Multivariate testing and model-checking for generalized order statistics with applications

The exponential family structure of the joint distribution of generalized order statistics is utilized to establish multivariate tests on the model parameters. For simple and composite null hypotheses, the likelihood ratio test (LR test), Wald's test, and Rao's score test are derived and turn out to have simple representations. The asymptotic distribution of the corresponding test statistics under the null hypothesis is stated, and, in case of a simple null hypothesis, asymptotic optimality of the LR test is addressed. Applications of the tests are presented; in particular, we discuss their use in reliability, and to decide whether a Poisson process is homogeneous. Finally, a power study is performed to measure and compare the quality of the tests for both, simple and composite null hypotheses.     

Inference in a two-parameter generalized order statistics model

As a submodel of generalized order statistics with two unknown model parameters, m-generalized order statistics may serve as a simple model for ordered quantities in a given application. It is shown that the joint distribution of m-generalized order statistics has a representation as a regular exponential family in the model parameters, as it is the case for the comprising model. Utilizing this finding, a minimal sufficient and complete statistic is obtained along with distributional properties. Joint maximum likelihood estimation of the parameters is considered, and strong consistency and asymptotic efficiency of the estimator are established. A test is provided to decide whether a restriction to the submodel is reasonable.     

On similarity and independence properties of composite edf goodness-of-fit tests

Many test statistics for classical simple goodness-of-fit hypothesis testing problems are distancemeasures between the distribution function of the null hypothesis distributipn and the empirical distribution function sometimes called EDF tests. If a composite parametric null hypothesis is considered in place of the simple null hypothesis, then a test statistic can be obtained from each EDF test by replacing the known distribution function of the simple problem by the Rao-Blackwell estimating distribution function. In this note we use known results to show that these Rao-Blackwell-EDF test statistics have distributions that do not depend upon parameter values, and hence that these tests are independent of a complete sufficient statistic for the parameters.     

Non Null Asymptotic Distributions of Some Criteria in Censored Exponential Regression

In this article, we consider the class of censored exponential regression models which is very useful for modeling lifetime data. Under a sequence of Pitman alternatives, the asymptotic expansions up to order n^{? 1/2} of the non null distribution functions of the likelihood ratio, Wald, Rao score, and gradient statistics are derive in this class of models. The non null asymptotic distribution functions of these statistics are obtained for testing a composite null hypothesis in the presence of nuisance parameters. The power of all four tests, which are equivalent to first order, are compared based on these non null asymptotic expansions. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, we consider Monte Carlo simulations. We also present an empirical application for illustrative purposes.     

Testing order restricted hypotheses with proportional hazards

Inferences for survival curves based on right censored continuous or grouped data are studied. Testing homogeneity with an ordered restricted alternative and testing the order restriction as the null hypothesis are considered. Under a proportional hazards model, the ordering on the survival curves corresponds to an ordering on the regression coefficients. Approximate likelihood methods are obtained by applying order restricted procedures to the estimates of the regression coefficients. Ordered analogues to the log rank test which are based on the score statistics are considered also. Chi-bar-squared distributions, which have been studied extensively, are shown to provide reasonable approximations to the null distributions of these tests statistics. Using Monte Carlo techniques, the powers of these two types of tests are compared with those that are available in the literature.     

Adjusted Supremum Score‐Type Statistics for Evaluating Non‐Standard Hypotheses

Supremum score test statistics are often used to evaluate hypotheses with unidentifiable nuisance parameters under the null hypothesis. Although these statistics provide an attractive framework to address non‐identifiability under the null hypothesis, little attention has been paid to their distributional properties in small to moderate sample size settings. In situations where there are identifiable nuisance parameters under the null hypothesis, these statistics may behave erratically in realistic samples as a result of a non‐negligible bias induced by substituting these nuisance parameters by their estimates under the null hypothesis. In this paper, we propose an adjustment to the supremum score statistics by subtracting the expected bias from the score processes and show that this adjustment does not alter the limiting null distribution of the supremum score statistics. Using a simple example from the class of zero‐inflated regression models for count data, we show empirically and theoretically that the adjusted tests are superior in terms of size and power. The practical utility of this methodology is illustrated using count data in HIV research.     

Tests of serial independence based on Kendall's process

The authors propose new rank statistics for testing the white noise hypothesis in a time series. These statistics are Cramér‐von Mises and Kolmogorov‐Smirnov functionals of an empirical distribution function whose mean is related to a serial version of Kendall's tau through a linear transform. The authors determine the asymptotic behaviour of the underlying serial process and the large‐sample distribution of the proposed statistics under the null hypothesis of white noise. They also present simulation results showing the power of their tests.     

Using point optimal test of a simple null hypothesis for testing a composite null hypothesis via maximized Monte Carlo approach

King’s Point Optimal (PO) test of a simple null hypothesis is useful in a number of ways, for example it can be used to trace the power envelope against which existing tests can be compared. However, this test cannot always be constructed when testing a composite null hypothesis. It is suggested in the literature that approximate PO (APO) tests can overcome this problem, but they also have some drawbacks. This paper investigates if King’s PO test can be used for testing a composite null in the presence of nuisance parameters via a maximized Monte Carlo (MMC) approach, with encouraging results.     

Empirical phi-divergence test statistics for testing simple and composite null hypotheses

The main purpose of this paper is to introduce first a new family of empirical test statistics for testing a simple null hypothesis when the vector of parameters of interest is defined through a specific set of unbiased estimating functions. This family of test statistics is based on a distance between two probability vectors, with the first probability vector obtained by maximizing the empirical likelihood (EL) on the vector of parameters, and the second vector defined from the fixed vector of parameters under the simple null hypothesis. The distance considered for this purpose is the phi-divergence measure. The asymptotic distribution is then derived for this family of test statistics. The proposed methodology is illustrated through the well-known data of Newcomb's measurements on the passage time for light. A simulation study is carried out to compare its performance with that of the EL ratio test when confidence intervals are constructed based on the respective statistics for small sample sizes. The results suggest that the ‘empirical modified likelihood ratio test statistic’ provides a competitive alternative to the EL ratio test statistic, and is also more robust than the EL ratio test statistic in the presence of contamination in the data. Finally, we propose empirical phi-divergence test statistics for testing a composite null hypothesis and present some asymptotic as well as simulation results for evaluating the performance of these test procedures.     

A New Goodness-of-Fit Test for Censored Data with an Application in Monitoring Processes

In this article, we propose a new goodness-of-fit test for Type I or Type II censored samples from a completely specified distribution. This test is a generalization of Michael's test for censored data, which is based on the empirical distribution and a variance stabilizing transformation. Using Monte Carlo methods, the distributions of the test statistics are analyzed under the null hypothesis. Tables of quantiles of these statistics are also provided. The power of the proposed test is studied and compared to that of other well-known tests also using simulation. The proposed test is more powerful in most of the considered cases. Acceptance regions for the PP, QQ, and Michael's stabilized probability plots are derived, which enable one to visualize which data contribute to the decision of rejecting the null hypothesis. Finally, an application in quality control is presented as illustration.     

Comparisons of tests for multivariate cointegration

This paper compares the small sample properties of different tests for multivariate cointegration like Johansen's trace test, stock &; Watson's common trend test, Phillips &; Ouliaris' principal component test, as well as cointegration rank decisions based on order selection criteria. Under the null hypothesis of non-cointegration we find a slow convergence rate of the test statistics. In bivariate models the Phillips &; Ouliaris test is extremely dependent on the specification and is outperformed by the other procedures. For trivariate processes we find dependence of the power results on the dynamic specification. The lag order is successfully estimated by order selection criteria.     

Gap-ratio goodness of fit tests for weibull or extreme value distribution assumptions with left or right censored data

With data collection in environmental science and bioassay, left censoring because of nondetects is a problem. Similarly in reliability and life data analysis right censoring frequently occurs. There is a need for goodness of fit tests that can adapt to left or right censored data and be used to check important distributional assumptions without becoming too difficult to regularly implement in practice. A new test statistic is derived from a plot of the standardized spacings between the order statistics versus their ranks. Any linear or curvilinear pattern is evidence against the null distribution. When testing the Weibull or extreme value null hypothesis this statistic has a null distribution that is approximately F for most combinations of sample size and censoring of practical interest. Our statistic is compared to the Mann-Scheuer-Fertig statistic which also uses the standardized spacings between the order statistics. The results of a simulation study show the two tests are competitive in terms of power. Although the Mann-Scheuer-Fertig statistic is somewhat easier to compute, our test enjoys advantages in the accuracy of the F approximation and the availability of a graphical diagnostic.     

Kendall's tau for serial dependence

The authors show how Kendall's tau can be adapted to test against serial dependence in a univariate time series context. They provide formulas for the mean and variance of circular and noncircular versions of this statistic, and they prove its asymptotic normality under the hypothesis of independence. They present also a Monte Carlo study comparing the power and size of a test based on Kendall's tau with the power and size of competing procedures based on alternative parametric and nonparametric measures of serial dependence. In particular, their simulations indicate that Kendall's tau outperforms Spearman's rho in detecting first‐order autoregressive dependence, despite the fact that these two statistics are asymptotically equivalent under the null hypothesis, as well as under local alternatives.     

Kernel estimators for the second order parameter in extreme value statistics

We develop and study in the framework of Pareto-type distributions a general class of kernel estimators for the second order parameter ρ$\rho$, a parameter related to the rate of convergence of a sequence of linearly normalized maximum values towards its limit. Inspired by the kernel goodness-of-fit statistics introduced in Goegebeur et al. (2008), for which the mean of the normal limiting distribution is a function of ρ$\rho$, we construct estimators for ρ$\rho$ using ratios of ratios of differences of such goodness-of-fit statistics, involving different kernel functions as well as power transformations. The consistency of this class of ρ$\rho$ estimators is established under some mild regularity conditions on the kernel function, a second order condition on the tail function 1−F of the underlying model, and for suitably chosen intermediate order statistics. Asymptotic normality is achieved under a further condition on the tail function, the so-called third order condition. Two specific examples of kernel statistics are studied in greater depth, and their asymptotic behavior illustrated numerically. The finite sample properties are examined by means of a simulation study.     

Simple and accurate one‐sided inference from signed roots of likelihood ratios

The authors propose two methods based on the signed root of the likelihood ratio statistic for one‐sided testing of a simple null hypothesis about a scalar parameter in the présence of nuisance parameters. Both methods are third‐order accurate and utilise simulation to avoid the need for onerous analytical calculations characteristic of competing saddlepoint procedures. Moreover, the new methods do not require specification of ancillary statistics. The methods respect the conditioning associated with similar tests up to an error of third order, and conditioning on ancillary statistics to an error of second order.     

Tests for the skewness parameter of two-piece double exponential distribution in the presence of nuisance parameters

In this paper, tests for the skewness parameter of the two-piece double exponential distribution are derived when the location parameter is unknown. Classical tests like Neyman structure test and likelihood ratio test (LRT), that are generally used to test hypotheses in the presence of nuisance parameters, are not feasible for this distribution since the exact distributions of the test statistics become very complicated. As an alternative, we identify a set of statistics that are ancillary for the location parameter. When the scale parameter is known, Neyman–Pearson's lemma is used, and when the scale parameter is unknown, the LRT is applied to the joint density function of ancillary statistics, in order to obtain a test for the skewness parameter of the distribution. Test for symmetry of the distribution can be deduced as a special case. It is found that power of the proposed tests for symmetry is only marginally less than the power of corresponding classical optimum tests when the location parameter is known, especially for moderate and large sample sizes.     

Goodness-of-fit tests via ϕ-measures of divergence for censored data

Measures of divergence or discrepancy are used extensively in statistics in various fields. In this article, we are focusing on divergence measures that are based on a class of measures known as Csiszar's divergence measures. In particular, we propose a class of goodness-of-fit tests based on Csiszar's class of measures designed for censored survival or reliability data. Further, we derive the asymptotic distribution of the test statistic under simple and composite null hypotheses as well as under contiguous alternative hypotheses. Simulations are furnished and real data are analysed to show the performance of the proposed tests for different ?-divergence measures.     

Precedence tests for equality of two distributions based on early failures of ranked set samples

In this article, two different types of precedence tests, each with two different test statistics, based on ranked set samples for testing the equality of two distributions are discussed. The exact null distributions of proposed test statistics are derived, critical values are tabulated for both set size and number of cycles up to 8, and the exact power functions of these two types of precedence tests under the Lehmann alternative are derived. Then, the power values of these two test procedures and their competitors based on simple random samples and based on ranked set samples are compared under the Lehmann alternative exactly and also under a location-shift alternative by means of Monte Carlo simulations. Finally, the impact of imperfect ranking is discussed and some concluding remarks are presented.     

Extensions of empirical likelihood and chi-squared-based tests for ordered alternatives

Several methods for comparing k populations have been proposed in the literature. These methods assess the same null hypothesis of equal distributions but differ in the alternative hypothesis they consider. We focus on two important alternative hypotheses: monotone and umbrella ordering. Two new families of test statistics are proposed, including two known tests, as well as two new powerful tests under monotone ordering. Furthermore, these families are adapted for testing umbrella ordering. We compare some members of the families with respect to power and Type I errors under different simulation scenarios. Finally, the methods are illustrated in several applications to real 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.