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.     

Assessing the lifetime performance index of Rayleigh products based on the Bayesian estimation under progressive type II right censored samples

Lifetime performance assessment is important in service (or manufacturing) industries. Hence, lifetime performance index C_L is used to measure the potential and performance of a process, where L is the lower specification limit. In this paper, assuming the conjugate prior distribution and squared-error loss function, this study constructs a Bayes estimator under the Rayleigh distribution with the progressive type II right censored sample. The Bayes estimator of C_L is then utilized to develop a credible interval in the condition of known L. Moreover, we also propose a Bayesian test to assess the lifetime performance of products. Finally, we give two examples and the Monte Carlo simulation to assess the behavior of the lifetime performance index C_L. Moreover, the purchasers can then employ the credible interval and the Bayesian test to determine whether the product performance adheres to the required level.     

Random Features of the Fatigue Limit

The classical fatigue limit is often an important characteristic in fatigue design regarding metallic material. The limit is usually obtained from a staircase test in combination with some assumption about the statistical distribution of the limit. This distribution can be of a normal, log-normal or of extreme value type and no particular physical argument gives favor to any specific distribution. This leads to a certain ambiguity in the evaluation of test results which forces the designer to introduce large safety factors. In order to find a physically based statistical distribution for use in staircase tests to determine the fatigue limit we present here a random model for the fatigue limit based on the following assumptions; (i) The square root area model according to Murakami and co-workers is valid, (ii) the randomness in the fatigue limit is induced by the randomness of the maximum defect size, (iii) the random maximum defect size has an extreme value distribution of Gumbel type. This leads to the fatigue limit distribution based on Gumbel (FLG), which is recommended to replace the normal distribution in the evaluation of staircase fatigue tests in case of hard materials. It turns out that the skewness of the resulting distribution depends on the coefficient of variation; with a normal-like non-skewed distribution at the coefficient of variation of five percent.     

Sign-based Test for Mean Vector in High-dimensional and Sparse Settings

In this article, we introduce a robust sparse test statistic which is based on the maximum type statistic. Both the limiting null distribution of the test statistic and the power of the test are analysed. It is shown that the test is particularly powerful against sparse alternatives. Numerical studies are carried out to examine the numerical performance of the test and to compare it with other tests available in the literature. The numerical results show that the test proposed significantly outperforms those tests in a range of settings, especially for sparse alternatives.     

删失数据下几种两样本检验的功效研究

本文讨论了删失数据下的两样本检验问题,并提出了一个新的检验统计量.在样本来自指数分布和Weibull分布及不同的删失水平下,我们把这种检验与其它检验的功效进行了比较.结果表明,在某些情况下,这种检验比其它检验好.     

Conditional tests for elliptical symmetry

In this paper, we suggest the conditional test procedures for testing elliptical symmetry of multivariate distribution. The conditional tests are exactly valid if the symmetric center and the shape matrix are given and are asymptotically valid if they are unknowns to be estimated. The equivalence, in the large sample sense, between the conditional tests and their unconditional counterparts is established. The power behavior of the tests under global as well as local alternatives is investigated theoretically. A small simulation study is performed.     

Permutation Tests for Multivariate Location Problems

The paper presents some permutation test procedures for multivariate location. The tests are based on projected univariate versions of multivariate data. For one-sample cases, the tests are affine invariant and strictly distribution-free for the symmetric null distribution with elliptical direction and their permutation counterparts are conditionally distribution-free when the underlying null distribution of the sample is angularly symmetric. For multi-sample cases, the tests are also affine invariant and permutation counterparts of the tests are conditionally distribution-free for any null distribution with certain continuity. Hence all of the tests in this paper are exactly valid. Furthermore, the equivalence, in the large sample sense, between the tests and their permutation counterparts are established. The power behavior of the tests and of their permutation counterparts under local alternative are investigated. A simulation study shows the tests to perform well compared with some existing tests in the literature, particularly when the underlying null distribution is symmetric whether light-tailed or heavy-tailed. For revealing the influence of data sparseness on the effect of the test, some simulations with different dimensions are also performed.     

Tests for normality based on density estimators of convolutions

Recent results show that densities of convolutions can be estimated by local U-statistics at the root-n rate in various norms. Motivated by this and the fact that convolutions of normal densities are normal, we introduce new tests for normality which use as test statistics weighted L₁-distances between the standard normal density and local U-statistics based on standardized observations. We show that such test statistics converge at the root-n rate and determine their limit distributions as functionals of Gaussian processes. We also address a choice of bandwidth. Simulations show that our tests are competitive with other tests of normality.     

Testing for additivity with B-splines

Regression splines are often used for fitting nonparametric functions, and they work especially well for additivity models. In this paper, we consider two simple tests of additivity: an adaptation of Tukey's one degree of freedom test and a nonparametric version of Rao's score test. While the Tukey-type test can detect most forms of the local non-additivity at the parametric rate of O(n-1/2), the score test is consistent for all alternative at a nonparametric rate. The asymptotic distribution of these test statistics is derived under both the null and local alternative hypotheses. A simulation study is conducted to compare their finite-sample performances with some existing kernel-based tests. The score test is found to have a good overall performance.     

极值分布参数检验的样本崩溃点

检验的样本崩溃点是样本中能逆转判决的离群值的最小比例.本文计算和分析了一类极值分布位置参数的似然比检验的样本崩溃点.并用截尾方法改进了这类检验的样本崩溃性质.     

Goodness-of-fit tests for multivariate Laplace distributions

Consistent goodness-of-fit tests are proposed for symmetric and asymmetric multivariate Laplace distributions of arbitrary dimension. The test statistics are formulated following the Fourier-type approach of measuring the weighted discrepancy between the empirical and the theoretical characteristic function, and result in computationally convenient representations. For testing the symmetric Laplace distribution, and in the particular case of a Gaussian weight function, a limit value of these test statistics is obtained when this weight function approaches a Dirac delta function. Interestingly, this limit value is related to a couple of well-known measures of multivariate skewness. A Monte Carlo study is conducted in order to compare the new procedures with standard tests based on the empirical distribution function. A real data application is also included.     

Goodness-of-Fit Tests for the Cauchy Distribution Based on the Empirical Characteristic Function

Let X ₁,...,X _n be independent observations on a random variable X. This paper considers a class of omnibus procedures for testing the hypothesis that the unknown distribution of X belongs to the family of Cauchy laws. The test statistics are weighted integrals of the squared modulus of the difference between the empirical characteristic function of the suitably standardized data and the characteristic function of the standard Cauchy distribution. A large-scale simulation study shows that the new tests compare favorably with the classical goodness-of-fit tests for the Cauchy distribution, based on the empirical distribution function. For small sample sizes and short-tailed alternatives, the uniformly most powerful invariant test of Cauchy versus normal beats all other tests under discussion.     

Tail fit and the Zipf–Pareto law

A limit theorem with bounds on the rate of convergence is proven. The joint distribution of a fixed number of relative decrements of the top order statistics from a random sample converges to the limit as the sample size increases if and only if the underlying distribution is in essence a Pareto. In conjunction with a chi-square test of fit, it provides an asymptotically distribution-free test of fit to the family of distributions with regularly varying tails at infinity. When the limit distribution holds, rank-size plots obey Zipf’s law. The test can be used to detect departures from this Zipf–Pareto law.      

On goodness-of-fit testing for ergodic diffusion process with shift parameter

A problem of goodness-of-fit test for ergodic diffusion processes is presented. In the null hypothesis the drift of the diffusion is supposed to be in a parametric form with unknown shift parameter. Two Cramer–von Mises type test statistics are studied. The first test uses the local time estimator of the invariant density, the second one uses the empirical distribution function. The unknown parameter is estimated via the maximum likelihood estimator. It is shown that the limit distribution of the two test statistics does not depend on the unknown parameter, thus both the tests are asymptotically parameter free. Some considerations on the consistency of the proposed tests and some simulation studies are also given.     

Testing Departures from Gamma, Rayleigh and Truncated Normal Distributions

This paper provides necessary and sufficient conditions for a solution to likelihood equations for an exponential family of distributions, which includes Gamma, Rayleigh and singly truncated normal distributions. Furthermore, the maximum likelihood estimator is obtained as a limit case when the equations have no solution. These results provide a way to test departures from Rayleigh and singly truncated normal distributions using the likelihood ratio test. A new easy way to test departures from a Gamma distribution is also introduced.     

ON TESTING FOR NO EFFECT OF THE PREDICTOR ON RESPONSE

1.Introduction'Givenasequenceofpairdata(yi1ti),'t(aam,t.),',itisimportanttocheckwhetherornotthepredictorvariablethasaneffectontheresponsevariabley,sincethisisessentialinstudyingthebehaviouroftheprocessandpredictingthefuturevalueofy.Alongthisdirection,therearetwokindsofmodels,namely,regressionmodelandtimeseriesmodelwhichareoftenconsidered:i)Regressionmodelyi~p g(ti) fi,i~1,...?n,(1.1)wheretiarefixeddesignedpointsorrandompoilltscorrespondingtovariousconcretesituations.n)Timeseriesmodelyi~P …     

On the distributions of some test criteria for a covariance matrix under local alternatives and bootstrap approximations

The asymptotic distribution of some test criteria for a covariance matrix are derived under local alternatives. Except for the existence of some higher moments, no assumption as to the form of the distribution function is made. As an illustration, a case of t distribution included normal model is considered and the power of the likelihood ratio test and Nagao's test for sphericity, as described in Srivastava and Khatri and Anderson, is computed. Also, the power is computed using the bootstrap method. In the case of t distribution, the bootstrap approximation does not appear to be as good as the one obtained by the asymptotic expansion method.     

Goodness-of-Fit Tests for the Inverse Gaussian Distribution Based on the Empirical Laplace Transform

This paper considers two flexible classes of omnibus goodness-of-fit tests for the inverse Gaussian distribution. The test statistics are weighted integrals over the squared modulus of some measure of deviation of the empirical distribution of given data from the family of inverse Gaussian laws, expressed by means of the empirical Laplace transform. Both classes of statistics are connected to the first nonzero component of Neyman's smooth test for the inverse Gaussian distribution. The tests, when implemented via the parametric bootstrap, maintain a nominal level of significance very closely. A large-scale simulation study shows that the new tests compare favorably with classical goodness-of-fit tests for the inverse Gaussian distribution, based on the empirical distribution function.     

Tests on categorical data from the unionintersection principle

Summary The union-intersection principle developed by S. N. Roy [13] has become an important tool in multivariate analysis. In this paper the union-intersection principle is applied to obtain some of the standard tests of hypothesis on categorical data, as well as a new test for homogeneity in anr×c table. In particular, tests of hypothesis on a single multinomial distribution and tests for the comparison of two multinomials are derived on the union-intersection principle and the corresponding simultaneous confidence intervals obtained. A test for homogeneity in anr×c table is derived on the union-intersection principle, and for the case of equal sample size from each of ther populations it is shown that the test statistic is distributed as the largest root of a central Wishart matrix.     

On nonparametric tests for symmetry

Summary This paper is concerned with an extension of the problem of testing symmetry about zero of a distribution function. In order to obtain the asymptotic null distribution of test statistics for the problem, a limit theorem is proved, which indeed plays an essential role in the asymptotic theory of testing, problem for symmetry. The Institute of Statistical Mathematics