Visual assessment of matrix-variate normality

In recent years, the analysis of three-way data has become ever more prevalent in the literature. It is becoming increasingly common to analyse such data by means of matrix-variate distributions, the most prevalent of which is the matrix-variate normal distribution. Although many methods exist for assessing multivariate normality, there is a relative paucity of approaches for assessing matrix-variate normality. Herein, a new visual method is proposed for assessing matrix-variate normality by means of a distance–distance plot. In addition, a testing procedure is discussed to be used in tandem with the proposed visual method. The proposed approach is illustrated via simulated data as well as an application on analysing handwritten digits.     

Percentage points for directional anderson-darling goodness-of-fit tests

This paper discusses the problem of assessing the asymptotic distribution when parameters of the hypothesized distribution are estimated from a sample, pointing out a common mistake included in the paper by Sinclair, Spurr, and Ahmad (1990) which introduced two modifications of the Anderson-Darling goodness-of-fit test statistic. Their two test statistics modify the popular Anderson-Darling test statistic to be sensitive to departures of the fitted distribution from the true distribution in one or the other of the tails. This paper uses these new test statistics to develop tests of fit for the normal and exponential distributions. Easy to use formulas are given so the reader can perform these tests at any sample size without consulting exhaustive tables of percentage points. Finally a power study is given to demonstrate the test statistics’ viability against a broad range of alternatives.     

A density based empirical likelihood approach for testing bivariate normality

Sample entropy based tests, methods of sieves and Grenander estimation type procedures are known to be very efficient tools for assessing normality of underlying data distributions, in one-dimensional nonparametric settings. Recently, it has been shown that the density based empirical likelihood (EL) concept extends and standardizes these methods, presenting a powerful approach for approximating optimal parametric likelihood ratio test statistics, in a distribution-free manner. In this paper, we discuss difficulties related to constructing density based EL ratio techniques for testing bivariate normality and propose a solution regarding this problem. Toward this end, a novel bivariate sample entropy expression is derived and shown to satisfy the known concept related to bivariate histogram density estimations. Monte Carlo results show that the new density based EL ratio tests for bivariate normality behave very well for finite sample sizes. To exemplify the excellent applicability of the proposed approach, we demonstrate a real data example.     

Multivariate extension of chi-squared univariate normality test

We propose a multivariate extension of the univariate chi-squared normality test. Using a known result for the distribution of quadratic forms in normal variables, we show that the proposed test statistic has an approximated chi-squared distribution under the null hypothesis of multivariate normality. As in the univariate case, the new test statistic is based on a comparison of observed and expected frequencies for specified events in sample space. In the univariate case, these events are the standard class intervals, but in the multivariate extension we propose these become hyper-ellipsoidal annuli in multivariate sample space. We assess the performance of the new test using Monte Carlo simulation. Keeping the type I error rate fixed, we show that the new test has power that compares favourably with other standard normality tests, though no uniformly most powerful test has been found. We recommend the new test due to its competitive advantages.     

A note on the asymptotic distribution of mardia's measure of multivariate kurtosis

Weak convergence results are used to investigate asymptotic properties of Mardia's measure of multivariate kurtosis in the context of assessing multivariate normality.     

Testing the proportional odds model for interval-censored data

This paper discusses the goodness-of-fit test for the proportional odds model for K-sample interval-censored failure time data, which frequently occur in, for example, periodic follow-up survival studies. The proportional odds model has a feature that allows the ratio of two hazard functions to be monotonic and converge to one and provides an important tool for the modeling of survival data. To test the model, a procedure is proposed, which is a generalization of the method given in Dauxois and Kirmani [Dauxois JY, Kirmani SNUA (2003) Biometrika 90:913–922]. The asymptotic distribution of the procedure is established and its properties are evaluated by simulation studies     

A class of test statistics for testing whether new is better than used

A class of test statistics Is proposed for testing that a life distribution Is exponential against that it is new better than used and not exponential, Consistency, unbiasedness and the asymptotic normality for the one class of class of test are proved. The efflication are compeled and compared with some otner statinties. The proposed test is shown to perform well against other tests.     

New tests for multivariate normality based on Small's and Srivastava's graphical methods

In this paper, we propose a new measure of fit which can be used in the case of quantile–quantile plots. This measure, when applied to Small's and Srivastava's graphical methods provides two new tests for assessing multivariate normality. For different sample sizes and numbers of variables, the critical values of these tests were evaluated via simulations. The power of the new tests and its comparison with some other tests for multivariate normality are presented herein.     

Improvement of goodness-of-fit test for normal distribution based on entropy and power comparison

Vasicek's entropy test for normality is based on sample entropy and a parametric entropy estimator. These estimators are known to have bias in small samples. The use of Vasicek's test could affect the capability of detecting non-normality to some extent. This paper presents an improved entropy test, which uses bias-corrected entropy estimators. A Monte Carlo simulation study is performed to compare the power of the proposed test under several alternative distributions with some other tests. The results report that as anticipated, the improved entropy test has consistently higher power than the ordinary entropy test in nearly all sample sizes and alternatives considered, and compares favorably with other tests.     

A new test for multivariate normality by combining extreme and nonextreme BHEP tests

In this article, we propose a new multiple test procedure for assessing multivariate normality, which combines BHEP (Baringhaus–Henze–Epps–Pulley) tests by considering extreme and nonextreme choices of the tuning parameter in the definition of the BHEP test statistic. Monte Carlo power comparisons indicate that the new test presents a reasonable power against a wide range of alternative distributions, showing itself to be competitive against the most recommended procedures for testing a multivariate hypothesis of normality. We further illustrate the use of the new test for the Fisher Iris dataset.     

A Goodness-of-fit Test for Exponentiality Based on the Memoryless Property

In this investigation a test of goodness of fit for exponentiality is proposed. This procedure applies equally whether the scale and/or the location parameters of the distribution are known or not. The limiting null and non-null distributions of the test statistic are normal under minimal conditions. Monte Carlo critical values for small sample sizes are given and the power of the test is calculated for various alternatives showing that it compares favourably relatively to other more complicated published procedures.     

Normality testing: two new tests using L-moments

Establishing that there is no compelling evidence that some population is not normally distributed is fundamental to many statistical inferences, and numerous approaches to testing the null hypothesis of normality have been proposed. Fundamentally, the power of a test depends on which specific deviation from normality may be presented in a distribution. Knowledge of the potential nature of deviation from normality should reasonably guide the researcher's selection of testing for non-normality. In most settings, little is known aside from the data available for analysis, so that selection of a test based on general applicability is typically necessary. This research proposes and reports the power of two new tests of normality. One of the new tests is a version of the R-test that uses the L-moments, respectively, L-skewness and L-kurtosis and the other test is based on normalizing transformations of L-skewness and L-kurtosis. Both tests have high power relative to alternatives. The test based on normalized transformations, in particular, shows consistently high power and outperforms other normality tests against a variety of distributions.     

Asymptotic Distribution for Symmetric Spacing Statistics

A new procedure for testing the H ₀: μ₁ = ··· = μ _k against the alternative H _u :μ₁ ≥ ··· ≥μ _r  ≤ ··· ≤ μ _k with at least one strict inequality, where μ _i is the location parameter of the ith two-parameter exponential distribution, i = 1,…, k, is proposed. Exact critical constants are computed using a recursive integration algorithm. Tables containing these critical constants are provided to facilitate the implementation of the proposed test procedure. Simultaneous confidence intervals for certain contrasts of the location parameters are derived by inverting the proposed test statistic. In comparison to existing tests, it is shown, by a simulation study, that the new test statistic is more powerful in detecting U-shaped alternatives when the samples are derived from exponential distributions. As an extension, the use of the critical constants for comparing Pareto distribution parameters is discussed.     

A re-appraisal of tests for normality

Tests for normality can be divided into two groups - those based upon a function of the empirical distribution function and those based upon a function of the original observations. The latter group of statistics test spherical symmetry and not necessarily normality. If the distribution is completely specified then the first group can be used to test for ‘spherical’ normality. However, if the distribution is incompletely specified and F‘‘x_i - x’/s’ is used these test statistics also test sphericity rather than normality. A Monte Carlo study was conducted for the completely specified case, to investigate the sensitivity of the distance tests to departures from normality when the alternative distributions are non-normal spherically symmetric laws. A “new” test statistic is proposed for testing a completely specified normal distribution     

Assessing interaction effects in linear measurement error models

Summary.  In a linear model, the effect of a continuous explanatory variable may vary across groups defined by a categorical variable, and the variable itself may be subject to measurement error. This suggests a linear measurement error model with slope-by-factor interactions. The variables that are defined by such interactions are neither continuous nor discrete, and hence it is not immediately clear how to fit linear measurement error models when interactions are present. This paper gives a corollary of a theorem of Fuller for the situation of correcting measurement errors in a linear model with slope-by-factor interactions. In particular, the error-corrected estimate of the coefficients and its asymptotic variance matrix are given in a more easily assessable form. Simulation results confirm the asymptotic normality of the coefficients in finite sample cases. We apply the results to data from the Seychelles Child Development Study at age 66 months, assessing the effects of exposure to mercury through consumption of fish on child development for females and males for both prenatal and post-natal exposure.     

Test of Normality Against Generalized Exponential Power Alternatives

The family of symmetric generalized exponential power (GEP) densities offers a wide range of tail behaviors, which may be exponential, polynomial, and/or logarithmic. In this article, a test of normality based on Rao's score statistic and this family of GEP alternatives is proposed. This test is tailored to detect departures from normality in the tails of the distribution. The main interest of this approach is that it provides a test with a large family of symmetric alternatives having non-normal tails. In addition, the test's statistic consists of a combination of three quantities that can be interpreted as new measures of tail thickness. In a Monte-Carlo simulation study, the proposed test is shown to perform well in terms of power when compared to its competitors.     

A powerful test for uniformity to detect highly peaked alternativedensities

A new class of statistical tests for uniformity based on sample ranges is proposed to detect a uniform density contaminated by a density with one or more high peaks. These kinds of alternatives occur quite frequently, especially in physics. It is shown that the proposed tests arc consistent and have high Pitman asymptotic relative efficiencies. Results of a Monte Carlo power study indicate that they, compared with other tests of uniformity, possess good power properties.A comparative study of various tests is also conducted using real data. An effcient algorithm for computing out test statistic and a table of percentage points are given, providing a practical guide for using the new test.     

非参数固定效应Panel Data模型的分位数回归推断

利用分位数回归方法,讨论了非参数固定效应Panel Data模型的估计和检验问题,得到了参数估计的渐近正态性及收敛速度。同时,建立一个秩得分(rank score)统计量来检验模型的固定效应,并证明了这个统计量渐近服从标准正态分布。     

Estimation of the area under ROC curve with censored data

The area under the receiver operating characteristic curve is the most commonly used summary measure of diagnostic accuracy for a continuous-scale diagnostic test. In this paper, we develop methods to estimate the area under the curve (AUC) with censored data. Based on two different integration representations of this parameter, two nonparametric estimators are defined by the “plug in” method. Both the proposed estimators are shown to be asymptotically normal based on counting process and martingale theory. A simulation study is conducted to evaluate the performances of the proposed estimators.     

A Necessary Power Divergence Type Family Tests of Multivariate Normality

In a recent article, Cardoso de Oliveira and Ferreira have proposed a multivariate extension of the univariate chi-squared normality test, using a known result for the distribution of quadratic forms in normal variables. In this article, we propose a family of power divergence type test statistics for testing the hypothesis of multinormality. The proposed family of test statistics includes as a particular case the test proposed by Cardoso de Oliveira and Ferreira. We assess the performance of the new family of test statistics by using Monte Carlo simulation. In this context, the type I error rates and the power of the tests are studied, for important family members. Moreover, the performance of significant members of the proposed test statistics are compared with the respective performance of a multivariate normality test, proposed recently by Batsidis and Zografos. Finally, two well-known data sets are used to illustrate the method developed in this article as well as the specialized test of multivariate normality proposed by Batsidis and Zografos.