testing for additivity in nonparametric regression additive models are one means of assuaging the curse of dimensionality when nonparametric smoothing methods are used to estimate multivariable regression functions it is important to have methods for testing the fit of such models especially in high dimensions where visual assessment of fit becomes difficult new tests of additivity are proposed in this paper that derive from fourier series estimators with data driven smoothing parameters other tests related to the classical tukey test for additivity are also considered while the new tests are consistent against essentially any smooth alternative to additivity the tukey type tests are found to be inconsistent in certain situations asymptotic power of both varieties of tests is studied under local alternatives that tend toward additivity at a parametric rate and small sample power comparisons are carried out by means of a simulation study

a geometrical method for removing edge effects from kernel type nonparametric regression estimators abstract we introduce a simple geometric method for removing edge effects from kernel type nonparametric regression estimators it involves reflecting the data set in two estimated points thereby generating a new data set with three times the range of the original data the usual kernel type estimator may be applied to the new enlarged data set without any danger of edge effects this technique is applicable generally to both regularly spaced and randomly spaced designs and admits a natural analog of leave one out cross validation the new cross validation algorithm may be extended to the very ends of the design interval unlike its more conventional counterpart which must be downweighted at the ends of the interval to avoid edge effects

classification of phase transitions in reaction diffusion models equilibrium phase transitions are associated with rearrangements of minima of a lagrangian potential treatment of nonequilibrium systems requires doubling of degrees of freedom which may be often interpreted as a transition from the coordinate to the phase space representation as a result one has to deal with the hamiltonian formulation of the field theory instead of the lagrangian one we suggest a classification scheme of phase transitions in reaction diffusion models based on the topology of the phase portraits of corresponding hamiltonians in models with an absorbing state such a topology is fully determined by intersecting curves of zero energy we identify four families of topologically distinct classes of phase portraits stable upon renormalization group transformations

acorn a new method for generating sequences of uniformly distributed pseudo random numbers a new family of pseudo random number generators the acorn additive congruential random number generators is proposed the resulting numbers are distributed uniformly in the interval 0 1 the acorn generators are defined recursively and the k 1 th order generator is easily derived from the kth order generator some theorems concerning the period length are presented and compared with existing results for linear congruential generators a range of statistical tests are applied to the acorn generators and their performance is compared with that of the linear congruential generators and the chebyshev generators the tests show the acorn generators to be statistically superior to the chebyshev generators while being statistically similar to the linear congruential generators however the acorn generators execute faster than linear congruential generators for the same statistical faithfulness the main advantages of the acorn generator are speed of execution long period length and simplicity of coding

random number generators in genetic algorithms for unconstrained and constrained optimization abstract presented here is a genetic algorithm that computes an approximate solution to constrained and unconstrained global optimization problems this technique has been implemented using several pseudo and quasi random number generators and the results of several test examples are presented the performance of this technique is based on a ranked comparison of relative error

nonparametric inference for periodic sequences this article proposes a nonparametric method for estimating the period and values of a periodic sequence when the data are evenly spaced in time the period is estimated by a leave out one cycle version of cross validation cv and complements the periodogram a widely used tool for period estimation the cv method is computationally simple and implicitly penalizes multiples of the smallest period leading to a virtually consistent estimator of integer periods this estimator is investigated both theoretically and by simulation we also propose a nonparametric test of the null hypothesis that the data have constant mean against the alternative that the sequence of means is periodic finally our methodology is demonstrated on three well known time series the sunspots and lynx trapping data and the el nino series of sea surface temperatures

on model diagnostics using varying coefficient models summary the paper motivates the use of varying coefficient models for diagnostics in regression models with continuous and factorial covariates the varying coefficient model which is restricted only by assumptions concerning the smoothness of the effects is considered as the alternative to a parametric generalised linear model estimation is based on local likelihood smoothing and asymptotic properties of the estimators are derived for the investigation of the discrepancy between the parametric model and the smooth alternative a simple graphical procedure is proposed moreover tests are derived which allow for overall and componentwise investigation the componentwise tests may be used to examine interaction effects of continuous and factorial regressors

alternative goodness of fit tests for linear models fan and huang 2001 presented a goodness of fit test for linear models based on fourier transformations of the residuals of the fitted model we present two more theoretically appealing tests in which the fourier transforms are incorporated into a fitted model we show that when suitably normalized the new test statistics have the same asymptotic distribution as fan and huang s test we propose modifications to the asymptotic normalization constants to improve the small sample sizes of our tests while retaining their asymptotic distributions small sample sizes and powers are examined via simulations an illustration is given

the mean field scaling function of the universality class of absorbing phase transitions with a conserved field we consider two mean field like models which belong to the universality class of absorbing phase transitions with a conserved field in both cases we analytically derive the order parameter as a function of the control parameter and of an external field conjugated to the order parameter this allows us to calculate the universal scaling function of the mean field behaviour the obtained universal function is in perfect agreement with recently obtained numerical data of the corresponding five and six dimensional models showing that four is the upper critical dimension of this particular universality class

sieve empirical likelihood ratio tests for nonparametric functions generalized likelihood ratio statistics have been proposed in fan zhang and zhang ann statist 29 2001 153 193 as a generally applicable method for testing norparametic hypotheses about nonparametric functions the likelihood ratio statistics are constructed based on the assumption that the distributions of stochastic errors are in a certain parametric family we extend their work to the case where the error distribution is completely unspecified via newly proposed sieve empirical likelihood ratio selr tests the approach is also applied to test conditional estimating equations on the distributions of stochastic errors it is shown that the proposed selr statistics follow asymptotically resealed chi 2 distributions with the scale constants and the degrees of freedom being independent of the nuisance parameters this demonstrates that the wilks phenomenon observed in fan zhang and zhang ann statist 29 2001 153 193 continues to hold under more relaxed models and a larger class of techniques the asymptotic power of the proposed test is also derived which achieves the optimal rate for nonparametric hypothesis testing the proposed approach has two advantages over the generalized likelihood ratio method it requires one only to specify some conditional estimating equations rather than the entire distribution of the stochastic error and the procedure adapts automatically to the unknown error distribution including heteroscedasticity a simulation study is conducted to evaluate our proposed procedure empirically

testing generalized linear and semiparametric models against smooth alternatives we propose goodness of fit tests for testing generalized linear models and semiparametric regression models against smooth alternatives the focus is on models having both continuous and factorial covariates as smooth extension of a parametric or semiparametric model we use generalized varying coefficient models as proposed by hastie tibshirani jrss b 1993 a likelihood ratio statistic is used for testing and asymptotic normality of the test statistic is proven due to a slow asymptotic convergence rate a bootstrap approach is pursued asymptotic expansions allow to write the estimates as linear smoothers which in turn guarantees simple and fast bootstrapping the test is shown to have sqrt n power but in contrast to parametric tests it is powerful against smooth alternatives in general

the centro invertible matrix a new type of matrix arising in pseudo random number generation abstract this paper defines a new type of matrix which will be called a centro invertible matrix with the property that the inverse can be found by simply rotating all the elements of the matrix through 180 degrees about the mid point of the matrix centro invertible matrices have been demonstrated in a previous paper to arise in the analysis of a particular algorithm used for the generation of uniformly distributed pseudo random numbers an involutory matrix is one for which the square of the matrix is equal to the identity it is shown that there is a one to one correspondence between the centro invertible matrices and the involutory matrices when working in modular arithmetic this result allows all possible k by k centro invertible matrices with integer entries modulo m to be enumerated by drawing on existing theoretical results for involutory matrices consider the k by k matrices over the integers modulo m if m takes any specified finite integer value greater than or equal to two then there are only a finite number of such matrices and it is valid to consider the likelihood of such a matrix arising by chance it is possible to derive both exact expressions and order of magnitude estimates for the number of k by k centro invertible matrices that exist over the integers modulo m it is shown that order n of the n m k 2 different k by k matrices modulo m are centro invertible so that the proportion of these matrices that are centro invertible is order 1 n

bias robust estimation in finite populations using nonparametric calibration a standard problem in sample survey inference is that of predicting the finite population total h of a function h y of a random variable y the model based approach to this problem first defines a working model for y and then predicts h by estimating its expectation under conditional on the sample values of y this approach leads to biased predictions if is incorrect we explore an automatic solution to this misspecification bias that uses nonparametric regression to define a robust but inefficient predictor of h and then calibrates this predictor for its bias under an application to prediction of the finite population distribution function of a population of australian beef farms is presented

smoothing parameter selection for power optimality in testing of regression curves abstract we consider selection of smoothing parameters to obtain optimal power in tests of regression curves we examine three tests and propose empirical smoothing parameters to maximize the power in each test we also show that the data based smoothing parameters converge to the optimal smoothing parameters as sample sizes get larger we conduct a simulation study for various classes of alternative showing the effectiveness of the proposed procedures

heteroscedastic one way anova and lack of fit tests recent articles have considered the asymptotic behavior of the one way analysis of variance anova f statistic when the number of levels or groups is large in these articles the results were obtained under the assumption of homoscedasticity and for the case when the sample or group sizes ni remain fixed as the number of groups a tends to infinity in this article we study both weighted and unweighted test statistics in the heteroscedastic case the unweighted statistic is new and can be used even with small group sizes we demonstrate that an asymptotic approximation to the distribution of the weighted statistic is possible only if the group sizes tend to infinity suitably fast in relation to a our investigation of local alternatives reveals a similarity between lack of fit tests for constant regression in the present case of replicated observations and the case of no replications which uses smoothing techniques the asymptotic theory uses a novel application of the projection principle to obtain

how far are automatically chosen regression smoothing parameters from their optimum abstract we address the problem of smoothing parameter selection for nonparametric curve estimators in the specific context of kernel regression estimation call the optimal bandwidth the minimizer of the average squared error we consider several automatically selected bandwidths that approximate the optimum how far are the automatically selected bandwidths from the optimum the answer is studied theoretically and through simulations the theoretical results include a central limit theorem that quantifies the convergence rate and gives the differences asymptotic distribution the convergence rate turns out to be excruciatingly slow this is not too disappointing because this rate is of the same order as the convergence rate of the difference between the minimizers of the average squared error and the mean average squared error in some simulations by john rice the selectors considered here performed quite differently from each other we anticipated that these differences would be reflected in differ

quasi random initial population for genetic algorithms the selection of the initial population in a population based heuristic optimizationmethod is important since it affects the search for several iterations and often has an influence on the final solution if no a priori information about the optima is available the initial population is often selected randomly using pseudorandom numbers usually however it is more important that the points are as evenly distributed as possible than that they imitate random points in this paper we study the use of quasi random sequences in the initial population of a genetic algorithm sample points in a quasi random sequence are designed to have good distribution properties here a modified genetic algorithm using quasi random sequences in the initial population is tested by solving a large number of continuous benchmark problems from the literature the numerical results of two implementations of genetic algorithms using different quasi random sequences are compared to those of a traditional implementation using pseudorandom numbers the results obtained are promising

testing lack of fit in multiple regression summary we study lack of fit tests based on orthogonal series estimators a common feature of these tests is that they are functions of score statistics that employ data driven model dimensions the criteria used to select the dimension are score based versions of aic and bic the tests can be applied in a wide variety of settings including both continuous and discrete data with two or more covariates a model sequence i e a path in the alternative models space has to be chosen critical points and p values of the lack of fit tests can be obtained via asymptotic distribution theory or by use of the bootstrap data examples and a simulation study illustrate the applicability of the tests

oblique projectors and group involutory matrices in this work conditions characterizing when a lineal combination of two projectors of the same order is a group involutory matrix are analyzed in particular a representation of an involutory matrix as a sum or difference of two projectors is given in addition some results are applied to control problems an application to obtain a state feedback control is given

linear smoothers and additive models we study linear smoothers and their use in building nonparametric regression models in the first part of this paper we examine certain aspects of linear smoothers for scatterplots examples of these are the running mean and running line kernel and cubic spline smoothers the eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother and several other topics such as the number of degrees of freedom of a smoother are discussed in the second part of the paper we describe how linear smoothers can be used to estimate the additive model a powerful nonparametric regression model using the backfitting algorithm we show that backfitting is the gauss seidel iterative method for solving a set of normal equations associated with the additive model we provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers

scaling behavior of the order parameter and its conjugated field in an absorbing phase transition around the upper critical dimension we analyze numerically the critical behavior of an absorbing phase transition in a conserved lattice gas in an external field the external field is realized as a spontaneous creation of active particles that drives the system away from criticality nevertheless the order parameter obeys certain scaling laws for sufficiently small external fields these scaling laws are investigated and the corresponding exponents are determined in various dimensions d 2 3 4 5 at the so called upper critical dimension d mathrm c 4 one has to modify the usual scaling laws by logarithmic corrections

universal scaling behaviour of directed percolation and the pair contact process in an external field we consider the scaling behaviour of directed percolation and of the pair contact process with a conjugated field in particular we determine numerically the equation of state and show that both models are characterized by the same universal scaling function furthermore we derive the equation of state for the pair contact process within a mean field approach which again agrees with the mean field equation of state of the directed percolation universality class

algorithms for solving the inverse problem associated with k a k a s 1 in previous papers the authors introduced and characterized a class of matrices called k s 1 potent also they established a method to construct these matrices the purpose of this paper is to solve the associated inverse problem several algorithms are developed in order to find all involutory matrices k satisfying k a s 1 k a for a given matrix a c n n and a given natural number s the cases s 0 and s 1 are separately studied since they produce different situations in addition some examples are presented showing the numerical performance of the methods

testing the parametric null model hypothesis in semiparametric partial and generalized spline models on etend les resultats de cox et koh 1986 aux modeles generalises de wahba 1985 et a divers autres modeles splines partiels on montre que la statistique de test a une relation etroite avec le comportement de la fonction de validation croisee generalisee

characterization and properties of matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a there are positive integers r and s with r s n and matrices p cn r and q cn s such that p p ir q q is rp p and rq q we give an explicit representation of an arbitrary r symmetric matrix a in terms of p and q and show that solving az w and the eigenvalue problem for a reduce to the corresponding problems for matrices app cr r and aqq cs s we also express a 1 in terms of app 1 and aqq 1 under the additional assumption that r r we show that moore penrose inversion and singular value decomposition of a reduce to the corresponding problems for app and aqq we give similar results for r skew symmetric matrices these results are known for the case where r j i n j 1 i j 1n however our proofs are simpler even in this case we say that a cn n is r conjugate if rar a where r rn n and r r 1 i in this case r a is r symmetric and i a is r skew symmetric so our results provide explicit representations for r conjugate matrices in terms of p and q which are now in rn r and rn s respectively we show that solving az w inverting a and the eigenvalue problem for a reduce to the corresponding problems for a related matrix s rn n if rt r this is also true for moore penrose inversion and singular value decomposition of a

data driven smooth tests when the hypothesis is composite in recent years several authors have recommended smooth tests for testing goodness of fit however the number of components in the smooth test statistic should be chosen well otherwise considerable loss of power may occur schwarz s selection rule provides one such good choice earlier results on simple null hypotheses are extended here to composite hypotheses which tend to be of more practical interest for general composite hypotheses consistency of the data driven smooth tests holds at essentially any alternative monte carlo experiments on testing exponentiality and normality show that the data driven version of neyman s test compares well to other even specialized tests over a wide range of alternatives

semiparametric double balancing score estimation for incomplete data with ignorable missingness when estimating the marginal mean response with missing observations a critical issue is robustness to model misspecification in this article we propose a semiparametric estimation method with extended double robustness that attains the optimal efficiency under less stringent requirement for model specifications than the doubly robust estimators in this semiparametric estimation covariate information is collapsed into a two dimensional score s with one dimension for i the pattern of missingness and the other for ii the pattern of response both estimated from some working parametric models the mean response e y is then estimated by the sample mean of e y s which is estimated via nonparametric regression the semiparametric estimator is consistent if either the core of i or the core of ii is captured by s and attains the optimal efficiency if both are captured by s as the cores can be obtained without correctly specifying the full parametric models for i or ii the proposed es

multiscale testing of qualitative hypotheses suppose that one observes a process y on the unit interval where dy t n1 2f t dt dw t with an unknown function parameter f given scale parameter n 1 sample size and standard brownian motion w we propose two classes of tests of qualitative nonparametric hypotheses about f such as monotonicity or concavity these tests are asymptotically optimal and adaptive in a certain sense they are constructed via a new class of multiscale statistics and an extension of levy s modulus of continuity of brownian motion

residual variance and residual pattern in nonlinear regression summary a nonparametric estimator of residual variance in nonlinear regression is proposed it is based on local linear fitting asymptotically the estimator has a small bias but a larger variance compared with the parametric estimator in linear regression finite sample properties are investigated in a simulation study including a comparison with other nonparametric estimators the method is also useful for spotting heteroscedasticity and outliers in the residuals at an early stage of the data analysis a further application is checking the fit of parametric models this is illustrated for longitudinal growth data

characterization and properties of matrices with k involutory symmetries we say that a matrix r cn n is k involutory if its minimal polynomial is xk 1 for some k 2 so rk 1 r 1 and the eigenvalues of r are 1 2 k 1 where e2 i k let 0 1 k 1 if r cm m a cm n s cn n and r and s are k involutory we say that a is r s symmetric if ras a we show that an r s symmetric matrix a can be usefully represented in terms of matrices fl cc l dl 0 l k 1 where cl and dl are respectively the dimensions of the l eigenspaces of r and s this continues a theme initiated in an earlier paper with the same title in which we assumed that 1 we say that a k involution is equidimensional with width d if all of its eigenspaces have dimension d we show that if r and s are equidimensional k involutions with widths d1 and d2 respectively then r s symmetric matrices are closely related to generalized circulants ras r r s 0k 1 where a0 a1 ak 1 cd1 d2 for this case our results are new even if 1 we also give an explicit formula for the moore penrose inverse of a unilevel block circulant as r r s 0k 1 for any 0 1 k 1 generalizing a result previously obtained for the case where gcd k 1

block boundary value methods for linear hamiltonian systems the problem of characterizing multistep methods suitable to efficiently approximate the solutions of linear hamiltonian systems is discussed showing that the appropriate methods should belong to the class of discrete boundary value methods bvms three families of such methods are proposed the presented methods have infinite regions of absolute stability and can be of any order in fact for every odd k there are k step methods of order up to 2k which is the maximum order reachable by a k step formula

efficient generation of random bits from finite state markov chains the problem of random number generation from an uncorrelated random source of unknown probability distribution dates back to von neumann s 1951 work elias 1972 generalized von neumann s scheme and showed how to achieve optimal efficiency in unbiased random bits generation hence a natural question is what if the sources are correlated both elias and samuelson proposed methods for generating unbiased random bits in the case of correlated sources of unknown probability distribution specifically they considered finite markov chains however their proposed methods are not efficient or have implementation difficulties blum 1986 devised an algorithm for efficiently generating random bits from degree 2 finite markov chains in expected linear time however his beautiful method is still far from optimality on information efficiency in this paper we generalize blum s algorithm to arbitrary degree finite markov chains and combine it with elias s method for efficient generation of unbiased bits as a result we provide the first known algorithm that generates unbiased random bits from an arbitrary finite markov chain operates in expected linear time and achieves the information theoretic upper bound on efficiency

model selection using wavelet decomposition and applications in this paper we discuss how to use wavelet decompositions to select a regression model the methodology relies on a minimum description length criterion which is used to determine the number of nonzero coefficients in the vector of wavelet coefficients consistency properties of the selection rule are established and simulation studies reveal information on the distribution of the minimum description length selector we then apply the selection rule to specific problems including testing for pure white noise the power of this test is investigated via simulation studies and the selection criterion is also applied to testing for no effect in nonparametric regression

theoretical and empirical convergence results for additive congruential random number generators additive congruential random number acorn generators represent an approach to generating uniformly distributed pseudo random numbers that is straightforward to implement efficiently for arbitrarily large order and modulus if it is implemented using integer arithmetic it becomes possible to generate identical sequences on any machine this paper briefly reviews existing results concerning acorn generators and relevant theory concerning sequences that are well distributed mod 1 in k dimensions it then demonstrates some new theoretical results for acorn generators implemented in integer arithmetic with modulus m 2 m showing that they are a family of generators that converge in a sense that is defined in the paper to being well distributed mod 1 in k dimensions as m log 2m tends to infinity by increasing k it is possible to increase without limit the number of dimensions in which the resulting sequences approximate to well distributed the paper concludes by applying the standard testu01 test suite to acorn generators for selected values of the modulus between 2 6 0 and 2 1 5 0 the order between 4 and 30 and various odd seed values on the basis of these and earlier results it is recommended that an order of at least 9 be used together with an odd seed and modulus equal to 2 3 0 p for a small integer value of p while a choice of p 2 should be adequate for most typical applications increasing p to 3 or 4 gives a sequence that will consistently pass all the tests in the testu01 test suite giving additional confidence in more demanding applications the results demonstrate that the acorn generators are a reliable source of uniformly distributed pseudo random numbers and that in practice as suggested by the theoretical convergence results the quality of the acorn sequences increases with increasing modulus and order

testing the fit of a parametric function abstract general methods for testing the fit of a parametric function are proposed the idea underlying each method is to accept the prescribed parametric model if and only if it is chosen by a model selection criterion several different selection criteria are considered including one based on a modified version of the akaike information criterion and others based on various score statistics the tests have a connection with nonparametric smoothing because they use orthogonal series estimators to detect departures from a parametric model an important aspect of the tests is that they can be applied in a wide variety of settings including generalized linear models spectral analysis the goodness of fit problem and longitudinal data analysis implementation using standard statistical software is straightforward asymptotic distribution theory for several test statistics is described and the tests are shown to be consistent against essentially any alternative hypothesis simulations and a data exampl

on the use of nonparametric regression for model checking summary the use of nonparametric regression is explored to check the fit of a parametric regression model the principal aim is to check the validity of the regression curve rather than necessarily to detect outliers a pseudo likelihood ratio test is developed to provide a global assessment of fit and simulation bands are used to indicate the nature of departures from the model the types of data considered include discrete response variables where standard diagnostic techniques are often not appropriate and first order autoregressive series several numerical examples are given nonparametric regression can be used in an informal graphical way to assess the relationship between a response and an explanatory variable in this paper we aim to develop more formal methods of assessing the assumptions of a parametric model in particular when regression diagnostics of the type developed for normal linear models are not readily available the principal aim is to check the validity of the systematic part of the model by comparing a nonparametric estimate of the regression curve with a parametric one such a comparison may also identify outliers although the distinction between outliers and model inadequacy is not always easy two techniques are used to assess the fit of a parametric model in 2 confidence bands are constructed around the fitted regression curve by simulation a comparison of these with the nonparametric curve gives an indication of the nature of any departures from the model in 3 a pseudo likelihood ratio test is developed this provides a quantitative global assessment of fit in applying these ideas special emphasis is given to discrete data and notably logistic regression because of the difficulty in applying standard residual based model checking techniques to this type of response variable a poisson regression example is discussed in 4 however the underlying ideas have wider applications autoregressive time series of order 1 are discussed in 6 sections 5 and 7 discuss general issues we first discuss the context of binary regression with a single covariate and the difficulties caused by the discreteness of the response variable the observed data are assumed to be of the form xi yi ni where xi is a covariate value and yi has a binomial

optimal bandwidth selection in nonparametric regression function estimation kernel estimators of an unknown multivariate regression function are investigated a bandwidth selection rule is considered which can be formulated in terms of cross validation under mild assumptions on the kernel and the unknown regression function it is seen that this rule is asymptotically optimal

testing for no effect when estimating a smooth function by nonparametric regression a randomization approach abstract when a linear regression function is estimated by ordinary least squares the null hypothesis of no relationship between the response and the design variable can be tested by the normal theory f test this article describes a new test for use in the case in which a smooth regression function is estimated by a nonparametric procedure such as kernel estimation or local regression the test is constructed as an approximation to an exact permutation test the test statistic is the ratio of sums of squares that are defined by analogy to the analysis of variance the null permutation distribution of the test statistic is approximated by matching its exact mean and variance to the moments of a gamma distribution a simulation study shows that the approximation is excellent for several regression procedures and for normal heavy tailed and skewed error distributions simulation results are also used to investigate the power of the test as a function of the smoothing parameter the test is applied to dat

model checking with nonparametric curves summary the use of local likelihood curve estimates in model checking is described test statistics are based on deviance differences and are interpreted using nonparametric bootstrap monte carlo some theoretical aspects are discussed methods are illustrated for regression with discrete data

bandwidth choice for nonparametric regression this paper is concerned with the problem of choosing a bandwidth parameter for nonparametric regression we analyze a tapered fourier series estimate and discuss the relationship of this estimate to a kernel estimate we first consider a method based on an unbiased estimate of mean square error and show that the bandwidth thus chosen is asymptotically optimal other methods are examined as well and are shown to be asymptotically equivalent a small simulation shows however that for small or moderate sample size the methods perform quite differently

semiparametric comparison of regression curves the comparison of nonparametric regression curves is considered it is assumed that there are parametric possibly nonlinear transformations of the axes which map one curve into the other estimation and testing of the parameters in the transformations are studied the rate of convergence is n 1 2 although the nonparametric components of the model typically have a rate slower than that a statistic is provided for testing the validity of a given completely parametric model

local likelihood estimation abstract a scatterplot smoother is applied to data of the form x 1 y 1 x 2 y 2 xn yn and uses local fitting to estimate the dependence of y on x a simple example is the running lines smoother which fits a least squares line to the y values falling in a window around each x value the value of the estimated function at x is given by the value of the least squares line at x a smoother generalizes the least squares line which assumes that the dependence of y on x is linear in this article we extend the idea of local fitting to likelihood based regression models one such application is to the class of generalized linear models nelder and wedderburn 1972 we enlarge this class by replacing the covariate form 0 x 1 with an unspecified smooth function s x this function is estimated from the data by a technique we call local likelihood estimation the method consists of maximum likelihood estimation for 0 and 1 applied in a window around each x value multiple covariates are incor

the additive congruential random number generator a special case of a multiple recursive generator this paper considers an approach to generating uniformly distributed pseudo random numbers which works well in serial applications but which also appears particularly well suited for application on parallel processing systems additive congruential random number acorn generators are straightforward to implement for arbitrarily large order and modulus if implemented using integer arithmetic it becomes possible to generate identical sequences on any machine previously published theoretical analysis has demonstrated that a kth order acorn sequence approximates to being uniformly distributed in up to k dimensions for any given k acorn generators can be constructed to give period lengths exceeding any given number for example with period length in excess of 2 3 0 p for any given p results of empirical tests have demonstrated that if p is greater than or equal to 2 then the acorn generator can be used successfully for generating double precision uniform random variates this paper demonstrates that an acorn generator is a particular case of a multiple recursive generator and therefore also a special case of a matrix generator both these latter approaches have been widely studied and it is to be hoped that the results given in the present paper will lead to greater confidence in using the acorn generators

goodness of fit tests for parametric regression models several new tests are proposed for examining the adequacy of a family of parametric models against large nonparametric alternatives these tests formally check if the bias vector of residuals from parametric fits is negligible by using the adaptive neyman test and other methods the testing procedures formalize the traditional model diagnostic tools based on residual plots we examine the rates of contiguous alternatives that can be detected consistently by the adaptive neyman test applications of the procedures to the partially linear models are thoroughly discussed our simulation studies show that the new testing procedures are indeed powerful and omnibus the power of the proposed tests is comparable to the f test statistic even in the situations where the f test is known to be suitable and can be far more powerful than the f test statistic in other situations an application to testing linear models versus additive models is also discussed

relations between k s 1 potent matrices and different classes of complex matrices in this paper k s 1 potent matrices are considered a matrix a cn n is called k s 1 potent when kas 1k a where k is an involutory matrix and s 1 2 3 specifically k s 1 potent matrices are analyzed considering their relations to different classes of complex matrices these classes of matrices are s 1 generalized projectors k hermitian matrices normal matrices and matrices b cn n anti commuting with k or such that kb is involutory hermitian or normal in addition some new relations for k generalized centrosymmetric matrices have been derived

testing the goodness of fit of a linear model via nonparametric regression techniques abstract this article investigates the use of nonparametric regression methodology to test the adequacy of a parametric linear model the large sample properties of parametric goodness of fit tests for linearity are considered the inadequacies of such tests lead to the proposal of new tests that are constructed from nonparametric regression fits to the residuals from linear regression large sample theory is derived for two variants of this type of statistic the results demonstrate that such tests are consistent against all fixed smooth alternatives to linearity but are incapable of detecting local alternatives converging to a linear model at the parametric rate n 1 2 simulation experiments involving a test based on fitting cubic smoothing splines to residuals reveals that this test has good power properties against several reasonable alternatives

a simple consistent bootstrap test for a parametric regression function abstract a simple consistent test is considered and a bootstrap method is proposed for testing a parametric regression functional form it is shown that the bootstrap method gives a more accurate approximation to the null distribution of the test than the asymptotic normal theory result we also propose a consistent test for testing a parametric partially linear model versus a semiparametric partially linear alternative monte carlo simulations suggest that the bootstrap test performs well based on wild bootstrap critical values

bootstrap test for difference between means in nonparametric regression abstract a bootstrap test is proposed for detecting a difference between two mean functions in the setting of nonparametric regression error distributions in the regression model are permitted to be arbitrary and unequal the test enjoys power properties akin to those in a parametric setting in the sense that it can distinguish between regression functions distant only n 1 2 apart where n is the sample size it has exceptional level accuracy with level error of only n 2 and uses a very accurate estimate of the critical point of an exact test being in error by only n 3 2 under the null hypothesis the test admits several generalizations for example to the case of testing for differences between several regression means this is a nonparametric regression analog of analysis of variance a simulation study using n as small as 15 corroborates the asymptotic result on level accuracy of the bootstrap test applications are illustrated with an example involving acid rain data

test of significance based on wavelet thresholding and neyman s truncation abstract traditional nonparametric tests such as the kolmogorov smirnov test and the cramer von mises test are based on the empirical distribution functions although these tests possess root n consistency they effectively use only information contained in the low frequencies this leads to low power in detecting fine features such as sharp and short aberrants as well as global features such as high frequency alternations the drawback can be repaired via smoothing based test statistics in this article we propose two such kind of test statistics based on the wavelet thresholding and the neyman truncation we provide extensive evidence to demonstrate that the proposed tests have higher power in detecting sharp peaks and high frequency alternations while maintaining the same capability in detecting smooth alternative densities as the traditional tests similar conclusions can be made for two sample nonparametric tests of distribution functions in that case the traditional linear rank tests such as th

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

commonality of cusum von neumann and smoothing based goodness of fit tests summary recent papers by munson jernigan 1989 and buckley 1991 propose nonparametric tests for the hypothesis of no predictor effect in regression the munson jernigan test is similar to the von neumann 1941 test while that of buckley is based on a functional of cusums these tests are shown to be special cases of a wider class of tests based on nonparametric function estimation ideas fourier analysis is used to qualitatively compare the munson jernigan and buckley tests with two new tests constructed from nonparametric smoothers their relative powers are then studied by means of large sample analysis and simulation the cusum test is the most powerful for very smooth departures from the no effect hypothesis while the new tests based on smoothing ideas are clearly superior when the alternative is high frequency

recursive identification of hammerstein systems convergence rate and asymptotic normality in this work recursive identification algorithms are developed for hammerstein systems under the conditions considerably weaker than those in the existing literature for example orders of linear subsystems may be unknown and no specific conditions are imposed on their moving average part the recursive algorithms for estimating both linear and nonlinear parts are based on stochastic approximation and kernel functions almost sure convergence and strong convergence rates are derived for all estimates in addition the asymptotic normality of the estimates for the nonlinear part is also established the nonlinearity considered in the paper is more general than those discussed in the previous papers a numerical example verifies the theoretical analysis with simulation results

locally weighted regression an approach to regression analysis by local fitting abstract locally weighted regression or loess is a way of estimating a regression surface through a multivariate smoothing procedure fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series with local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions such as polynomials the goal of this article is to show through applications how loess can be used for three purposes data exploration diagnostic checking of parametric models and providing a nonparametric regression surface along the way the following methodology is introduced a a multivariate smoothing procedure that is an extension of univariate locally weighted regression b statistical procedures that are analogous to those used in the least squares fitting of parametric functions c several graphical methods that are useful tools for understanding loess estimates and checking the a

goodness of fit tests based on kullback leibler discrimination information we present a general methodology for developing asymptotically distribution free goodness of fit tests based on the kullback leibler discrimination information the tests are shown to be omnibus within an extremely large class of nonparametric global alternatives and to have good local power the proposed test procedure is a nonparametric extension of the classical neyman pearson log likelihood ratio test and is based on mth order spacings between order statistics cross validated by the observed log likelihood the developed method also generalizes cox s procedure of testing separate families and covers virtually all parametric families of distributions encountered in statistics it can also be viewed as a procedure based on sum log functionals of nonparametric density quantile estimators cross validated by the log likelihood with its good power properties the method provides an extremely simple and potentially much better alternative to the classical empirical distribution function edf based test procedures the important problem of selecting the order of spacings m in practice is also considered and a method based on maximizing the sample entropy constrained by the observed log likelihood is proposed this data driven method of choosing m is demonstrated by monte carlo simulations to be more powerful than deterministic choices of m and thus provides a practically useful tool for implementing our test procedure

characterizations of k s 1 potent matrices and applications abstract recently situations where a matrix coincides with some of its powers have been studied this kind of matrices is related to the generalized inverse matrices on the other hand it is possible to introduce another class of matrices that involve an involutory matrix generalizing the well known idempotent matrix widely useful in many applications in this paper we introduce a new kind of matrices called k s 1 potent as an extension of the aforementioned ones first different properties of k s 1 potent matrices have been developed later the main result developed in this paper is the characterization of this kind of matrices from a spectral point of view in terms of powers of the matrix by means of the group inverse and via a block representation of a matrix of index 1 finally an application of the above results to study linear combinations of k s 1 potent matrices is derived

semiparametric analysis of heterogeneous data using varying scale generalized linear models this article describes a class of heteroscedastic generalized linear regression models in which a subset of the regression parameters are rescaled nonparametrically and develops efficient semiparametric inferences for the parametric components of the models such models provide a means to adapt for heterogeneity in the data due to varying exposures varying levels of aggregation and so on the class of models considered includes generalized partially linear models and nonparametrically scaled link function models as special cases we present an algorithm to estimate the scale function nonparametrically and obtain asymptotic distribution theory for regression parameter estimates in particular we establish that the asymptotic covariance of the semiparametric estimator for the parametric part of the model achieves the semiparametric lower bound we also describe bootstrap based goodness of scale test we illustrate the methodology with simulations published data and data from collaborative research on ultrasound safety

algorithms for k s 1 potent matrix constructions in this paper we deal with k s 1 potent matrices these matrices generalize all the following classes of matrices k potent matrices periodic matrices idempotent matrices involutory matrices centrosymmetric matrices mirrorsymmetric matrices circulant matrices among others several applications of these classes of matrices can be found in the literature we develop algorithms in order to compute k s 1 potent matrices and k s 1 potent linear combinations of k s 1 potent matrices in addition some examples are presented in order to show the numerical performance of the method

on the combination of self organized systems to generate pseudo random numbers cellular automata ca is a self organizing structure with complex behavior which can be used to generating pseudo random numbers pure ca has a simple structure and has no ability to produce long sequences of random numbers but in order to resolve this problem a programmable controllable ca pcca using actuating factor or combination of different self organizing criticality phenomena can be used the purpose of the aforementioned methods is the use of the ca speed while preventing the automata self organizing factor for specific use in generating random numbers in this paper a pcca by using sandpile model is proposed the sandpile model is a complex system that operates at a critical state between chaos and order this state is known as self organized criticality soc and is characterized by displaying invariant scale behavior in the precise case of the sandpile model by randomly and continuously dropping sand grains on top of a two dimensional grid lattice a power law relationship between the frequency and size of sand avalanches is observed the avalanche behavior and the pure ca behavior are combined in a novel method which can be used as the pseudo random number generator prng experimental results show that this generator is able to reach a random behavior from a pseudo chaotic one by combining two self organized systems in addition to independency of the initial state it has been also indicated that the generated numbers are independent and its probability distribution is likely to be uniform accordingly some tests of prngs such as entropy chi square diehard and several basic statistical tests have been performed and all of them have been successfully passed

theory and applications of cellular automata in cryptography this paper deals with the theory and application of cellular automata ca for a class of block ciphers and stream ciphers based on ca state transitions certain fundamental transformations are defined which are block ciphering functions of the proposed enciphering scheme these fundamental transformations are found to generate the simple alternating group of even permutations which in turn is a subgroup of the permutation group these functions are implemented with a class of programmable cellular automata pca built around rules 51 153 and 195 further high quality pseudorandom pattern generators built around rule 90 and 150 programmable cellular automata with a rule selector i e combining function has been proposed as running key generators in stream ciphers both the schemes provide better security against different types of attacks with a simple regular modular and cascadable structure of ca hardware implementation of such schemes idealy suit vlsi implementation

matrices that commute with a permutation matrix abstract let p be an n n permutation matrix and let p be the corresponding permutation let a be a matrix such that ap pa it is well known that when p is an n cycle a is permutation similar to a circulant matrix we present results for the band patterns in a and for the eigenstructure of a when p consists of several disjoint cycles these results depend on the greatest common divisors of pairs of cycle lengths

universal finite size scaling behavior and universal dynamical scaling behavior of absorbing phase transitions with a conserved field we analyze numerically three different models exhibiting an absorbing phase transition we focus on the finite size scaling as well as the dynamical scaling behavior an accurate determination of several critical exponents allows one to validate certain hyperscaling relations using these hyperscaling relations it is possible to express the avalanche exponents of a self organized critical system in terms of the ordinary exponents of a continuous absorbing phase transition

asymptotically optimal difference based estimation of variance in nonparametric regression summary we define and compute asymptotically optimal difference sequences for estimating error variance in homoscedastic nonparametric regression our optimal difference sequences do not depend on unknowns such as the mean function and provide substantial improvements over the suboptimal sequences commonly used in practice for example in the case of normal data the usual variance estimator based on symmetric second order differences is only 64 efficient relative to the estimator based on optimal second order differences the efficiency of an optimal mth order difference estimator relative to the error sample variance is 2m 2m 1 again this is for normal data and increases as the tails of the error distribution become heavier

on the existence of a common eigenvector for all matrices in the commutant of a single matrix the main purpose of this paper is to study common invariant subspaces of any matrix in the centralizer of a given matrix a mn f where f denotes an algebraically closed field in particular we obtain a necessary and sufficient condition for the existence of a common eigenvector for all the matrices in this set

structural tests in additive regression we consider the component analysis problem for a regression model with an additive structure the problem is to test whether some of the additive components are of polynomial structure e g linear without specifying the structure of the remaining components a particular case is the problem of selecting the significant covariates the method that we present is based on the wavelet transform using the haar basis which allows for applications under mild conditions on the design and smoothness of the regression function the results demonstrate that each component of the model can be tested with the rate corresponding to the case if all of the remaining components were known the proposed procedure is also computationally straightforward simulation results and a real data example about female labor supply demonstrate the test s good performance

local polynomial estimation in multiparameter likelihood models abstract the nonparametric regression technique of local polynomial fitting is extended to multiparameter likelihood models some well known appealing features of local polynomial smoothers such as the behavior at the boundary are shown to carry over to the multiparameter case asymptotic consistency and normality of the resulting estimators are derived under suitable regularity conditions this work is motivated by the need for a nonparametric alternative to parametric dose response models for clustered binary data probability models for clustered binary response data include a success probability parameter and one or more correlation parameters the proposed local polynomial estimators can play an important role as a diagnostic tool or to suggest the form of the functional relationships in parametric likelihood models as an illustration it is shown how the local likelihood estimation procedure can be implemented for fitting a dose response curve based on the beta binomial model a data example and

shadowing and iterative interpolation for cebysev mixing transformations abstract the iteration of cebysev polynomials generates mixing transformations that model canonical features of chaotic systems these include pseudo random evolution ergodicity fading memory and the irreversible dispersal of any set of positive measure throughout the mixing region mixing processes are also analytically and numerically unstable nevertheless iterative interpolation or numerical retrodiction demonstrates that the computer generated trajectories are shadowed within strict error bounds by exact cebysev iterates pervasive shadowing is however not sufficient to ensure a generic correspondence between computer simulations and true dynamics this latitude is illustrated by several basic distinctions between the computer generated orbit structures and the exact analytic orbits of the cebysev mixing transformations

the choice of weights in kernel regression estimation summary for kernel regression estimation a weighting scheme due to nadaraya and watson has been associated with random design and a convolution type weighting scheme with fixed design based on integrated mean square error none of the estimators is uniformly optimal in either design however the convolution type weights are minimax optimal further advantages of this estimator can be seen in the structure of the bias

phase transitions in a lattice population model we introduce a model for a population on a lattice with diffusion and birth death according to and for a particle a we find that the model displays a phase transition from an active to an absorbing state which is continuous in 1 1 dimensions and of first order in higher dimensions in agreement with the mean field equation for the 1 1 dimensional case we examine the critical exponents and a scaling function for the survival probability and show that it belongs to the universality class of directed percolation in higher dimensions we look at the first order phase transition by plotting a histogram of the population density and use the presence of phase coexistence to find an accurate value for the critical point in 2 1 dimensions

true random number generator based on mouse movement and chaotic hash function a true random number generator trng makes use of a non deterministic source to produce randomness it is considered as more secure than a pseudorandom number generator as the degree of randomness is higher in this letter we propose a novel trng which generates random bits by moving the mouse casually it is convenient universal and low cost for the personal computer pc platform to eliminate the regular patterns of mouse movements caused by the habit of the same user we propose three trngs based on chaotic hash functions experiments are conducted to evaluate the speed diffusion and randomness performance of these trngs the results show that two of them possess satisfactory performance and can be implemented on common pc platform

on the generation of high quality random numbers by two dimensional cellular automata finding good random number generators rngs is a hard problem that is of crucial import in several fields ranging from large scale statistical physics simulations to hardware self test in this paper we employ the cellular programming evolutionary algorithm to automatically generate two dimensional cellular automata ca rngs applying an extensive suite of randomness tests to the evolved cas we demonstrate that they rapidly produce high quality random number sequences moreover based on observations of the evolved cas we are able to handcraft even better rngs which not only outperform previously demonstrated high quality rngs but can be potentially tailored to satisfy given hardware constraints

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

spectral study of r s 1 k and r s 1 k potent matrices abstract the r s 1 k and r s 1 k potent matrices have been studied in several recent papers we continue these investigations from a spectral point of view specifically a spectral study of r s 1 k potent matrices is developed using characterizations involving an associated matrix pencil a r the corresponding spectral study for r s 1 k potent matrices involves the pencil a r in order to present some properties the relevance of the projector i a a where a is the group inverse of a is highlighted in addition some applications and numerical examples are given particularly involving pauli matrices and the quaternions

kernel regression when the boundary region is large with an application to testing the adequacy of polynomial models abstract it is well known that kernel regression estimators are subject to so called boundary or edge effects a phenomenon in which the bias of an estimator increases near the endpoints of the estimation interval when the regression curve is linear or nearly linear the requisite amount of smoothing is so great that the boundary region is effectively the entire estimation interval special boundary kernels are proposed here to deal with such cases it is shown that the proposed kernel estimator has a property also enjoyed by cubic smoothing splines namely as the estimator s smoothing parameter becomes large the estimator tends to a straight line the limiting straight line is essentially the least squares line when the design points are equally spaced a simple generalization of ideas in the linear case leads to kernel estimates that are polynomials of any given degree for large bandwidths such estimates are an important component of a proposed test for the adequacy of a polynomial model the test s