robust bounded influence tests in general parametric models abstract we introduce robust tests for testing hypotheses in a general parametric model these are robust versions of the wald scores and likelihood ratio tests and are based on general m estimators their asymptotic properties and influence functions are derived it is shown that the stability of the level is obtained by bounding the self standardized sensitivity of the corresponding m estimator furthermore optimally bounded influence tests are derived for the wald and scores type tests applications to real and simulated data sets are given to illustrate the tests performance

clarke coderivatives of efficient point multifunctions in parametric vector optimization abstract the paper is devoted to the study of the clarke circatangent coderivatives of the efficient point multifunction of parametric vector optimization problems in banach spaces we provide inner outer estimates for evaluating the clarke circatangent coderivative of this multifunction in a broad class of conventional vector optimization problems in the presence of geometrical operator and finite and infinite functional constraints examples are given for analyzing and illustrating the obtained results

the proximal normal formula in banach space approximation by proximal normals to the clarke generalized subdifferential for a distance function generated by a nonempty closed set and the normal cone to the set generated by the proximal normals are important tools in nonsmooth analysis we give simple general versions of such formulae in infinite dimensional banach spaces which satisfy different geometrical conditions our first class of spaces with uniformly g teaux differentiable norm includes the hilbert space case and the formulae is attained through dense subsets our second class of reflexive kadec smooth spaces is the most general for which such formulae can be obtained for all nonempty closed sets in the space our technique also allows us to establish the existence of solutions for a class of optimization problems substantially extending similar work of ekeland and lebourg

subdifferential regularities of perturbed distance functions outside the target set in banach spaces abstract this paper is devoted to study the frechet and proximal regularities at points outside the target set of perturbed distance functions d s j determined by a closed subset s and a lipschitz function j also we provide some important results on the frechet proximal and clarke subdifferentials of d s j at those points in banach spaces

comparison of local power of alternative tests of non nested regression models the paper derives and compares the local power of three different methods of testing non nested regression models that are available in the literature it shows that the asymptotic power of the orthodox f test against local alternatives is strictly less than that of cox s non nested test or the j test recently proposed by davidson and mackinnon 5 unless the number of non overlapping variables of the alternative hypothesis over the null hypothesis is unity in which case all three tests are shown to be asymptotically equivalent the final section of the paper gives results of monte carlo experiments designed to check the validity of the theoretical findings of the paper and also to shed light on the small sample properties of the cox test and the orthodox test

smoothness of the metric projection onto nonconvex bodies in hilbert spaces abstract based on a fundamental work of r b holmes from 1973 we study differentiability properties of the metric projection onto prox regular sets we show that if the set is a nonconvex body with a c p 1 smooth boundary then the projection is c p smooth near suitable open truncated normal rays which are determined only by the function of prox regularity a local version of the same result is established as well namely when the smoothness of the boundary and the prox regularity of the set are assumed only near a fixed point finally similar results are derived when the prox regular set is itself a c p 1 submanifold

tests for model specification in the presence of alternative hypotheses some further results abstract in davidson and mackinnon 1981 two of the present authors proposed a novel and very simple procedure for testing the specification of a nonlinear regression model against the evidence provided by a non nested alternative in this paper we extend their results in several directions first we relax a number of the assumptions of the previous paper we admit the possibility that the nonlinear regression functions may depend on lagged dependent variables and we do not require that the error terms be normally distributed second we show how the earlier procedure may straightforwardly be generalized to the case where the two non nested models involve different transformations of the dependent variable finally we propose a simple procedure for testing non nested linear regression models which have endogenous variables on the right hand side and have therefore been estimated by two stage least squares

non regular maximum likelihood problems four non regular estimation problems are reviewed and discussed one the unbounded likelihood problem involves distributions with infinite spikes for which maximum likelihood can fail to give consistent estimators a comparison is made with modified likelihood and spacings methods which do give efficient estimators in this case an application to the box cox shifted power transform is given the other three problems occur when the true parameter lies in some special subregion in one the constrained parameter problem the subregion is a boundary the other two the embedded model and the indeterminate parameters problems occur when the model takes on a special form in the subregion these last two problems have previously been investigated separately we show that they are equivalent in some situations both often arise in non linear models and we give a directed graph approach which allows for their occurrence in nested model building it is argued that many non regular problems can be handled systematically without having to resort to elaborate technical assumptions relatively uncomplicated methods may be used provided that the underlying nature of the non regularity is understood

relaxation of an optimal control problem involving a perturbed sweeping process we establish first in the setting of infinite dimensional hilbert space a result concerning the existence of solutions for perturbed sweeping processes whose perturbations are lipschitz single valued maps then we use this result to extend to the infinite dimensional setting a relaxation result concerning optimal control problems involving such processes

subdifferentials of a minimum time function in banach spaces in general banach space setting we study the minimum time function determined by a closed convex set k and a closed set s this function is simply the usual minkowski function of k if s is the singleton consisting of the origin in particular we show that various subdifferentials of a minimum time function are representable by virtue of corresponding normal cones of sublevel sets of the function

some aspects of testing non nested hypotheses abstract this paper examines some aspects of testing non nested hypotheses an inequality between the cox and atkinson statistics is noted and the necessary and sufficient condition for the atkinson test to be consistent is derived a new test procedure is also outlined the rest of the paper illustrates the various test statistics their properties and relationships for competing linear regression models

integration of subdifferentials of nonconvex functions in nonsmooth analysis and optimization subgradients come in many different flavors e g approximate dini proximal clarke generalized see 2 5 6 12 14 191 these subgradients are important and valuable tools however many questions remain unsolved concerning the exact link between the function and its subgradients for instance can two functions not differing by an additive constant have the same subgradients in this paper we study the fundamental problem of determining functions that can be recovered up to an additive constant from the knowledge of their subgradients this integration problem is not very well understood and very few functions or classes of functions are known to be recoverable from their subgradients in section 4 we show that if the basic constraint qualification holds at r then the composition of a closed i e lowersemicontinuous proper convex function with a twice continuously differentiable mapping is determined up to an additive constant by its generalized subgradients actually in this case all above mentioned subgradients are the same beside the obvious theoretical interest of this integration problem it is our hope or perhaps our long term goal that once this problem is better understood we can then tackle the question of uniqueness of solutions to generalized differential equations involving subgradients in place of partial derivatives an example of such an equation that well deserves study is the extended hamilton jacobi equation used in optimal control see clarke 2 let us also mention that a problem similar to the integration problem is the one of determining the set valued mappings that are in fact subgradient set valued mappings uniqueness is not mandatory for a contribution to this problem see janin 8 before we look at some of the known cases where the function can be recovered from its subgradients let us look at some negative examples it is clear that not every function can be recovered up to an additive constant from its subgradients we only need to look at the following two functions 0 x50 0 x10 f x 1 x o g x 2 x o

a classical approach to cox s test for non nested hypotheses abstract this paper argues that in econometrics the outcome of a significance testshould be carefully interpreted cox s statistic for non nested hypotheses is then derived as a lagrange multiplier test hence the cox procedure can also be viewed as a classical test

subsmooth sets functional characterizations and related concepts prox regularity of a set poliquin rockafellar thibault 2000 or its global version proximal smoothness clarke stern wolenski 1995 plays an important role in variational analysis not only because it is associated with some fundamental properties as the local continuous differentiability of the function dist c or the local uniqueness of the projection mapping but also because in the case where c is the epigraph of a locally lipschitz function it is equivalent to the weak convexity lower c 2 property of the function in this paper we provide an adapted geometrical concept called sub smoothness which permits an epigraphic characterization of the approximate convex functions or lower c 1 property subsmooth sets turn out to be naturally situated between the classes of prox regular and of nearly radial sets this latter class has been recently introduced by lewis in 2002 we hereby relate it to the mifflin semismooth functions

monte carlo methodology and the finite sample properties of instrumental variables statistics for testing nested and non nested hypotheses using monte carlo methodology this paper investigates the effect of dynamics and simultaneity on the finite sample properties of instrumental variables statistics for testing nested and nonnested hypotheses simple numerical analytical formulae response surfaces are obtained which closely approximate the statistics unknown size and power functions for a dynamic simultaneous equations model the analysis illustrates the value and limitations of asymptotic theory in interpreting finite sample properties two practical results arise the f form and the wald statistic is favored over its chi squared form and large sigma and small effective sample size strongly affect the test of over identifying restrictions and the cox type test copyright 1991 by the econometric society

prox regular sets and epigraphs in uniformly convex banach spaces various regularities and other properties we continue the study of prox regular sets that we began in a previous work in the setting of uniformly convex banach spaces endowed with a norm both uniformly smooth and uniformly convex e g l p w m p spaces we prove normal and tangential regularity properties for these sets and in particular the equality between mordukhovich and proximal normal cones we also compare in this setting the proximal normal cone with different holderian normal cones depending on the power types s q of moduli of smoothness and convexity of the norm in the case of sets that are epigraphs of functions we show that j primal lower regular functions have prox regular epigraphs and we compare these functions with poliquin s primal lower nice functions depending on the power types s q of the moduli the preservation of prox regularity of the intersection of finitely many sets and of the inverse image is obtained under a calmness assumption a conical derivative formula for the metric projection mapping of prox regular sets is also established among other results of the paper it is proved that the attouch wets convergence preserves the uniform r prox regularity property and that the metric projection mapping is in some sense continuous with respect to this convergence for such sets

a comparison of nonnested tests for misspecified models using the method of approximate slopes abstract comparisons of the cox wald and j tests are conducted under a standard framework and in a misspecified setting the data generation process includes a regressor that is not in either model most power comparisons have been carried out for local alternatives but comparisons under a nonlocal alternatives framework using the method of approximate slopes developed by bahadur 1960 1967 are made in this paper monte carlo experiments are conducted and the cox test is shown to have the highest small sample power and approximate slope for most of the cases considered the results from response surface regressions provide strong evidence that the approximate slope is a good predictor of the small power of nonnested tests it is also useful for directly comparing the small sample power of one degree of freedom tests of nonnested hypotheses

goodness of fit statistics for age specific reference intervals the age specific reference interval is a commonly used screening tool in medicine it involves estimation of extreme quantile curves such as the 5th and 95th centiles of a reference distribution of clinically normal individuals it is crucial that models used to estimate such intervals fit the data extremely well however few procedures to assess goodness of fit have been proposed in the literature and even fewer have been evaluated systematically here we consider procedures based on the distribution of the z scores standardized residuals from a model and on pearson chi 2 statistics for observed and expected counts in groups defined by age and the estimated reference centile curves two of the procedures q and grid tests are mainly inferential whereas the third permutation bands and b tests is essentially graphical we obtain approximations to the null distributions of several relevant test statistics and examine their size and power for a range of models based on real data sets we recommend q tests in all situations where z scores are available since they are general simple to calculate and usually have the highest power among the three classes of test considered for the cases considered the grid tests are always inferior to the q and b tests

regularity conditions for cox s test of non nested hypotheses abstract in this article we provide for the first time general regularity conditions and a rigorous proof of the asymptotic normality of cox s statistic for testing separate families of hypotheses the cox test for choosing between competing linear regression models is discussed as an example

statistical inference in non nested econometric models the purpose of this paper is to discuss some procedures that are available for testing non nested or separate hypotheses in the statistics and econometrics literature since many of these techniques may also be exploited in other disciplines it is hoped that an elaboration of the principal theoretical findings may make them more readily accessible to researchers in other disciplines several simple examples are used to illustrate the concepts of nested and non nested hypotheses and within the latter category global and partial non nested hypotheses two alternative methods of testing non nested hypotheses are discussed and contrasted the first of these is cox s modification of the likelihood ratio statistic and the second is atkinson s comprehensive model approach a major emphasis is placed on the role of the cox principle of hypothesis testing which enables a broad range of hypotheses to be tested within the same framework the problem associated with the application of the comprehensive model approach to composite non nested hypotheses is also highlighted roy s union intersection principle is presented as a viable method of dealing with this problem simulation results concerning the finite sample properties of various tests are discussed together with an analysis of some attempts to correct the poor size of the cox and related tests

statistical analysis of the main parameters in the fuzzy inference process abstract as there are many possibilities to select the set of basic operators used in the fuzzy inference process the search for the fuzzy operators that are most suitable for the different steps of a fuzzy system their characterization and evaluation can be included among the most important topics in the field of fuzzy logic a better insight into the performances of the alternative operators would make it easier to develop a fuzzy application in the present paper the relevancy and relative importance of the operators involved in the fuzzy inference process are investigated by using a powerful statistical tool the analysis of the variance anova 8 the results obtained show that the defuzzifier and the t norm operator are the most relevant factors in the fuzzy inference process moreover this statistical analysis is able to establish a classification of the defuzzifiers and t norms according to their intrinsic characteristics the conclusions here obtained justify the present interest observed in many current papers in studying both operators 6 22 24 33 64 67 futhermore our results are confirmed by some experiments dealing with a real control application

infimal convolution and optimal time control problem iii minimal time projection set we continue the investigation of a general minimal time problem with a convex constant dynamics and a lower semicontinuous extended real valued target function defined on a banach space in this paper we obtain an explicit description of the minimal time projection set in terms of the legendre fenchel conjugate function and provide some sufficient conditions for this set to be nonempty the single valuedness and lipschitz property of the minimal time projection are also investigated

subdifferentials of a minimal time function in normed spaces abstract in a general normed vector space we study the minimal time function determined by a differential inclusion where the set valued mapping involved has constant values of a bounded closed convex set u and by a closed target set s we show that proximal and frechet subdifferentials of a minimal time function are representable by virtue of corresponding normal cones of sublevel sets of the function and level or suplevel sets of the support function of u the known results in the literature require the set u to have the origin as an interior point or u be compact in particular if the set u is the unit closed ball the results obtained reduce to the subdifferential of the distance function defined by s

proximal analysis in reflexive smooth banach spaces abstract in the present paper we introduce and study a new proximal normal cone in reflexive banach spaces in terms of a generalized projection operator two new variants of generalized proximal subdifferentials are also introduced in reflexive smooth banach spaces the density theorem for both proximal subdifferentials has been proved in p uniformly convex and q uniformly smooth banach spaces various important properties and applications of our concepts are also proved

uniqueness and frechet differentiability of functional solutions to maximum likelihood type equations solutions of simultaneous equations of the maximum likelihood type or m estimators can be represented as functionals existence and uniqueness of a root in a local region of the parameter space are proved under conditions that are easy to check if only one root of the equation exists the resulting statistical functional is frechet differentiable and robust when several solutions exists conditions on the loss criterion used to select the root for the statistic ensure frechet differentiability an interesting example of a frechet differentiable functional is the solution of the maximum likelihood equations for location and scale parameters in a cauchy distribution the estimator is robust and asymptotically efficient

a note on regularity of sets and of distance functions in banach space abstract we show that in every infinite dimensional banach space there is a compact set c and x in c such that c is regular at x in the sense of nonsmooth analysis but such that the associated distance function d c is not regular at x and neither is d c

tests of non nested regression models small sample adjustments and monte carlo evidence abstract it is known that the small sample significance levels of cox type tests of non nested regression models can be much greater than the nominal level adjustments designed to overcome this problem are discussed and two tests are proposed monte carlo evidence on the performance of the tests derived in this paper the davidson mackinnon j test and the fisher mcaleer test is presented the f test applied to the comprehensive model is also included in the simulation experiments

the encompassing principle and its application to testing nonnested hypotheses the encompassing principle is used to develop a testing framework which unifies the literature on non nested testing allowing analysis of the relationship between alternative tests and in particular enabling asymptotic and finite sample equivalences and identities to be established easily when they exist as well as embracing nested hypothesis testing the concept of the implicit null hypothesis of a test is introduced to show that the effective acceptance region for some tests extends beyond the acceptance region corresponding to the null of interest and so such tests can be inconsistent against fixed alternatives closely related to the nominal null and alternative the analysis is illustrated by an application to two non nested linear regression models and it is shown that the conventional f test as well as all the one degree of freedom non nested tests has an encompassing interpretation and that the f test is a complete parametric encompassing test

testing nested or non nested hypotheses this paper presents a test procedure for nested or non nested hypotheses the test statistic is based on the difference between two estimators of the pseudo true value as defined for instance by sawa this statistic is similar to the usual wald statistic in the case of nested hypotheses and it can be replaced by an asymptotically equivalent one deduced from the score function

weak regularity of functions and sets in asplund spaces abstract in this paper we study a new concept of weak regularity of functions and sets in asplund spaces we show that this notion includes prox regular functions functions whose subdifferential is weakly submonotone and amenable functions in infinite dimension we establish also that weak regularity is equivalent to mordukhovich regularity in finite dimension finally we give characterizations of the weak regularity of epi lipschitzian sets in terms of their local representations

limiting subdifferentials of perturbed distance functions in banach spaces abstract we explore in arbitrary banach spaces the frechet type e subdifferentials and the limiting subdifferentials for the perturbed distance function d s j determined by a closed subset s and a lower semicontinuous function j defined on s in particular upper and lower estimates for the frechet type e subdifferentials and for the limiting subdifferentials are provided in terms of the corresponding subdifferentials of the sum of the associated functions j and s

a robust test for non nested hypotheses we propose a robust version of cox type test statistics for the choice between two non nested hypotheses we first show that the influence of small amounts of contamination in the data on the test decision can be very large secondly we build a robust test statistic by using the results on robust parametric tests that are available in the literature and show that the level of the robust test is stable finally we show numerically not only the robustness of this new test statistic but also that its asymptotic distribution is a good approximation of its sample distribution unlike for the classical test statistic we apply our results to the choice between a pareto and an exponential distribution as well as between two competing regressors in the simple linear regression model without intercept

linear regularity and ϕ regularity of nonconvex sets in this paper we discuss some sufficient conditions for the linear regularity and bounded linear regularity and their variations of finitely many closed not necessarily convex sets in a normed vector space the accompanying necessary conditions are also given in the setting of asplund spaces

bv solutions of nonconvex sweeping process differential inclusion with perturbation abstract this paper is devoted to the study of differential inclusions particularly discontinuous perturbed sweeping processes in the infinite dimensional setting on the one hand the sets involved are assumed to be prox regular and to have a variation given by a function which is of bounded variation and right continuous on the other hand the perturbation satisfies a linear growth condition with respect to a fixed compact subset finally the case where the sets move in an absolutely continuous way is recovered as a consequence

can approximate reasoning be consistent abstract a fuzzy expert system with one block of rules is said to be consistent if when the input data exactly matches the antecedent of a certain rule then the final conclusion of the fuzzy expert system exactly matches the consequence of that rule we investigate the two types of approximate reasoning 1 first infer then aggregate fita and 2 first aggregate then infer fati we argue that fita can be consistent but fati is in general not consistent

softness sleekness and regularity properties in nonsmooth analysis abstract we study continuity properties of tangent and normal cones to a closed subset of a banach space such a study can be seen as a nonsmooth set theoretic analogue of the study of continuously differentiable functions we give particular attention to the case where different concepts coincide a desirable feature such properties are able to offer we identify a wide class of sets and functions for which this property occurs

a new method for testing separate families of hypotheses abstract theoretical and empirical evidence are reported showing that the tests given in cox 1961 and williams 1970a b for the problem of testing separate families of hypotheses suffer from fundamental difficulties neither test is guaranteed to be asymptotically level a even for very smooth problems a new method proposed here as an alternative works adaptively by repeated application of the parametric bootstrap idea over slowly shrinking confidence regions theoretical properties of the method are derived and its finite sample behavior is studied in a number of examples

a numerical scheme for a class of sweeping processes the aim of this paper is to study a whole class of first order differential inclusions which fit into the framework of perturbed sweeping process by uniformly prox regular sets after obtaining well posedness results we propose a numerical scheme based on a prediction correction algorithm and we prove its convergence finally we apply these results to a problem coming from the modelling of crowd motion

alternative procedures and associated tests of significance for non nested hypotheses this paper attempts to synthesize various procedures for testing non nested hypotheses within the framework of artificial nesting and establishes the result that different tests correspond to different treatments of its identification problem numerical and a priori identification methods are distinguished we derive two modified likelihood ratio tests that are asymptotically equivalent to the cox test and explain the relationship between them the tests are consistent this abstract was borrowed from another version of this item

subdifferentiation of integral functionals we examine how the subdifferentials of nonconvex integral functionals can be deduced from the subdifferentials of the corresponding integrand or at least be estimated with the help of them in fact assuming some regularity properties of the integrands we obtain exact expressions for the subdifferentials of the integral functionals we draw some consequences in terms of duality for such integral functionals extending in this way the early work of rockafellar to the nonconvex case

local differentiability of distance functions recently clarke stern and wolenski characterized in a hilbert space the closed subsets c for which the distance function dc is continuously differentiable everywhere on an open tube of uniform thickness around c here a corresponding local theory is developed for the property of dc being continuously differentiable outside of c on some neighborhood of a point x c this is shown to be equivalent to the prox regularity of c at x which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation additional characterizations are provided in terms of dc being locally of class c 1 or such that dc 2 is convex around x for some 0 prox regularity of c at x corresponds further to the normal cone mapping nc having a hypomonotone truncation around x and leads to a formula for pc by way of nc the local theory also yields new insights on the global level of the clarke stern wolenski results and on a property of sets introduced by shapiro as well as on the concept of sets with positive reach considered by federer in the finite dimensional setting

sweeping process with regular and nonregular sets abstract differential inclusions involving the normal cone to a moving set are investigated a special attention is paid to the sweeping process associated with sets for which no regularity assumption is required

fuzzy implication in fuzzy systems control abstract different interpretations of fuzzy conditional statements in decision making for fuzzy systems control are considered some of their attributes which embarrass their application in this area are analysed a list of demands for fuzzy implication and a compositional rule of inference is suggested it is proposed to apply mamdani s fuzzy implication with mizumoto s composition some properties are proved and an example is discussed