on strong invariance principles under dependence assumptions on obtient des principes d invariance forts avec un ordre d approximation o t 1 2 k pour des suites de variables aleatoires dependantes

a reexamination of diffusion estimators with applications to financial model validation time homogeneous diffusion models have been widely used for describing the stochastic dynamics of the underlying economic variables recently stanton proposed drift and diffusion estimators based on a higher order approximation scheme and kernel regression method he claimed that higher order approximations must outperform lower order approximations and concluded nonlinearity in the instantaneous return function of short term interest rates to examine the impact of higher order approximations we develop general and explicit formulas for the asymptotic behavior of both drift and diffusion estimators we show that these estimators will reduce the numerical approximation errors in asymptotic biases but their asymptotic variances escalate nearly exponentially with the order of approximation simulation studies also confirm our asymptotic results this variance inflation problem arises not only from nonparametric fitting but also from parametric fitting stanton s work also postulates the interesting qu

the jackknife and the bootstrap for general stationary observations we extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence we do not attempt a reduction to i i d values the jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of l consecutive data once in the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero under appropriate conditions consistency is obtained if l l n rightarrow infty and l n n rightarrow 0 general statistics are approximated by an arithmetic mean in regular cases this approximation determines the asymptotic behavior bootstrap replicates are constructed by selecting blocks of length l randomly with replacement among the blocks of observations the procedures are illustrated by using the sunspot numbers and some simulated data

linear bounds for stochastic dispersion it has been suggested that stochastic flows might be used to model the spread of passive tracers in a turbulent fluid we define a stochastic flow by the equations begin eqnarray phi 0 x x d phi t x f dt phi t x end eqnarray where f t x is a field of semimartingales on x in mathbb r d for d geq 2 whose local characteristics are bounded and lipschitz the particles are points in a bounded set mathscr x and we ask how far the substance has spread in a time t that is we define phi t sup x in mathscr x sup 0 leq t leq t phi t x and seek to bound p phi t z without drift when f cdot x are required to be martingales although single points move on the order of sqrt t it is easy to construct examples in which the supremum phi t still grows linearly in time that is lim inf t to infty phi t t 0 almost surely we show that this is an upper bound for the growth that is we compute a finite constant k 0 depending on the bounds for the local characteristics such that limsup t to infty frac phi t t leq k 0 text almost surely a linear bound on growth holds even when the field itself includes a drift term

the use of subseries values for estimating the variance of a general statistic from a stationary sequence let z i infty i infty be a strictly stationary infty mixing sequence without specifying the dependence model giving rise to z i and without specifying the marginal distribution of z i we address the question of variance estimation for a general statistic t n t n z 1 z n for estimating var t n from just the available data z 1 z n we propose computing subseries values t m z i 1 z i 2 z i m 0 leq i i m leq n these subseries values are used as replicates to model the sampling variability of t n in particular we use adjacent nonoverlapping subseries of length m m n with m rightarrow infty and m n n rigtharrow 0 our variance estimator is just the usual sample variance computed amongst these subseries values after appropriate standardization this estimator is shown to be consistent under mild integrability conditions we present optimal i e minimum m s e choices of m n for the special case where t n overset z n and z i is a normal ar 1 sequence a simulation study is conducted showing that those same choices of m n are effective when t n is a robust estimator of location and z i is subject to contamination

simultaneous confidence bands for linear regression and smoothing suppose we observe y i f x i varepsilon i i 1 ldots n we wish to find approximate 1 alpha simultaneous confidence regions for f x x in mathscr x our regions will be centered around linear estimates hat f x of nonparametric or nonparametric f x there is a large amount of previous work on this subject substantial restrictions have been usually placed on some or all of the dimensionality of x the class of functions f that can be considered the class of linear estimates hat f and the region mathscr x the method we present is an approximation to the tube formula dn can be used for multidimensional x and a wide class of linear estimates by considering the effect of bias we are able to relax assumptions on the class of functions f which are considered simultaneous and numerical computations are used to illustrate the performance

bias corrected confidence bands in nonparametric regression summary bias corrected confidence bands for general nonparametric regression models are considered we use local polynomial fitting to construct the confidence bands and combine the cross validation method and the plug in method to select the bandwidths related asymptotic results are obtained our simulations show that confidence bands constructed by local polynomial fitting have much better coverage than those constructed by using the nadaraya watson estimator the results are also applicable to nonparametric autoregressive time series models

testing for parallelism among trends in multiple time series this paper considers the inference of trends in multiple nonstationary time series to test whether trends are parallel to each other we use a parallelism index based on the l2 distances between nonparametric trend estimators and their average a central limit theorem is obtained for the test statistic and the test s consistency is established we propose a simulation based approximation to the distribution of the test statistic which significantly improves upon the normal approximation the test is also applied to devise a clustering algorithm finally the finite sample properties of the test are assessed through simulations and the test methodology is illustrated by a cell phone download data collected in the united states

discontinuous versus smooth regression given measurements x i y i i 1 n we discuss methods to assess whether an underlying regression function is smooth continuous or differentiable or whether it has discontinuities the variance of the measurements is assumed to be unknown and is estimated simultaneously by regressing squared differences of the data formed with various span sizes on the span size itself we obtain an asymptotic linear model with dependent errors the parameters of this asymptotic linear model include the sum of the squared jump sizes as well as the variance of the measurements both parameters can be consistently estimated with mean squared error rates of convergence of n 2 3 for the sum of squared jump sizes and n 1 for the error variance we derive the asymptotic constants of the mean squared error mse and discuss the dependence of mse on the maximum span size l the test for the existence of jumps is formulated for the null hypothesis that the sum of squared jump sizes is 0 the asymptotic distribution of the test statistic is obtained essentially via a central limit theorem for u statistics we motivate and illustrate the methods with data surrounded by a scientific controversy concerning the question whether the growth of children occurs smoothly or rather in jumps

nonlinear system theory another look at dependence based on the nonlinear system theory we introduce previously undescribed dependence measures for stationary causal processes our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems the proposed dependence measures provide a natural framework for a limit theory for stationary processes in particular under conditions with quite simple forms we present limit theorems for partial sums empirical processes and kernel density estimates the conditions are mild and easily verifiable because they are directly related to the data generating mechanisms

confidence bands for regression functions abstract a method for obtaining conservative confidence bands for regression functions in low dimensions is given of particular importance is the applicability of the method to nonparametric regression computations examples and comparisons indicate the effectiveness of the method

confidence bands in nonparametric regression abstract new bias corrected confidence bands are proposed for nonparametric kernal regression these bands are constructed using only a kernel estimator of the regression curve and its data selected bandwidth they are shown to have asymptotically correct coverage properties and to behave well in a small sample study one consequence of the large sample developments is that bonferroni type bands for the regression curve at the design points also have conservative asymptotic coverage behavior with no bias correction

a uniqueness theorem for viscous fluid motions in exterior domains as is known one of the most interesting still open questions in the dynamics of viscous incompressible flows governed by the navier stokes equations is that of the uniqueness of three dimensional motions an early answer to this problem is found in the paper 5 of e fo subsequently the uniqueness question acquired new importance when it was recognized by j leray 18 that when formulated in a suitable class it is tightly linked with that of the existence of global in time and regular solutions to the navier stokes equations el also 17 22 23 the above class is called the hopf class which consists of those weak solutions satisfying suitable summability conditions along with the energy inequality in particular they and their first spatial derivatives belong to the lebesgue space l2 2 q is the domain of motion however it was also recognized by j kamp de f riet that in general an existence and uniqueness theory in l2 classes is not well suited to studying homogeneous turbulence 16 when q is unbounded motivated by these and other considerations d graffi in a remarkable paper of 1960 13 proved uniqueness for classical solutions in an arbitrary smooth unbounded domain without assuming that the velocity field v tends to a limit at infinity rather he required v and vv only bounded in space and time however he needed suitable assumptions on the behavior of the pressure p at infinity this type of problem has been reconsidered by several authors who improved graffi s theorem by reducing the assumptions on both v and p but who still required that the pressure tends to a limit po suitably at great distances 2 4 6 12 21 26 actually it is worth remarking that one of the main goals of papers 3 4 8 21 was just to avoid or at least to weaken as far as possible the hypothesis about p po for such an assumption is always an undesired feature for flows of incompressible fluids this kind of results which hold for

inference of trends in time series summary we consider statistical inference of trends in mean non stationary models a test statistic is proposed for the existence of structural breaks in trends on the basis of a strong invariance principle of stationary processes we construct simultaneous confidence bands with asymptotically correct nominal coverage probabilities the results are applied to global warming temperature data and nile river flow data our confidence band of the trend of the global warming temperature series supports the claim that the trend is increasing over the last 150 years

a local polynomial jump detection algorithm in nonparametric regression we suggest a one dimensional jump detection algorithm based on local polynomial fitting for jumps in regression functions zero order jumps or jumps in derivatives first order or higherorder jumps if jumps exist in the mth order derivative of the underlying regression function then an m 1 order polynomial is fitted in a neighborhood of each design point we then characterize the jump information in the coefficients of the highest order terms of the fitted polynomials and suggest an algorithm for jump detection this method is introduced briefly for the general setup and then presented in detail for zero order and first order jumps several simulation examples are discussed we apply this method to the bombay india sea level pressure data

confidence intervals for nonparametric curve estimates numerous nonparametric regression methods exist that yield consistent estimators of function curves often one is also interested in constructing confidence intervals for the unknown function when a function estimate is based on a single global smoothing parameter the resulting confidence intervals may hold their desired confidence level 1 on average but because bias in nonparametric estimation is not uniform they do not hold the desired level uniformly at all design points most research in this area has focused on mean squared error properties of the estimator for example mise itself a global measure in addition measures like mise are one step removed from the practical issue of coverage probability recent work that focuses on coverage probability has considered only coverage in an average sense ignoring the important issue of uniformity of coverage across the design space to deal with this problem a new estimator is developed which uses a local cross validation criterion lcv to determi

the effect of serial correlation on tests for parameter change at unknown time it is shown that serial correlation can produce striking effects in distributions of change point statistics failure to account for these effects is shown to invalidate change point tests either through increases in the type 1 error rates if low frequency spectral mass predominates in the spectrum of the noise process or through diminution of the power of the tests when high frequency mass predominates these effects are characterized by the expression 2i f o fjr f a da where f is the spectral density of the noise process in sample survey work this is known as the design effect or deff simple precise adjustments to change point test statistics which account for serial correlation are provided the same adjustment applies to all commonly used regression models residual processes are derived for both stationary time series satisfying a moment condition and for general linear regression models with stationary error structure 1 introduction stochastic models for time sequenced data are generally characterized by several unknown parameters these parameters may change over time and if the changes when they occur do so unannounced and at unknown time points then the associated inferential problem is referred to as the change point problem various important application areas of statistics involve change detection in a central way two of these areas are quality assurance and environmental monitoring most of the statistics commonly applied to the change point problem involve cumulative sums or partial sums of regression residuals the distribution theory for these statistics has been computed under the assumption that the error process for the regression model is white noise in this paper we consider linear regression of a random variable against general nonstochastic functions of time but with error variables that form a serially correlated time series we then examine the large sample properties of the stochastic processes defined by the partial sums of the regression residuals large sample distribution theory for fixed sample size statistics used to detect changes in regression parameters is usually derived by computing the distributions of various functionals on

kernel type estimators of jump points and values of a regression function in the fixed design nonparametric regression model kernel type estimators of the locations of jump points and the corresponding sizes of jump values of the regression function are proposed these kernel type estimators are analyzed with almost sure results and limiting distributions using the limiting distributions we are able to test the number of jump points and give asymptotic confidence intervals for the sizes of jump values of the regression function simulation studies demonstrate that the asymptotic results hold for reasonable sample sizes

trend function hypothesis testing in the presence of serial correlation in this paper test statistics are proposed that can be used to test hypotheses about the parameters of the deterministic trend function of a univariate time series the tests are valid in the presence of general forms of serial correlation in the errors and can be used without having to estimate the serial correlation parameters either parametrically or nonparametrically the tests are valid for i 0 and i 1 errors trend functions that are permitted include general linear polynomial trend functions that may have breaks at either known or unknown locations asymptotic distributions are derived and consistency of the tests is established the general results are applied to a model with a simple linear trend a local asymptotic analysis is used to compute asymptotic size and power of the tests for this example size is well controlled and is relatively unaffected by the variance of the initial condition asymptotic power curves are computed for the simple linear trend model and are compared to existing tests it is shown that the new tests have nontrivial asymptotic power a simulation study shows that the asymptotic approximations are adequate for sample sizes typically used in economics the tests are used to construct confidence intervals for average gnp growth rates for eight industrialized countries using post war 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

structural break estimation for nonstationary time series models this article considers the problem of modeling a class of nonstationary time series using piecewise autoregressive ar processes the number and locations of the piecewise ar segments as well as the orders of the respective ar processes are assumed unknown the minimum description length principle is applied to compare various segmented ar fits to the data the goal is to find the best combination of the number of segments the lengths of the segments and the orders of the piecewise ar processes such a best combination is implicitly defined as the optimizer of an objective function and a genetic algorithm is implemented to solve this difficult optimization problem numerical results from simulation experiments and real data analyses show that the procedure has excellent empirical properties the segmentation of multivariate time series is also considered assuming that the true underlying model is a segmented autoregression this procedure is shown to be consistent for estimating the location of

jump detection in a regression curve and its derivative curve estimation from observed noisy data has broad applications in certain applications the underlying regression curve may have singularities including jumps and roofs valleys i e jumps in the first order derivative of the regression curve at some unknown positions representing structural changes of the related process detection of such singularities is important for understanding the structural changes in the literature a number of jump detection procedures have been proposed most of which are based on estimation of the one sided first order derivatives of the true regression curve in this paper motivated by certain related research in image processing we propose an alternative jump detection procedure besides the first order derivatives we suggest using helpful information about jumps in the second order derivatives as well theoretical justifications and numerical studies show that this jump detector works well in applications this procedure is then extended for detecting roofs v

on the uniqueness of the unbounded classical solutions of the navier stokes and associated equations abstract in this note we present a new theorem on the uniqueness of unbounded classical solutions to some class of equations which include the navier stokes equations and the associated leray and foias temam equations as special cases

a priori estimates in terms of the maximum norm for the solutions of the navier stokes equations in this paper we consider the cauchy problem for the incompressible navier stokes equations with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial velocity field for illustrative purposes we first derive corresponding a priori estimates for certain parabolic systems because of the pressure term the case of the navier stokes equations is more difficult however

estimation of a function with discontinuities via local polynomial fit with an adaptive window choice we propose a method of adaptive estimation of a regression function which is near optimal in the classical sense of the mean integrated error at the same time the estimator is shown to be very sensitive to discontinuities or change points of the underlying function f or its derivatives for instance in the case of a jump of a regression function beyond the intervals of length in order n 1 log n around change points the quality of estimation is essentially the same as if locations of jumps were known the method is fully adaptive and no assumptions are imposed on the design number and size of jumps the results are formulated in a nonasymptotic way and can therefore be applied for an arbitrary sample size

inference for monotone functions under short and long range dependence confidence intervals and new universal limits abstractwe introduce new point wise confidence interval estimates for monotone functions observed with additive dependent noise our methodology applies to both short and long range dependence regimes for the errors the interval estimates are obtained via the method of inversion of certain discrepancy statistics this approach avoids the estimation of nuisance parameters such as the derivative of the unknown function which previous methods are forced to deal with the resulting estimates are therefore more accurate stable and widely applicable in practice under minimal assumptions on the trend and error structure the dependence of the errors especially long range dependence leads to new phenomena where new universal limits based on convex minorant functionals of drifted fractional brownian motion emerge some extensions to uniform confidence bands are also developed supplementary materials for this article are available online

bandwidth selection for kernel regression with long range dependent errors summary we investigate the effect of long range dependence on bandwidth selection for kernel regression with the plug in method of herrmann gasser kneip 1992 a new bandwidth estimator is proposed to allow for long range dependence properties of the proposed estimator are investigated theoretically and via simulation we find that the proposed estimator performs well in terms of integrated squared error of the estimated trend allowing us to incorporate both deterministic nonlinear features having an unknown structure and long range dependence into a single model the method is illustrated using biweekly measurements of the volume of the great salt lake

testing for trends in high dimensional time series the article considers statistical inference for trends of high dimensional time series based on a modified l2 distance between parametric and nonparametric trend estimators we propose a de diagon

change points in nonparametric regression analysis an otherwise smooth regression model are proposed the assumptions needed are much weaker than those made in parametric models the proposed estimators apply as well to the detection of discontinuities in derivatives and therefore to the detection of change points of slope and of higher order curvature the proposed estimators are based on a comparison of left and right one sided kernel smoothers weak convergence of a stochastic process in local differences to a gaussian process is established for properly scaled versions of estimators of the location of a change point the continuous mapping theorem can then be invoked to obtain asymptotic distributions and corresponding rates of convergence for change point estimators these rates are typically faster than n 1 2 rates of global lp convergence of curve estimates with appropriate kernel modifications adapting to estimated change points are derived as a consequence it is shown that these rates of convergence are the same as if the location of the change point was known the methods are illustrated by means of the well known data on the annual flow volume of the nile river between 1871 and 1970 1 introduction nonparametric regression methods are usually applied in order to obtain a smooth fit of a regression curve without having to specify a parametric class of regression functions sometimes a generally smooth curve might contain an isolated discontinuity or change point in the curve or in a possibly higher order derivative and in many cases interest focuses on the occurrence of such change points in parametric approaches to the regression change point problem simple linear regressions before and after a possible change point are assumed and then the possibility of a discontinuity in the form of a jump or of a jump in the first derivative or equivalently a slope change is incorporated into the model see for instance hinkley 1969 and brown durbin and evans 1975 the analysis of change points which describe sudden localized changes typically occurring in economics medicine and the physical sciences has recently found increasing interest general smoothness assumptions allowing for a large class of regression functions to be considered seem to be more appropriate in a variety of applied problems than parametric modelling an

bootstrap simultaneous error bars for nonparametric regression simultaneous error bars are constructed for nonparametric kernel estimates of regression functions the method is based on the bootstrap where resampling is done from a suitably estimated residual distribution the error bars are seen to give asymptotically correct coverage probabilities uniformly over any number of gridpoints applications to an economic problem are given and comparison to both pointwise and bonferroni type bars is presented through a simulation study

nonparametric quasi likelihood the quasi likelihood function proposed by wedderburn broadened the scope of generalized linear models by specifying the variance function instead of the entire distribution however complete specification of variance functions in the quasi likelihood approach may not be realistic we define a nonparametric quasi likelihood by replacing the specified variance function in the conventional quasi likelihood with a nonparametric variance function estimate this nonparametric variance function estimate is based on squared residuals from an initial model fit the rate of convergence of the nonparametric variance function estimator is derived it is shown that the asymptotic limiting distribution of the vector of regression parameter estimates is the same as for the quasi likelihood estimates obtained under correct specification of the variance function thus establishing the asymptotic efficiency of the nonparametric quasi likelihood estimates we propose bandwidth selection strategies based on deviance and pearson s chi square statistic it is demonstrated in simulations that for finite samples the proposed nonparametric quasi likelihood method can improve upon extended quasi likelihood or pseudo likelihood methods where the variance function is assumed to fall into a parametric class with unknown parameters we illustrate the proposed methods with applications to dental data and cherry tree data

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

semiparametric model building for regression models with time varying parameters abstract this paper considers the problem of semiparametric model building for linear regression models with potentially time varying coefficients by allowing the response variable and explanatory variables be jointly a nonstationary process the proposed methods are widely applicable to nonstationary and dependent observations we propose a local linear shrinkage method that can simultaneously achieve parameter estimation and variable selection its selection consistency and the favorable oracle property are established due to the fear of losing efficiency an information criterion is further proposed for distinguishing between time varying and time constant components numerical examples are presented to illustrate the proposed methods

approximate periodic solutions for the helmholtz duffing equation approximate periodic solutions for the helmholtz duffing oscillator are obtained in this paper he s energy balance method hebm and he s frequency amplitude formulation hfaf are adopted as the solution methods oscillation natural frequencies are analytically analyzed error analysis is carried out and accuracy of the solution methods is evaluated

choice of bandwidth for kernel regression when residuals are correlated summary standard techniques for selecting the bandwidth of a kernel estimator from the data in a nonparametric regression model perform badly when the errors are correlated in this paper we propose a modified version of an existing plug in bandwidth selector the method is generalized to stationary error variables by estimating a functional of the residual covariance function the proposed bandwidth selector shows good properties in asymptotic theory and in simulations without assuming a parametric model for the error process

sieve bootstrap for smoothing in nonstationary time series we propose a sieve bootstrap procedure for time series with a deterministic trend the sieve for constructing the bootstrap is based on nonparametric trend estimation and autoregressive approximation for some noise process the bootstrap scheme itself does i i d resampling of estimated innovations from fitted autoregressive models we show the validity and indicate second order correctness of such sieve bootstrap approximations for the limiting distribution of nonparametric linear smoothers the resampling can then be used to construct nonparametric confidence intervals for the underlying trend in particular we show asymptotic validity for constructing confidence bands which are simultaneous within a neighborhood of size of the order of the smoothing bandwidth our resampling procedure yields satisfactory results in a simulation study for finite sample sizes we also apply it to the longest series of total ozone measurements from arosa switzerland and find a significant decreasing trend

isotonic regression another look at the changepoint problem a test based on isotonic regression is developed for monotonic trends in short range dependent sequences and is applied to argentina rainfall data and global warming data this test provides another perspective for changepoint problems the isotonic test is shown to be more powerful than some existing tests for trend

confidence bands in nonparametric time series regression we consider nonparametric estimation of mean regression and conditional variance or volatility functions in nonlinear stochastic regression models simultaneous confidence bands are constructed and the coverage probabilities are shown to be asymptotically correct the imposed dependence structure allows applications in many linear and nonlinear auto regressive processes the results are applied to the s p 500 index data

an effective bandwidth selector for local least squares regression abstract local least squares kernel regression provides an appealing solution to the nonparametric regression or scatterplot smoothing problem as demonstrated by fan for example the practical implementation of any scatterplot smoother is greatly enhanced by the availability of a reliable rule for automatic selection of the smoothing parameter in this article we apply the ideas of plug in bandwidth selection to develop strategies for choosing the smoothing parameter of local linear squares kernel estimators our results are applicable to odd degree local polynomial fits and can be extended to other settings such as derivative estimation and multiple nonparametric regression an implementation in the important case of local linear fits with univariate predictors is shown to perform well in practice a by product of our work is the development of a class of nonparametric variance estimators based on local least squares ideas and plug in rules for their implementation

strong invariance principles for dependent random variables we establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds applications to linear and some nonlinear processes are discussed strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions

a note on jackknifing kernel regression function estimators corresp estimation of the value of a regression function at a point of continuity using a kernel type estimator is considered and improvements by a jackknife technique are discussed it is seen that a so called generalized jackknife estimator asymptotically improves upon an ordinary kernel type estimator however for a fixed sample size the generalized jackknife method may inflate the mean square error

simultaneous inference of linear models with time varying coefficients summary the paper considers construction of simultaneous confidence tubes for time varying regression coefficients in functional linear models using a gaussian approximation result for non stationary multiple time series we show that the constructed simultaneous confidence tubes have asymptotically correct nominal coverage probabilities our results are applied to the problem of testing whether the regression coefficients are of certain parametric forms which is a fundamental problem in the inference of functional linear models as an application we analyse an environmental data set and study the association between levels of pollutants and hospital admissions