Parametric bootstrap and approximate tests for two Poisson variates

The parametric bootstrap tests and the asymptotic or approximate tests for detecting difference of two Poisson means are compared. The test statistics used are the Wald statistics with and without log-transformation, the Cox F statistic and the likelihood ratio statistic. It is found that the type I error rate of an asymptotic/approximate test may deviate too much from the nominal significance level α under some situations. It is recommended that we should use the parametric bootstrap tests, under which the four test statistics are similarly powerful and their type I error rates are all close to α. We apply the tests to breast cancer data and injurious motor vehicle crash data.     

Goodness-of-fit tests for the Pareto distribution based on its characterization

A new characterization of the Pareto distribution is proposed, and new goodness-of-fit tests based on it are constructed. Test statistics are functionals of U-empirical processes. The first of these statistics is of integral type, it is similar to the classical statistics \(\omega _n^1\). The second one is a Kolmogorov type statistic. We show that the kernels of our statistics are non-degenerate. The limiting distribution and large deviations asymptotics of the new statistics under null hypothesis are described. Their local Bahadur efficiency for parametric alternatives is calculated. This type of efficiency is mostly appropriate for the solution of our problem since the Kolmogorov type statistic is not asymptotically normal, and the Pitman approach is not applicable to this statistic. For the second statistic we evaluate the critical values by using Monte-Carlo methods. Also conditions of local optimality of new statistics in the sense of Bahadur are discussed and examples of such special alternatives are given. For small sample size we compare the power of those tests with some common goodness-of-fit tests.     

RAO'S SCORE TESTS IN SURVIVAL ANALYSIS: EXAMPLES AND BARTLETT TYPE ADJUSTMENTS

Score method in hypothesis testing is one of Professor C. R. Rao's great contributions to statistics. It provides a simple and unified way to test some simple and composite hypotheses in many statistical problems. Some popular tests in statistical practice derived with the help of intuitions can be shown as score tests under some statistical models. The subject-years test and log-rank test in survival analysis are two of the examples. In this paper, we first introduce these two examples. After formulating these two tests as score tests, we then review some recent results on the Bartlett type adjustments for these tests.     

面板数据单位根似然比检验研究

本文采用似然比类检验统计量进行面板单位根检验（简称为LR检验）研究,在局部备择假设成立的条件下,推导了其在无确定项、仅含截距项以及存在线性时间趋势项三种模型下所对应的渐近分布与局部渐近势函数。Monte Carlo模拟结果显示,当面板数据中含确定项（截距项或时间趋势项）时,LR检验水平比LLC和IPS检验水平更接近于给定的显著性检验水平;此外,当面板数据中包含发散个体时,经水平修正后的LR检验势要远远高于经水平修正后的LLC与IPS检验势,其中,经水平修正后的LLC与IPS检验势接近于零。     

A note on Cochran test for homogeneity in one-way ANOVA and meta-analysis

In this paper we provide a formal yet simple and straightforward proof of the asymptotic χ² distribution for Cochran test statistic. Then, we show that the general form of this type of test statistics is invariant for the choice of weights. This fact is important since in practice many such test statistics are constructed with more complicated forms which usually require calculating generalized inverse matrices. Based on our results, we can simplify the construction of the test statistics. More importantly, properties such as anti-conservativeness of this type of test statistics can be drawn from Cochran test statistic. Furthermore, one can improve the performance of the tests by using some modified statistics with correction for small sample size situations.     

WILCOXON-TYPE RANK-SUM PRECEDENCE TESTS

This paper introduces Wilcoxon‐type rank‐sum precedence tests for testing the hypothesis that two life‐time distribution functions are equal. They extend the precedence life‐test first proposed by Nelson in 1963. The paper proposes three Wilcoxon‐type rank‐sum precedence test statistics—the minimal, maximal and expected rank‐sum statistics—and derives their null distributions. Critical values are presented for some combinations of sample sizes, and the exact power function is derived under the Lehmann alternative. The paper examines the power properties of the Wilcoxon‐type rank‐sum precedence tests under a location‐shift alternative through Monte Carlo simulations, and it compares the power of the precedence test, the maximal precedence test and Wilcoxon rank‐sum test (based on complete samples). Two examples are presented for illustration.     

Equivalence of two tests in varying coefficient partially linear errors in variable model with missing responses

This paper considers two tests on varying coefficient partially linear errors-in-variables models (VCPLM-EV) with missing responses under the linear constraint. The restricted estimator for the parametric component is derived and proven to share asymptotically normal distribution. In order to test the linear constraint, two statistics based on the profile Lagrange multiplier method and the corrected residual sum of squares method respectively, are proposed. It is of interest to obtain that the magnitudes of the two statistics are equal exactly and follow the asymptotical chi-square distribution. This reveals a new type of Wilk’s phenomenon in VCPLM-EV models with missing response. Finally, some numerical examples are carried out to illustrate relevant performances.     

More powerful modifications of unit root tests allowing structural change

It is well known that more powerful variants of Dickey–Fuller unit root tests are available. We apply two of these modifications, on the basis of simple maximum statistics and weighted symmetric estimation, to Perron tests allowing for structural change in trend of the additive outlier type. Local alternative asymptotic distributions of the modified test statistics are derived, and it is shown that their implementation can lead to appreciable finite sample and asymptotic gains in power over the standard tests. Also, these gains are largely comparable with those from GLS-based modifications to Perron tests, though some interesting differences do arise. This is the case for both exogenously and endogenously chosen break dates. For the latter choice, the new tests are applied to the Nelson–Plosser data.     

New tests based on Rubin's empirical distribution function

In this paper we derive some new tests for goodness-of-fit based on Rubin's empirical distribution function (EDF). Substituting Rubin's EDF for the classical EDF in the Kolmogorov–Smirnov, Cramér–von Mises, Anderson–Darling statistics, since Rubin's EDF for a given sample is a randomized distribution function, randomized statistics are derived, of which the qth quantile and the expectation are chosen as test statistics. We show that the new tests are consistent under simple hypothesis. Several power comparisons are also performed to show that the new tests are generally more powerful than the classical ones.     

Lack-of-fit testing of a regression model with response missing at random

This paper proposes a class of lack-of-fit tests for fitting a linear regression model when some response variables are missing at random. These tests are based on a class of minimum integrated square distances between a kernel type estimator of a regression function and the parametric regression function being fitted. These tests are shown to be consistent against a large class of fixed alternatives. The corresponding test statistics are shown to have asymptotic normal distributions under null hypothesis and a class of nonparametric local alternatives. Some simulation results are also presented.     

Simultaneous tests for homogeneity of two zero-inflated (beta) populations

ABSTRACT

Motivated by an example in marine science, we use Fisher’s method to combine independent likelihood ratio tests (LRTs) and asymptotic independent score tests to assess the equivalence of two zero-inflated Beta populations (mixture distributions with three parameters). For each test, test statistics for the three individual parameters are combined into a single statistic to address the overall difference between the two populations. We also develop non parametric and semiparametric permutation-based tests for simultaneously comparing two or three features of unknown populations. Simulations show that the likelihood-based tests perform well for large sample sizes and that the statistics based on combining LRT statistics outperforms the ones based on combining score test statistics. The permutation-based tests have overall better performance in terms of both power and type I error rate. Our methods are easy to implement and computationally efficient, and can be expanded to more than two populations and to other multiple parameter families. The permutation tests are entirely generic and can be useful in various applications dealing with zero (or other) inflation.     

Exponentiality test based on alpha-divergence and gamma-divergence

In the present paper, we use the already defined alpha-divergence and gamma-divergence for constructing some goodness of fit tests for exponentiality. These divergence measures are very robust with respect to outliers. Since the existence of outliers among statistical data can be lead to misleading results, therefore utilizing these divergence measures can be of importance. In order to construct test statistics, two estimators are used for alpha-divergence and gamma-divergence. In the first one, we consider the alpha-divergence and gamma-divergence of the equilibrium distribution function, which is well defined on the empirical distribution function (EDF) and is proposed as an EDF-based goodness of fit test statistic. The second one is an estimator in manner of Vasicek entropy estimator. Simulation results indicate that in comparison with the other tests statistics, our mentioned test statistics almost in most of the cases have higher power. Finally, two examples containing outliers illustrate the importance and use of the proposed tests.     

Moments of nonparametric probability density functions of entropy estimators applied to testing the inverse Gaussian distribution

The inverse Gaussian (IG) distribution is widely used to model data and then it is important to develop efficient goodness of fit tests for this distribution. In this article, we introduce some new test statistics for examining the IG goodness of fit based on correcting moments of nonparametric probability density functions of entropy estimators. These tests are consistent against all alternatives. Critical points and power of the tests are explored by simulation. We show that the proposed tests are more powerful than competitor tests. Finally, the proposed tests are illustrated by real data examples.     

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     

Between stability and higher-order asymptotics

We discuss the effects of model misspecifications on higher-order asymptotic approximations of the distribution of estimators and test statistics. In particular we show that small deviations from the model can wipe out the nominal improvements of the accuracy obtained at the model by second-order approximations of the distribution of classical statistics. Although there is no guarantee that the first-order robustness properties of robust estimators and tests will carry over to second-order in a neighbourhood of the model, the behaviour of robust procedures in terms of second-order accuracy is generally more stable and reliable than that of their classical counterparts. Finally, we discuss some related work on robust adjustments of the profile likelihood and outline the role of computer algebra in this type of research.     

Average Most Powerful Tests for a Segmented Regression

To improve the goodness of fit between a regression model and observations, the model can be complicated; however, that can reduce the statistical power when the complication does not lead significantly to an improved model. In the context of two-phase (segmented) logistic regressions, the model evaluation needs to include testing for simple (one-phase) versus two-phase logistic regression models. In this article, we propose and examine a class of likelihood ratio type tests for detecting a change in logistic regression parameters that splits the model into two-phases. We show that the proposed tests, based on Shiryayev–Roberts type statistics, are on average the most powerful. The article argues in favor of a new approach for fixing Type I errors of tests when the parameters of null hypotheses are unknown. Although the suggested approach is partly based on Bayes–Factor-type testing procedures, the classical significance levels of the proposed tests are under control. We demonstrate applications of the average most powerful tests to an epidemiologic study entitled “Time to pregnancy and multiple births.”     

Entropy-based tests of uniformity: A Monte Carlo power comparison

In this article, we consider different entropy estimators and propose some entropy-based tests of uniformity. Critical values of the proposed test statistics are obtained by Monte Carlo simulation. Then the power values of the tests for various alternatives and sample sizes are compared. Finally, some recommendations for the application of the proposed tests in practice are presented.     

Conditional tests in a competing risks model

Testing for equality of competing risks based on their cumulative incidence functions (CIFs) or their cause specific hazard rates (CSHRs) has been considered by many authors. The finite sample distributions of the existing test statistics are in general complicated and the use of their asymptotic distributions can lead to conservative tests. In this paper we show how to perform some of these tests using the conditional distributions of their corresponding test statistics instead (conditional on the observed data). The resulting conditional tests are initially developed for the case of k = 2 and are then extended to k > 2 by performing a sequence of two sample tests and by combining several risks into one. A simulation study to compare the powers of several tests based on their conditional and asymptotic distributions shows that using conditional tests leads to a gain in power. A real life example is also discussed to show how to implement such conditional tests.     

NONPARAMETRIC TWO SAMPLE TESTS BASED ON AREA STATISTICS

ABSTRACT

Area statistics are sample versions of areas occurring in a probability plot of two distribution functions F and G. This paper presents a unified basis for five statistics of this type. They can be used for various testing problems in the framework of the two sample problem for independent observations, such as testing equality of distributions against inequality or testing stochastic dominance of distributions in one or either direction against nondominance. Though three of the statistics considered have already been suggested in literature, two of them are new and deserve our interest. The finite sample distributions of the statistics (under F=G) can be calculated via recursion formulae. Two tables with critical values of the new statistics are included. The asymptotic distribution of the properly normalized versions of the area statistics are functionals of the Brownian bridge. The distribution functions and quantiles thereof are obtained by Monte Carlo simulation. Finally, the power functions of the two new tests based on area statistics are compared to the power functions of the tests based on the corresponding supremum statistics, i.e., statistics of the Kolmogorov–Smirnov type.     

Some tests for a poisson process and their powers with respect to trend in the intensity

Recently many authors have worked on Wei bull process in the area of modelling and analysis. Much less work is done in the area of testing of hypothesis. In this article, some tests for testing the Poisson process against a class of Wei bull process based on the conditional distribution of observations given the sufficient statistic, are proposed. The percentage points of the distributions of the proposed test statistics are simulated. The powers of the tests under alternatives are computed by Monte Carlo method. It is seen that the suggested tests perform well for decreasing intensities.