Energy statistics: A class of statistics based on distances

Energy distance is a statistical distance between the distributions of random vectors, which characterizes equality of distributions. The name energy derives from Newton's gravitational potential energy, and there is an elegant relation to the notion of potential energy between statistical observations. Energy statistics are functions of distances between statistical observations in metric spaces. Thus even if the observations are complex objects, like functions, one can use their real valued nonnegative distances for inference. Theory and application of energy statistics are discussed and illustrated. Finally, we explore the notion of potential and kinetic energy of goodness-of-fit.     

Cross-correlating wavelet coefficients with applications to high-frequency financial time series

This paper uses a new concept in wavelet analysis to explore a financial transaction data set including returns, durations, and volume. The concept is based on a decomposition of the Allan covariance of two series into cross-covariances of wavelet coefficients, which allows a natural interpretation of cross-correlations in terms of frequencies. It is applied to financial transaction data including returns, durations between transactions, and trading volume. At high frequencies, we find significant spillover from durations to volume and a strong contemporaneous relation between durations and returns, whereas a strong causality between volume and volatility exists at various frequencies.     

Effect of heteroscedasticity between treatment groups on mixed‐effects models for repeated measures

Mixed‐effects models for repeated measures (MMRM) analyses using the Kenward‐Roger method for adjusting standard errors and degrees of freedom in an “unstructured” (UN) covariance structure are increasingly becoming common in primary analyses for group comparisons in longitudinal clinical trials. We evaluate the performance of an MMRM‐UN analysis using the Kenward‐Roger method when the variance of outcome between treatment groups is unequal. In addition, we provide alternative approaches for valid inferences in the MMRM analysis framework. Two simulations are conducted in cases with (1) unequal variance but equal correlation between the treatment groups and (2) unequal variance and unequal correlation between the groups. Our results in the first simulation indicate that MMRM‐UN analysis using the Kenward‐Roger method based on a common covariance matrix for the groups yields notably poor coverage probability (CP) with confidence intervals for the treatment effect when both the variance and the sample size between the groups are disparate. In addition, even when the randomization ratio is 1:1, the CP will fall seriously below the nominal confidence level if a treatment group with a large dropout proportion has a larger variance. Mixed‐effects models for repeated measures analysis with the Mancl and DeRouen covariance estimator shows relatively better performance than the traditional MMRM‐UN analysis method. In the second simulation, the traditional MMRM‐UN analysis leads to bias of the treatment effect and yields notably poor CP. Mixed‐effects models for repeated measures analysis fitting separate UN covariance structures for each group provides an unbiased estimate of the treatment effect and an acceptable CP. We do not recommend MMRM‐UN analysis using the Kenward‐Roger method based on a common covariance matrix for treatment groups, although it is frequently seen in applications, when heteroscedasticity between the groups is apparent in incomplete longitudinal data.     

The General Multivariate Gauss—Markov Model of the Incomplete Block Design

The aim of the paper is to generalize testing and estimation for the multivariate standard incomplete block model (Rao & Mitra, 1971a) to the general multivariate Gauss—Markov incomplete block model with singular covariance matrix. The results of this paper can be applied to particular cases of the multivariate Gauss—Markov incomplete block model, including the Zyskind—Martin model.     

Determinants of Trust in Industry, Government, and Citizen''s Groups in Japan

The causal structure of the determinants of trust in industry, government, and citizen's groups in Japan was investigated on the basis of Peters et al. (1997). A preliminary survey of the adequacy of the hypotheses proposed by Peters et al. in Japan was made. A set of hypothesized determinants of trust in Japan was proposed based on results of the preliminary survey. Questionnaires concerning perceptions of trust in the organizations and the proposed determinants were sent by mail to residents in the area where environmental risk problems had emerged. The data were analyzed by covariance structure analysis to construct models of trust in industry, government, and citizen's groups. As a result, "openness and honesty," "concern and care," "competence," "people's concern with risks," and "consensual values" were found to be factors directly determining trust. Suggested in particular is that "openness" of an organization is not attained merely by information disclosure, but also by bi-directional communication with the people. Moreover, these models include "consensual values," which do not appear in the model proposed by Peters et al.     

Asymptotic distribution of hotelling's trace for two unequal covariance matrices and robustness study of test of equality of mean vectors

Asymptotic expansions for the percentiles and c.d.f., up to terms of order $\text{1}\text{n}{}^2$ of the statistic $\text{T =m}\text{T}\text{r}\text{S}{}_1\text{S}{}^{-1}{}_2$, where mS₁ and nS₂ independently distributed W(m, p, Σ₁) and W(n, p, Σ₂) respectively, are obtained using methods similar to those of Ito [4], Chattopadhyay and Pillai [2]. These expansions hold when $\text{Σ}{}_1\text{Σ}{}^{-1}{}_2\text{=}\text{I}\text{+}\text{F}$ and$\text{|}\text{Ch}{}_{\mathrm{i}}\text{(}\text{F}\text{)| < 1}$. Tables of powers of T for p = 3 and p = 4 for m = 4 and various values of n are given and comparison made with the exact powers for p = 3. These powers are useful for the study of (i) the test of equality of covariance matrices in two p-variate normal populations and (ii) robustness of test of equality of mean vectors of l normal populations against the violation of the assumption of equality of covariance matrices.     

Estimation of patterned covariance in the multivariate linear models: an outer product least-squares approach

In this article, we present a framework of estimating patterned covariance of interest in the multivariate linear models. The main idea in it is to estimate a patterned covariance by minimizing a trace distance function between outer product of residuals and its expected value. The proposed framework can provide us explicit estimators, called outer product least-squares estimators, for parameters in the patterned covariance of the multivariate linear model without or with restrictions on regression coefficients. The outer product least-squares estimators enjoy the desired properties in finite and large samples, including unbiasedness, invariance, consistency and asymptotic normality. We still apply the framework to three special situations where their patterned covariances are the uniform correlation, a generalized uniform correlation and a general q-dependence structure, respectively. Simulation studies for three special cases illustrate that the proposed method is a competent alternative of the maximum likelihood method in finite size samples.     

Likelihood Ratio Tests for High‐Dimensional Normal Distributions

In their recent work, Jiang and Yang studied six classical Likelihood Ratio Test statistics under high‐dimensional setting. Assuming that a random sample of size n is observed from a p‐dimensional normal population, they derive the central limit theorems (CLTs) when p and n are proportional to each other, which are different from the classical chi‐square limits as n goes to infinity, while p remains fixed. In this paper, by developing a new tool, we prove that the mentioned six CLTs hold in a more applicable setting: p goes to infinity, and p can be very close to n. This is an almost sufficient and necessary condition for the CLTs. Simulations of histograms, comparisons on sizes and powers with those in the classical chi‐square approximations and discussions are presented afterwards.     

Linear discrimination for multi-level multivariate data with separable means and jointly equicorrelated covariance structure

In this article we study a linear discriminant function of multiple m-variate observations at u-sites and over v-time points under the assumption of multivariate normality. We assume that the m-variate observations have a separable mean vector structure and a “jointly equicorrelated covariance” structure. The new discriminant function is very effective in discriminating individuals in a small sample scenario. No closed-form expression exists for the maximum likelihood estimates of the unknown population parameters, and their direct computation is nontrivial. An iterative algorithm is proposed to calculate the maximum likelihood estimates of these unknown parameters. A discriminant function is also developed for unstructured mean vectors. The new discriminant functions are applied to simulated data sets as well as to a real data set. Results illustrating the benefits of the new classification methods over the traditional one are presented.