adjusted estimates and wald statistics for the ar 1 model with constant abstract adjusted profile likelihood of cox and reid 1993 journal of the royal statistical society b 55 467 471 for the autoregressive coefficient of the ar 1 model with constant yields an estimator which can be more accurate than the maximum likelihood estimator in small samples whether the process is stationary or a random walk and in the latter case also asymptotically a related estimator is asymptotically distributed compactly and symmetrically around zero under a random walk but is not consistent in general the adjusted wald statistic can follow the asymptotic standard normal distribution much better than the non adjusted under stationarity or a drifted random walk

on the influence of sampling and observation times on estimation of the bandwidth parameter of a gauss markov process the statistical problem of estimating the bandwidth parameter of a gauss markov gm process from a realization of fixed and finite duration t at selectable sampling interval spl delta is addressed in this paper as the observation time t is fixed and finite the variance of estimated autocorrelation and continuous time parameter does not vanish as spl delta approaches 0 this necessitates a second order taylor expansion in deriving the parameter estimator bias and variance the second order taylor expansion produces better bias and variance results than a first order one does the distribution of the estimator is also discussed according to the gradient change of the variance a key result is that three sample size regions which are termed finite large and very large corresponding to substantial gradual and very slight decrease in the variance of the parameter estimator respectively are quantified in terms of analysis bw the three regions are 23 35 35 55 and 55 spl infin db the characterization of the tradeoff between the variance decrease and sampling rate results in a practical guideline for choosing sampling rate these results are applied to the prediction problems of a time invariant gm process to show their value

interpolation and approximation of 3 d and 4 d scattered data abstract given a set of 3 d or 4 d scattered data methods are presented that yield a bivariate or trivariate function that interpolate or approximate the given data the subroutine package includes hardy s multiquadric interpolant and multistage methods with many options available to the user thus several different interpolation and approximation methods can be generated the options available in the multistage methods can be effectively used if the data set is noisy rapidly varying or nonuniformly distributed the computational and storage requirements for the multistage methods are linear in the number of data points thus they are efficient on large data sets

the map and blend scattered data interpolant on a sphere abstract the map and blend technique constructs a function defined over the sphere that interpolates to a discrete sample of measurements at arbitrary locations on the sphere this technique consists of two different mappings of the sphere to planar domains solving two corresponding scattered data interpolation problems on the planar domains and then the interpolant on the sphere is formed by blending these two planar interpolants if the user has software for solving the scattered data interpolation problem on a planar domain only a small programming effort is needed to implement the map and blend interpolant on the sphere although any interpolant to scattered data on a planar domain can be used in our general technique we use the multiquadric radial basis method

noninformative priors and frequentist risks of bayesian estimators of vector autoregressive models in this study we examine posterior properties and frequentist risks of bayesian estimators based on several noninformative priors in vector autoregressive var models we prove existence of the posterior distributions and posterior moments under a general class of priors using a variety of priors in this class we conduct numerical simulations of posteriors we find that in most examples bayesian estimators with a shrinkage prior on the var coefficients and the reference prior of yang and berger ann statist 22 1994 1195 on the var covariance matrix dominate mle bayesian estimators with the diffuse prior and bayesian estimators with the prior used in rats we also examine the informative minnesota prior and find that its performance depends on the nature of the data sample and on the tightness of the minnesota prior a tightly set minnesota prior is better when the data generating processes are similar to random walks but the shrinkage prior or constant prior can be better otherwise

smooth interpolation of scattered data by local thin plate splines abstract an algorithm and the corresponding computer program for solution of the scattered data interpolation problem is described given points x k y k f k k 1 n a locally defined function f x y which has the property f x k y k f k k 1 n is constructed the algorithm is based on a weighted sum of locally defined thin plate splines and yields an interpolation function which is differentiable the program is available from the author

regression based methods for using control variates in monte carlo experiments abstract methods based on linear regression provide an easy way to use the information in control variates to improve the efficiency with which certain features of the distributions of estimators and test statistics are estimated in monte carlo experiments we propose a new technique that allows these methods to be used when the quantities of interest are quantiles we also propose new ways to obtain approximately optimal control variates in many cases of interest these methods seem to work well in practice and can greatly reduce the number of replications required to obtain a given level of accuracy

bias assessment and reduction in linear error correction models abstract the small sample bias of the ordinary least squares coefficient estimator for dynamic regression models with innovation errors and lagged dependent and strongly exogenous explanatory variables is approximated through both small disturbance and large sample asymptotics results for the standard armax p 0 k model are obtained and also for such models under linear parameter constraints and variable transformations these approximations are then used to construct corrected estimators for the parameters of interest in higher order dynamic models including the empirically highly relevant linear error correction model by simulating two empirical cases the corrected estimators obtained via large sample asymptotics are shown to have more attractive location and efficiency properties than ordinary least squares

evolving space filling curves to distribute radial basis functions over an input space an evolutionary neural network training algorithm is proposed for radial basis function rbf networks the locations of basis function centers are not directly encoded in a genetic string but are governed by space filling curves whose parameters evolve genetically this encoding causes each group of codetermined basis functions to evolve to fit a region of the input space a network produced from this encoding is evaluated by training its output connections only networks produced by this evolutionary algorithm appear to have better generalization performance on the mackey glass time series than corresponding networks whose centers are determined by k means clustering

a method for interpolating scattered data based upon a minimum norm network a method for interpolating scattered data is described ctiven xi yi zi i 1 n a bivariate function s with continuous first order partial derivatives is defined which has the property that s xi yi zi i 1 n the method is based upon a triangulation of the domain and a curve network which has certain minimum pseudonorm properties algorithms and examples are included

a mirror image invariance for m estimators much research into the performance of econometric estimators utilizes monte carlo experiments either as a primary tool or as a check on the accuracy of first or higher order asymptotic approximations it is important to design efficient experiments and to extract all the information they provide but it is also important to avoid constructing experiments which give an overly optimistic view of the performance of an estimator or test this note gives a theorem concerning choice of covariate distribution which is relevant to both these considerations it is shown that when a subset of covariates x has a symmetric conditional distribution given the remaining covariates z a mirror image invariance applies to m estimators huber 1977 of the coefficients that measure the impact of x on the conditional distribution of the response variates namely that reversal of the signs of the coefficients causes the marginal2 sampling distributions of their m estimators to be reflected around the origin while the marginal sampling distributions of estimators of other parameters are unaffected it follows that when covariate distributions are symmetric results for one half of the parameter space can be deduced from results for the other half for example if bias b 3 and mse b p are respectively the bias and mean squared error assumed for the moment to exist of an estimator 3 when the true value of 3 is b then bias b 13 bias b 3 and mse b 3 mse b p it follows that bias o 13 0 and more generally the result implies that sampling distributions are symmetric when coefficients associated with symmetrically distributed covariates are zero this typically removes the n 1 2 term from the edgeworth expansion of the estimator and associated t statistics therefore choosing a symmetric covariate distribution in a monte carlo experiment may lead to a false view of the accuracy of first order approximations to distributions of certain estimators and test statistics the theorem highlights the potential sensitivity of the results of monte carlo experiments to the choice of covariate distribution the mirror image result and the associated symmetry result are unusual because they do not require strong assumptions about the distribution of the response variate in contrast to the symmetry results described by andrews 1986 and the mirror image invariance given by cryer nankervis and savin 1989 apart from the symmetry of the conditional distribution of a subset of covariates the main requirements are that the coefficients of interest 3 satisfy two related single index restrictions namely that they

second order conservative schemes and the entropy condition we consider numerical approximations to solutions of systems of hyperbolic conservation laws of the form au at af u ax 0 u e r andf r r smooth we show that conservative three point second order accurate methods cannot satisfy a local entropy inequality

new distribution theory for the estimation of structural break point in mean abstract based on the girsanov theorem this paper obtains the exact distribution of the maximum likelihood estimator of structural break point in a continuous time model the exact distribution is asymmetric and tri modal indicating that the estimator is biased these two properties are also found in the finite sample distribution of the least squares ls estimator of structural break point in the discrete time model suggesting the classical long span asymptotic theory is inadequate the paper then builds a continuous time approximation to the discrete time model and develops an in fill asymptotic theory for the ls estimator the in fill asymptotic distribution is asymmetric and tri modal and delivers good approximations to the finite sample distribution to reduce the bias in the estimation of both the continuous time and the discrete time models a simulation based method based on the indirect estimation ie approach is proposed monte carlo studies show that ie achieves substantial bias reductions

approximate bias correction in econometrics abstract this paper discusses methods for reducing the bias of consistent estimators that are biased in finite samples these methods are available whenever the bias function which relates the bias of the parameter estimates to the values of the parameters can be estimated by computer simulation or by some other method if so bias can be reduced by one full order in the sample size and in some cases that may not be unrealistic virtually eliminated unfortunately reducing bias may increase the variance or even the mean squared error of an estimator whether it does so depends on the shape of the bias function the results of the paper are illustrated by applying them to two problems estimating the autoregressive parameter in an ar 1 model with a constant term and estimating a logic model

a collocation meshless method based on local optimal point interpolation this paper deals with the use of the local optimal point interpolating lopi formula in solving partial differential equations pdes with a collocation method lopi is an interpolating formula constructed by localization of optimal point interpolation formulas that reproduces polynomials and verifies the delta kronecker property this scheme results in a truly meshless method that produces high quality output and accurate solutions

prepivoting to reduce level error of confidence sets summary approximate confidence sets for a parameter 0 may be obtained by referring a function of 0 and of the sample to an estimated quantile of that function s sampling distribution we call this function the root of the confidence set either asymptotic theory or bootstrap methods can be used to estimate the desired quantile when the root is not a pivot in the sense of classical statistics the actual level of the approximate confidence set may differ substantially from the intended level prepivoting is the transformation of a confidence set root by its estimated bootstrap cumulative distribution function prepivoting can be iterated bootstrap confidence sets generated from a root prepivoted one or more times have smaller error in level than do confidence sets based on the original root the first prepivoting is nearly equivalent to studentizing when that operation is appropriate further iterations of prepivoting make higher order corrections automatically

the finite sample effects of var dimensions on ols bias ols variance and minimum mse estimators vector autoregressions vars are important tools in time series analysis however relatively little is known about the finite sample behaviour of parameter estimators we address this issue by investigating ordinary least squares ols estimators given a data generating process that is a purely nonstationary first order var specifically we use monte carlo simulation and numerical optimisation to derive response surfaces for ols bias and variance in terms of var dimensions given correct specification and several types of over parameterisation of the model we include a constant and a constant and trend and introduce excess lags we then examine the correction factors that are required for the least squares estimator to attain the minimum mean squared error mse our results improve and extend one of the main finite sample multivariate analytical bias results of abadir hadri and tzavalis abadir k m hadri k tzavalis e 1999 the influence of var dimensions on estimator biases econometrica 67 163 181 generalise the univariate variance and mse findings of abadir abadir k m 1995 unbiased estimation as a solution to testing for random walks economics letters 47 263 268 to the multivariate setting and complement various asymptotic studies

cooperative competitive genetic evolution of radial basis function centers and widths for time series prediction in a radial basis function rbf network the rbf centers and widths can be evolved by a cooperative competitive genetic algorithm the set of genetic strings in one generation of the algorithm represents one rep network not a population of competing networks this leads to moderate computation times for the algorithm as a whole selection operates on individual rbfs rather than on whole networks selection therefore requires a genetic fitness function that promotes competition among rbfs which are doing nearly the same job while at the same time promoting cooperation among rbfs which cover different parts of the domain of the function to be approximated niche creation resulting from a fitness function of the form w sub i sup spl beta e w sub i sup spl beta 1 spl beta 2 can facilitate the desired cooperative competitive behavior the feasibility of the resulting algorithm to evolve networks of gaussian inverse multiquadric and thin plate spline rbfs is demonstrated by predicting the mackey glass time series for each type of rbf and for networks of 25 50 75 100 125 and 150 rbf units prediction errors for the evolved gaussian rbf networks are 50 70 lower than rbf networks obtained by k means clustering

one sided difference approximations for nonlinear conservation laws we analyze one sided or upwind finite difference approximations to hyperbolic partial differential equations and in particular nonlinear conservation laws second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions we show that no such stable approximation of order higher than two is possible these one sided schemes have desirable properties for shock calculations we show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance new and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles existence uniqueness and stability of discrete shocks numerical examples are given

estimating functions in indirect inference summary there are models for which the evaluation of the likelihood is infeasible in practice for these models the metropolis hastings acceptance probability cannot be easily computed this is the case for instance when only departure times from a g g 1 queue are observed and inference on the arrival and service distributions are required indirect inference is a method to estimate a parameter in models whose likelihood function does not have an analytical closed form but from which random samples can be drawn for fixed values of first an auxiliary model is chosen whose parameter can be directly estimated next the parameters in the auxiliary model are estimated for the original data leading to an estimate the parameter is also estimated by using several sampled data sets simulated from the original model for different values of the original parameter finally the parameter which leads to the best match to is chosen as the indirect inference estimate we analyse which properties an auxiliary model should have to give satisfactory indirect inference we look at the situation where the data are summarized in a vector statistic t and the auxiliary model is chosen so that inference on is drawn from t only under appropriate assumptions the asymptotic covariance matrix of the indirect estimators is proportional to the asymptotic covariance matrix of t and componentwise inversely proportional to the square of the derivative with respect to of the expected value of t we discuss how these results can be used in selecting good estimating functions we apply our findings to the queuing problem

application of the multiquadric method for numerical solution of elliptic partial differential equations abstract we have used the multiquadric mq approximation scheme for the solution of elliptic partial differential equations with dirichlet and or neumann boundary conditions the scheme has the advantage of using the data points in arbitrary locations with an arbitrary ordering two dimensional laplace poisson and biharmonic equations describing the various physical processes have been taken as the test examples the agreement is found to be very good between the computed and exact solutions the method also provides an excellent approximation with a curved boundary

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

bias in the estimation of the mean reversion parameter in continuous time models abstract it is well known that for continuous time models with a linear drift standard estimation methods yield biased estimators for the mean reversion parameter both in finite discrete samples and in large in fill samples in this paper we obtain two expressions to approximate the bias of the least squares maximum likelihood estimator of the mean reversion parameter in the ornstein uhlenbeck process with a known long run mean when discretely sampled data are available the first expression mimics the bias formula of marriott and pope 1954 for the discrete time model simulations show that this expression does not work satisfactorily when the speed of mean reversion is slow slow mean reversion corresponds to the near unit root situation and is empirically realistic for financial time series an improvement is made in the second expression where a nonlinear correction term is included into the bias formula it is shown that the nonlinear term is important in the near unit root situation simulations indicate that the second expression captures the magnitude the curvature and the non monotonicity of the actual bias better than the first expression