Change point problems in the model of logistic regression

The paper considers generalized maximum likelihood asymptotic power one tests which aim to detect a change point in logistic regression when the alternative specifies that a change occurred in parameters of the model. A guaranteed non-asymptotic upper bound for the significance level of each of the tests is presented. For cases in which the test supports the conclusion that there was a change point, we propose a maximum likelihood estimator of that point and present results regarding the asymptotic properties of the estimator. An important field of application of this approach is occupational medicine, where for a lot chemical compounds and other agents, so-called threshold limit values (or TLVs) are specified.We demonstrate applications of the test and the maximum likelihood estimation of the change point using an actual problem that was encountered with real data.     

Testing the homogeneity of risk differences with sparse count data

ABSTRACT

In this paper, we consider testing the homogeneity of risk differences in independent binomial distributions especially when data are sparse. We point out some drawback of existing tests in either controlling a nominal size or obtaining powers through theoretical and numerical studies. The proposed test is designed to avoid the drawbacks of existing tests. We present the asymptotic null distribution and asymptotic power function for the proposed test. We also provide numerical studies including simulations and real data examples showing the proposed test has reliable results compared to existing testing procedures.     

A class of exact UMP unbiased tests for conditional symmetry in small-sample square contingency tables

Testing conditional symmetry against various alternative diagonals-parameter symmetry models often provides a point of departure in studies of square contingency tables with ordered categories. Typically, chi-square or likelihood-ratio tests are used for such purposes. Since these tests depend on the validity of asymptotic approximation, they may be inappropriate in small-sample situations where exact tests are required. In this paper, we apply the theory of UMP unbiased tests to develop a class of exact tests for conditional symmetry in small samples. Oesophageal cancer and longitudinal income data are used to illustrate the approach.     

Tests for the total time on test transform order

In this paper we consider two test statistics for testing the strict TTT transform order between two life distributions of interest. We give their asymptotic distributions and compare our tests with some other related tests in terms of Pitman's asymptotic efficiency. Also we present some results to show the performance and the asymptotic normality of our tests.     

Non-parametric Tests for the Comparison of Point Processes Based on Incomplete Data

We consider the comparison of point processes in a discrete observation situation in which each subject is observed only at discrete time points and no history information between observation times is available. A class of non-parametric test statistics for the comparison of point processes based on this kind of data is presented and their asymptotic distributions are derived. The proposed tests are generalizations of the corresponding tests for continuous observations. Some results from a simulation study for evaluating the proposed tests are presented and an illustrative example from a clinical trial is discussed.     

On False Discovery and Non‐discovery Proportions of the Dynamic Adaptive Procedure

This paper studies the asymptotic behaviour of the false discovery and non‐discovery proportions of the dynamic adaptive procedure under some dependence structure. A Bahadur‐type representation of the cut point in simultaneously performing a large scale of tests is presented. The asymptotic bias decompositions of the false discovery and non‐discovery proportions are given under some dependence structure. In addition to existing literatures, we find that the randomness due to the dynamic selection of the tuning parameter in estimating the true null rate serves as a source of the approximation error in the Bahadur representation and enters into the asymptotic bias term of the false discovery proportion and those of the false non‐discovery proportion. The theory explains to some extent why some seemingly attractive dynamic adaptive procedures do not outperform the competing fixed adaptive procedures substantially in some situations. Simulations justify our theory and findings.     

A New Test for Testing Non Inferiority in Matched-Pairs Design

Non inferiority of one diagnostic method to another is a common issue in medical research. This article proposes a new test using an approximate p-value, which is based on only one point of the two-dimension nuisance parameter space. The sizes and powers of our test, the asymptotic normal test,Sidik and Hsueh's unconditional exact tests are considered. Simulation results suggest that our test can definitely control the Type I error rates with reasonable powers under all studied conditions while the asymptotic normal test cannot for most cases. Compared to Sidik and Hsueh's tests, our test is much easier to implement.     

Near exogeneity,weak identification and specification testing: Some asymptotic results

The paper studies the asymptotic size property of various specification tests in linear structural models where instrumental variables may locally violate the exclusion restrictions. Our results provide some new insights and extensions of earlier studies. In particular, we derive an explicit formula of the asymptotic size of the tests which shows clearly the factors that influence their size under instrument endogeneity. We show that all tests have correct asymptotic size when the usual orthogonality condition holds, but their asymptotic size can be arbitrary large even if only one instrument is slightly correlated with the error term. We present a Monte Carlo experiment that confirms our theoretical findings.     

Combining asymptotically normal tests: case studies in comparison of two groups

When several candidate tests are available for a given testing problem, and each has nice properties with respect to different criteria such as efficiency and robustness, it is desirable to combine them. We discuss various combined tests based on asymptotically normal tests. When the means of two standardized tests under contiguous alternatives are close, we show that the maximum of the two tests appears to have an overall best performance compared with other forms of combined tests considered, and that it retains most power compared with the better one of the two tests combined. As an application, for testing zero location shift between two groups, we studied the normal, Wilcoxon, median tests and their combined tests. Because of their structural differences, the joint convergence and the asymptotic correlation of the tests are not easily derived from the usual asymptotic arguments of the tests. We developed a novel application of martingale theory to obtain the asymptotic correlations and their estimators. Simulation studies were also performed to examine the small sample properties of these combined tests. Finally we illustrate the methods by a real data example.     

Measuring the association of stationary point processes using spectral analysis techniques

In this work we focus on relationships between stationary point process using spectral analysis techniques. The evaluation of these relationships is accomplished with the help of the product ratio of association (PRA), which is based on the cumulant densities of the point processes. The estimation procedure is obtained by smoothing the periodogram statistic, a function of the frequency domain. It is proved that the asymptotic distribution of the square root of the estimated PRA is Normal with a constant variance. Statistical tests for hypotheses concerning the independence of two point processes and the characterization of a Poisson process are proposed. Furthermore, approximate 95% pointwise confidence interval can be obtained for the estimated PRA. These results can be applied on stochastic systems involving as input and output stationary point processes. An illustrative example from the framework of neurophysiology is presented.     

BOOTSTRAP TESTS FOR THE ERROR DISTRIBUTION IN LINEAR AND NONPARAMETRIC REGRESSION MODELS

In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and non‐parametric regression model, which are based on empirical processes of residuals. It is well known that tests in this context are not asymptotically distribution‐free and the parametric bootstrap is applied to deal with this problem. The performance of the resulting bootstrap test is investigated from an asymptotic point of view and by means of a simulation study. The results demonstrate that even for moderate sample sizes the parametric bootstrap provides a reliable and easy accessible solution to the problem of goodness‐of‐fit testing of assumptions regarding the error distribution in linear and non‐parametric regression models.     

Assessing the Precision of Turning Point Estimates in Polynomial Regression Functions

Researchers often report point estimates of turning point(s) obtained in polynomial regression models but rarely assess the precision of these estimates. We discuss three methods to assess the precision of such turning point estimates. The first is the delta method that leads to a normal approximation of the distribution of the turning point estimator. The second method uses the exact distribution of the turning point estimator of quadratic regression functions. The third method relies on Markov chain Monte Carlo methods to provide a finite sample approximation of the exact distribution of the turning point estimator. We argue that the delta method may lead to misleading inference and that the other two methods are more reliable. We compare the three methods using two data sets from the environmental Kuznets curve literature, where the presence and location of a turning point in the income-pollution relationship is the focus of much empirical work.     

On the asymptotic behaviour of location-scale invariant Bickel–Rosenblatt tests

Location-scale invariant Bickel–Rosenblatt goodness-of-fit tests (IBR tests) are considered in this paper to test the hypothesis that f, the common density function of the observed independent d-dimensional random vectors, belongs to a null location-scale family of density functions. The asymptotic behaviour of the test procedures for fixed and non-fixed bandwidths is studied by using an unifying approach. We establish the limiting null distribution of the test statistics, the consistency of the associated tests and we derive its asymptotic power against sequences of local alternatives. These results show the asymptotic superiority, for fixed and local alternatives, of IBR tests with fixed bandwidth over IBR tests with non-fixed bandwidth.     

Assessing the Precision of Turning Point Estimates in Polynomial Regression Functions

Researchers often report point estimates of turning point(s) obtained in polynomial regression models but rarely assess the precision of these estimates. We discuss three methods to assess the precision of such turning point estimates. The first is the delta method that leads to a normal approximation of the distribution of the turning point estimator. The second method uses the exact distribution of the turning point estimator of quadratic regression functions. The third method relies on Markov chain Monte Carlo methods to provide a finite sample approximation of the exact distribution of the turning point estimator. We argue that the delta method may lead to misleading inference and that the other two methods are more reliable. We compare the three methods using two data sets from the environmental Kuznets curve literature, where the presence and location of a turning point in the income-pollution relationship is the focus of much empirical work.     

Comparing tests judged asymptotically equally efficient

For comparing two competing testing procedures whose power functions cannot be calculated exactly, the Pitman asymptotic relative efficiency is commonly used. This measure offers the advantages of simplicity in computation and interpretation, but is asymptotic in nature. In particular, it cannot distinguish between certain pairs of tests, namely, those with an asymptotic relative efficiency of 1. In this paper, we offer an alternative basis for making finite sample comparisons between tests judged asymptotically equally efficient. The criteria for comparison is based on the ratio of efficacies. Using this new criterion, we consider a series of examples in which two tests with an ARE of 1 are compared for finite n. In each case, either exact or simulated power curves are used to evaluate the effectiveness of this criterion in properly discriminating between the competing test procedures. Lastly, conditions are specified where the proposed criterion should be most effective.     

Generalized information matrix tests for copulas

We propose a family of goodness-of-fit tests for copulas. The tests use generalizations of the information matrix (IM) equality of White and so relate to the copula test proposed by Huang and Prokhorov. The idea is that eigenspectrum-based statements of the IM equality reduce the degrees of freedom of the test’s asymptotic distribution and lead to better size-power properties, even in high dimensions. The gains are especially pronounced for vine copulas, where additional benefits come from simplifications of score functions and the Hessian. We derive the asymptotic distribution of the generalized tests, accounting for the nonparametric estimation of the marginals and apply a parametric bootstrap procedure, valid when asymptotic critical values are inaccurate. In Monte Carlo simulations, we study the behavior of the new tests, compare them with several Cramer–von Mises type tests and confirm the desired properties of the new tests in high dimensions.     

Guaranteed testing for epidemic changes of a linear regression model

The objective of this paper is to propose and examine a class of generalized maximum likelihood asymptotic power one tests for detection of various types of changes in a linear regression model. The proposed retrospective tests are based on martingales structures Shiryayev–Roberts statistics. This approach is widely known in a sequential analysis of change point problems as an optimal method of detecting a change in distribution. Guaranteed non-asymptotic upper bounds for the significance levels of the considered tests are presented.Simulated data sets are used to demonstrate that the proposed tests can give good results in practice.     

A class of rank tests for interaction in two-way layouts

A class of aligned rank order tests for interaction in two-way layouts is considered. The technique is based on application of the existing nonparametric techniques for one-way layouts to the transformed observations. It is shown that the asymptotic properties of these techniques are preserved after alignment. Furthermore, asymptotic relative efficiency comparisons are made with some of the competing tests. Finally, we demonstrate that our technique can be used to test for higher-order interactions in general multi-factor layouts.     

Large sample behavior of entropy measures when parameters are estimated

In this paper, we study the asymptotic behavior of (h,π)-entropy statistics when the parameters are replaced by some consistent and asymptotically normal (CAN) estimates. In the case of stratified sampling, asymptotic distribution is obtained and the optimum allocation is derived for a fixed cost and for a fixed variance. Some tests of hypotheses are constructed on the basis of these asymptotic distributions and a numerical example is presented.     

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.