A Monte Carlo evaluation of the performance of two new tests for symmetry

We propose two new tests for symmetry based on well-known characterisations of symmetric distributions. The performance of the new tests is evaluated and compared to that of other existing tests by means of a Monte Carlo study. All tests are carried out in a regression setup where we test whether the error distribution in a linear regression model is symmetric. It is found that the newly proposed tests perform favourably compared to the other tests.     

Statistical tests based on empirical processes and related questions

We consider goodness-of-fit tests for hypotheses about the forms of distributions and their membership in prescribed families of distributions. We first describe the classical tests based on empirical processes such as the omega-square tests of Cramér-von Mises-Smirnov and the Kolmogorov-Smirnov tests. We also consider Shapiro-Wilk tests. We devote a considerable amount of attention to testing the hypothesis that a random variable or vector is normal. We describe tests based on transformations of the empirical process, minimal distance tests and estimates, tests for symmetry, uniformity, and independence, and tests based on spacings. At the end we study methods of computing and the distribution functions of quadratic forms of normal random variables connected with tests of omega-square type. Bibliography: 372 titles.Translated fromItogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 30, pp. 3–112, 1992.     

上界型拟合优度检验

对简单零假设情况,构造出一类上界型拟合优度检验.取不同的参数λ和不同的权函数,这类检验不仅包含许多已存在的检验,如Kolmogorov-Smirov检验,Berk-Jones检验等,而且还给出一些新的检验.众所周知,对不同的问题,"最优"的检验是不同的,有必要对这类检验的性质进行讨论.该文对任意给定的λ和较一般的权函数q(·),在较弱的条件下,导出了相应上界型检验统计量在零假设下的渐近分布,研究了它们的局部渐近功效;在若干固定备择假设下,对该类检验的功效进行了模拟研究.模拟结果表明,在不同的备择假设下,功效较优的检验是不同的,不存在对所有情况一致最优的检验.     

An ‘apples to apples’ comparison of various tests for exponentiality

The exponential distribution is a popular model both in practice and in theoretical work. As a result, a multitude of tests based on varied characterisations have been developed for testing the hypothesis that observed data are realised from this distribution. Many of the recently developed tests contain a tuning parameter, usually appearing in a weight function. In this paper we compare the powers of 20 tests for exponentiality—some containing a tuning parameter and some that do not. To ensure a fair ‘apples to apples’ comparison between each of the tests, we employ a data-dependent choice of the tuning parameter for those tests that contain these parameters. The comparisons are conducted for various samples sizes and for a large number of alternative distributions. The results of the simulation study show that the test with the best overall power performance is the Baringhaus and Henze test, followed closely by the test by Henze and Meintanis; both tests contain a tuning parameter. The score test by Cox and Oakes performs the best among those tests that do not include a tuning parameter.     

Nonlinearity tests for bilinear systems

The purpose of this paper is to develop nonlinearity tests for open-loop bilinear systems. Lagrange multiplier tests of linear systems against a bilinear alternative are proposed. A simulation study is performed to check the validity of the asymptotic null distributions of the test statistics and to investigate the power characteristics of the tests. Two recent nonlinearity tests in the time-series context are adapted to linear systems and compared with Lagrange multiplier tests. Simulation results show that the proposed Lagrange multiplier tests are more powerful than the other tests.     

Tests of fit for the Rayleigh distribution based on the empirical Laplace transform

In this paper a class of goodness-of-fit tests for the Rayleigh distribution is proposed. The tests are based on a weighted integral involving the empirical Laplace transform. The consistency of the tests as well as their asymptotic distribution under the null hypothesis are investigated. As the decay of the weight function tends to infinity the test statistics approach limit values. In a particular case the resulting limit statistic is related to the first nonzero component of Neyman’s smooth test for this distribution. The new tests are compared with other omnibus tests for the Rayleigh distribution.     

Similar tests for covariance structures in multivariate linear models

Nyblom (J. Multivariate Anal. 76 (2001) 294) has derived locally best invariant test for the covariance structure in a multivariate linear model. The class of invariant tests obtained by Nyblom [9] does not coincide with the class of similar tests for this testing set-up. This paper extends some of the results of Nyblom [9] by deriving the locally best similar tests for the covariance structure. Moreover, it develops a saddlepoint approximation to optimal weighted average power similar tests (i.e. tests which maximize a weighted average power).     

Goodness-Of-Fit Tests Based on Estimated Expectations of Probability Integral Transformed Order Statistics

New goodness-of-fit tests, based on bootstrap estimated expectations of probability integral transformed order statistics, are derived for the location-scale model. The resulting test statistics are location and scale invariant, and are sensitive to discrepancies at the tails of the hypothesized distribution. The limiting null distributions of the test statistics are derived in terms of functionals of a certain Gaussian process, and the tests are shown to be consistent against a broad family of alternatives. Critical points for all sample sizes are provided for tests of normality. A simulation study shows that the proposed tests are more powerful than established tests such as Shapiro-Wilk, Cramér-von Mises and Anderson-Darling, for a wide range of alternative distributions.     

Diagnostic checking for conditional heteroscedasticity models

We suggest the score type tests for goodness-of-fit of conditional heteroscedasticity models in both univariate and multivariate time series. The tests can detect the alternatives converging to the null at a parametric rate. Weight functions are involved in the construction of the tests, which provides us with the flexibility to choose scores, especially under directional alternatives, for enhancing power performance. Furthermore, when the alternatives are not directional, we construct asymptotically distribution-free maximin tests for a large class of alternatives. A possibility to construct score-based omnibus tests is discussed when the alternative is saturated. The power performance is also investigated. A simulation study is carried out and a real data is analyzed.     

Comparative analysis of the power of goodness-of-fit tests for near competing hypotheses. I. The verification of simple hypotheses

Using statistical modeling methods, we analyze the power of a series of goodness-of-fit tests for simple and complex hypotheses. The estimates we give for the power of the tests for simple hypotheses versus some near competing hypotheses enable us to rank the goodness-of-fit tests.     

多元正态的两个均匀性检验

本文提出两个均匀性统计量用于检验多元正态性．该检验建立在多元正态分布的一个特征性质基础上，模拟研究和实例分析显示该均匀性统计量可以帮助解释来自已有正态性检验统计量的结论．     

一类上界型拟合优度检验统计量的精确分布

对于简单假设的拟合优度检验,Zhang (2002)构造出一类上界型检验.取不同的参数$\lambda$和不同的权函数$q(t)$,这类检验包含了Kolmogorov-Smirov检验, Berk and Jones(1979)检验等已有的上界型检验.文献中仅对极少数$\lambda$和$q(t)$所对应的检验给出了零假设下的精确分布.然而, 针对不同的问题, ``好'的检验是不同的,因此有必要对任意给定的$\lambda$和$q(t)$情况, 讨论该类检验.本文对任意给定的$\lambda$和$q(t)\equiv 1$情况,导出了相应上界型检验统计量在零假设下的精确分布. 当样本容量$n$较大时,精确分布的计算时间较长, 本文还通过模拟比较得到了在不同样本量下,应采用的计算方法. 最后, 给出一个实际例子对前述方法加以简单说明.     

Diversification,protection of liability holders and regulatory arbitrage

Any solvency regime for financial institutions should be aligned with the fundamental objectives of regulation: protecting liability holders and securing the stability of the financial system. The first objective leads to consider surplus-invariant capital adequacy tests, i.e. tests that do not depend on the surplus of a financial institution. We provide a complete characterization of closed, convex, surplus-invariant capital adequacy tests that highlights an inherent tension between surplus-invariance and the desire to give credit for diversification. The second objective leads to requiring consistency of capital adequacy tests across jurisdictions. Of particular importance in this respect are capital adequacy tests that remain invariant under a change of numéraire. We establish an intimate link between surplus- and numéraire invariant tests.     

Ratio-Like and Recurrence Relation Tests for Convergence of Series

Three tests for convergence of series are given which are wellsuited for the automatic determination of the radius of convergenceof a series. The first test is similar to the standard ratiotest, while the other two tests use a recurrence relation forthe terms of the series. Together, these tests determine theradius of convergence of many series which arise as solutionsto ordinary differential equations. Numerical examples usingthese tests are given.     

Asymptotic optimality of nonparametric tests for restricted alternatives in some multivariate mixed models

For some mixed models (involving both stochastic and nonstochastic predictors), a general class of permutationally distribution-free rank tests for some restricted alternative problems is considered. The proposed tests are asymptotically optimal in the light of the restricted likelihood ratio tests. For an ordered alternative problem in a two-way analysis of covariance model, the proposed tests are asymptotically optimal.     

DEA-based hypothesis tests for comparing two groups of decision making units

In this paper, we develop five statistical tests to compare the efficiencies of different groups of DMUs. We consider a data generating process (DGP) that models the deviation of the output from the best practice frontier as the sum of two components, a one-sided inefficiency term and a two-sided random noise term. We use simulation to evaluate the performance of the five tests against the Banker tests (Banker, 1993) that were designed for DGPs containing a single one-sided error term. It is found that while the Banker tests are very effective when efficiency dominates noise, the tests developed in this paper perform better than the Banker tests when noise levels are significant.     

Weighted Kolmogrov–Smirnov type tests for grouped Rayleigh data

We consider goodness of fit tests for the Rayleigh distribution with grouped data. New Kolmogrov–Smirnov type tests are suggested and compared with the traditional chi-square and likelihood ratio tests. The results show that some of the suggested tests have a good power performance as compared with the traditional ones.     

On score-functions and goodness-of-fit tests for stochastic processes

The problems of the construction of asymptotically distribution free goodness-of-fit tests for two diffusion processes are considered. The null hypothesis is composite parametric. All tests are based on the score-function processes, where the unknown parameter is replaced by the maximum likelihood estimators. We show that a special change of time transforms the limit score-function processes into the Brownian bridge. This property allows us to construct asymptotically distribution-free tests for dynamical systems with small noise and ergodic diffusion processes. The proposed tests are in some sense universal. We discuss the possibilities of the construction of asymptotically distribution free tests for inhomogeneous Poisson processes and nonlinear AR time series.     

Some implications of the union-intersection principle for tests of sphericity

The sphericity hypothesis may be expressed as an intersection of simpler hypotheses on the invariant subspaces of the variance matrix. Applying the union-intersection principle to dissections of this type establishes a link between tests of independence and tests of sphericity. We use some recent results of Bloomfield and Watson [2] and Knott [4] to derive a class of union-intersection tests for sphericity from likelihood ratio tests of independence of two sets of variates. As well, we show that the ordinary likelihood ratio test for sphericity has a natural union-intersection interpretation.     

Mathematical Reasoning Requirements in Swedish Upper Secondary Level Assessments

We investigate the mathematical reasoning required to solve the tasks in the Swedish national tests and a random selection of Swedish teacher-made tests. The results show that only a small proportion of the tasks in the teacher-made tests require the students to produce new reasoning and to consider the intrinsic mathematical properties involved in the tasks. In contrast, the national tests include a large proportion of tasks for which memorization of facts and procedures are not sufficient. The conditions and constraints under which the test development takes place indicate some of the reasons for this discrepancy and difference in alignment with the reform documents.