on the critical periods of perturbed isochronous centers abstract consider a family of planar systems x x x e having a center at the origin and assume that for e 0 they have an isochronous center firstly we give an explicit formula for the first order term in e of the derivative of the period function we apply this formula to prove that up to first order in e at most one critical period bifurcates from the periodic orbits of isochronous quadratic systems when we perturb them inside the class of quadratic reversible centers moreover necessary and sufficient conditions for the existence of this critical period are explicitly given from the tools developed in this paper we also provide a new characterization of planar isochronous centers

testing for weak identification in possibly nonlinear models in this paper we propose a chi square test for identification our proposed test statistic is based on the distance between two shrinkage extremum estimators the two estimators converge in probability to the same limit when identification is strong and their asymptotic distributions are different when identification is weak the proposed test is consistent not only for the alternative hypothesis of no identification but also for the alternative of weak identification which is confirmed by our monte carlo results we apply the proposed technique to test whether the structural parameters of a representative taylor rule monetary policy reaction function are identified

variance component testing in generalised linear models with random effects summary there is considerable interest in testing for overdispersion correlation and heterogeneity across groups in biomedical studies in this paper we cast the problem in the framework of generalised linear models with random effects we propose a global score test for the null hypothesis that all the variance components are zero this test is a locally asymptotically most stringent test and is robust in the special sense that the test does not require specifying the joint distribution of the random effects we also propose individual score tests and their approximations for testing the variance components separately both tests can be easily implemented using existing statistical software we illustrate these tests with an application to the study of heterogeneity of mating success across males and females in an experiment on salamander matings and evaluate their performance through simulation

the period functionsʼ higher order derivatives abstract we prove a formula for the n th derivative of the period function t in a period annulus of a planar differential system for n 1 we obtain freire gasull and guillamon formula for the period s first derivative chicone and dumortier 1993 17 we apply such a result to hamiltonian systems with separable variables and other systems we give some sufficient conditions for the period function of conservative second order ode s to be convex

a class of statistical estimators related to principal components abstract in this paper we investigate the algebraic relationships between some of the more familiar estimation and testing procedures employed in multivariate econometrics and the principal components and continuum regression techniques of multivariate statistics

a general approach to lagrange multiplier model diagnostics abstract a diagnostic test can be formulated by taking the current model as the null hypothesis and a particular form of misspecification as the alternative the general approach of this paper is to calculate the score for some alternative and base a test on its distribution under the null this distribution is derived in two theorems corresponding to different models such a test is a langrange multiplier test and is asymptotically optimal these tests can be formulated as ways of looking at the residuals for particular types of non randomness the tests are simple to calculate even when the alternative may be very complicated examples are given of tests for omitted variables and block recursive structures

asymptotic robustness of tests of overidentification and predeterminedness abstract many statistical test procedures have been proposed for identification restrictions on one or several equations and the econometric predeterminedness of one or several variables in a system of structural equations this study is devoted to unifying many test procedures in a systematic way and to deriving the asymptotic distributions of the test statistics under a set of local alternative hypotheses and very general conditions on the disturbances by making use of a new martingale central limit theorem and a martingale convergence theorem we show that the limiting distributions of test statistics are noncentral 2 distributions under the local alternative hypotheses and central 2 distributions under the null hypotheses these limiting distributions are robust in the sense that they hold for a variety of disturbance distributions and models our results show that many tests already known among econometricians can be carried out without making the usual relatively restrictive assumptions

a note on permutation tests for variance components in multilevel generalized linear mixed models in many applications of generalized linear mixed models to multilevel data it is of interest to test whether a random effects variance component is zero it is well known that the usual asymptotic chi square distribution of the likelihood ratio and score statistics under the null does not necessarily hold in this note we propose a permutation test based on randomly permuting the indices associated with a given level of the model that has the correct type i error rate under the null results from a simulation study suggest that it is more powerful than tests based on mixtures of chi square distributions the proposed test is illustrated using data on the familial aggregation of sleep disturbance

a unified approach to estimation and orthogonality tests in linear single equation econometric models abstract maximum likelihood estimation is considered for a generalisation of the model of anderson and rubin 1949 in which the exogenous variables in the structural equation may not be included in the reduced form equations classical and specification tests are derived for orthogonality hypotheses a necessary and sufficient condition for their equivalence is presented the classical tests are compared using bahadur s asymptotic relative efficiency criterion it is shown that a generalisation of the durbin wu hausman t 2 statistics is asymptotically bahadur efficient

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v

generalized method of moments specification testing abstract this paper analyzes the asymptotic power properties of specification tests which are based on a finite set of moment conditions it shows that any such test may fail against general misspecification that causes estimator inconsistency the mutual asymptotic equivalence of maximal degree of freedom tests is shown and the form of optimal tests against specific forms of misspecification is derived applications to testing for exogeneity of a set of instrumental variables are presented

perturbed normalizers and melnikov functions abstract given m 1 and a smooth family of planar vector fields x e e that is a perturbation of a period annulus we provide a characterization in terms of lie brackets of the property that the first m 1 melnikov functions of x e e vanish identically the equivalent condition is the existence of a smooth family of planar vector fields u e e called here perturbed normalizers of order m we also provide an effective procedure for computing u e when the first m 1 melnikov functions of x e e vanish identically a formula for the derivative of the m th order melnikov function is given

a score test against one sided alternatives abstract a score type statistic t s is introduced for testing h 0 against k 0 and more general one sided hypotheses when nuisance parameters may be present is a vector parameter the main advantages of t s are that it requires estimation of the model only under the null hypothesis and that it is asymptotically equivalent to the likelihood ratio statistic these are precisely the reasons for the popularity of the score tests for testing against two sided alternatives in this sense t s preserves the main attractive features of the classical two sided score test the theoretical results are presented in a general framework where the likelihood based score function is replaced by an estimating function so that the test is applicable even if the exact population distribution is unknown computation of t s is simplified by the fact that it can be computed easily once the corresponding two sided statistic has been computed the relevance and simplicity of t s are illustrated by discussing

first derivative of the period function with applications abstract given a centre of a planar differential system we extend the use of the lie bracket to the determination of the monotonicity character of the period function as far as we know there are no general methods to study this function and the use of commutators and lie bracket was restricted to prove isochronicity we give several examples and a special method which simplifies the computations when a first integral is known

tests of overidentification and predeterminedness in simultaneous equation models abstract a set of simultaneous equations that is a part of a larger complete system is considered one hypothesis concerning this set of equations is that it is overidentified as a block by a block of zero restrictions another hypothesis is that a subset of the current endogenous variables is uncorrelated with the disturbances in the set of simultaneous equations econometric predeterminedness likelihood ratio lagrange multiplier and wald tests are derived for the overidentification hypothesis against unrestricted alternatives for the predeterminedness hypothesis against unrestricted alternatives and for the predeterminedness hypothesis against overidentified alternatives the tests are derived on the basis of normality but are asymptotically valid under very general conditions this study generalizes and unifies many test procedures proposed previously some new tests are developed

variance components testing in the longitudinal mixed effects model this article discusses the asymptotic behavior of likelihood ratio tests for nonzero variance components in the longitudinal mixed effects linear model described by laird and ware 1982 biometrics 38 963 974 our discussion of the large sample behavior of likelihood ratio tests for nonzero variance components is based on the results for nonstandard testing situations by self and liang 1987 journal of the american statistical association 82 605 610

a test in the presence of nuisance parameters abstract we are interested in testing 0 against an alternative in the presence of some nuisance parameter the usual procedure for such problems is to use a test statistic that is a function of the data only let q denote the p value at a given value if q does not depend on then in principle we can apply this procedure however a major difficulty that arises in many situations is that q depends on and therefore cannot be used as a p value in such cases the usual approach is to define the p value as the supremum of q over the nuisance parameter space because this approach ignores sample information about it may be unnecessarily conservative this is a serious problem in order restricted inference to overcome this i propose the following obtain say a 99 confidence region for under the null hypothesis now for a given let t be a test statistic and r be the p value the test procedure is to reject the null hypothesis if 0 01 supremum of r over the 99 c

specification tests in simultaneous equations systems abstract this paper makes two contributions a it provides a new derivation of 2sls and 3sls estimators in which they are represented as restricted least squares and restricted generalized least squares respectively b in the convext thus created it provides a perfectly routine and flexible vehicle for carrying out identifiability and specification tests utilizing 2sls or 3sls estimators the tests in question are simple lagrange multiplier tests lmt carried out directly on the prior restrictions in contrast to the current literature which approaches the problem indirectly through tests on the implications of such prior restrictions on the reduced form in addition it provides a limited monte carlo study comparing the performance of the newly created tests to that of the hausman test ht

on the critical points of the flight return time function of perturbed closed orbits abstract we deal here with planar analytic systems x x x e which are small perturbations of a period annulus for each transversal section to the unperturbed orbits we denote by t q e the time needed by a perturbed orbit that starts from q to return to we call this the flight return time function we say that the closed orbit of x x x 0 is a continuable critical orbit in a family of the form x x x e if for any q and any that passes through q there exists q e a critical point of t e such that q e q as e 0 in this work we study this new problem of continuability in particular we prove that a simple critical periodic orbit of x x x 0 is a continuable critical orbit in any family of the form x x x e we also give sufficient conditions for the existence of a continuable critical orbit of an isochronous center x x x 0

tests for detecting overdispersion in poisson regression models abstract poisson regression models are widely used in analyzing count data this article develops tests for detecting extra poisson variation in such situations the tests can be obtained as score tests against arbitrary mixed poisson alternatives and are generalizations of tests of fisher 1950 and collings and margolin 1985 accurate approximations for computing significance levels are given and the power of the tests against negative binomial alternatives is compared with those of the pearson and deviance statistics one way to test for extra poisson variation is to fit models that parametrically incorporate and then test for the absence of such variation within the models for example negative binomial models can be used in this way cameron and trivedi 1986 lawless 1987a the tests in this article require only the poisson model to be fitted two test statistics are developed that are motivated partly by a desire to have good distributional approximations for computing significance levels simu

tests for variation over groups in survival data abstract when survival data can be divided into natural groups such as in a clinical trial with patients from a number of different institutions the question arises whether effects are different for the different groups for situations where one is concerned with a random pattern of variation over groups this article proposes testing the null hypothesis of no variation using statistics based on the sums of martingale residuals over subjects within groups these tests are closely related to the locally most powerful tests of liang for testing for variation across a large number of strata the tests are applied to data from a lung cancer trial conducted by the eastern cooperative oncology group