A Note on Sufficient Conditions for Valid Unmodified t Testing in Correlation Analysis with Autocorrelated and Heteroscedastic Sample Data

Traditionally, sphericity (i.e., independence and homoscedasticity for raw data) is put forward as the condition to be satisfied by the variance–covariance matrix of at least one of the two observation vectors analyzed for correlation, for the unmodified t test of significance to be valid under the Gaussian and constant population mean assumptions. In this article, the author proves that the sphericity condition is too strong and a weaker (i.e., more general) sufficient condition for valid unmodified t testing in correlation analysis is circularity (i.e., independence and homoscedasticity after linear transformation by orthonormal contrasts), to be satisfied by the variance–covariance matrix of one of the two observation vectors. Two other conditions (i.e., compound symmetry for one of the two observation vectors; absence of correlation between the components of one observation vector, combined with a particular pattern of joint heteroscedasticity in the two observation vectors) are also considered and discussed. When both observation vectors possess the same variance–covariance matrix up to a positive multiplicative constant, the circularity condition is shown to be necessary and sufficient. “Observation vectors” may designate partial realizations of temporal or spatial stochastic processes as well as profile vectors of repeated measures. From the proof, it follows that an effective sample size appropriately defined can measure the discrepancy from the more general sufficient condition for valid unmodified t testing in correlation analysis with autocorrelated and heteroscedastic sample data. The proof is complemented by a simulation study. Finally, the differences between the role of the circularity condition in the correlation analysis and its role in the repeated measures ANOVA (i.e., where it was first introduced) are scrutinized, and the link between the circular variance–covariance structure and the centering of observations with respect to the sample mean is emphasized.     

Extrapolation of subsampling distribution estimators: The i.i.d. and strong mixing cases

Politis & Romano (1994) proposed a general subsampling methodology for the construction of large‐sample confidence regions for an arbitrary parameter under minimal conditions. Nevertheless, the subsampling distribution estimators may sometimes be inefficient (in the case of the sample mean of i.i.d. data, for instance) as compared to alternative estimators such as the bootstrap and/or the asymptotic normal distribution (with estimated variance). The authors investigate here the extent to which the performance of subsampling distribution estimators can be improved by interpolation and extrapolation techniques, while at the same time retaining the robustness property of consistent distribution estimation even in nonregular cases; both i.i.d. and weakly dependent (mixing) observations are considered.     

Best unbiased estimation of fixed effects in mixed ANOVA models

In a linear model with an arbitrary variance–covariance matrix, Zyskind (Ann. Math. Statist. 38 (1967) 1092) provided necessary and sufficient conditions for when a given linear function of the fixed-effect parameters has a best linear unbiased estimator (BLUE). If these conditions hold uniformly for all possible variance–covariance parameters (i.e., there is a UBLUE) and if the data are assumed to be normally distributed, these conditions are also necessary and sufficient for the parametric function to have a uniformly minimum variance unbiased estimator (UMVUE). For mixed-effects ANOVA models, we show how these conditions can be translated in terms of the incidence array, which facilitates verification of the UBLUE and UMVUE properties and facilitates construction of designs having such properties.     

Correlation is first order independent of transformation

We show that the correlation between the estimates of two parameters is almost unchanged if they are each transformed in an arbitrary way. To be more specific, the correlation of two estimates is invariant (except for a possible sign change) up to a first order approximation, to smooth transformations of the estimates. There is a sign change if exactly one of the transformations is decreasing in a neighborhood of its parameter. In addition, we approximate the variance, covariance and correlation between functions of sample means and moments.     

Independence distribution preserving joint covariance structures for the multivariate two-group case

We characterize the general nonnegative-definite and positive-definite joint observation covariance structures for the two-group case such that the two sample mean vectors are independent of the two corresponding sample covariance matrices. Also, the sample covariance matrices are distributed as independent noncentral or central Wishart random matrices. We derive and utilize a representation of the general common non-negative-definite solution to a particular system of matrix equations with idempotent coefficient matrices.     

On the effect of correlation and unequal variances in detecting a spurious observation

Srivastava (1980) has shown that Grubbs's (1950) test for a univariate outlier is robust against the effect of equicorrelation. In this note we extend Srivastava's result by giving a more general covariance structure, which relaxes both the covariance structure and the assumption of equal variances. We also show that under the more general covariance structure, the power of Grubbs's test, as well as the significance level, is identical to the independently and identically distributed case.     

Comparison of combinatoric and likelihood ratio procedures for classifying samples

A random sample is to be classified as coming from one of two normally distributed populations with known parameters. Combinatoric procedures which classify the sample based upon the sample mean(s) and variance(s) are described for the univariate and multivariate problems. Comparisons of misclassification probabilities are made between the combinatoric and the likelihood ratio procedure in the univariate case and between two alternative combinatoric procedures in the bivariate case.     

Bias Reduction using Stochastic Approximation

The paper studies stochastic approximation as a technique for bias reduction. The proposed method does not require approximating the bias explicitly, nor does it rely on having independent identically distributed (i.i.d.) data. The method always removes the leading bias term, under very mild conditions, as long as auxiliary samples from distributions with given parameters are available. Expectation and variance of the bias-corrected estimate are given. Examples in sequential clinical trials (non-i.i.d. case), curved exponential models (i.i.d. case) and length-biased sampling (where the estimates are inconsistent) are used to illustrate the applications of the proposed method and its small sample properties.     

Subsampling,symmetrization, and robust interpolation

The recently developed subsampling methodology has been shown to be valid for the construction of large-sample confidence regions for a general unknown parameter 9 under very minimal conditions. Nevertheless, in some specific cases—e.g. in the case of the sample mean of i.i.d. data—it has been noted that the subsampling distribution estimator underperforms as compared to alternative estimators such as the bootstrap or the asymptotic normal distribution (with estimated variance). In the present report we introduce a (partially) symmetrized.     

Variance estimation for measures of change in probability sampling

Variance estimation of changes requires estimates of variances and covariances that would be relatively straightforward to make if the sample remained the same from one wave to the next, but this is rarely the case in practice as successive waves are usually different overlapping samples. The author proposes a design‐based estimator for covariance matrices that is adapted to this situation. Under certain conditions, he shows that his approach yields non‐negative definite estimates for covariance matrices and therefore positive variance estimates for a large class of measures of change.     

Critical point and power calculations for the studentized range test for generally correlated means

The studentized range test is a widely applied statistical procedure to compare several normal means within the analysis of variance. However, up to now no general methodology is available to perform the all-pair comparisons precisely, such as the computation of p-values or quantiles in the simple unbalanced one-way layout. Instead, a variety of approximations have been proposed in the past. This article focuses on exact computations of simultaneous confidence intervals and exact sample size determinations for all-pair comparisons in the analysis of variance involving arbitrary variance-covariance matrices. General power expressions in closed form are developed and numerical issues concerning the arising multivariate central and noncentral t-distributions are discussed. An application to the usual fixed effects analysis of covariance illustrates the use of the obtained results.     

Estimation of parameters in a generalized GMANOVA model based on an outer product analogy and least squares

This paper investigates estimation of parameters in a combination of the multivariate linear model and growth curve model, called a generalized GMANOVA model. Making analogy between the outer product of data vectors and covariance yields an approach to directly do least squares to covariance. An outer product least squares estimator of covariance (COPLS estimator) is obtained and its distribution is presented if a normal assumption is imposed on the error matrix. Based on the COPLS estimator, two-stage generalized least squares estimators of the regression coefficients are derived. In addition, asymptotic normalities of these estimators are investigated. Simulation studies have shown that the COPLS estimator and two-stage GLS estimators are alternative competitors with more efficiency in the sense of sample mean, standard deviations and mean of the variance estimates to the existing ML estimator in finite samples. An example of application is also illustrated.     

Maximum variance of order statistics from symmetric populations revisited

We consider i.i.d. samples of size n with symmetric non-degenerate parent distributions and finite variances. Papadatos [A note on maximum variance of order statistics from symmetric populations, Ann. Inst. Statist. Math. 48 (1997), pp. 117–121] proved that the maximal variance of each non-extreme order statistic, expressed in the population variance units, is attained in a one-parametric family of symmetric two- and three-point distributions. The parameters of the extreme variance distributions coincide with the arguments maximizing some polynomials of degree 2n?1 over a finite interval. The bounds for variances are equal to the maximal values of the polynomials. We present a more precise solution to the problem by applying the variation diminishing property of Bernstein polynomials.     

Conditionally unbiased estimation in the normal setting with unknown variances

To efficiently and completely correct for selection bias in adaptive two-stage trials, uniformly minimum variance conditionally unbiased estimators (UMVCUEs) have been derived for trial designs with normally distributed data. However, a common assumption is that the variances are known exactly, which is unlikely to be the case in practice. We extend the work of Cohen and Sackrowitz (Statistics & Probability Letters, 8(3):273-278, 1989), who proposed an UMVCUE for the best performing candidate in the normal setting with a common unknown variance. Our extension allows for multiple selected candidates, as well as unequal stage one and two sample sizes.     

Simultaneous estimation of the multivariate normal mean under balanced loss function

This paper considers simultaneous estimation of multivariate normal mean vector using Zellner's(1994) balanced loss function which is defined as follows:

where 0 < w < 1 and for i = 1,…,p and j = 1,…,n, X_ij is distributed as normal with mean θi and variance 1. It is shown that the sample mean, X, is admissible when p <3. For p ≥3, we obtain that James-Stein type estimator which has uniformly smaller risk than that of sample mean X.     

Quasi-likelihood estimation for GLM with random scales

This paper uses random scales similar to random effects used in the generalized linear mixed models to describe “inter-location” population variation in variance components for modeling complicated data obtained from applications such as antenna manufacturing. Our distribution studies lead to a complicated integrated extended quasi-likelihood (IEQL) for parameter estimations and large sample inference derivations. Laplace's expansion and several approximation methods are employed to simplify the IEQL estimation procedures. Asymptotic properties of the approximate IEQL estimates are derived for general structures of the covariance matrix of random scales. Focusing on a few special covariance structures in simpler forms, the authors further simplify IEQL estimates such that typically used software tools such as weighted regression can compute the estimates easily. Moreover, these special cases allow us to derive interesting asymptotic results in much more compact expressions. Finally, numerical simulation results show that IEQL estimates perform very well in several special cases studied.     

On the sample characterization criterion for normal distributions

Conventionally, it was shown that the underlying distribution is normal if and only if the sample mean and sample variance from a random sample are independent. This paper focusses on the normal population characterization theorem by showing that, if the joint distribution of a skew normal sample follows certain multivariate skew normal distribution, the sample mean and sample variance are still independent.     

Hill's estimator for the tail index of an ARMA model

This paper investigates Hill's estimator for the tail index of an ARMA model with i.i.d. residuals. Based on the estimated residuals, it is shown that Hill's estimator is asymptotically normal. This method can achieve a smaller asymptotic variance than applying Hill's estimator to the original data. These results are the same as those in Resnick and Starica (Commun. Statist.—Stochastic Models 13 (4) (1997) 703) for an AR model. However, Resnick and Starica (Commun. Statist.—Stochastic Models 13 (4) (1997) 703) imposed one more condition on the choice of sample fraction than the i.i.d. case. This condition is removed in this paper so that data-driven methods for choosing optimal sample fraction based on i.i.d. data can be applied to our case. As an auxiliary theorem, we establish the weak convergence of the tail empirical process of the estimated residuals, which may be of independent interest.     

Distribution of multivariate quadratic forms under certain covariance structures

Necessary and sufficient conditions are given for the covariance structure of all the observations in a multivariate factorial experiment under which certain multivariate quadratic forms are independent and distributed as a constant times a Wishart. It is also shown that exact multivariate test statistics can be formed for certain covariance structures of the observations when the assumption of equal covariance matrices for each normal population is relaxed. A characterization is given for the dependency structure between random vectors in which the sample mean and sample covariance matrix have certain properties.     

Optimal sample size allocation for multi-level stress testing with Weibull regression under Type-II censoring

We discuss the optimal allocation problem in a multi-level stress test with Type-II censoring and Weibull (extreme value) regression model. We derive the maximum-likelihood estimators and their asymptotic variance–covariance matrix through the Fisher information. Four optimality criteria are used to discuss the optimal allocation problem. Optimal allocation of units, both exactly for small sample sizes and asymptotically for large sample sizes, for two- and four-stress-level situations are determined numerically. Conclusions and discussions are provided based on the numerical studies.