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.     

Testing the Homogeneity of Two Survival Functions Against a Mixture Alternative Based on Censored Data

When the survival distribution in a treatment group is a mixture of two distributions of the same family, traditional parametric methods that ignore the existence of mixture components or the nonparametric methods may not be very powerful. We develop a modified likelihood ratio test (MLRT) for testing homogeneity in a two sample problem with censored data and compare the actual type I error and power of the MLRT with that nonparametric log-rank test and parametric test through Monte-Carlo simulations. The proposed test is also applied to analyze data from a clinical trial on early breast cancer.     

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.     

Likelihood ratio tests for multivariate normality

This paper presents some powerful omnibus tests for multivariate normality based on the likelihood ratio and the characterizations of the multivariate normal distribution. The power of the proposed tests is studied against various alternatives via Monte Carlo simulations. Simulation studies show our tests compare well with other powerful tests including multivariate versions of the Shapiro–Wilk test and the Anderson–Darling test.     

Bartlett corrections in beta regression models

We consider the issue of performing accurate small-sample testing inference in beta regression models, which are useful for modeling continuous variates that assume values in (0,1), such as rates and proportions. We derive the Bartlett correction to the likelihood ratio test statistic and also consider a bootstrap Bartlett correction. Using Monte Carlo simulations we compare the finite sample performances of the two corrected tests to that of the standard likelihood ratio test and also to its variant that employs Skovgaard's adjustment; the latter is already available in the literature. The numerical evidence favors the corrected tests we propose. We also present an empirical application.     

On the Relationship between Directional and Omnibus Statistical Tests

Abstract.  A common statistical problem involves the testing of a K -dimensional parameter vector. In both parametric and semiparametric settings, two types of directional tests – linear combination and constrained tests – are frequently used instead of omnibus tests in hopes of achieving greater power for specific alternatives. In this paper, we consider the relationship between these directional tests, as well as their relationship to omnibus tests. Every constrained directional test is shown to be asymptotically equivalent to a specific linear combination test under a sequence of contiguous alternatives and vice versa. Even when the direction of the alternative is known, the constrained test in general will not be optimal unless the objective function used to derive it is efficient. For an arbitrary alternative, insight into the power characteristics of directional tests in comparison to omnibus tests can be gained by a chi-square partition of the omnibus test.     

Testing equality of mean vectors in a one-way MANOVA with monotone missing data

In this study, testing the equality of mean vectors in a one-way multivariate analysis of variance (MANOVA) is considered when each dataset has a monotone pattern of missing observations. The likelihood ratio test (LRT) statistic in a one-way MANOVA with monotone missing data is given. Furthermore, the modified test (MT) statistic based on likelihood ratio (LR) and the modified LRT (MLRT) statistic with monotone missing data are proposed using the decomposition of the LR and an asymptotic expansion for each decomposed LR. The accuracy of the approximation for the Chi-square distribution is investigated using a Monte Carlo simulation. Finally, an example is given to illustrate the methods.     

Modified inference about the mean of the exponential distribution using moving extreme ranked set sampling

The maximum likelihood estimator (MLE) and the likelihood ratio test (LRT) will be considered for making inference about the scale parameter of the exponential distribution in case of moving extreme ranked set sampling (MERSS). The MLE and LRT can not be written in closed form. Therefore, a modification of the MLE using the technique suggested by Maharota and Nanda (Biometrika 61:601–606, 1974) will be considered and this modified estimator will be used to modify the LRT to get a test in closed form for testing a simple hypothesis against one sided alternatives. The same idea will be used to modify the most powerful test (MPT) for testing a simple hypothesis versus a simple hypothesis to get a test in closed form for testing a simple hypothesis against one sided alternatives. Then it appears that the modified estimator is a good competitor of the MLE and the modified tests are good competitors of the LRT using MERSS and simple random sampling (SRS).     

Hypotheses testing for generalized order statistics with simple order restrictions on model parameters under the alternative

In applications of generalized order statistics as, for instance, reliability analysis of engineering systems, prior knowledge about the order of the underlying model parameters is often available and may therefore be incorporated in inferential procedures. Taking this information into account, we establish the likelihood ratio test, Rao's score test, and Wald's test for test problems arising from the question of appropriate model selection for ordered data, where simple order restrictions are put on the parameters under the alternative hypothesis. For simple and composite null hypothesis, explicit representations of the corresponding test statistics are obtained along with some properties and their asymptotic distributions. A simulation study is carried out to compare the order restricted tests in terms of their power. In the set-up considered, the adapted tests significantly improve the power of the associated omnibus versions for small sample sizes, especially when testing a composite null hypothesis.     

Local and omnibus goodness-of-fit tests in classical measurement error models

We consider functional measurement error models, i.e. models where covariates are measured with error and yet no distributional assumptions are made about the mismeasured variable. We propose and study a score-type local test and an orthogonal series-based, omnibus goodness-of-fit test in this context, where no likelihood function is available or calculated-i.e. all the tests are proposed in the semiparametric model framework. We demonstrate that our tests have optimality properties and computational advantages that are similar to those of the classical score tests in the parametric model framework. The test procedures are applicable to several semiparametric extensions of measurement error models, including when the measurement error distribution is estimated non-parametrically as well as for generalized partially linear models. The performance of the local score-type and omnibus goodness-of-fit tests is demonstrated through simulation studies and analysis of a nutrition data set.     

X2 and its components as tests of normality for grouped data

We consider testing for an unobservable normal distribution with unspecified mean and variance. It is only possible to observe the counts in groups with boundaries specified before sighting the data. On the basis of a small power study, we recommend the usual X² test be used as an omnibus test, augmented by informal examination of the first two non-zero components of X². We also recommend use of maximum likelihood and method of moments estimation.     

The equivalence between likelihood ratio test and F-test for testing variance component in a balanced one-way random effects model

In mixed linear models, it is frequently of interest to test hypotheses on the variance components. F-test and likelihood ratio test (LRT) are commonly used for such purposes. Current LRTs available in literature are based on limiting distribution theory. With the development of finite sample distribution theory, it becomes possible to derive the exact test for likelihood ratio statistic. In this paper, we consider the problem of testing null hypotheses on the variance component in a one-way balanced random effects model. We use the exact test for the likelihood ratio statistic and compare the performance of F-test and LRT. Simulations provide strong support of the equivalence between these two tests. Furthermore, we prove the equivalence between these two tests mathematically.     

Testing Equality of Means Under Order Restrictions Using Multiple Contrasts

The likelihood ratio test for equality of ordered means is known to have power characteristics that are generally superior to those of competing procedures. Difficulties in implementing this test have led to the development of alternative approaches, most of which are based on contrasts. While orthogonal contrasts can be chosen to simplify the distribution theory, we propose a class of tests that is easy to implement even if the contrasts used are not orthogonal. An overall measure of significance may be obtained by using Fisher's combination statistic to combine the dependent p-values arising from these contrasts. This method can be easily implemented for testing problems involving unequal sample sizes and any partial order, and has power properties that compare well with those of the likelihood ratio test and other contrast-based tests.     

Bootstrap versus traditional hypothesis testing procedures for coefficients in least absolute value regression

Several approaches to hypothesis testing for coefficients in least absolute value regression are compared using a Monte Carlo simulation: likelihood ratio test, Lagrange multiplier test, and three versions of the bootstrap hypothesis test. Factors considered that might influence test performance include the disturbance distribution, the type of independent variable, and the sample size. Overall, the likelihood ratio and the bootstrap tests perform best, with the likelihood ratio test being marginally more powerful. Least absolute value tests are also compared to the standard t test and three versions of the bootstrapped t test for least squares regression.     

Omnibus test of normality using the Q statistic

A new statistical procedure for testing normality is proposed. The Q statistic is derived as the ratio of two linear combinations of the ordered random observations. The coefficients of the linear combinations are utilizing the expected values of the order statistics from the standard normal distribution. This test is omnibus to detect the deviations from normality that result from either skewness or kurtosis. The statistic is independent of the origin and the scale under the null hypothesis of normality, and the null distribution of Q can be very well approximated by the Cornish-Fisher expansion. The powers for various alternative distributions were compared with several other test statistics by simulations.     

A TEST OF HOMOGENEITY AGAINST SCALE ALTERNATIVES USING SUBSAMPLE EXTREMA1

In this paper a new class of non-parametric tests for testing homogeneity of several populations against scale alternatives is proposed. For this, independent samples of fixed sizes are drawn from each population and from these samples, all possible sub-samples of the same size are drawn and their maxima and minima are computed. Using these extreme the class of tests is obtained. Tests of this type have been offered for the two-sample slippage problem by Kochar (1978). Under certain conditions, this class of tests is shown to be consistent against ‘difference in scale’ alternatives. The test has been compared with Bhapkar's V-test (1961), Deshpande's D-test (1965), Sugiura's D_rs-test (1965) and with a classical test given by Lehmann (1959, pp. 273–275). It is shown that some members of this proposed class of tests are more efficient than the first three tests in the case of uniform, Laplace and normal distributions, when the number of populations compared is small.     

Permutation tests of scale using deviances

Robust tests for comparing scale parameters, based on deviances—absolute deviations from the median—are examined. Higgins (2004) proposed a permutation test for comparing two treatments based on the ratio of deviances, but the performance of this procedure has not been investigated. A simulation study examines the performance of Higgins’ test relative to other tests of scale utilizing deviances that have been shown in the literature to have good properties. An extension of Higgins’ procedure to three or more treatments is proposed, and a second simulation study compares its performance to other omnibus tests for comparing scale. While no procedure emerged as a preferred choice in every scenario, Higgins’ tests are found to perform well overall with respect to Type I error rate and power.     

An adaptive-to-model test for partially parametric single-index models

Residual marked empirical process-based tests are commonly used in regression models. However, they suffer from data sparseness in high-dimensional space when there are many covariates. This paper has three purposes. First, we suggest a partial dimension reduction adaptive-to-model testing procedure that can be omnibus against general global alternative models although it fully use the dimension reduction structure under the null hypothesis. This feature is because that the procedure can automatically adapt to the null and alternative models, and thus greatly overcomes the dimensionality problem. Second, to achieve the above goal, we propose a ridge-type eigenvalue ratio estimate to automatically determine the number of linear combinations of the covariates under the null and alternative hypotheses. Third, a Monte-Carlo approximation to the sampling null distribution is suggested. Unlike existing bootstrap approximation methods, this gives an approximation as close to the sampling null distribution as possible by fully utilising the dimension reduction model structure under the null model. Simulation studies and real data analysis are then conducted to illustrate the performance of the new test and compare it with existing tests.     

A modified likelihood ratio test for homogeneity in finite mixture models

Testing for homogeneity in finite mixture models has been investigated by many researchers. The asymptotic null distribution of the likelihood ratio test (LRT) is very complex and difficult to use in practice. We propose a modified LRT for homogeneity in finite mixture models with a general parametric kernel distribution family. The modified LRT has a χ-type of null limiting distribution and is asymptotically most powerful under local alternatives. Simulations show that it performs better than competing tests. They also reveal that the limiting distribution with some adjustment can satisfactorily approximate the quantiles of the test statistic, even for moderate sample sizes.