on the ideal theory of graphs we study algebras defined by finite sets g m1 mq of monomials of a polynomial ring r there are two basic algebras i k g k m1 mq the k subalgebra of r spanned by the mi and ii the quotient ring r i g where i g m1 mq they come together in the construction of the rees algebra r i g of the ideal i g the emphasis is almost entirely on sets of squarefree monomials of degree two and their attached graphs the main results are assertions about the cohen macaulay behaviour of the koszul homology of i g and how normality or cohen macaulayness of one of the algebras can be read off the properties of the graph or in the other algebra

symbolic powers of sums of ideals let i and j be nonzero ideals in two noetherian algebras a and b over a field k let i j denote the ideal generated by i and j in a otimes k b we prove the following expansion for the symbolic powers begin aligned i j n sum i j n i i j j end aligned if a and b are polynomial rings and if mathrm char k 0 or if i and j are monomial ideals we give exact formulas for the depth and the castelnuovo mumford regularity of i j n which depend on the interplay between the symbolic powers of i and j the proof involves a result of independent interest which states that the induced map mathrm tor i a k i n rightarrow mathrm tor i r k i n 1 is zero for any homogeneous ideal i and i ge 0 n ge 0 we also investigate other properties and invariants of i j n such as the equality between ordinary and symbolic powers the waldschmidt constant and the cohen macaulayness

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

a test of separate families of distributions based on the empirical moment generating function summary in testing an hypothesis that a population has a specified distribution against the alternative that its distribution belongs to a separate family it is well known that the likelihood ratio test does not apply we propose a test based on the difference between the sample moment generating function and its theoretical counterpart under ho the argument of the moment generating function is chosen to maximize the test s largesample power small sample properties of the test in two applications are studied by simulation

combinatorial symbolic powers abstract symbolic powers are studied in the combinatorial context of monomial ideals when the ideals are generated by quadratic squarefree monomials the generators of the symbolic powers are obstructions to vertex covering in the associated graph and its blowups as a result perfect graphs play an important role in the theory dual to the role played by perfect graphs in the theory of secants of monomial ideals we use grobner degenerations as a tool to reduce questions about symbolic powers of arbitrary ideals to the monomial case among the applications are a new unified approach to the grobner bases of symbolic powers of determinantal and pfaffian ideals

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

a return to an old paper tests of separate families of hypotheses summary two papers of 50 years ago concerning the comparison of non nested models are reviewed and some subsequent developments and problems are sketched

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

the strong perfect graph theorem a graph g is perfect if for every induced subgraph h the chromatic number of h equals the size of the largest complete subgraph of h and g is berge if no induced subgraph of g is an odd cycle of length at least five or the complement of one the strong perfect graph conjecture berge 1961 asserts that a graph is perfect if and only if it is berge a stronger conjecture was made recently by conforti cornu ejols and vuiskovi c that every berge graph either falls into one of a few basic classes or admits one of a few kinds of separation designed so that a minimum counterexample to berge s conjecture cannot have either of these properties in this paper we prove both of these conjectures

sobolev s spaces on time scales and its applications to a class of second order hamiltonian systems on time scales abstract in this paper we study the sobolev s spaces on time scales and their properties as applications we present a recent approach via variational methods and the critical point theory to obtain the existence of solutions for a class of second order hamiltonian systems on time scales by establishing a proper variational setting three existence results for systems under consideration are obtained finally three examples are presented to illustrate the feasibility and effectiveness of the existence results

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

a simulation approach to the problem of computing cox s statistic for testing nonnested models abstract this paper proposes a new procedure for computing the cox statistic for tests of nonnested hypotheses using the method of stochastic simulation the procedure is applicable to a wide class of probability distributions is relatively simple to implement and does not require an analytic derivation of the pseudo true estimators that enter the cox test statistic the paper also contains an application of the proposed method to the test of probit versus logistic formulations of the univariate binary choice models the empirical results show that the simulation method works reasonably well even for a moderate number of replications

existence results for dynamic sturm liouville boundary value problems via variational methods abstract several conditions ensuring existence of solutions of a dynamic sturm liouville boundary value problem are derived variational methods are utilized in the proofs an example illustrating the main results is given

bahadur efficiency of tests of separate hypotheses and adaptive test statistics the notion of separability of hypotheses was first introduced by cox 1961 two families of hypotheses are separate if no distribution in one family can be obtained as a limit of distributions from the other family for testing separate hypotheses cox 1961 1962 suggested the test based on the comparison of the logarithm of the likelihood ratio with its expected value or estimate thereof under each hypothesis the tests of such hypotheses are required to have high power against the specified alternatives when nuisance parameters are present this concept leads to the notion of an adaptive test which by definition must be asymptotically efficient for any value of the unknown nuisance parameter a testing problem of separate hypotheses for an exponential family is studied from the standpoint of bahadur asymptotic optimality a formula for the bahadur exact slope of any smooth test statistic is obtained and in the example of testing lognormality versus exponentiality it is shown that cox s test can

symbolic powers of monomial ideals and vertex cover algebras abstract we introduce and study vertex cover algebras of weighted simplicial complexes these algebras are special classes of symbolic rees algebras we show that symbolic rees algebras of monomial ideals are finitely generated and that such an algebra is normal and cohen macaulay if the monomial ideal is squarefree for a simple graph the vertex cover algebra is generated by elements of degree 2 and it is standard graded if and only if the graph is bipartite we also give a general upper bound for the maximal degree of the generators of vertex cover algebras

detection of dependent heavy tailed signals this paper examines the problem of detection of dependent alpha stable signals measurements of several phenomena exhibit non gaussian heavy tailed behavior in their probability density functions p d f we use the class of alpha stable distributions to characterize these signals when two sensors make simultaneous measurements of such phenomena these heavy tailed realizations are dependent across sensors the intersensor dependence is modeled using copulas we consider a two sided test in the neyman pearson framework and present an asymptotic analysis of the generalized likelihood test glrt both nested and non nested models are considered in the analysis the performance of the proposed scheme is evaluated numerically on simulated data as well as indoor seismic data with appropriately selected models our results demonstrate that a high probability of detection can be achieved for false alarm probabilities of the order of 10 4

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

weak maximum principle and accessory problem for control problems on time scales abstract in this paper we derive the first and second variations for a nonlinear time scale optimal control problem with control and state endpoints equality constraints using the first variation a first order necessary condition for weak local optimality is obtained under the form of a weak maximum principle generalizing the dubois reymond lemma to the optimal control setting and time scales a second order necessary condition in terms of the accessory problem is derived by using the nonnegativity of the second variation at all admissible directions the control problem is studied under a controllability assumption and with or without the shift in the state variable these two forms of the problem are shown to be equivalent

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

non nested testing of spatial correlation we develop non nested tests in a general spatial spatio temporal or panel data context the spatial aspect can be interpreted quite generally in either a geographical sense or employing notions of economic distance or when parametric modelling arises in part from a common factor or other structure in the former case observations may be regularly spaced across one or more dimensions as is typical with much spatio temporal data or irregularly spaced across all dimensions both isotropic models and non isotropic models can be considered and a wide variety of correlation structures in the second case models involving spatial weight matrices are covered such as spatial autoregressive models the setting is sufficiently general to potentially cover other parametric structures such as certain factor models and vector valued observations and here our preliminary asymptotic theory for parameter estimates is of some independent value the test statistic is based on a gaussian pseudo likelihood ratio and is shown to have an asymptotic standard normal distribution under the null hypothesis that one of the two models is correct this limit theory rests strongly on a central limit theorem for the gaussian pseudo maximum likelihood parameter estimates a small monte carlo study of finite sample performance is included

selecting the best linear regression model a classical approach in this paper we apply the model selection approach based on likelihood ratio lr tests developed in vuong 1985 to the problem of choosing between two normal linear regression models which are not nested in each other first we compare our model selection procedure to other model selection criteria then we explicitly derive the procedure when the competing linear models are non nested and neither one is correctly specified some simplifications are seen to arise when both models are contained in a larger correctly specified linear regression model or when at least one competing linear model is correctly specified a comparison of our model selection tests and previous non nested hypothesis tests concludes the paper

tests of non nested regression models some results on small sample behaviour and the bootstrap the most popular procedures for testing a regression model against a single non nested alternative can be substantially oversized in small samples also when a regression model is to be tested in the presence of several non nested alternatives the null model is sometimes accepted only if testing against each alternative in turn produces no significant outcomes this approach leads to an implicit overall test with a significance level that is not known even asymptotically it is shown that the bootstrap can be used to control significance levels in both types of situation power estimates for various tests using bootstrap critical values are also obtained and compared

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

the approximate slopes of econometric tests in this paper the concept of approximate slope introduced by r r bahadur is used to make asymptotic global power comparisons of econometric tests the approximate slope of a test is the rate at which the logarithm of the asymptotic marginal significance level of the test decreases as sample size increases under a given alternative a test with greater approximate slope may therefore be expected to reject the null hypothesis more frequently under that alternative than one with smaller approximate slope two theorems which facilitate the computation and interpretation of the approximate slopes of most econometric tests are established these results are used to undertake some illustrative comparisons sampling experiments and an empirical illustration suggest that the comparison of approximate slopes may provide an adequate basis for evaluating the actual performance of alternative tests of the same hypothesis

a boundary value problem for fourth order elastic beam equations multiplicity results for a fourth order nonlinear boundary value problem are presented the proof of the main result is based on the critical point theory

asymptotic power comparisons of tests of separate parametric families by bahadur s approach summary the paper investigates asymptotic power comparisons of tests of separate parametric families when the size of the test is allowed to tend to zero as the sample size tends to infinity it shows that bahadur s method of asymptotic comparison by means of the approximate slope of the tests can be extended to the case of separate parametric families the paper contains a number of applications

symbolic powers of edge ideals abstract let i r k x k x 1 x n be an ideal in a polynomial ring over the field k we define the essential symbolic module of i to be the r i module f i r 2 i r r i where r i i 1 r 1 i i i r i and i m stands for the mth symbolic power of i we will mainly focus on the case where i is generated by square free monomials of degree two among our main results are optimal bounds for the degrees of the minimal generators of f i several criteria for a monomial to be such a generator and an upper bound for the generation type of the symbolic rees algebra of i as a byproduct we recapture the result of simis vasconcelos and villarreal on when such an ideal is normally torsionfree

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 critical point approach for a second order dynamic sturm liouville boundary value problem with p laplacian abstract in this paper we give conditions guaranteeing the existence of at least three solutions for a second order dynamic sturm liouville boundary value problem involving two parameters in the proofs of the results we utilize critical point theory and variational methods in addition an example is given in order to illustrate our results

a systematic look at the gamma process capability indices abstract process capability indices pcis are widely used to measure whether an in control process conforms to manufacturing specifications the normal distribution is assumed in most traditional applications of pcis nevertheless it is not uncommon that some quality characteristics have skewed distributions in such cases the gamma distribution is an appropriate model and percentile based pcis for the gamma process have been studied in the literature in practical applications of pcis it is important to select an appropriate distribution between the normal and the gamma distributions based on historical data in this study we first construct a hypothesis test for model discrimination between the normal and the gamma distributions asymptotic distribution of the test statistic under the gamma process is derived we then consider statistical inference for the percentile based pcis under the gamma process the maximum likelihood method is used for point estimation and the method of generalized pivotal quantities is used for interval estimation we demonstrate the proposed methods by a practical example

straightening law and powers of determinantal ideals of hankel matrices in this paper we establish a standard monomial theory for generic hankel matrices by a generic hankel matrix we mean a matrix y yij with yij x i j 1 where the x i are indeterminates over a field k we use this structure to determine the symbolic powers and the primary decomposition of the powers of the determinantal ideals it of y further we prove that the symbolic and ordinary rees algebras of it are cohen macaulay normal domains the first standard monomial theory was developed by hodge h to study the homogeneous coordinate ring of the grassmannian variety later standard monomial theories were established for generic matrices by doubilet rota and stein drs and for generic symmetric and generic skew symmetric matrices by de concini and procesi dp these are all examples of algebras with straightening law asl for short over a poset or over a doset the abstract notion of asl was introduced and developed by eisenbud e1 and by de concini eisenbud and procesi dep2 see also bv these structures turned out to be an extremely powerful tool in studying determinantal rings and ideals arising from the above mentioned generic matrices let now x1 xn be indeterminates over an arbitrary field k for j 1 n we denote by xj the j n 1 j hankel matrix with entries x1 xn that is

non differentiable functionals and applications to elliptic problems with discontinuous nonlinearities abstract multiple critical points theorems for non differentiable functionals are established applications both to elliptic variational hemivariational inequalities and eigenvalue problems with discontinuous nonlinearities are then presented

quantifying singularities with differential operators abstract the f signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties for example this invariant detects non singularity and strong f regularity however it is very difficult to compute motivated by different aspects of the f signature we define a numerical invariant for rings of characteristic zero or p 0 that exhibits many of the useful properties of the f signature we also compute many examples of this invariant including cases where the f signature is not known we also obtain a number of results on symbolic powers and bernstein sato polynomials

sturm liouville eigenvalue problems on time scales for sturm liouville eigenvalue problems on time scales with separated boundary conditions we give an oscillation theorem and establish rayleigh s principle our results not only unifly the corresponding theories for differential and difference equations but are also new in the discrete case

exact tests of a model against nonnested alternatives the results of milliken graybill 1970 are used to develop a general procedure for obtaining exact tests of a linear model against several nonnested and nonlinear alternatives simultaneously the procedure may be extended to test a null model against alternatives in which the dependent variable is subjected to different data transformations arguments are presented in favour of multiple tests

a nonnested approach to testing continuous time models against discrete alternatives abstract this paper considers a test of two competing models when one is formulated in continuous time as a system of stochastic differential equations and the other is formulated in discrete time as a system of nonlinear equations since the competing hypotheses are nonnested the appropriate formulae are derived for the implementation of the cox test this approach is illustrated with an application to the demand for food and related products in the united kingdom