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Utilizing the energy distance and energy statistics, Sang and Dang (2020) proposed a test statistic as a difference of two U -statistics for the diagonal symmetry test of a p -vector X. Under the regular setting where the dimensionality of the random vector is fixed, the test statistic is a degenerate U -statistic and hence converges to a mixture of chi-squared distributions. In this paper, we test the diagonal symmetry of X in a more realistic setting where both the sample size and the dimensionality are diverging to infinity. Our theoretical results reveal that the degenerate U -statistic admits a central limit theorem in the high dimensional setting and the accuracy of normal approximation can increase with dimensionality. We then construct a powerful and consistent test for the diagonal symmetry problem based on the asymptotic normality. Simulation studies are conducted to illustrate the performances of the test. [ABSTRACT FROM AUTHOR]

Abstract Let Z be a standard normal random variable and let H n denote the n th Hermite polynomial. In this note, we obtain Stein equations for the random variables H 3 (Z) and H 4 (Z) , which represent a first step towards developing Stein's method for distributional limits from the third and fourth Wiener chaoses. Perhaps surprisingly, these Stein equations are fifth and third order linear ordinary differential equations, respectively. As a warm up, we obtain a Stein equation for the random variable a Z 2 + b Z + c , a , b , c ∈ R , which leads us to a Stein equation for the non-central chi-square distribution. We also provide a discussion as to why obtaining Stein equations for H n (Z) , n ≥ 5 , is more challenging. [ABSTRACT FROM AUTHOR]

The procedure based on the modified information criterion is proposed to detect the change points in the piecewise linear failure rate function. Consistency of the test is established and the asymptotic null distribution of the test statistic is also derived as the standard chi-square distribution. The performance of the test statistic is studied through the simulations. The proposed method is applied to a real example to illustrate the detecting procedure. [ABSTRACT FROM AUTHOR]

We show that the distribution of the scalar Schur complement in a noncentral Wishart matrix is a mixture of central chi-square distributions with different degrees of freedom. For the case of a rank-1 noncentrality matrix, we reveal that the mixture weights arise from a noncentral beta mixture of Poisson distributions. [ABSTRACT FROM AUTHOR]

Under the stratified Cox model, an empirical likelihood based simultaneous confidence band for the difference of two individualized survival functions is proposed. Simulation studies demonstrate its superiority over the normal approximation based competitors. [ABSTRACT FROM AUTHOR]

Abstract: We revisit the problem of sequential fixed-width interval estimation of the mean of a normal population. For a two-stage procedure, we derive an approximation to the coverage probability, when the variance has a known lower bound. Our approximation is more explicit than a recent result. [Copyright &y& Elsevier]

Abstract: ROC surface analysis was recently developed as an extension of ROC curve analysis to accommodate tests with three outcomes. The volume under ROC surface (VUS) is the most commonly used summary measure of the diagnostic accuracy of a test. In this paper, we develop an empirical likelihood approach for the inference on VUS. First we define an empirical likelihood ratio for VUS and show that its limiting distribution is a scaled chi-square distribution. We then obtain an empirical likelihood confidence interval for VUS. A small simulation study as well as analysis of a real example is also presented in this paper. [Copyright &y& Elsevier]

Abstract: We introduce an adjusted likelihood ratio procedure for computing pointwise confidence intervals for survival functions from censored data. The test statistic, scaled by a ratio of two variance quantities, is shown to converge to a chi-squared distribution with one degree of freedom. The confidence intervals are seen to be a neighborhood of a semiparametric survival function estimator and are shown to have correct empirical coverage. Numerical studies also indicate that the proposed intervals have smaller estimated mean lengths in comparison to the ones that are produced as a neighborhood of the Kaplan–Meier estimator. We illustrate our method using a lung cancer data set. [Copyright &y& Elsevier]

Abstract: We begin with a new, general, interesting, and useful identity () based on recursive convolutions. Next, we extend the basic message from by generalizing it under broadened set of assumptions. Interesting examples are provided involving multivariate Cauchy as well as multivariate equi-correlated normal and equi-correlated distributions. Then, we discuss yet another approach () that also works in the evaluation of the expectation of a ratio of suitable random variables. especially stands out because it cannot be handled by the previous approaches, but steps in to help. [ABSTRACT FROM AUTHOR]

Abstract: Much attention has been given in recent years to adaptive designs of clinical trials as ethical alternatives when the traditional randomization becomes ethically infeasible. But such designs create dependency among the collected data, and hence statistical methods for adaptive clinical trials are more complex than those for traditional randomized clinical trials. In this paper, we examine some extensions of common statistical methods for independent data. Under regularity conditions, the logarithm of likelihood ratio statistic for dependent data is shown to be asymptotically chi-square distributed, providing a foundation for asymptotic analysis of adaptive clinical trials with k treatments. We also discuss both the consistency and the asymptotic normality of the maximum likelihood estimators for a wide class of adaptive designs. [Copyright &y& Elsevier]

Abstract: Following the paper by Gupta and Chang (Multivariate skew-symmetric distributions. Appl. Math. Lett. 16, 643–646 2003.) we generate a multivariate skew normal-symmetric distribution with probability density function of the form , where , is the p-dimensional normal p.d.f. with zero mean vector and correlation matrix , and G is taken to be an absolutely continuous function such that is symmetric about 0. First we obtain the moment generating function of certain quadratic forms. It is interesting to find that the distributions of some quadratic forms are independent of G. Then the joint moment generating functions of a linear compound and a quadratic form, and two quadratic forms, and conditions for their independence are given. Finally we take G to be one of normal, Laplace, logistic or uniform distribution, and determine the distribution of a special quadratic form for each case. [Copyright &y& Elsevier]

In this note, we establish an equivalence between chi-square and generalized skew-normal distributions. This result is based on a distributional invariance property of even functions in generalized skew-normal random vectors. It extends the chi-square properties related to univariate and multivariate skew-normal distributions. [Copyright &y& Elsevier]

A subclass of the scale-parameter exponential family is considered and for the rth power of the scale parameter, which is lower bounded, an admissible minimax estimator under scale-invariant squared-error loss is presented. Also, an admissible minimax estimator of a lower-bounded parameter in the family of transformed chi-square distributions is given. These estimators are the pointwise limits of a sequence of Bayes estimators. Some examples are given. [Copyright &y& Elsevier]

The exact distribution of a linear combination with positive coefficients of inverted chi-square variables with odd degrees of freedom is derived. This distribution function could be expressed as another linear combination of distribution functions of chi-square random variables. [Copyright &y& Elsevier]

In this paper, jackknife empirical likelihood is proposed to be applied in stationary time series models. By applying the jackknife method to Whittle estimator, we obtain new asymptotically independent pseudo samples which will be used to construct linear constraints for empirical likelihood. The jackknife empirical log-likelihood ratio is shown to follow a chi-square limiting distribution, which validates the corresponding confidence regions asymptotically. However, similar to the drawbacks of empirical likelihood, this method suffers from the non-definition problem and the inaccurate coverage probability in constructing confidence regions. So we further develop the adjusted jackknife empirical likelihood borrowing the idea of Chen et al. (2008) to improve the performance of the jackknife empirical likelihood. With a specific adjustment level, the adjusted jackknife empirical likelihood achieves a more high-order coverage precision than the classical jackknife empirical likelihood does and our simulations corroborate this point. [ABSTRACT FROM AUTHOR]

Energy distance is a statistical distance between the distributions of random variables, which characterizes the equality of the distributions. Utilizing the energy distance, we develop a nonparametric test for the diagonal symmetry, which is consistent against any fixed alternatives. The test statistic developed in this paper is based on the difference of two U -statistics. By applying the jackknife empirical likelihood approach, the standard limiting chi-square distribution with degree freedom of one is established and is used to determine critical value and p -value of the test. Simulation studies show that our method is competitive in terms of empirical sizes and empirical powers. [ABSTRACT FROM AUTHOR]