Exact Likelihood Ratio Testing for Homogeneity of the Exponential Distribution

The aim of this article is to discuss homogeneity testing of the exponential distribution. We introduce the exact likelihood ratio test of homogeneity in the subpopulation model, ELR, and the exact likelihood ratio test of homogeneity against the two-components subpopulation alternative, ELR2. The ELR test is asymptotically optimal in the Bahadur sense when the alternative consists of sampling from a fixed number of components. Thus, in some setups the ELR is superior to frequently used tests for exponential homogeneity which are based on the EM algorithm (like the MLRT, ADDS, and D-tests). One important example of superiority of ELR and ELR2 tests is the case of lower contamination. We demonstrate this fact by both theoretical comparisons and simulations.     

Empirical-likelihood-based Test for Partially Linear Single-index Models with Error-prone Linear Covariates

In this article, we consider whether the empirical likelihood ratio (ELR) test is applicable to testing for serial correlation in the partially linear single-index models (PLSIM) with error-prone linear covariates. It is shown that under the null hypothesis the proposed ELR statistic follows asymptotically a χ²-distribution with the scale constant and the degrees of freedom. A comparison between the ELR and the normal approximation method is also considered. Both simulated and real data examples are used to illustrate our proposed methodology.     

Simple and Exact Empirical Likelihood Ratio Tests for Normality Based on Moment Relations

The empirical likelihood (EL) technique is a powerful nonparametric method with wide theoretical and practical applications. In this article, we use the EL methodology in order to develop simple and efficient goodness-of-fit tests for normality based on the dependence between moments that characterizes normal distributions. The new empirical likelihood ratio (ELR) tests are exact and are shown to be very powerful decision rules based on small to moderate sample sizes. Asymptotic results related to the Type I error rates of the proposed tests are presented. We present a broad Monte Carlo comparison between different tests for normality, confirming the preference of the proposed method from a power perspective. A real data example is provided.     

On the robustness of empirical likelihood ratio confidence intervals for location

The authors examine the robustness of empirical likelihood ratio (ELR) confidence intervals for the mean and M‐estimate of location. They show that the ELR interval for the mean has an asymptotic breakdown point of zero. They also give a formula for computing the breakdown point of the ELR interval for M‐estimate. Through a numerical study, they further examine the relative advantages of the ELR interval to the commonly used confidence intervals based on the asymptotic distribution of the M‐estimate.     

Evaluating the relative merits of competing models based on empirical likelihood ratio test

Competing models arise naturally in many research fields, such as survival analysis and economics, when the same phenomenon of interest is explained by different researcher using different theories or according to different experiences. The model selection problem is therefore remarkably important because of its great importance to the subsequent inference; Inference under a misspecified or inappropriate model will be risky. Existing model selection tests such as Vuong's tests [26 Q.H. Vuong, Likelihood ratio test for model selection and non-nested hypothesis, Econometrica 57 (1989), pp. 307–333. doi: 10.2307/1912557[Crossref], [Web of Science ®] , [Google Scholar]] and Shi's non-degenerate tests [21 X. Shi, A non-degenerate Vuong test, Quant. Econ. 6 (2015), pp. 85–121. doi: 10.3982/QE382[Crossref], [Web of Science ®] , [Google Scholar]] suffer from the variance estimation and the departure of the normality of the likelihood ratios. To circumvent these dilemmas, we propose in this paper an empirical likelihood ratio (ELR) tests for model selection. Following Shi [21 X. Shi, A non-degenerate Vuong test, Quant. Econ. 6 (2015), pp. 85–121. doi: 10.3982/QE382[Crossref], [Web of Science ®] , [Google Scholar]], a bias correction method is proposed for the ELR tests to enhance its performance. A simulation study and a real-data analysis are provided to illustrate the performance of the proposed ELR tests.     

Methods for testing subblock patterns

A number of statistical tests have been recommended over the last twenty years for assessing the randomness of long binary strings used in cryptographic algorithms. Several of these tests include methods of examining subblock patterns. These tests are the uniformity test, the universal test and the repetition test. The effectiveness of these tests are compared based on the subblock length, the limitations on data requirements, and on their power in detecting deviations from randomness. Due to the complexity of the test statistics, the power functions are estimated by simulation methods. The results show that for small subblocks the uniformity test is more powerful than the universal test, and that there is some doubt about the parameters of the hypothesised distribution for the universal test statistic. For larger subblocks the results show that the repetition test is the most effective test, since it requires far less data than either of the other two tests and is an efficient test in detecting deviations from randomness in binary strings.     

Assessing the fit of finite mixture distributions

Mixture distributions have become a very flexible and common class of distributions, used in many different applications, but hardly any literature can be found on tests for assessing their goodness of fit. We propose two types of smooth tests of goodness of fit for mixture distributions. The first test is a genuine smooth test, and the second test makes explicit use of the mixture structure. In a simulation study the tests are compared to some traditional goodness of fit tests that, however, are not customised for mixture distributions. The first smooth test has overall good power and generally outperforms the other tests. The second smooth test is particularly suitable for assessing the fit of each component distribution separately. The tests are applicable to both continuous and discrete distributions and they are illustrated on three medical data sets.     

An adaptive two-sample location-scale test of lepage type for symmetric distributions

For the two-sample location and scale problem we propose an adaptive test which is based on so called Lepage type tests. The well known test of Lepage (1971) is a combination of the Wilcoxon test for location alternatives and the Ansari-Bradley test for scale alternatives and it behaves well for symmetric and medium-tailed distributions. For the cae of short-, medium- and long-tailed distributions we replace the Wilcoxon test and the .Ansari-Bradley test by suitable other two-sample tests for location and scale, respectively, in oder to get higher power than the classical Lepage test for such distribotions. These tests here are called Lepage type tests. in practice, however, we generally have no clear idea about the distribution having generated our data. Thus, an adaptive test should be applied which takes the the given data set inio consideration. The proposed adaptive test is based on the concept of Hogg (1974), i.e., first, to classify the unknown symmetric distribution function with respect to a measure for tailweight and second, to apply an appropriate Lepage type test for this classified type of distribution. We compare the adaptive test with the three Lepage type tests in the adaptive scheme and with the classical Lepage test as well as with other parametric and nonparametric tests. The power comparison is carried out via Monte Carlo simulation. It is shown that the adaptive test is the best one for the broad class of distributions considered.     

Applying estimated score tests in econometrics

The estimated score test is a hypothesis testing procedure that can improve on the standard score, or Lagrange multiplier (LM) test. The score for a parameter of interest will usually contain nuisance parameters. Essentially, the test replaces nuisance parameters by estimates and then takes the difference between this estimated score and its expectation as the critical region for a test. If the expectation is zero the test coincides with the standard score, or LM, test, but if the expectation is non-zero the small sample properties of the tests differ. In some cases even asymptotic properties differ. This paper examines the scope for the application of estimated score tests in econometrics and illustrates with examples. Comparisons with the standard tests emphasize differences between the tests in terms of the true, as distinct from nominal, sizes of tests.     

Confidence intervals and tests on the variance ratio in random models with two variance components

The LM test is modified to test any value of the ratio of two variance components in a mixed effects linear model with two variance components. The test is exact, so it can be used to construct exact confidence intervals on this ratio.Exact Neyman-Pearson (NP) tests on the variance ratio are described.Their powers provide attainable upper bounds on powers of tests on the variance ratio.Efficiencies of LM tests, which include ANOVA tests, and NP tests are compared for unbalanced, random, one-way ANOVA models.Confidence intervals corresponding to LM tests and NP tests are described.     

Nonparametric two-sample tests for dispersion differences based on placements

Two different two-sample tests for dispersion differences based on placement statistics are proposed. The means and variances of the test statistics are derived, and asymptotic normality is established for both. Variants of the proposed tests based on reversing the X and Y labels in the test statistic calculations are shown to have different small-sample properties; for both pairs of tests, one member of the pair will be resolving, the other nonresolving. The proposed tests are similar in spirit to the dispersion tests of both Mood and Hollander; comparative simulation results for these four tests are given. For small sample sizes, the powers of the proposed tests are approximately equal to the powers of the tests of both Mood and Hollander for samples from the normal, Cauchy and exponential distributions. The one-sample limiting distributions are also provided, yielding useful approximations to the exact tests when one sample is much larger than the other. A bootstrap test may alternatively be performed. The proposed test statistics may be used with lightly censored data by substituting Kaplan-Meier estimates for the empirical distribution functions.     

Power studies of tests for uniformity,II

Results from a power study of six statistics for testing that a sample is from a uniform distribution on the unit interval (0,1) are reported. The test statistics are all well-known and each of them was originally proposed because they should have high power against some alternative distributions. The tests considered are the Pearson probability product test, the Neyman smooth test, the Sukhatme test, the Durbin-Kolmogorov test, the Kuiper test, and the Sherman test. Results are given for each of these tests against each of four classes of alternatives. Also, the most powerful test against each member of the first three alternatives is obtained, and the powers of these tests are given for the same sample sizes as for the six general "omnibus" test statistics. These values constitute a "power envelope" against which all tests can be compared. The Neyman smooth tests with 2nd and 4th degree polynomials are found to have good power and are recommended as general tests for uniformity.     

A maxmin linear test of normal means and its application to lachin’s data

For stochastic ordering tests for normal distributions there exist two well known types of tests. One of them is based on the maximum likelihood ratio principle, the other is the most stringent somewhere most powerful test of Schaafsma and Smid(for a comprehensive treatment see Robertson, Wright and Dykstra(1988), for the latter test also Shi and Kudo(1987)). All these tests are in general numerically tedious. Wei, Lachin(1984)and particularly Lachin(1992)formulate a simple and easily computable test. However, it is not known so far for which sort of ordered alternatives his test is optimal

In this paper it is shown that his procedure is a maxmin test for reasonable subalternatives, provided the covariance matrix has nonnegative row sums. If this property is violated then his procedure can be altered in such a manner that the resul ting test again is a maxmin test. An example is glven where the modified procedure even in the least favourable case leads to a nontrifling increase in power. The fact that Lachins test resp. the modified version are maxmin tests on appropriate subalternatives amounts to the property that they are maxmin tests on subhypotheses which are relevant in practical applications.     

Simple Tests for Exogeneity of a Binary Explanatory Variable in Count Data Regression Models

This article investigates power and size of some tests for exogeneity of a binary explanatory variable in count models by conducting extensive Monte Carlo simulations. The tests under consideration are Hausman contrast tests as well as univariate Wald tests, including a new test of notably easy implementation. Performance of the tests is explored under misspecification of the underlying model and under different conditions regarding the instruments. The results indicate that often the tests that are simpler to estimate outperform tests that are more demanding. This is especially the case for the new test.     

A class of distribution-free tests for two-way layout

In this article a class of distribution-free tests for the hypothesis of no row (treatment) effect in a two-way layout design, with several observations per cell, is proposed. The tests are based on U-statistics, constructed by considering minima of all possible subsamples of same size from each cell.The proposed class of tests is compared with the parametric test, Mack and Skillings test and Yate's test for two-way layout, in terms of Pitman ARE sense. It is seen that for the case of equal number of observations per cell, the proposed tests have better efficiency for exponential and uniform error distributions.     

Evaluation of three lack of fit tests in linear regression models

A key diagnostic in the analysis of linear regression models is whether the fitted model is appropriate for the observed data. The classical lack of fit test is used for testing the adequacy of a linear regression model when replicates are available. While many efforts have been made in finding alternative lack of fit tests for models without replicates, this paper focuses on studying the efficacy of three tests: the classical lack of fit test, Utts' (1982) test, Burn & Ryan's (1983) test. The powers of these tests are computed for a variety of situations. Comments and conclusions on the overall performance of these tests are made, including recommendations for future studies.     

Testing hypotheses about the common mean of two normal populations

Two simple tests which allow for unequal sample sizes are considered for testing hypothesis for the common mean of two normal populations. The first test is an exact test of size a based on two available t-statistics based on single samples made exact through random allocation of α among the two available t-tests. The test statistic of the second test is a weighted average of two available t-statistics with random weights. It is shown that the first test is more efficient than the available two t-tests with respect to Bahadur asymptotic relative efficiency. It is also shown that the null distribution of the test statistic in the second test, which is similar to the one based on the normalized Graybill-Deal test statistic, converges to a standard normal distribution. Finally, we compare the small sample properties of these tests, those given in Zhou and Mat hew (1993), and some tests given in Cohen and Sackrowitz (1984) in a simulation study. In this study, we find that the second test performs better than the tests given in Zhou and Mathew (1993) and is comparable to the ones given in Cohen and Sackrowitz (1984) with respect to power..     

A new jackknife test for homogeneity of variances

A new jackknife test is proposed to test the equality of variances in several populations. The new test is based on jackknifing one group of observations at a time, instead of one observation in each group as recommended by Miller for a two sample case, and by Layard for several samples. The proposed test is examined, and compared with other tests, in terms of power and robustness with respect to a wide variety of non-normal distributions. It is found that the new test is robust and has reasonably high power for normal as well as for non-normal observations, irrespective of the sample size. Furthermore, the proposed test is certainly superior to all other tests considered here in small to moderate size samples, and is as good as or better than the other tests in large samples, irrespective of the distribution of sampling observations.     

Monte carlo sampling approach to testing nonnested hypothesis: monte carlo results

Alternative ways of using Monte Carlo methods to implement a Cox-type test for separate families of hypotheses are considered. Monte Carlo experiments are designed to compare the finite sample performances of Pesaran and Pesaran's test, a RESET test, and two Monte Carlo hypothesis test procedures. One of the Monte Carlo tests is based on the distribution of the log-likelihood ratio and the other is based on an asymptotically pivotal statistic. The Monte Carlo results provide strong evidence that the size of the Pesaran and Pesaran test is generally incorrect, except for very large sample sizes. The RESET test has lower power than the other tests. The two Monte Carlo tests perform equally well for all sample sizes and are both clearly preferred to the Pesaran and Pesaran test, even in large samples. Since the Monte Carlo test based on the log-likelihood ratio is the simplest to calculate, we recommend using it.     

An omnibus two-sample test for ranked-set sampling data

We develop an omnibus two-sample test for ranked-set sampling (RSS) data. The test statistic is the conditional probability of seeing the observed sequence of ranks in the combined sample, given the observed sequences within the separate samples. We compare the test to existing tests under perfect rankings, finding that it can outperform existing tests in terms of power, particularly when the set size is large. The test does not maintain its level under imperfect rankings. However, one can create a permutation version of the test that is comparable in power to the basic test under perfect rankings and also maintains its level under imperfect rankings. Both tests extend naturally to judgment post-stratification, unbalanced RSS, and even RSS with multiple set sizes. Interestingly, the tests have no simple random sampling analog.