the time dependent quartic oscillator a feynman path integral approach the schrodinger equation with a time dependent quadratic plus quartic hamiltonian is considered a rigorous feynman path integral representation for its solution is given in terms of a well defined infinitedimensional oscillatory integral

path integration of the duffing rayleigh oscillator subject to harmonic and stochastic excitations this paper is focused on the duffing rayleigh oscillator subject to harmonic and stochastic excitations using path integration based on the gauss legendre integration scheme first applying methods of harmonic balance and multiple scales to the deterministic case the stabilities of the responses can be analyzed then the steady state periodic solution of probability density can be captured via path integration at the same time the changes of probability density induced by the intensities of harmonic and stochastic excitations are discussed in three cases

a semiparametric single index risk score across populations abstractwe consider a problem motivated by issues in nutritional epidemiology across diseases and populations in this area it is becoming increasingly common for diseases to be modeled by a single diet score such as the healthy eating index the mediterranean diet score etc for each disease and for each population a partially linear single index model is fit the partially linear aspect of the problem is allowed to differ in each population and disease however and crucially the single index itself having to do with the diet score is common to all diseases and populations and the nonparametrically estimated functions of the single index are the same up to a scale parameter using b splines with an increasing number of knots we develop a method to solve the problem and display its asymptotic theory an application to the nih aarp study of diet and health is described where we show the advantages of using multiple diseases and populations simultaneously rather than one at a time in understand

local asymptotics for polynomial spline regression in this paper we develop a general theory of local asymptotics for least squares estimates over polynomial spline spaces in a regression problem the polynomial spline spaces we consider include univariate splines tensor product splines and bivariate or multivariate splines on triangulations we establish asymptotic normality of the estimate and study the magnitude of the bias due to spline approximation the asymptotic normality holds uniformly over the points where the regression function is to be estimated and uniformly over a broad class of design densities error distributions and regression functions the bias is controlled by the minimum l norm of the error when the target regression function is approximated by a function in the polynomial spline space that is used to define the estimate the control of bias relies on the stability in l norm of l 2 projections onto polynomial spline spaces asymptotic normality of least squares estimates over polynomial or trigonometric polynomial spaces is also treated by the general theory in addition a preliminary analysis of additive models is provided

estimation and variable selection for generalized additive partial linear models we study generalized additive partial linear models proposing the use of polynomial spline smoothing for estimation of nonparametric functions and deriving quasi likelihood based estimators for the linear parameters we establish asymptotic normality for the estimators of the parametric components the procedure avoids solving large systems of equations as in kernel based procedures and thus results in gains in computational simplicity we further develop a class of variable selection procedures for the linear parameters by employing a nonconcave penalized quasi likelihood which is shown to have an asymptotic oracle property monte carlo simulations and an empirical example are presented for illustration

generalized partially linear single index models abstract the typical generalized linear model for a regression of a response y on predictors x z has conditional mean function based on a linear combination of x z we generalize these models to have a nonparametric component replacing the linear combination t 0x t 0z by 0 t 0x t 0z where 0 is an unknown function we call these generalized partially linear single index models gplsim the models include the single index models which have 0 0 using local linear methods we propose estimates of the unknown parameters 0 0 and the unknown function 0 and obtain their asymptotic distributions examples illustrate the models and the proposed estimation methodology

a path integration algorithm for stochastic structural dynamic systems a numerical integration scheme based on closed newton cotes formulas is first proposed for calculating the response statistics of a dynamic system under gaussian white noise excitation accurate system response results down to very low probability levels have been obtained by utilizing the high degree closed newton cotes formulas for the numerical integration algorithm the computational efficiency of the numerical integration scheme is found to be very high the proposed newton cotes scheme is also compared with the gauss legendre scheme in the existing literature it is found that the newton cotes algorithm is easy to develop because newton cotes quadrature uses values of the integrand at equally spaced abscissas and the values of the quadrature coefficients are easily calculated

doubly robust and efficient estimators for heteroscedastic partially linear single index models allowing high dimensional covariates we study the heteroscedastic partially linear single index model with an unspecified error variance function which allows for high dimensional covariates in both the linear and the single index components of the mean function we propose a class of consistent estimators of the parameters by using a proper weighting strategy an interesting finding is that the linearity condition which is widely assumed in the dimension reduction literature is not necessary for methodological or theoretical development it contributes only to the simplification of non optimal consistent estimation we also find that the performance of the usual weighted least square type of estimators deteriorates when the non parametric component is badly estimated however estimators in our family automatically provide protection against such deterioration in that the consistency can be achieved even if the baseline non parametric function is completely misspecified we further show that the most efficient estimator is a member of this family and can be easily obtained by using non parametric estimation properties of the estimators proposed are presented through theoretical illustration and numerical simulations an example on gender discrimination is used to demonstrate and to compare the practical performance of the estimators

correlated binary regression using a quadratic exponential model summary a model of quadratic exponential form is parameterized in terms of marginal means and pairwise correlations for the regression analysis of correlated binary data pseudo maximum likelihood methods using a special case termed the multiplicative model are proposed but are noted to be computationally unattractive if the blocks of correlated responses are at all large on the other hand score estimating functions for mean and correlation parameters are shown to be of a particularly simple form under the quadratic exponential family special cases of such estimating functions having attractive computational properties are identified and illustrated

estimation for a partial linear single index model in this paper we study the estimation for a partial linear single index model a two stage estimation procedure is proposed to estimate the link function for the single index and the parameters in the single index as well as the parameters in the linear component of the model asymptotic normality is established for both parametric components for the index a constrained estimating equation leads to an asymptotically more efficient estimator than existing estimators in the sense that it is of a smaller limiting variance the estimator of the nonparametric link function achieves optimal convergence rates and the structural error variance is obtained in addition the results facilitate the construction of confidence regions and hypothesis testing for the unknown parameters a simulation study is performed and an application to a real dataset is illustrated the extension to multiple indices is briefly sketched

a note on borel summability in this article nevanlinna s theorem about borel summability is improved moreover the borel summability of the asymptotic expansion in the method of stationary phase in infinitely many dimensions is discussed

identifiability of single index models and additive index models summary we provide a proof for the identifiability for both single index models and partially linear single index models assuming only the continuity of the regression function a condition much weaker than the differentiability conditions assumed in the existing literature our discussion is then extended to the identifiability of the additive index models

identification of non linear additive autoregressive models summary we propose a lag selection method for non linear additive autoregressive models that is based on spline estimation and the bayes information criterion the additive structure of the autoregression function is used to overcome the curse of dimensionality whereas the spline estimators effectively take into account such a structure in estimation a stepwise procedure is suggested to implement the method proposed a comprehensive monte carlo study demonstrates good performance of the method proposed and a substantial computational advantage over existing local polynomial based methods consistency of the lag selection method based on the bayes information criterion is established under the assumption that the observations are from a stochastic process that is strictly stationary and strongly mixing which provides the first theoretical result of this kind for spline smoothing of weakly dependent data

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

numerical meshfree path integration method for non linear dynamic systems abstract we present a new numerical meshfree path integration mpi method for non linear dynamic systems the obtained mpi method can be performed in the irregular computational domain and the probability density values of the random nodes in the domain can be calculated via the mpi method and the ordinary differential equations for the first and second order moments on the basis of gaussian closure method the piecewise linear interpolation based on adaptive least squares is utilized as a post processor to approximate the probability density values on arbitrary positions the good performance of the resulting method is finally shown in the numerical examples by using three specific non linear dynamic systems duffing oscillator subjected to both harmonic and stochastic excitations duffing rayleigh oscillator subjected to both harmonic and stochastic excitations and chen system driven by three different gaussian white noises

direct estimation of the index coefficient in a single index model single index modeling is widely applied in for example econometric studies as a compromise between too restrictive parametric models and flexible but hardly estimable purely nonparametric models by such modeling the statistical analysis usually focuses on estimating the index coefficients the average derivative estimator ade of the index vector is based on the fact that the average gradient of a single index function f x t is proportional to the index vector unfortunately a straightforward application of this idea meets the so called curse of dimensionality problem if the dimensionality d of the model is larger than 2 however prior information about the vector can be used for improving the quality of gradient estimation by extending the weighting kernel in a direction of small directional derivative the method proposed in this paper consists of such iterative improvements of the original ade the whole procedure requires at most 2 log n iterations and the resulting estimator is n consistent under relatively mild assumptions on the model independently of the dimensionality d

efficiency of generalized estimating equations for binary responses summary using standard correlation bounds we show that in generalized estimation equations gees the so called working correlation matrix r for analysing binary data cannot in general be the true correlation matrix of the data methods for estimating the correlation param eter in current gee software for binary responses disregard these bounds to show that the gee applied on binary data has high efficiency we use a multivariate binary model so that the covariance matrix from estimating equation theory can be compared with the inverse fisher information matrix but r should be viewed as the weight matrix and it should not be confused with the correlation matrix of the binary responses we also do a comparison with more general weighted estimating equations by using a matrix cauchy schwarz inequality our analysis leads to simple rules for the choice of in an exchangeable or autoregressive ar 1 weight matrix r based on the strength of dependence between the binary variables an example is given to illustrate the assessment of dependence and choice of

the method of stationary phase for oscillatory integrals on hilbert spaces we develop the method of stationary phase for the normalized oscillatory integral on hilbert space giving borel summable expansions the developments that we obtain hold for more general situations than the ones of previous papers on the same subject

profile kernel versus backfitting in the partially linear models for longitudinal clustered data summary we study the profile kernel and backfitting methods in partially linear models for clustered longitudinal data for independent data despite the potential root n inconsist ency of the backfitting estimator noted by rice 1986 the two estimators have the same asymptotic variance matrix as shown by opsomer ruppert 1999 in this paper theoretical comparisons of the two estimators for multivariate responses are investigated we show that for correlated data backfitting often produces a larger asymptotic variance than the profile kernel method that is for clustered data in addition to its bias problem the backfitting estimator does not have the same asymptotic efficiency as the profile kernel estimator consequently the common practice of using the backfitting method to compute profile kernel estimates is no longer advised we illustrate this in detail by following zeger diggle 1994 and lin carroll 2001 with a working independence covariance structure for nonparametric estimation and a correlated covariance structure for para metric estimation numerical performance of the two estimators is investigated through a simulation study their application to an ophthalmology dataset is also described

optimal smoothing in single index models single index models generalize linear regression they have applications to a variety of fields such as discrete choice analysis in econometrics and dose response models in biometrics where high dimensional regression models are often employed single index models are similar to the first step of projection pursuit regression a dimension reduction method in both cases the orientation vector can be estimated root n consistently even if the unknown univariate function or nonparametric link function is assumed to come from a large smoothness class however as we show in the present paper the similarities end there in particular the amount of smoothing necessary for root n consistent orientation estimation is very different in the two cases we suggest a simple empirical rule for selecting the bandwidth appropriate to single index models this rule is studied in a small simulation study and an application in binary response models

penalized spline estimation for partially linear single index models single index models are potentially important tools for multivariate nonparametric regression they generalize linear regression by replacing the linear combination 0tx with a nonparametric component 0 0tx where 0 is an unknown univariate link function by reducing the dimensionality from that of a general covariate vector x to a univariate index 0tx single index models avoid the so called curse of dimensionality we propose penalized spline p spline estimation of 0 in partially linear single index models where the mean function has the form 0 0tx 0tz the p spline approach offers a number of advantages over other fitting methods for single index models all parameters in the p spline single index model can be estimated simultaneously by penalized nonlinear least squares as a direct least squares fitting method our approach is rapid and computationally stable standard nonlinear least squares software can be used moreover joint inference for 0 0 and 0 is possible by

variable selection and function estimation in additive nonparametric regression using a data based prior abstract a hierarchical bayesian approach is proposed for variable selection and function estimation in additive nonparametric gaussian regression models and additive nonparametric binary regression models the prior for each component function is an integrated wiener process resulting in a posterior mean estimate that is a cubic smoothing spline each of the explanatory variables is allowed to be in or out of the model and the regression functions are estimated by model averaging to allow variable selection and model averaging data based priors are used for the smoothing parameter and the slope at 0 of each component function a two step markov chain monte carlo method is used to efficiently obtain the data based prior and to carry out variable selection and function estimation it is shown by simulation that significant improvements in the function estimators can be obtained over an approach that estimates all the unknown functions simultaneously the methodology is illustrated for a binary regressio

local averaged path integration method approach for nonlinear dynamic systems abstract traditional path integration method always has a limited time step length whose minimum is based on the subdivision degree of integral space in order to numerically analyze a smaller time step probability evolution without significant increase in computing time a new local averaged path integration lapi method is investigated the basic idea of this method is giving a local averaged value of transition probability density function tpdf then the numerical integration of chapman kolmogorov ck equation to the local averaged probability density function pdf will be calculated in lower space subdivision degree second order partial derivative of pdf can be used to correct the deviation between local averaged pdf and the real one it is demonstrated that the proposed method can give a more efficient calculation compared with traditional gauss legendre path integration glpi in the case of our method smaller time step value is permitted with the same space subdivision degree and smaller space subdivision degree is needed to keep the accuracy of stationary pdf with the same time step value this merits also exist when analyzing transient pdf which leads to a better capability of capturing the evolution of the pdf

smoothing parameter selection in nonparametric regression using an improved akaike information criterion summary many different methods have been proposed to construct nonparametric estimates of a smooth regression function including local polynomial convolution kernel and smoothing spline estimators each of these estimators uses a smoothing parameter to control the amount of smoothing performed on a given data set in this paper an improved version of a criterion based on the akaike information criterion aic termed aicc is derived and examined as a way to choose the smoothing parameter unlike plug in methods aicc can be used to choose smoothing parameters for any linear smoother including local quadratic and smoothing spline estimators the use of aicc avoids the large variability and tendency to undersmooth compared with the actual minimizer of average squared error seen when other classical approaches such as generalized cross validation or the aic are used to choose the smoothing parameter monte carlo simulations demonstrate that the aicc based smoothing parameter is competitive with a plug in method assuming that one exists when the plug in method works well but also performs well when the plug in approach fails or is unavailable

semiparametric regression using variational approximations semiparametric regression offers a flexible framework for modeling nonlinear relationships between a response and covariates a prime example are generalized additive models gams where splines s

empirical likelihood confidence regions in a partially linear single index model summary empirical likelihood based inference for the parameters in a partially linear singleindex model is investigated unlike existing empirical likelihood procedures for other simpler models if there is no bias correction the limit distribution of the empirical likelihood ratio cannot be asymptotically tractable to attack this difficulty we propose a bias correction to achieve the standard x2 limit the bias corrected empirical likelihood ratio shares some of the desired features of the existing least squares method the estimation of the parameters is not needed when estimating nonparametric functions in the model undersmoothing for ensuring in consistency of the estimator of the parameters is avoided the bias corrected empirical likelihood is self scale invariant and no plug in estimator for the limiting variance is needed furthermore since the index is of norm 1 we use this constraint as information to increase the accuracy of the confidence regions smaller regions at the same nominal level as a by product our approach of using bias correction may also shed light on nonparametric estimation in model checking for other semiparametric regression models a simulation study is carried out to assess the performance of the bias corrected empirical likelihood an application to a real data set is illustrated

generalized partially linear single index model for zero inflated count data count data often arise in biomedical studies while there could be a special feature with excessive zeros in the observed counts the zero inflated poisson model provides a natural approach to accounting for the excessive zero counts in the semiparametric framework we propose a generalized partially linear single index model for the mean of the poisson component the probability of zero or both we develop the estimation and inference procedure via a profile maximum likelihood method under some mild conditions we establish the asymptotic properties of the profile likelihood estimators the finite sample performance of the proposed method is demonstrated by simulation studies and the new model is illustrated with a medical care dataset

on the asymptotics of penalized splines summavry we study the asymptotic behaviour of penalized spline estimators in the univariate case we use b splines and a penalty is placed on mth order differences of the coefficients the number of knots is assumed to converge to infinity as the sample size increases we show that penalized splines behave similarly to nadaraya watson kernel estimators with equivalent kernels depending upon m the equivalent kernels we obtain for penalized splines are the same as those found by silverman for smoothing splines the asymptotic distribution of the penalized spline estimator is gaussian and we give simple expressions for the asymptotic mean and variance provided that it is fast enough the rate at which the number of knots converges to infinity does not affect the asymptotic distribution the optimal rate of convergence of the penalty parameter is given penalized splines are not design adaptive

robustified maximum likelihood estimation in generalized partial linear mixed model for longitudinal data summary in this article we study the robust estimation of both mean and variance components in generalized partial linear mixed models based on the construction of robustified likelihood function under some regularity conditions the asymptotic properties of the proposed robust estimators are shown some simulations are carried out to investigate the performance of the proposed robust estimators just as expected the proposed robust estimators perform better than those resulting from robust estimating equations involving conditional expectation like sinha 2004 journal of the american statistical association99 451 460 and qin and zhu 2007 journal of multivariate analysis98 1658 1683 in the end the proposed robust method is illustrated by the analysis of a real data set

the efm approach for single index models single index models are natural extensions of linear models and circumvent the so called curse of dimensionality they are becoming increasingly popular in many scientific fields including biostatistics medicine economics and financial econometrics estimating and testing the model index coefficients is one of the most important objectives in the statistical analysis however the commonly used assumption on the index coefficients 1 represents a nonregular problem the true index is on the boundary of the unit ball in this paper we introduce the efm approach a method of estimating functions to study the single index model the procedure is to first relax the equality constraint to one with d 1 components of lying in an open unit ball and then to construct the associated d 1 estimating functions by projecting the score function to the linear space spanned by the residuals with the unknown link being estimated by kernel estimating functions the root n consistency and asymptotic normality for the estimator obtained from solving the resulting estimating equations are achieved and a wilks type theorem for testing the index is demonstrated a noticeable result we obtain is that our estimator for has smaller or equal limiting variance than the estimator of carroll et al j amer statist assoc 92 1997 447 489 a fixed point iterative scheme for computing this estimator is proposed this algorithm only involves one dimensional nonparametric smoothers thereby avoiding the data sparsity problem caused by high model dimensionality numerical studies based on simulation and on applications suggest that this new estimating system is quite powerful and easy to implement

a dynamic structure for high dimensional covariance matrices and its application in portfolio allocation abstractestimation of high dimensional covariance matrices is an interesting and important research topic in this article we propose a dynamic structure and develop an estimation procedure for high dimensional covariance matrices asymptotic properties are derived to justify the estimation procedure and simulation studies are conducted to demonstrate its performance when the sample size is finite by exploring a financial application an empirical study shows that portfolio allocation based on dynamic high dimensional covariance matrices can significantly outperform the market from 1995 to 2014 our proposed method also outperforms portfolio allocation based on the sample covariance matrix the covariance matrix based on factor models and the shrinkage estimator of covariance matrix supplementary materials for this article are available online

semiparametric gee analysis in partially linear single index models for longitudinal data in this article we study a partially linear single index model for longitudinal data under a general framework which includes both the sparse and dense longitudinal data cases a semiparametric estimation method based on a combination of the local linear smoothing and generalized estimation equations gee is introduced to estimate the two parameter vectors as well as the unknown link function under some mild conditions we derive the asymptotic properties of the proposed parametric and nonparametric estimators in different scenarios from which we find that the convergence rates and asymptotic variances of the proposed estimators for sparse longitudinal data would be substantially different from those for dense longitudinal data we also discuss the estimation of the covariance or weight matrices involved in the semiparametric gee method furthermore we provide some numerical studies including monte carlo simulation and an empirical application to illustrate our methodology and theory

variance function partially linear single index models we consider heteroscedastic regression models where the mean function is a partially linear single index model and the variance function depends upon a generalized partially linear single index model we do not insist that the variance function depend only upon the mean function as happens in the classical generalized partially linear single index model we develop efficient and practical estimation methods for the variance function and for the mean function asymptotic theory for the parametric and nonparametric parts of the model is developed simulations illustrate the results an empirical example involving ozone levels is used to further illustrate the results and is shown to be a case where the variance function does not depend upon the mean function

feynman path integrals and the trace formula for the schrodinger operators we study schrodinger operators of the form h frac h 2 2 delta frac 1 2 x cdot a 2 x v x on d wherea2 is a strictly positive symmetricd d matrix andv x is a continuous real function which is the fourier transform of a bounded measure if n are the eigenvalues ofh we show that the theta function theta t sum limits n exp left frac i h t lambda n right is explicitly expressible in terms of infinite dimensional oscillatory integrals feynman path integrals over the hilbert space of closed trajectories we use these explicit expressions to give the asymptotic behaviour of t for smallh in terms of classical periodic orbits thus obtaining a trace formula for the schrodinger operators this then yields an asymptotic expansion of the spectrum ofh in terms of the periodic orbits of the corresponding classical mechanical system these results extend to the physical case the recent work on poisson and trace formulae for compact manifolds

semiparametric least squares sls and weighted sls estimation of single index models abstract for the class of single index models i construct a semiparametric estimator of coefficients up to a multiplicative constant that exhibits 1 n consistency and asymptotic normality this class of models includes censored and truncated tobit models binary choice models and duration models with unobserved individual heterogeneity and random censoring i also investigate a weighting scheme that achieves the semiparametric efficiency bound

varying coefficient models and basis function approximations for the analysis of repeated measurements summary a global smoothing procedure is developed using basis function approximations for estimating the parameters of a varying coefficient model with repeated measurements inference procedures based on a resampling subject bootstrap are proposed to construct confidence regions and to perform hypothesis testing conditional biases and variances of our estimators and their asymptotic consistency are developed explicitly finite sample properties of our procedures are investigated through a simulation study application of the proposed approach is demonstrated through an example in epidemiology in contrast to the existing methods this approach applies whether or not the covariates are timeinvariant and does not require binning of the data when observations are sparse at distinct observation times

parameter estimation of partial differential equation models partial differential equation pde models are commonly used to model complex dynamic systems in applied sciences such as biology and finance the forms of these pde models are usually proposed by experts based on their prior knowledge and understanding of the dynamic system parameters in pde models often have interesting scientific interpretations but their values are often unknown and need to be estimated from the measurements of the dynamic system in the presence of measurement errors most pdes used in practice have no analytic solutions and can only be solved with numerical methods currently methods for estimating pde parameters require repeatedly solving pdes numerically under thousands of candidate parameter values and thus the computational load is high in this article we propose two methods to estimate parameters in pde models a parameter cascading method and a bayesian approach in both methods the underlying dynamic process modeled with the pde model is represented via basis function

on the numerical evaluation of fredholm determinants some significant quantities in mathematics and physics are most naturally expressed as the fredholm determinant of an integral operator most notably many of the distribution functions in random matrix theory though their numerical values are of interest there is no systematic numerical treatment of fredholm determinants to be found in the literature instead the few numerical evaluations that are available rely on eigenfunction expansions of the operator if expressible in terms of special functions or on alternative numerically more straightforwardly accessible analytic expressions e g in terms of painleve transcendents that have masterfully been derived in some cases in this paper we close the gap in the literature by studying projection methods and above all a simple easily implementable general method for the numerical evaluation of fredholm determinants that is derived from the classical nystrom method for the solution of fredholm equations of the second kind using gauss legendre or clenshaw curtis as the underlying quadrature rule we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel in particular we get exponential convergence for analytic kernels which are typical in random matrix theory the application of the method to the distribution functions of the gaussian unitary ensemble gue in the bulk scaling limit and the edge scaling limit is discussed in detail after extending the method to systems of integral operators we evaluate the two point correlation functions of the more recently studied airy and airy 1 processes

local asymptotics for regression splines and confidence regions in this paper we study the local behavior of regression splines in particular explicit expressions for the asymptotic pointwise bias and variance of regression splines are obtained in addition asymptotic normality for regression splines is established leading to the construction of approximate confidence intervals and confidence bands for the regression function

an improved path integration method for nonlinear systems under poisson white noise excitation abstract in order to overcome some unsatisfactory trends and limitations of the traditional path integration pi method for poisson white noise a novel pi method is proposed in this paper which includes two improved schemes the first one is a new transition probability density function tpdf approximation which considers the randomness of the impulse happening time during each time intervals the second one is a transformation of chapman kolmogorov ck equation by a variable substitution instead of directly using it whose numerical calculation is based on the back stepping runge kutta scheme and the triangulation based interpolation monte carlo simulations mcs are utilized to measure the accuracy of the improved algorithm with three illustrative nonlinear systems the results show that compared with the traditional pi method the improved pi method can give a more accurate description of the tpdf values and provide more precise stationary probability density function pdf results whenever the mean arrival rate is large or small the improved algorithm has a wider range of choices in time interval values to maintain the accuracy of stationary pdf results besides it is discovered that cubic interpolation deserves to be applied in the improved pi method more than linear and natural interpolations

a multiple index model and dimension reduction dimension reduction can be used as an initial step in statistical modeling further specification of model structure is imminent and important when the reduced dimension is still greater than 1 in this article we investigate one method of specification that involves separating the linear component from the nonlinear components leading to further dimension reduction in the unknown link function and thus better estimation and easier interpretation of the model the specified model includes the popular econometric multiple index model and the partially linear single index model as its special cases a criterion is developed to validate the model specification an algorithm is proposed to estimate the model directly asymptotic distributions for the estimators of the parameters and the nonparametric link function are derived air pollution data in chicago are used to illustrate the modeling procedure and to demonstrate its advantages over the existing dimension reduction approaches

feynman integrals and fredholm determinants in this article simple properties of feynman integrals such as the translation invariance fubini theorem and cameron martin formula are used in order to derive a product formula for fredholm determinants this formula enables us to compute explicitly some of such determinants as well as the index of the related fredholm operators some examples are done which have applications in the semiclassical approximation 0

study of the duffing rayleigh oscillator subject to harmonic and stochastic excitations by path integration in this paper the duffing rayleigh oscillator subject to harmonic and stochastic excitations is investigated via path integration based on the gauss legendre integration formula the method can successfully capture the steady state periodic solution of probability density function this path integration method using the periodicity of the coefficient of associated fokker planck kolmogorov equation is extended to deal with the averaged stationary probability density and is efficient to computation meanwhile the changes of probability density caused by the intensities of harmonic and stochastic excitations are discussed in three cases through the instantaneous probability density and the averaged stationary probability density

bayesian smoothing and regression splines for measurement error problems in the presence of covariate measurement error estimating a regression function nonparametrically is extremely difficult the problem being related to deconvolution various frequentist approaches exist for this problem but to date there has been no bayesian treatment in this article we describe bayesian approaches to modeling a flexible regression function when the predictor variable is measured with error the regression function is modeled with smoothing splines and regression p splines two methods are described for exploration of the posterior the first called the iterative conditional modes icm is only partially bayesian icm uses a componentwise maximization routine to find the mode of the posterior it also serves to create starting values for the second method which is fully bayesian and uses markov chain monte carlo mcmc techniques to generate observations from the joint posterior distribution use of the mcmc approach has the advantage that interval estimates that directly model and

adapting for the missing link we consider the fitting of generalized linear models in wich the link function is assumed to be unknown and propose the following computational method first estimate regression coefficients using the canonical link then estimate the link via a kernel smoother treating the direction in the predictor space determined by the regression coefficients as known then reestimate the direction using the estimated link and alternate between these two steps we show that under fairly general conditions n 1 2 consistent estimates of the direction are obtained a small monte carlo study is presented

spline based tests in survival analysis this paper examines a method for testing hypotheses on covariate effects in a proportional hazards model and also on how effects change over time in regression analysis of survival data the technique used is very general and can be applied to testing many other aspects of parametric and semiparametric models the basic idea is to formulate a flexible parametric alternative using fixed knot splines together with a penalty function that penalizes noisy alternatives more than smooth ones to focus the power of the tests toward smooth alternatives the test statistics are the analogs of ordinary likelihood based statistics only computed from a penalized likelihood formed by subtracting the penalty function from the ordinary log likelihood large sample approximations to the distributions are found when the number of knots is held fixed as the sample size increases numerical results suggest these approximations may be adequate with moderate sized samples

estimating the interest rate term structure of corporate debt with a semiparametric penalized spline model this article provides a new methodology for estimating the term structure of corporate debt using a semiparametric penalized spline model the method is applied to a case study of at t bonds typically very few data are available on individual corporate bond prices too little to find a nonparametric estimate of term structure from these bonds alone this problem is solved by borrowing strength from treasury bond data more specifically we combine a nonparametric model for the term structure of treasury bonds with a parametric component for the credit spread our methodology generalizes the work of fisher nychka and zervos in several ways first their model was developed for treasury bonds only and cannot be applied directly to corporate bonds second we more fully investigate the problem of choosing the smoothing parameter a problem that is complicated because the forward rate is the derivative log d t where the discount function d is the function fit to the data in our case study estimatio

miscellanea on the optimal amount of smoothing in penalised spline regression summary the optimal amount of smoothing in penalised spline regression is investigated in particular a simple closed form approximation to the optimal smoothing parameter is derived comparisons with its exact counterpart show it to be a useful starting point for measuring the optimal amount of smoothing in penalised spline regression it also lends itself to the development of quick and simple rules for automatic smoothing parameter selection

on partial sufficient dimension reduction with applications to partially linear multi index models partial dimension reduction is a general method to seek informative convex combinations of predictors of primary interest which includes dimension reduction as its special case when the predictors in the remaining part are constants in this article we propose a novel method to conduct partial dimension reduction estimation for predictors of primary interest without assuming that the remaining predictors are categorical to this end we first take the dichotomization step such that any existing approach for partial dimension reduction estimation can be employed then we take the expectation step to integrate over all the dichotomic predictors to identify the partial central subspace as an example we use the partially linear multi index model to illustrate its applications for semiparametric modeling simulations and real data examples are given to illustrate our methodology

variable selection in semiparametric regression modeling in this paper we are concerned with how to select significant variables in semiparametric modeling variable selection for semiparametric regression models consists of two components model selection for nonparametric components and select significant variables for parametric portion thus it is much more challenging than that for parametric models such as linear models and generalized linear models because traditional variable selection procedures including stepwise regression and the best subset selection require model selection to nonparametric components for each submodel this leads to very heavy computational burden in this paper we propose a class of variable selection procedures for semiparametric regression models using nonconcave penalized likelihood the newly proposed procedures are distinguished from the traditional ones in that they delete insignificant variables and estimate the coefficients of significant variables simultaneously this allows us to establish the sampling properties of the resulting estimate we first establish the rate of convergence of the resulting estimate with proper choices of penalty functions and regularization parameters we then establish the asymptotic normality of the resulting estimate and further demonstrate that the proposed procedures perform as well as an oracle procedure semiparametric generalized likelihood ratio test is proposed to select significant variables in the nonparametric component we investigate the asymptotic behavior of the proposed test and demonstrate its limiting null distribution follows a chi squared distribution which is independent of the nuisance parameters extensive monte carlo simulation studies are conducted to examine the finite sample performance of the proposed variable selection procedures