization step. This article focuses on the reduction step and introduces two new reduction algorithms: the Tree algorithm and the HeightMap algorithm. The Tree algorithm is mentioned only briefly. The HeightMap algorithm is discussed in detail and also given in pseudo code. It is a fast and simple algorithm of time complexityO(n²). This is an order faster than the best known algorithm thus far by Bogaerts and Lesaffre. We compare the new algorithms to earlier algorithms in a simulation study, and demonstrate that the new algorithms are significantly faster. Finally, we discuss how the HeightMap algorithm can be generalized to d-dimensional data with d > 2. Such a multivariate version of the HeightMap algorithm has time complexity O(n^d).     

Average case complexity of linear multivariate problems I. Theory

We study the average case complexity of linear multivariate problems, that is, the approximation of continuous linear operators on functions of d variables. The function spaces are equipped with Gaussian measures. We consider two classes of information. The first class Λ^std consists of function values, and the second class Λ^all consists of all continuous linear functionals. Tractability of a linear multivariate problem means that the average case complexity of computing an ε-approximation is O((1/)^p) with p independent of d. The smallest such p is called the exponent of the problem. Under mild assumptions, we prove that tractability in Λ^all is equivalent to tractability in Λ^std and that the difference of the exponents is at most 2. The proof of this result is not constructive. We provide a simple condition to check tractability in Λ^all. We also address the issue of how to construct optimal (or nearly optimal) sample points for linear multivariate problems. We use relations between average case and worst case settings. These relations reduce the study of the average case to the worst case for a different class of functions. In this way we show how optimal sample points from the worst case setting can be used in the average case. In Part II we shall apply the theoretical results to obtain optimal or almost optimal sample points, optimal algorithms, and average case complexity functions for linear multivariate problems equipped with the folded Wiener sheet measure. Of particular interest will be the multivariate function approximation problem.     

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.     

Multivariate Polynomial Minimization and Its Application in Signal Processing

We make a conjecture that the number of isolated local minimum points of a 2n-degree or (2n+1)-degree r-variable polynomial is not greater than n ^r when n 2. We show that this conjecture is the minimal estimate, and is true in several cases. In particular, we show that a cubic polynomial of r variables may have at most one local minimum point though it may have 2^r critical points. We then study the global minimization problem of an even-degree multivariate polynomial whose leading order coefficient tensor is positive definite. We call such a multivariate polynomial a normal multivariate polynomial. By giving a one-variable polynomial majored below a normal multivariate polynomial, we show the existence of a global minimum of a normal multivariate polynomial, and give an upper bound of the norm of the global minimum and a lower bound of the global minimization value. We show that the quartic multivariate polynomial arising from broad-band antenna array signal processing, is a normal polynomial, and give a computable upper bound of the norm of the global minimum and a computable lower bound of the global minimization value of this normal quartic multivariate polynomial. We give some sufficient and necessary conditions for an even order tensor to be positive definite. Several challenging questions remain open.     

Two tests for multivariate normality based on the characteristic function

We present two tests for multivariate normality. The presented tests are based on the Lévy characterization of the normal distribution and on the BHEP tests. The tests are affine invariant and consistent. We obtain the asymptotic null distribution of the test statistics using some results about generalized one-sample U-statistics, which are of independent interest.      

Tests for standardized generalized variances of multivariate normal populations of possibly different dimensions

In many practical problems, one needs to compare variabilities of several multidimensional populations. The concept of standardized generalized variance (SGV) is introduced as an extension of the concept of GV. Considering multivariate normal populations of possibly different dimensions and general covariance matrices, LRTs are derived for SGVs. The criteria turn out to be elegant multivariate analogs to those for tests for variances in the univariate cases. The null and nonnull distributions of the test criteria are deducdd in computable forms in terms of Special Functions, e.g., Pincherle'sH-function, by exploiting the theory of calculus of residues (Mathai and Saxena,Ann. Math. Statist.40, 1439–1448).     

A procedure for assessing vector correlations

Three known measures of multivariate relationship are presented. Under the null hypothesis of lack of multivariate relationship between K random vectors, the asymptotic joint distributions of the % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa% qabeGabaaabaGaam4saaqaaiaaikdaaaaacaGLOaGaayzkaaaaaa!390F!\[\left( {\begin{array}{*{20}c}K \\2 \\\end{array} } \right)\]values taken by these measures for all possible pairs (X ⁽ⁱ⁾, X^(j)), 1i<jK, is used to construct tests of the null hypothesis based on the maximum and more generally, on the greatest values of the measures. The asymptotic power of the tests is also obtained under a sequence of alternatives.The author was partially supported by a grant of the Natural Sciences and Engineering Research Council of Canada.     

On generating multivariate Poisson data in management science applications

Generating multivariate Poisson random variables is essential in many applications, such as multi echelon supply chain systems, multi‐item/multi‐period pricing models, accident monitoring systems, etc. Current simulation methods suffer from limitations ranging from computational complexity to restrictions on the structure of the correlation matrix, and therefore are rarely used in management science. Instead, multivariate Poisson data are commonly approximated by either univariate Poisson or multivariate Normal data. However, these approximations are often not adequate in practice. In this paper, we propose a conceptually appealing correction for NORTA (NORmal To Anything) for generating multivariate Poisson data with a flexible correlation structure and rates. NORTA is based on simulating data from a multivariate Normal distribution and converting it into an arbitrary continuous distribution with a specific correlation matrix. We show that our method is both highly accurate and computationally efficient. We also show the managerial advantages of generating multivariate Poisson data over univariate Poisson or multivariate Normal data. Copyright © 2011 John Wiley & Sons, Ltd.     

Two-sample inference for normal mean vectors based on monotone missing data

Inferential procedures for the difference between two multivariate normal mean vectors based on incomplete data matrices with different monotone patterns are developed. Assuming that the population covariance matrices are equal, a pivotal quantity, similar to the Hotelling T² statistic, is proposed, and its approximate distribution is derived. Hypothesis testing and confidence estimation of the difference between the mean vectors based on the approximate distribution are outlined. The validity of the approximation is investigated using Monte Carlo simulation. Monte Carlo studies indicate that the approximate method is very satisfactory even for small samples. A multiple comparison procedure is outlined and the proposed methods are illustrated using an example.     

Forecasting retained earnings of privately held companies with PCA and L1 regression

We use proprietary data collected by SVB Analytics, an affiliate of Silicon Valley Bank, to forecast the retained earnings of privately held companies. Combining methods of principal component analysis (PCA) and L¹/quantile regression, we build multivariate linear models that feature excellent in‐sample fit and strong out‐of‐sample predictive accuracy. The combined PCA and L¹ technique effectively deals with multicollinearity and non‐normality of the data, and also performs favorably when compared against a variety of other models. Additionally, we propose a variable ranking procedure that explains which variables from the current quarter are most predictive of the next quarter's retained earnings. We fit models to the top five variables identified by the ranking procedure and thereby, discover interpretable models with excellent out‐of‐sample performance. Copyright © 2013 John Wiley & Sons, Ltd.     

Robin Approximation of Dirichlet Boundary Value Problems

This paper describes the rate of convergence of solutions of Robin boundary value problems of an elliptic equation to the solution of a Dirichlet problem as a boundary parameter decreases to zero. The results are found using representations for solutions of the equations in terms of Steklov eigenfunctions. Particular interest is in the case where the Dirichlet data is only in L²(?Ω,dσ). Various approximation bounds are obtained and the rate of convergence of the Robin approximations in the H¹ and L² norms are shown to have convergence rates that depend on the regularity of the Dirichlet data.     

Third-order asymptomic properties of a class of test statistics under a local alternative

Suppose that {X_i; I = 1, 2, …,} is a sequence of p-dimensional random vectors forming a stochastic process. Let p_n, θ(X_n), X_n ^np, be the probability density function of X_n = (X₁, …, X_n) depending on θ Θ, where Θ is an open set of ¹. We consider to test a simple hypothesis H : θ = θ₀ against the alternative A : θ ≠ θ₀. For this testing problem we introduce a class of tests , which contains the likelihood ratio, Wald, modified Wald, and Rao tests as special cases. Then we derive the third-order asymptotic expansion of the distribution of T under a sequence of local alternatives. Using this result we elucidate various third-order asymptotic properties of T (e.g., Bartlett's adjustments, third-order asymptotically most powerful properties). Our results are very general, and can be applied to the i.i.d. case, multivariate analysis, and time series analysis. Two concrete examples will be given. One is a Gaussian ARMA process (dependent case), and the other is a nonlinear regression model (non-identically distributed case).     

Certain estimation problems for multivariate hypergeometric models

Certain estimation problems associated with the multivariate hypergeometric models: the property of completeness, maximum likelihood estimates of the parameters of multivariate negative hypergeometric, multivariate negative inverse hypergeometric, Bayesian estimation of the parameters of multivariate hypergeometric and multivariate inverse hypergeometrics are discussed in this paper. A two stage approach for generating the prior distribution, first by setting up a parametric super population and then choosing a prior distribution is followed. Posterior expectations and variances of certain functions of the parameters of the finite population are provided in cases of direct and inverse sampling procedures. It is shown that under extreme diffuseness of prior knowledge the posterior distribution of the finite population mean has an approximate mean and variance (N-n)S ²/Nn, providing a Bayesian interpretation for the classical unbiased estimates in traditional sample survey theory.     

Testing significance of a mean vector—A possible alternative to Hotelling'sT 2

In problems involving multivariate measurements experimental considerations often indicate grouping of variables into subsets ordered according to their importance. In such situations, the problems such as comparison of two mean vectors and profile analysis may be treated by Hotelling'sT ²-test adapted along the lines of the step-wise procedure of J. Roy [10], or the well known test for additional information due to Rao [9]. In this paper we study a modification of the step-wise procedure obtained by combining the component tests. The exact Bahadur slopes of resulting procedures are computed and it is shown that the procedure based upon Fisher's combination method is asymptotically equivalent to Hotelling'sT ². A Monte Carlo study suggests that even in small samples the power functions of the new method and Hotelling'sT ²-test are practically equivalent. Research sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF under Grant No. AFOSR-77-3360. The United States Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation hereon.     

On Dual Wavelet Tight Frames

A characterization of multivariate dual wavelet tight frames for any general dilation matrix is presented in this paper. As an application, Lawton's result on wavelet tight frames inL²( ) is generalized to then-dimensional case. Two ways of constructing certain dual wavelet tight frames inL²( ⁿ) are suggested. Finally, examples of smooth wavelet tight frames inL²( ) andH²( ) are provided. In particular, an example is given to demonstrate that there is a function ψ whose Fourier transform is positive, compactly supported, and infinitely differentiable which generates a non-MRA wavelet tight frame inH²( ).