Testing the validity of the exponential model for hybrid Type-I censored data

The exponential distribution has been used in life-testing and reliability studies. In this article, we first express the entropy of Type-I hybrid censoring scheme in terms of hazard function and provide an estimate of the entropy of Type-I hybrid censored data. Then, we construct a goodness-of-fit test statistic based on Kullback–Leibler information for Type-I hybrid censored data. The test statistic is used to test for exponentiality. A Monte Carlo simulation is conducted to obtain the power of the proposed test against various alternatives. Finally, a data example is presented for illustrative purpose.     

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.     

A semiparametric maximum likelihood ratio test for the change point in copula models

In the present paper, a semiparametric maximum-likelihood-type test statistic is proposed and proved to have the same limit null distribution as the classical parametric likelihood one. Under some mild conditions, the limiting law of the proposed test statistic, suitably normalized and centralized, is shown to be double exponential, under the null hypothesis of no change in the parameter of copula models. We also discuss the Gaussian-type approximations for the semiparametric likelihood ratio. The asymptotic distribution of the proposed statistic under specified alternatives is shown to be normal, and an approximation to the power function is given. Simulation results are provided to illustrate the finite sample performance of the proposed statistical tests based on the double exponential and Gaussian-type approximations.     

Testing a linear relationship in varying coefficient spatial autoregressive models

In this article, we propose a test to check a linear relationship in varying coefficient spatial autoregressive models, in which a residual-based bootstrap procedure is suggested to approximate the null distribution of the resulting test statistic. We conduct simulation studies to assess the performance of the test, including the validity of the bootstrap approximation to the null distribution of the test statistic and the power of the test. The simulation results demonstrate that the residual-based bootstrap procedure gives very accurate estimate of the null distribution of the test statistic and the test is of satisfactory power. Furthermore, a real example is given to demonstrate the application of the proposed test.     

On Variance Components in Semiparametric Mixed Models for Longitudinal Data

Abstract. First, to test the existence of random effects in semiparametric mixed models (SMMs) under only moment conditions on random effects and errors, we propose a very simple and easily implemented non‐parametric test based on a difference between two estimators of the error variance. One test is consistent only under the null and the other can be so under both the null and alternatives. Instead of erroneously solving the non‐standard two‐sided testing problem, as in most papers in the literature, we solve it correctly and prove that the asymptotic distribution of our test statistic is standard normal. This avoids Monte Carlo approximations to obtain p ‐values, as is needed for many existing methods, and the test can detect local alternatives approaching the null at rates up to root n. Second, as the higher moments of the error are necessarily estimated because the standardizing constant involves these quantities, we propose a general method to conveniently estimate any moments of the error. Finally, a simulation study and a real data analysis are conducted to investigate the properties of our procedures.     

Repeated chi-square testing

A sequentialized version of the x²; goodness of fit test, called repeated x,²; test, is introduced. The form of the asymptotic distribution of the repeated x² test statistic is given under the null hypothesis as well as under local alternatives. For various numbers of cells Monte Carlo results are given for critical values, power and distribution of stopping time. Finally, the perfor-mance of the repeated and the fixed sample x² test are compared.     

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 robust test of exogeneity based on quantile regressions

In this paper, we propose a robust test of exogeneity. The test statistics is constructed from quantile regression estimators, which are robust to heavy tails of errors. We derive the asymptotic distribution of the test statistic under the null hypothesis of exogeneity at a given quantile. The finite sample properties of the test are investigated through Monte Carlo simulations that exhibit not only good size and power properties, but also good robustness to outliers.     

An extension of Monte Carlo hypothesis tests

There are many hypothesis testing settings in which one can calculate a “reasonable” test statistic, but in which the null distribution of the statistic is unknown or completely intractable. Fortunately, in many such situations, it is possible to simulate values of the test statistic under the null hypothesis, in which case one can conduct a Monte Carlo test. A difficulty however arises in that Monte Carlo tests, as they are currently structured, are applicable only if ties cannot occur among the values of the test statistics. There is a frequently occurring scenario in which there are lots of ties, namely that in which the null distribution of the test statistic has a (single) point mass. It turns out that one can modify the current form of Monte Carlo tests so as to accommodate such settings. Developing this modification leads to an intriguing identity involving the binomial probability function and its derivatives. In this article, we will briefly explain the modified procedure, discuss simulation studies which demonstrate its efficacy, and provide a proof of the identity referred to above.     

On a data-dependent choice of the tuning parameter appearing in certain goodness-of-fit tests

We propose a data-dependent method for choosing the tuning parameter appearing in many recently developed goodness-of-fit test statistics. The new method, based on the bootstrap, is applicable to a class of distributions for which the null distribution of the test statistic is independent of unknown parameters. No data-dependent choice for this parameter exists in the literature; typically, a fixed value for the parameter is chosen which can perform well for some alternatives, but poorly for others. The performance of the new method is investigated by means of a Monte Carlo study, employing three tests for exponentiality. It is found that the Monte Carlo power of these tests, using the data-dependent choice, compares favourably to the maximum achievable power for the tests calculated over a grid of values of the tuning parameter.     

Tests for Structural Changes in Time Series of Counts

We propose methods for detecting structural changes in time series with discrete‐valued observations. The detector statistics come in familiar L2‐type formulations incorporating the empirical probability generating function. Special emphasis is given to the popular models of integer autoregression and Poisson autoregression. For both models, we study mainly structural changes due to a change in distribution, but we also comment for the classical problem of parameter change. The asymptotic properties of the proposed test statistics are studied under the null hypothesis as well as under alternatives. A Monte Carlo power study on bootstrap versions of the new methods is also included along with a real data example.     

Tests of fit for the Gumbel distribution: EDF-based tests against entropy-based tests

In this article, we propose some tests of fit based on sample entropy for the composite Gumbel (Extreme Value) hypothesis. The proposed test statistics are constructed using different entropy estimates. Through a Monte Carlo simulation, critical values of the test statistics for various sample sizes are obtained. Since the tests based on the empirical distribution function (EDF) are commonly used in practice, the power values of the entropy-based tests with those of the EDF tests are compared against various alternatives and different sample sizes. Finally, two real data sets are modeled by the Gumbel distribution.KEYWORDS: Entropy estimator, Gumbel distribution, Monte Carlo simulation, test power     

A sequential approximation to a permutation test

A sequential method for approximating a general permutation test (SAPT) is proposed and evaluated. Permutations are randomly generated from some set G, and a sequential probability ratio test (SPRT) is used to determine whether an observed test statistic falls sufficiently far in the tail of the permutation distribution to warrant rejecting some hypothesis. An estimate and bounds on the power function of the SPRT are used to find bounds on the effective significance level of the SAPT. Guidelines are developed for choosing parameters in order to obtain a desired significance level and minimize the number of permutations needed to reach a decision. A theoretical estimate of the average number of permutations under the null hypothesis is given along with simulation results demonstrating the power and average number of permutations for various alternatives. The sequential approximation retains the generality of the permutation test,- while avoiding the computational complexities that arise in attempting to computer the full permutation distribution exactly     

Distribution-free comparison of hazard rates of two distributions under progressive type-II censoring

Some distribution-free tests have been discussed in the literature with regard to the comparison of hazard rates of two distributions when the available samples are complete. We generalize here Kochar's [S.C. Kochar, A new distribution-free test for the equality of two failure rates, Biometrika 68 (1981), pp. 423–426] test statistic to the case when one available sample is progressively Type-II censored, and then derive its exact null distribution and examine its power properties by means of a Monte Carlo simulation study.     

Testing Exponentiality Against Laplace Order Dominance-

We present a statistical procedure to test that a life distribution belongs to the class of exponential distributions against that it belongs to a class of alternatives based on the Laplace transform. The test has been shown to be consistent and the asymptotic distribution of the test statistic has been obtained. The performance of the test against various classes of alternatives has been studied by means of Monte Carlo simulation. An interesting characterization theorem for exponentials, which motivates our test procedure, has been proved.     

Data driven chi-square test for uniformity with unequal cells

The power of Pearson's chi-square test for uniformity depends heavily on the choice of the partition of the unit interval involved in the form of the test statistic. We propose a selection rule which chooses a proper partition based on the data. This selection rule leads usually to essentially unequal cells well suited to the observed distribution. We investigate the corresponding data driven chi-square test and present a Monte Carlo simulation study. The conclusion is that this test achieves a high and very stable power for a large class of alternatives, and is much more stable than any other test we compare to.     

Tests for Symmetric Error Distribution in Linear and Nonparametric Regression Models

In linear and nonparametric regression models, the problem of testing for symmetry of the distribution of errors is considered. We propose a test statistic which utilizes the empirical characteristic function of the corresponding residuals. The asymptotic null distribution of the test statistic as well as its behavior under alternatives is investigated. A simulation study compares bootstrap versions of the proposed test to other more standard procedures.     

部分函数型线性变系数模型的序列相关检验

部分函数型线性变系数模型(PFLVCM)是近几年出现的一个比较灵活、应用广泛的新模型。在实际应用中,搜集到的经济和金融数据往往存在序列相关性。如果不考虑数据间的相关性直接对其进行建模,会影响模型中参数估计的精度和有效性。本文主要研究了PFLVCM中误差的序列相关性的检验问题,基于经验似然,把标量时间序列数据相关性检验的方法拓展到函数型数据中,提出了经验对数似然比检验统计量,并在零假设下得到了检验统计量的近似分布。通过蒙特卡洛数值模拟说明该统计量在有限样本下有良好的水平和功效。最后,把该方法用于检验美国商业用电消费数据是否有序列相关性,证明该统计量的有效性和实用性。     

Distribution-free test for symmetry based on Bonferroni's measure

We propose a test based on Bonferroni's measure of skewness. The test detects the asymmetry of a distribution function about an unknown median. We study the asymptotic distribution of the given test statistic and provide a consistent estimate of its variance. The asymptotic relative efficiency of the proposed test is computed along with Monte Carlo estimates of its power. This allows us to perform a comparison of the test based on Bonferroni's measure with other tests for symmetry.