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We revisit the well-known Behrens–Fisher problem in an original and challenging high-dimensional framework, and propose a testing procedure which accommodates a low-dimensional latent factor model. The developed inferential framework is general, as it applies to problems where the underlying populations may be non-normal, the dimension of the population mean vectors may highly exceed the sample size, the design may be unbalanced, and the loading factor dimensions may be different. Under a high-dimensional asymptotic regime, combined with fairly weak technical conditions, we show that null limiting distributions of the test statistics follow a weighted mixture of chi-square distributions, which depends only on the spectrum of the noise covariance matrix and the number of latent factors. As these latter are usually unknown in practice, we exploit an estimation procedure which builds on recent advances in random matrix theory. The asymptotic power of the proposed test is established. A numerical study confirms good analytical properties of the new test that compares favorably to existing procedures used in a similar context. Real data applications are demonstrated with a study of a leukemia data set. [ABSTRACT FROM AUTHOR]

Likelihood ratio tests (LRTs) for separability of a covariance structure for doubly multivariate data are widely studied in the literature. There are three types of LRT: biased tests based on an asymptotic chi-square null distribution; unbiased/unmodified tests based on an empirical null distribution; and unbiased/modified tests with a test statistic modified to follow a theoretical chi-square null distribution. The Rao’s score test (RST) statistic, an alternative for both biased and unbiased/unmodified versions of the corresponding LRT test statistics, is derived for a common case. In this paper the separability of a covariance structure with the first component as a compound symmetric correlation matrix under the assumption of multivariate normality is tested. For this purpose Monte Carlo simulation studies, which compare the biased LRT to biased RST, and unbiased/unmodified LRT to unbiased/unmodified RST, are conducted. It is shown that the RSTs outperform their corresponding LRTs in the sense of empirical Type I error as well as empirical null distribution. Moreover, since the RST does not require estimation of a general variance–covariance matrix (the alternative hypothesis), RST can be performed for small sample sizes, where the variance–covariance matrix could not be estimated for the corresponding LRT, making the LRT infeasible. Three examples are presented to illustrate and compare statistical inference based on LRT and RST. [ABSTRACT FROM AUTHOR]

Let X ∼ N v ( 0 , Λ ) be a normal vector in v ( ≥ 1 ) dimensions, where Λ is diagonal. With reference to the truncated distribution of X on the interior of a v -dimensional Euclidean ball, we completely prove a variance inequality and a covariance inequality that were recently discussed by Palombi and Toti (2013). These inequalities ensure the convergence of an algorithm for the reconstruction of Λ only on the basis of the covariance matrix of X truncated to the Euclidean ball. The concept of monotone likelihood ratio is useful in our proofs. Moreover, we also prove and utilize the fact that the cumulative distribution function of any positive linear combination of independent chi-square variates is log-concave, even though the same may not be true for the corresponding density function. [ABSTRACT FROM AUTHOR]

In this paper, we consider the application of the empirical likelihood method to a partly linear model with measurement errors in possibly all the variables. It is shown that the empirical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. Also, a class of estimators for the parameter are constructed, and the asymptotic distributions of the proposed estimators are obtained. Some simulations and an application are conducted to illustrate the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: The Bartlett-type adjustment is a higher-order asymptotic method for improving the chi-squared approximation to the null distributions of various test statistics. Though three influential papers were published in 1991—Chandra and Mukerjee (1991) , Cordeiro and Ferrari (1991) and Taniguchi (1991) in alphabetical order, the only CF-approach has been frequently applied in the literature during the last two decades, provided that asymptotic expansion for the null distribution of a given test statistic is available. Revisiting the CM/T-approaches developed in the absence of a nuisance parameter, this paper suggests general adjustments for a class of test statistics that includes, in particular, the likelihood ratio, Rao’s and Wald’s test statistics in the presence of a nuisance parameter. [Copyright &y& Elsevier]

Abstract: This article deals with the inference on a right-censored partially linear single-index model (RCPLSIM). The main focus is the local empirical likelihood-based inference on the nonparametric part in RCPLSIM. With a synthetic data approach, an empirical log-likelihood ratio statistic for the nonparametric part is defined and it is shown that its limiting distribution is not a central chi-squared distribution. To increase the accuracy of the confidence interval, we also propose a corrected empirical log-likelihood ratio statistic for the nonparametric function. The resulting statistic is proved to follow a standard chi-squared limiting distribution. Simulation studies are undertaken to assess the finite sample performance of the proposed confidence intervals. A real example is also considered. [Copyright &y& Elsevier]

Abstract: Double-sampling designs are commonly used in real applications when it is infeasible to collect exact measurements on all variables of interest. Two samples, a primary sample on proxy measures and a validation subsample on exact measures, are available in these designs. We assume that the validation sample is drawn from the primary sample by the Bernoulli sampling with equal selection probability. An empirical likelihood based approach is proposed to estimate the parameters of interest. By allowing the number of constraints to grow as the sample size goes to infinity, the resulting maximum empirical likelihood estimator is asymptotically normal and its limiting variance–covariance matrix reaches the semiparametric efficiency bound. Moreover, the Wilks-type result of convergence to chi-squared distribution for the empirical likelihood ratio based test is established. Some simulation studies are carried out to assess the finite sample performances of the new approach. [Copyright &y& Elsevier]

Abstract: In logistic regression models, we consider the deviance statistic (the log likelihood ratio statistic) as a goodness-of-fit test statistic. In this paper, we show the derivation of an expression of asymptotic expansion for the distribution of under a null hypothesis. Using the continuous term of the expression, we obtain a Bartlett-type transformed statistic that improves the speed of convergence to the chi-square limiting distribution of . By numerical comparison, we find that the transformed statistic performs much better than . We also give a real data example of being more reliable than for testing a hypothesis. [Copyright &y& Elsevier]

Abstract: This article proposes the efficient empirical-likelihood-based inferences for the single component of the parameter and the link function in the single-index model. Unlike the existing empirical likelihood procedures for the single-index model, the proposed profile empirical likelihood for the parameter is constructed by using some components of the maximum empirical likelihood estimator (MELE) based on a semiparametric efficient score. The empirical-likelihood-based inference for the link function is also considered. The resulting statistics are proved to follow a standard chi-squared limiting distribution. Simulation studies are undertaken to assess the finite sample performance of the proposed confidence intervals. An application to real data set is illustrated. [Copyright &y& Elsevier]

Abstract: One important step in regression analysis is to identify significant predictors from a pool of candidates so that a parsimonious model can be obtained using these significant predictors only. However, most of the existing methods assume linear relationships between response and predictors, which may be inappropriate in some applications. In this article, we discuss a link-free method that avoids specifying how the response depends on the predictors. Therefore, this method has no problem of model misspecification, and it is suitable for selecting significant predictors at the preliminary stage of data analysis. A test statistic is suggested and its asymptotic distribution is derived. Examples are used to demonstrate the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: Empirical likelihood (EL) ratio tests are developed for testing for or against the hypothesis that -population means are isotonic with respect to some quasi-order on . The null asymptotic distributions are derived and are shown to be of chi-bar squared type. The asymptotic power of the proposed test for testing for equality of these means against the order restriction is derived under contiguous alternatives and a simulation study is carried out to investigate the finite sample behaviors of this test. In addition, an adjusted EL test is used to improve the small size performance of our test and an example is also discussed to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the empirical likelihood for constructing a confidence region of the parameter of interest in a multi-link semiparametric model when an infinite-dimensional nuisance parameter exists. The new model covers the commonly used varying coefficient, generalized linear, single-index, multi-index, hazard regression models and their generalizations, as its special cases. Because of the existence of the infinite-dimensional nuisance parameter, the classical empirical likelihood with plug-in estimation cannot be asymptotically distribution-free, and the existing bias correction is not extendable to handle such a general model. We then propose a link-based correction approach to solve this problem. This approach gives a general rule of bias correction via an inner link, and consists of two parts. For the model whose estimating equation contains the score functions that are easy to estimate, we use a centering for the scores to correct the bias; for the model of which the score functions are of complex structure, a bias-correction procedure using simpler functions instead of the scores is given without loss of asymptotic efficiency. The resulting empirical likelihood shares the desired features: it has a chi-square limit and, under-smoothing technique, high order kernel and parameter estimation are not needed. Simulation studies are carried out to examine the performance of the new method. [Copyright &y& Elsevier]

Abstract: For a class of multivariate skew normal distributions, the noncentral skew chi-square distribution is studied. The necessary and sufficient conditions under which a sequence of quadratic forms is generalized noncentral skew chi-square distributed random variables are obtained. Several examples are given to illustrate the results. [Copyright &y& Elsevier]

A general linear hypothesis testing (GLHT) problem in heteroscedastic one-way MANOVA for high-dimensional data is considered and a normal reference L 2 -norm based test for the problem is proposed. Different from a few existing methodologies on the GLHT problem which impose strong assumptions on the underlying covariance matrices so that the associated tests' null distributions are asymptotically normal, it is shown that under some regularity conditions, the proposed test statistic under the null hypothesis and a chi-square type mixture have the same normal or non-normal limiting distributions. It is then suggested to approximate the test's null distribution using the distribution of the chi-square type mixture, which can be further approximated by the Welch–Satterthwaite chi-square-approximation with approximation parameters consistently estimated. Several simulation studies and a real data application are presented to demonstrate the good performance of the proposed test. [ABSTRACT FROM AUTHOR]

Abstract: The normal distribution based likelihood ratio (LR) statistic is widely used in structural equation modeling. Under a sequence of local alternative hypotheses, this statistic has been shown to asymptotically follow a noncentral chi-square distribution. In practice, the population mean vector and covariance matrix as well as the model and sample size are always fixed. It is hard to justify the validity of the noncentral chi-square distribution for the resulting LR statistic even when data are normally distributed and sample size is large. By extending results in the literature, this paper develops normal distributions to describe the behavior of the LR statistic for mean and covariance structure analysis. A sequence of local alternative hypotheses is not necessary for the proposed distributions to be asymptotically valid. When the effect size is medium and above or when the model is not trivially misspecified, empirical results indicate that a refined normal distribution describes the behavior of the LR statistic better than the commonly used noncentral chi-square distribution, as measured by the Kolmogorov–Smirnov distance. Quantile–quantile plots are also provided to better understand the different distributions. [Copyright &y& Elsevier]

We propose a significance test to determine if data on a regular d -dimensional grid can be assumed to be a realization of Gaussian process. By accounting for the spatial dependence of the observations, we derive statistics analogous to sample skewness and kurtosis. We show that the sum of squares of these two statistics converges to a chi-square distribution with two degrees of freedom. This leads to a readily applicable test. We examine two variants of the test, which are specified by two ways the spatial dependence is estimated. We provide a careful theoretical analysis, which justifies the validity of the test for a broad class of stationary random fields. A simulation study compares several implementations. While some implementations perform slightly better than others, all of them exhibit very good size control and high power, even in relatively small samples. An application to a comprehensive data set of sea surface temperatures further illustrates the usefulness of the test. [ABSTRACT FROM AUTHOR]