lade based inferences for autoregressive models with heavy tailed g garch 1 1 noise abstract this paper explores the least absolute deviation lad estimator of the autoregressive model with heavy tailed g garch 1 1 noise when the tail index 1 2 it is shown that the lad estimator asymptotically converges to a linear function of a series of stable random vectors with a rate of convergence n 1 1 the result is significantly different from that of the corresponding least square estimator which is not consistent and partially solves the problem on the asymptoticity of the lad estimator when the tail index is less than 2 a simulation study is carried out to assess the performance of the lad estimator and a real example is given to illustrate this approach

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

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

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

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

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

unbiased bits from sources of weak randomness and probabilistic communication complexity a new model for weak random physical sources is presented the new model strictly generalizes previous models e g the santha and vazirani model 27 the sources considered output strings according to probability distributions in which no single string is too probable the new model provides a fruitful viewpoint on problems studied previously such as extracting almost perfect bits from sources of weak randomness the question of possibility as well as the question of efficiency of such extraction schemes are addressed probabilistic communication complexity it is shown that most functions have linear communication complexity in a very strong probabilistic sense robustness of bpp with respect to sources of weak randomness generalizing a result of vazirani and vazirani 32 33

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

space lower bounds for online pattern matching we present space lower bounds for online pattern matching under a number of different distance measures given a pattern of length m and a text that arrives one character at a time the online pattern matching problem is to report the distance between the pattern and a sliding window of the text as soon as the new character arrives we require that the correct answer is given at each position with constant probability we give w m bit space lower bounds for l 1 l 2 l hamming edit and swap distances as well as for any algorithm that computes the cross correlation convolution we then show a dichotomy between distance functions that have wildcard like properties and those that do not in the former case which includes as an example pattern matching with character classes we give w m bit space lower bounds for other distance functions we show that there exist space bounds of w logm and o log 2m bits finally we discuss space lower bounds for non binary inputs and show how in some cases they can be improved

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

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

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

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

limit theory for the sample autocorrelations and extremes of a garch 1 1 process the asymptotic theory for the sample autocorrelations and extremes of a garch i 1 process is provided special attention is given to the case when the sum of the arch and garch parameters is close to 1 that is when one is close to an infinite variance marginal distribution this situation has been observed for various financial log return series and led to the introduction of the igarch model in such a situation the sample autocorrelations are unreliable estimators of their deterministic counterparts for the time series and its absolute values and the sample autocorrelations of the squared time series have nondegenerate limit distributions we discuss the consequences for a foreign exchange rate series

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

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

string matching under a general matching relation in standard string matching each symbol matches only itself in other string matching problems e g the string matching with don t cares problem a symbol may match several symbols in general an arbitrary many to many matching relation might hold between symbols we consider a general string matching problem in which such a matching relation is specified and those positions in a text t of length n are sought at which the pattern p of length m matches under this relation depending upon the existence of a simple and easily recognizable property in the given matching relation we show that string matching either requires linear i e o n m time or is at least as hard as boolean convolution as an application we show that the matching relations of several independently studied string matching problems do indeed fall into the latter hard category we also give a generic string matching algorithm that works far any matching relation and has complexity o nm except for very large matching relations

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

maintaining stream statistics over sliding windows we consider the problem of maintaining aggregates and statistics over data streams with respect to the last n data elements seen so far we refer to this model as the sliding window model we consider the following basic problem given a stream of bits maintain a count of the number of 1 s in the last n elements seen from the stream we show that using o frac 1 epsilon log 2 n bits of memory we can estimate the number of 1 s to within a factor of 1 epsilon we also give a matching lower bound of omega frac 1 epsilon log 2 n memory bits for any deterministic or randomized algorithms we extend our scheme to maintain the sum of the last n positive integers and provide matching upper and lower bounds for this more general problem as well we also show how to efficiently compute the lp norms p in 1 2 of vectors in the sliding window model using our techniques using our algorithm one can adapt many other techniques to work for the sliding window model with a multiplicative overhead of o frac 1 epsilon log n in memory and a 1 epsilon factor loss in accuracy these include maintaining approximate histograms hash tables and statistics or aggregates such as sum and averages

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