on the solvability of periodic boundary value problems with impulse in this work we obtain some new results concerning the existence of solutions to an impulsive first order nonlinear ordinary differential equation with periodic boundary conditions the ideas involve differential inequalities and schaefer s fixed point theorem

positive solutions of periodic boundary value problems for nonlinear first order impulsive differential equations abstract motivated by liu y liu further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations j math anal appl 327 2007 435 452 li et al x li x lin d jiang x zhang existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects nonlinear anal 62 2005 683 701 and liu and ge y liu w ge stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients nonlinear anal 57 2004 363 399 this paper is concerned with the periodic boundary value problem pbvp for short for the nonlinear impulsive differential equation x t a t x t f t x t a e t 0 t t 1 t p x t k x t k x t k i k x t k k 1 p x t x 0 we obtain new sufficient conditions for the existence of at least one two or three positive solutions of the above pbvp respectively examples are presented to illustrate the main results

a size correction to the lagrange multiplier test for heteroskedasticity abstract making use by the recent work by harris this paper provides the formula for size correction to the lagrange multiplier test for heteroskedasticity monte carlo experiments indicate that our size correction is effective in improving the accuracy of the size of the test since the lagrange multiplier test tends to have a true size less than the nominal size in our experiments the application of the test at a higher true significance level tends to increase the power of the test as well reasonable results in our numerical experiments also give a partial check on the validity of the derived formula

power and robustness of jackknife and likelihood ratio tests for grouped heteroscedasticity abstract the jackknife technique is used to develop robust tests for heteroscedasticity in regression analysis consistent with time series cross sectional studies the variance is assumed constant within subgroups of observations but may vary from group to group we also consider the likelihood ratio lr test and develop a number of modified lr statistics similar to those of bartlett each test is developed in the context of regression where the vector of coefficients may be constant for all groups or may vary between groups a number of sampling experiments reveal that if the disturbance terms are normally distributed bartlett s test is slightly more powerful than either of the jackknife tests however in nonnormal situations jack2 the test based on jackknifing the regression is very robust under h 0 and has substantial power against a number of alternative hypotheses

variational methods to sturm liouville boundary value problem for impulsive differential equations abstract in this paper we study a linear sturm liouville impulsive problem and a nonlinear sturm liouville impulsive problem by applying a new approach via critical point theory and variational methods the existence results of positive solutions are obtained

the statistical foundations of a class of parametric tests for heteroscedasticity abstract in this paper we examine the statistical foundations of the five tests for heteroscedasticity considered by griffiths and surekha 1986 we identify the precise specification of the alternative hypothesis as an important determinant of the power of the tests

testing multiple dispersion effects in unreplicated fractional factorial designs in unreplicated 2k p designs the assumption of constant variance is commonly made when the variance of the response differs between the two levels of a column in the effect matrix that column produces a dispersion effect in this article we show that two active dispersion effects may create a spurious dispersion effect in their interaction column most existing methods for dispersion effect testing in unreplicated fractional factorial designs are subject to these spurious effects we propose a method of dispersion effect testing based on geometric means of residual sample variances we show through examples from the literature and simulations that the proposed test has many desirable properties that are lacking in other tests

marginal likelihood score based tests of regression disturbances in the presence of nuisance parameters abstract this paper is concerned with tests of the covariance matrix of the disturbances in the linear regression model that involves nuisance parameters which cannot be eliminated by the usual invariance arguments score based tests namely lagrange multiplier lm and locally most mean powerful lmmp tests are derived from the marginal likelihood applications considered include i testing for random regression coefficients ii testing for second order autoregressive ar 2 disturbances and iii testing for arma 1 1 disturbances each in the presence of ar 1 disturbances an empirical size and power comparison shows that the new tests typically have more accurate asymptotic critical values and slightly more power than their respective conventional counterparts

further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations abstract this paper is concerned with the solvability of periodic boundary value problems for nonlinear impulsive functional differential equations x t a t x t f t x t x 1 t x n t t j t 1 t p x t k x t k i k x t k k 1 2 p x 0 x t we obtain sufficient conditions for the existence of at least one solution of problem at resonance and nonresonance cases respectively examples are presented to illustrate the main results

signal to noise ratios performance criteria and transformations for the analysis of designed experiments taguchi uses performance criteria that he calls signal to noise sn ratios three such criteria are here denoted by sn t sn l and sn s the criterion sn t was to be used in preference to the standard deviation for the problem of achieving for some quality characteristic y the smallest mean squared error about an operating target value leon shoemaker and kacker 1987 showed how sn t was appropriate to solve this problem only when y was proportional to y on that assumption the same result could be obtained more simply by conducting the analysis in terms of log y rather than y a more general transformation approach is here introduced for other commonly met kinds of dependence between y and y including no dependence and a lambda plot is presented that uses the data to suggest an appropriate transformation the criteria sn l and sn s were for problems in which the objective was to make the response as large or as small as possible it is arg

a monte carlo evaluation of the power of some tests for heteroscedasticity abstract szroeter s asymptotically normal test outperforms the goldfeld qu2ndt test the breusch pagan lagrange multiplier test and bamset when it is possible to order the observations according to increasing variance with no prior information on variance ordering bamset is best some observations concerning degree of heteroscedasticity and model specification are made

a further class of tests for heteroscedasticity abstract this paper considers the problem of testing for heteroscedasticity in the linear regression model when one is willing to postulate only the ranking of the disturbance variances under the alternative hypothesis a new class of tests is proposed and methods of finding the tests critical values are discussed an empirical power comparison with other tests that have found favor in the literature demonstrates the potential power advantage of these new tests and results in particular versions being recommended for practical use

the impact of unidentified location effects on dispersion effects identification from unreplicated factorial designs there have been several publications on identifying dispersion effects from an unreplicated fractional or full factorial experiment some of these methods implicitly assume that unknown location effects are first identified correctly from the data in practice however an empirical location effects identification procedure may leave some small to moderate location effects undetected it is shown in this study that such unidentified location effects can cumulatively impair subsequent dispersione effects identification numerical evidence based on a welding experiment and results from a simulation study are provided a mathematical explanation of the impact of unidentified location effects on dispersion effects identification is given the dispersion effects identification methods initially proposed for unreplicated data can be naturally extended to replicated experiments with the impact of unidentified location effects eliminated a fluorescent lamp experiment is analyzed to illustrate these methods

analyzing dispersion effects from replicated factorial experiments recent developments in quality engineering methods have led to considerable interest in the analysis of dispersion effects from designed experiments a commonly used method for identifying important dispersion effects from replicated experiments is based on least squares analysis of the logarithm of the within replication variance bartlett and kendall 1946 box and meyer 1986 introduced a pooling technique for unreplicated two level experiments we extend this to replicated two level experiments and compare its performance with the least squares analysis we show that both of these methods can be obtained as special cases of maximum likelihood estimation under normal theory the pooling technique is generally biased and is not recommended for model identification the least squares analysis performs well as a model identification tool but the estimators can be inefficient in such cases we recommend that the parameters of the identified submodel be estimated by maximum likelihood we derive some prop

multiplicity positive solutions to periodic problems for first order impulsive differential equations this paper is devoted to studying the existence of single and multiple positive solutions to periodic problems for first order impulsive differential equations in addition we apply our existence theorems to nonlinear periodic problems with singularity at the origin our proofs are based on krasnoselskii fixed point theorem in cones

boundary value problems for a class of impulsive functional equations this paper is related to the existence and approximation of solutions for impulsive functional differential equations with periodic boundary conditions we study the existence and approximation of extremal solutions to different types of functional differential equations with impulses at fixed times by the use of the monotone method some of the options included in this formulation are differential equations with maximum and integro differential equations in this paper we also prove that the lipschitzian character of the function which introduces the functional dependence in a differential equation is not a necessary condition for the development of the monotone iterative technique to obtain a solution and to approximate the extremal solutions to the equation in a given functional interval the corresponding results are established for the impulsive case the general formulation includes several types of functional dependence delay equations equations with maxima integro differential equations finally we consider the case of functional dependence which is given by nonincreasing and bounded functions

the monotone method for first order problems with linear and nonlinear boundary conditions in this paper we develop the monotone method for the first order problem u t f t u t for a e t a b when f is a caratheodory function and u w 1 1 a b we consider the nonlinear boundary conditions l u a u b 0 with l c r 2 r nondecreasing in x or nonincreasing in y and the linear boundary conditions a 0u 0 b 0u t l 0 with a 0 b 0 and l 0 r we prove the existence of solutions and the validity of the monotone method if there exists a lower solution a and an upper solution b with either a b for the linear conditions we obtain eight new concepts of lower and upper solutions which generalize previous known cases

diagnostics for heteroscedasticity in regression summary for the usual regression model without replication we provide a diagnostic test for heteroscedasticity based on the score statistic a graphical procedure to complement the score test is also presented some key u ords influence linear model residual score test diagnostic methods in linear regression are used to examine the appropriateness of assumptions underlying the modelling process and to locate unusual characteristics of the data that may influence conclusions the recent literature on diagnostics is dominated by studies of methods for the detection of influential observations cook weisberg 1982 provide a review diagnostics for the relevance of specific assumptions however have not received the same degree of attention even though these may be of equal importance our purpose here is to provide appropriate diagnostic techniques to aid in an assessment of the validity of the usual assumption of homoscedasticity when little or no replication is present available methods for studying this assumption include both graphical and nongraphical procedures the usual graphical procedure consists of plotting the ordinary least squares residuals against fitted values or an explanatory variable a megaphone shaped pattern is taken as evidence that the variance depends on the quantity plotted on tlle abscissa weisberg 1980 chapter 6 in 3 we suggest several ways in which this

joint estimation of location dispersion and random effects in robust design this article presents a methodology for simultaneously modeling three components of a general mixed model approach to robust design location fixed effects dispersion effects and random effects control noise and random factors can be accommodated with this approach parameters associated with all three are estimated jointly using residual maximum likelihood and assuming normality simulated and real datasets illustrate the key concepts and advantages over previously proposed approaches

stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients abstract we consider the following nonlinear impulsive delay differential equation with variable coefficients and forcing term x t a t x t f t x e t t 0 t t k x t k x t k i k x t k k n using new methods different from those in tang et al appl math comput 131 2002 373 nieto j comput appl math lett 15 2002 489 franco and nieto j comput appl math 88 1998 144 nieto j math anal appl 205 1997 423 he and yu j math anal appl 272 2002 67 we establish stability theorems for when e t 0 under the assumption that there is bk such that bkx2 x x ik x x2 for k n and x r theorems 2 1 2 4 the existence results for three positive periodic solutions and nonexistence results for periodic solution of when e t 0 under the assumption exp 0 t a u d u 0 t k 1 b k and bk 1 for all k theorems 3 1 3 2 and the existence results for periodic solution of at resonance which is caused by impulses i e exp 0 t a u d u 0 t k 1 b k under the assumption bk 1 for all k theorems 4 2 4 3 some results obtained improve and generalize the known theorems and some other results are new examples are presented to illustrate the main results

robustness of size of tests of autocorrelation and heteroscedasticity to nonnormality abstract a comprehensive empirical examination is made of the sensitivity of tests of disturbance covariance in the linear regression model to nonnormal disturbance behaviour tests of autocorrelation appear to be quite robust except for extreme nonnormality but tests for heteroscedasticity are highly susceptible to kurtosis

variance function estimation abstract heteroscedastic regression models are used in fields including economics engineering and the biological and physical sciences often the heteroscedasticity is modeled as a function of the covariates or the regression and other structural parameters standard asymptotic theory implies that how one estimates the variance function in particular the structural parameters has no effect on the first order properties of the regression parameter estimates there is evidence however both in practice and higher order theory to suggest that how one estimates the variance function does matter further in some settings estimation of the variance function is of independent interest or plays an important role in estimation of other quantities in this article we study variance function estimation in a unified way focusing on common methods proposed in the statistical and other literature to make both general observations and compare different estimation schemes we show that there are significant di

three solutions for a p laplacian boundary value problem with impulsive effects in this paper a p laplacian boundary value problem with impulsive effects is considered multiplicity of solutions is obtained by three critical points theorem an example is presented to illustrate main result

existence of solutions of boundary value problems for differential equations with delayed arguments we study the existence and approximation of solution of boundary value problems for differential equations with delayed arguments sufficient conditions are established for the existence of a unique solution or extremal ones of the given problem a monotone iterative technique is applied

simulation based finite sample tests for heteroskedasticity and arch effects tests for heteroskedasticity in linear regressions are typically based on asymptotic approximations we show that the size of such tests can be perfectly controlledin nite samples through monte carlo test techniques with both gaussian and non gaussian heavy tailed disturbance distributions the procedures studied include standard heteroskedasticity tests e g glejser bartlett cochran hartley breusch pagan godfrey white szroeter as well as tests for arch type heterosked asticity sup type andcombinedtests are also proposedto allow for unknown breakpoints in the variance the fact that the proposedproced ures achieve size control andhave goodpower is demonstrated in a monte carlo simulation c 2003 elsevier b v all rights reserved

dispersion effects from 7 unreplicated designs in the 2 k p series quality improvement requires reduction of undesired variation design of experiments haa become a useful tool for discovering such minimum variance conditions but most existing techniques require a large experimental effort involving replicated experiments in this article we develop a method for significance testing of dispersion effects from unreplicated two level fractional factorial experiments using principles close to those for identifying location effects the method provides an f distributed test statistic and appears to be useful for identification of dispersion effects as early as the screening stage when fractionated unreplicated experiments are frequently employed

methods for identifying dispersion effects in unreplicated factorial experiments a critical analysis and proposed strategies there has been considerable interest recently in the use of statistically designed experiments to identify both location and dispersion effects for quality improvement analysis of dispersion effects usually requires replications that can be expensive or time consuming several recent articles have considered identification of both location and dispersion effects from unreplicated fractional factorial experiments in this article we provide a systematic study of various methods that are commonly used or have been proposed recently both theoretical and simulation results are used to characterize the properties although all methods suffer from some degree of bias some have serious problems when the bias remains large even as the design run size increases to infinity based on these analyses we propose some iterative strategies for model selection and estimation of the dispersion effects a real example and simulations as well are used to illustrate the results

existence results for an impulsive abstract partial differential equation with state dependent delay in this paper we establish the existence of mild solutions for a class of impulsive abstract partial functional differential equation with state dependent delay

joint identification of location and dispersion effects in unreplicated two level factorials most procedures that have been proposed to identify dispersion effects in unreplicated factorial designs assume that location effects have been identified correctly incorrect identification of location effects may impair subsequent identification of dispersion effects we develop a method for joint identification of location and dispersion effects that can reliably identify active effects of both types a normal based model containing parameters for effects in both the mean and variance is used parameters are estimated using maximum likelihood and subsequent effect selection is done using a specially derived information criterion an exhaustive search through a limited version of the space of possible models is conducted both a single model output and model averaging are considered the method is shown to be capable of identifying sensible location dispersion models that are missed by methods that rely on sequential estimation of location and dispersion effects supplementary materials for this articl

a point optimal test for heteroscedastic disturbances abstract a new test for additive heteroscedasticity in the disturbances of the linear regression model is proposed power functions of various forms of the new test are compared empirically with those of currently favoured tests for a range of heteroscedastic models the results highlight the power advantage of a test which is mp1 at a central point in the alternative hypothesis parameter space the main conclusion is that the recommended version of the new test is generally more powerful than existing tests against medium and severe heteroscedasticity whereas the king and szroeter tests perform better against weak heteroscedasticity

a generalized linear modeling approach to robust design this article presents an extension of the response surface approach to taguchi s parameter design by accommodating generalized linear modeling the approach of myers khuri and vining is taken as a starting point treating noise variables as fixed effects it is however assumed that process variability is caused not only by the presence of noise factors but also by residual variation therefore in addition to a model with a linear link function that relates the process mean to design and noise factors a second model with a log link function is proposed that relates the variance to a linear predictor in the design factors the conditional response model with noise factors taken as fixed is fitted by iteratively reweighted least squares unconditionally response surfaces for the process mean and process variance are produced through the use of a mean and variance operator on the conditional response model as an illustration data from an injection molding experiment are reanalyzed and residual disp

a study of several new and existing tests for heteroscedasticity in the general linear model abstract several optimum non parametric tests for heteroscedasticity are proposed and studied along with the tests introduced in the literature in terms of power and robustness properties it is found that all tests are reasonably robust to the ordinary least squares ols residual estimates number and character of the regressors only a few are robust to both the distributional and independence assumptions about the errors the power of tests can be improved with the ols residual estimates the increased sample size and the variability of the regressors it can be substantially reduced if the observations are not normally distributed and may increase or decrease if the errors are dependent each test is optimum to detect a specific form of heteroscedasticity and a serious power loss may occur if the underlying heteroscedasticity assumption in the data generation deviates from it

maximum principles for periodic impulsive first order problems we present some comparison results for the periodic boundary value problem for first order ordinary differential equations with impulses at fixed moments then the upper and lower solution method and the monotone iterative scheme are presented