a direct approach to the exit problem this paper considers the problem of exit for a dynamical system driven by small white noise from the domain of attraction of a stable state a direct singular perturbation analysis of the forward equation is presented based on kramers approach in which the solution to the stationary fokker planck equation is constructed for a process with absorption at the boundary and a source at the attractor in this formulation the boundary and matching conditions fully determine the uniform expansion of the solution without resorting to external selection criteria for the expansion coefficients such as variational principles or the lagrange identity as in our previous theory the exit density and the mean first passage time to the boundary are calculated from the solution of the stationary fokker planck equation as the probability current density and as the inverse of the total flux on the boundary respectively as an application a uniform expansion is constructed for the escape rate in kramers problem o

splines as local smoothers a smoothing spline is a nonparametric curve estimate that is defined as the solution to a minimization problem one problem with this representation is that it obscures the fact that a spline like most other nonparametric estimates is a local weighted average of the observed data this property has been used extensively to study the limiting properties of kernel estimates and it is advantageous to apply similar techniques to spline estimates although equivalent kernels have been identified for a smoothing spline these functions are either not accurate enough for asymptotic approximations or are restricted to equally spaced points this paper extends this previous work to understand a spline estimate s local properties it is shown that the absolute value of the spline weight function decreases exponentially away from its center this result is not asymptotic the only requirement is that the empirical distribution of the observation points be sufficiently close to a continuous distribution with a strictly positive density function these bounds are used to derive the asymptotic form for the bias and variance of a first order smoothing spline estimate the arguments leading to this result can be easily extended to higher order splines

analytical approach to the stochastic fitzhugh nagumo system and coherence resonance we consider the fitzhugh nagumo system under the influence of white gaussian noise in the excitable regime we present an analytical approximation in the limit of fast activator time scale marginal probability densities of a reduced system and dynamical quantities such as the pulse rate are found and the mean interspike interval and its relative standard deviation are investigated the latter quantities allow a quantitative description of the phenomenon of coherence resonance as comparisons with simulations show

solution of the boundary value problem for optimal escape in continuous stochastic systems and maps topologies of invariant manifolds and optimal trajectories are investigated in stochastic continuous systems and maps a topological method is introduced that simplifies the solution of boundary value problems the activation energy is calculated as a function of a set of parameters characterizing the initial conditions of the escape path the method is applied explicitly to compute the optimal escape path and the activation energy for a variety of dynamical systems and maps

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

functional parallel factor analysis for functions of one and two dimensional arguments parallel factor analysis parafac is a useful multivariate method for decomposing three way data that consist of three different types of entities simultaneously this method estimates trilinear components each of which is a low dimensional representation of a set of entities often called a mode to explain the maximum variance of the data functional parafac permits the entities in different modes to be smooth functions or curves varying over a continuum rather than a collection of unconnected responses the existing functional parafac methods handle functions of a one dimensional argument e g time only in this paper we propose a new extension of functional parafac for handling three way data whose responses are sequenced along both a two dimensional domain e g a plane with x and y axis coordinates and a one dimensional argument technically the proposed method combines parafac with basis function expansion approximations using a set of piecewise quadratic finite element basis functions for estimating two dimensional smooth functions and a set of one dimensional basis functions for estimating one dimensional smooth functions in a simulation study the proposed method appeared to outperform the conventional parafac we apply the method to eeg data to demonstrate its empirical usefulness

automatic estimation of multivariate spectra via smoothing splines summary the classical method for estimating the spectral density of a multivariate time series is first to calculate the periodogram and then to smooth it to obtain a consistent estimator typically to ensure the estimate is positive definite all the elements of the periodogram are smoothed the same way there are however many situations for which different components of the spectral matrix have different degrees of smoothness we propose a bayesian approach that uses markov chain monte carlo techniques to fit smoothing splines to each component real and imaginary of the cholesky decomposition of the periodogram matrix the spectral estimator is then obtained by reconstructing the spectral estimator from the smoothed cholesky decomposition components our technique produces an automatically smoothed spectral matrix estimator along with samples from the posterior distributions of the parameters to facilitate inference

on gibbs sampling for state space models summary we show how to use the gibbs sampler to carry out bayesian inference on a linear state space model with errors that are a mixture of normals and coefficients that can switch over time our approach simultaneously generates the whole of the state vector given the mixture and coefficient indicator variables and simultaneously generates all the indicator variables conditional on the state vectors the states are generated efficiently using the kalman filter we illustrate our approach by several examples and empirically compare its performance to another gibbs sampler where the states are generated one at a time the empirical results suggest that our approach is both practical to implement and dominates the gibbs sampler that generates the states one at a time

hierarchical mixtures of experts and the em algorithm we present a tree structured architecture for supervised learning the statistical model underlying the architecture is a hierarchical mixture model in which both the mixture coefficients and the mixture components are generalized linear models glim s learning is treated as a maximum likelihood problem in particular we present an expectation maximization em algorithm for adjusting the parameters of the architecture we also develop an on line learning algorithm in which the parameters are updated incrementally comparative simulation results are presented in the robot dynamics domain

a penalized spline approach to functional mixed effects model analysis summary in this work we propose penalized spline based methods for functional mixed effects models with varying coefficients we decompose longitudinal outcomes as a sum of several terms a population mean function covariates with time varying coefficients functional subject specific random effects and residual measurement error processes using penalized splines we propose nonparametric estimation of the population mean function varying coefficient random subject specific curves and the associated covariance function which represents between subject variation and the variance function of the residual measurement errors which represents within subject variation proposed methods offer flexible estimation of both the population level and subject level curves in addition decomposing variability of the outcomes as a between subject and a within subject source is useful in identifying the dominant variance component therefore optimally model a covariance function we use a likelihood based method to select multiple smoothing parameters furthermore we study the asymptotics of the baseline p spline estimator with longitudinal data we conduct simulation studies to investigate performance of the proposed methods the benefit of the between and within subject covariance decomposition is illustrated through an analysis of berkeley growth data where we identified clearly distinct patterns of the between and within subject covariance functions of children s heights we also apply the proposed methods to estimate the effect of anti hypertensive treatment from the framingham heart study data

local spectral analysis via a bayesian mixture of smoothing splines in many practical problems time series are realizations of nonstationary random processes these processes can often be modeled as processes with slowly changing dynamics or as piecewise stationary processes in these cases various approaches to estimating the time varying spectral density have been proposed our approach in this article is to estimate the log of the dahlhaus local spectrum using a bayesian mixture of splines the basic idea of our approach is to first partition the data into small sections we then assume that the log spectral density of the evolutionary process in any given partition is a mixture of individual log spectra we use a mixture of smoothing splines model with time varying mixing weights to estimate the evolutionary log spectrum the mixture model is fit using markov chain monte carlo techniques that yield estimates of the log spectra of the individual subsections in addition to an estimate of the local log spectral density the method yields pointwise credible intervals

on convergence rates of mixtures of polynomial experts in this letter we consider a mixture of experts structure where m experts are mixed with each expert being related to a polynomial regression model of order k we study the convergence rate of the maximum likelihood estimator in terms of how fast the hellinger distance of the estimated density converges to the true density when the sample size n increases the convergence rate is found to be dependent on both m and k while certain choices of m and k are found to produce near optimal convergence rates

nonparametric regression using bayesian variable selection abstract this paper estimates an additive model semiparametrically while automatically selecting the significant independent variables and the appropriate power transformation of the dependent variable the nonlinear variables are modeled as regression splines with significant knots selected from a large number of candidate knots the estimation is made robust by modeling the errors as a mixture of normals a bayesian approach is used to select the significant knots the power transformation and to identify outliers using the gibbs sampler to carry out the computation empirical evidence is given that the sampler works well on both simulated and real examples and that in the univariate case it compares favorably with a kernel weighted local linear smoother the variable selection algorithm in the paper is substantially faster than previous bayesian variable selection algorithms

approximation of conditional densities by smooth mixtures of regressions this paper shows that large nonparametric classes of conditional multivariate densities can be approximated in the kullback leibler distance by different specifications of finite mixtures of normal regressions in which normal means and variances and mixing probabilities can depend on variables in the conditioning set covariates these models are a special case of models known as mixtures of experts in statistics and computer science literature flexible specifications include models in which only mixing probabilities modeled by multinomial logit depend on the covariates and in the univariate case models in which only means of the mixed normals depend flexibly on the covariates modeling the variance of the mixed normals by flexible functions of the covariates can weaken restrictions on the class of the approximable densities obtained results can be generalized to mixtures of general location scale densities rates of convergence and easy to interpret bounds are also obtained for different model specifications these approximation results can be useful for proving consistency of bayesian and maximum likelihood density estimators based on these models the results also have interesting implications for applied researchers

bayesian regression with multivariate linear splines summary we present a bayesian analysis of a piecewise linear model constructed using basis functions which generalizes the univariate linear spline to higher dimensions prior distributions are adopted on both the number and the locations of the splines which leads to a model averaging approach to prediction with predictive distributions that take into account model uncertainty conditioning on the data produces a bayes local linear model with distributions on both predictions and local linear parameters the method is spatially adaptive and covariate selection is achieved by using splines of lower dimension than the data

data driven bandwidth selection in local polynomial fitting variable bandwidth and spatial adaptation when estimating a mean regression function and its derivatives locally weighted least squares regression has proven to be a very attractive technique the present paper focuses on the important issue of how to select the smoothing parameter or bandwidth in the case of estimating curves with a complicated structure a variable bandwidth is desirable furthermore the bandwidth should be indicated by the data themselves recent developments in nonparametric smoothing techniques inspired us to propose such a data driven bandwidth selection procedure which can be used to select both constant and variable bandwidths the idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator the procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives the resulting estimation procedure has the necessary flexibility for capturing complicated shapes of curves this is illustrated via a large variety of testing examples including examples with a large spatial variability the results are also compared with wavelet thresholding techniques and it seems that our results are at least comparable i e local polynomial regression using our data driven variable bandwidth has spatial adaptation properties that are similar to wavelets

noise induced escape in an excitable system we consider the stochastic dynamics of escape in an excitable system the fitzhugh nagumo fhn neuronal model for different classes of excitability we discuss first the threshold structure of the fhn model as an example of a system without a saddle state we then develop a nonlinear nonlocal stability approach based on the theory of large fluctuations including a finite noise correction to describe noise induced escape in the excitable regime we show that the threshold structure is revealed via patterns of most probable optimal fluctuational paths the approach allows us to estimate the escape rate and the exit location distribution we compare the responses of a monostable resonator and monostable integrator to stochastic input signals and to a mixture of periodic and stochastic stimuli unlike the commonly used local analysis of the stable state our nonlocal approach based on optimal paths yields results that are in good agreement with direct numerical simulations of the langevin equation

fuzzy ideals in semigroups abstract the set of all fuzzy ideals of a semigroup is shown to be a convex set having the crisp ideals as extreme points some connections between fuzzy ideals and green s relations are discussed

hybrid adaptive splines abstract an adaptive spline method for smoothing is proposed that combines features from both regression spline and smoothing spline approaches one of its advantages is the ability to vary the amount of smoothing in response to the inhomogeneous curvature of true functions at different locations this method can be applied to many multivariate function estimation problems which is illustrated by an application to smoothing temperature data on the globe the method s performance in a simulation study is found to be comparable to the wavelet shrinkage methods proposed by donoho and johnstone the problem of how to count the degrees of freedom for an adaptively chosen set of basis functions is addressed this issue arises also in the mars procedure proposed by friedman and other adaptive regression spline procedures

smoothing splines with varying smoothing parameter this paper considers the development of spatially adaptive smoothing splines for the estimation of a regression function with nonhomogeneous smoothness across the domain two challenging issues arising in this context are the evaluation of the equivalent kernel and the determination of a local penalty the penalty is a function of the design points in order to accommodate local behaviour of the regression function we show that the spatially adaptive smoothing spline estimator is approximately a kernel estimator and that the equivalent kernel is spatially dependent the equivalent kernels for traditional smoothing splines are a special case of this general solution with the aid of the green s function for a two point boundary value problem explicit forms of the asymptotic mean and variance are obtained for any interior point thus the optimal roughness penalty function is obtained by approximately minimizing the asymptotic integrated mean squared error simulation results and an application illustrate the performance of the proposed estimator copyright 2013 oxford university press

model selection in spline nonparametric regression a bayesian approach is presented for model selection in nonparametric regression with gaussian errors and in binary nonparametric regression a smoothness prior is assumed for each component of the model and the posterior probabilities of the candidate models are approximated using the bayesian information criterion we study the model selection method by simulation and show that it has excellent frequentist properties and gives improved estimates of the regression surface all the computations are carried out efficiently using the gibbs sampler

bayesian mixture of splines for spatially adaptive nonparametric regression summary a bayesian approach is presented for spatially adaptive nonparametric regression where the regression function is modelled as a mixture of splines each component spline in the mixture has associated with it a smoothing parameter which is defined over a local region of the covariate space these local regions overlap such that individual data points may lie simultaneously in multiple regions consequently each component spline has attached to it a weight at each point of the covariate space and by allowing the weight of each component spline to vary across the covariate space a spatially adaptive estimate of the regression function is obtained the number of mixing components is chosen using a modification of the bayesian information criteria we study the procedure analytically and show by simulation that it compares favourably to three competing techniques these techniques are the bayesian regression splines estimator of smith kohn 1996 the hybrid adaptive spline estimator of luo wahba 1997 and the automatic bayesian curve fitting estimator of denison et al 1998 the methodology is illustrated by modelling global air temperature anomalies all the computations are carried out efficiently using markov chain monte carlo

bayesian analysis of binary and polychotomous response data abstract a vast literature in statistics biometrics and econometrics is concerned with the analysis of binary and polychotomous response data the classical approach fits a categorical response regression model using maximum likelihood and inferences about the model are based on the associated asymptotic theory the accuracy of classical confidence statements is questionable for small sample sizes in this article exact bayesian methods for modeling categorical response data are developed using the idea of data augmentation the general approach can be summarized as follows the probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data values of the latent data can be simulated from suitable truncated normal distributions if the latent data are known then the posterior distribution of the parameters can be computed using standard results for normal linear models draws from this posterior are used to sample new latent data and t

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

regression density estimation using smooth adaptive gaussian mixtures we model a regression density flexibly so that at each value of the covariates the density is a mixture of normals with the means variances and mixture probabilities of the components changing smo

bayesian modeling of joint and conditional distributions in this paper we study a bayesian approach to exible modeling of conditional distributions the approach uses a exible model for the joint distribution of the dependent and independent variables and then extracts the conditional distributions of interest from the estimated joint distribution we use a nite mixture of multivariate normals fmmn to estimate the joint distribution the conditional distributions can then be assessed analytically or through simulations the discrete variables are handled through the use of latent variables the estimation procedure employs an mcmc algorithm we provide a characterization of the kullback leibler closure of fmmn and show that the joint and conditional predictive densities implied by fmmn model are consistent estimators for a large class of data generating processes with continuous and discrete observables the method can be used as a robust regression model with discrete and continuous dependent and independent variables and as a bayesian alternative to semi and non parametric models such as quantile and kernel regression in experiments the method compares favorably with classical nonparametric and alternative bayesian methods

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

locally adaptive bayes nonparametric regression via nested gaussian processes we propose a nested gaussian process ngp as a locally adaptive prior for bayesian nonparametric regression specified through a set of stochastic differential equations sdes the ngp imposes a gaussian process prior for the function s mth order derivative the nesting comes in through including a local instantaneous mean function which is drawn from another gaussian process inducing adaptivity to locally varying smoothness we discuss the support of the ngp prior in terms of the closure of a reproducing kernel hilbert space and consider theoretical properties of the posterior the posterior mean under the ngp prior is shown to be equivalent to the minimizer of a nested penalized sum of squares involving penalties for both the global and local roughness of the function using highly efficient markov chain monte carlo for posterior inference the proposed method performs well in simulation studies compared to several alternatives and is scalable to massive data illustrated through a proteomics appli

spatially adaptive smoothing splines summary we use a reproducing kernel hilbert space representation to derive the smoothing spline solution when the smoothness penalty is a function a t of the design space t thereby allowing the model to adapt to various degrees of smoothness in the structure of the data we propose a convenient form for the smoothness penalty function and discuss computational algorithms for automatic curve fitting using a generalised crossvalidation measure of order m where m is typically taken to be two the smoothing spline solution uses a global smoothing parameter il which implies that the true underlying mean process has a constant degree of smoothness in this paper we suggest a more general spatially adaptive framework that accommodates varying degrees of roughness by seeking solutions where the smoothness penalty a t is now a function of the design t we derive the solution within a reproducing kernel hilbert space gu 2002 ch 2 wahba 1990 ch 1 when applying the method to data we propose a piecewise constant model for the smoothing function a t this provides a convenient computational framework with closed form solutions for the corresponding reproducing kernels of the hilbert space these kernels show similarities to regression splines in that the use of a piecewise constant i t induces breaks in the mth 2m l1 th derivatives at the changepoints in l t

generalized smooth finite mixtures we propose a general class of models and a unified bayesian inference methodology for flexibly estimating the density of a response variable conditional on a possibly high dimensional set of covari

bayesian regression with nonparametric heteroskedasticity abstract this paper studies large sample properties of a semiparametric bayesian approach to inference in a linear regression model the approach is to model the distribution of the regression error term by a normal distribution with the variance that is a flexible function of covariates the main result of the paper is a semiparametric bernstein von mises theorem under misspecification even when the distribution of the regression error term is not normal the posterior distribution of the properly recentered and rescaled regression coefficients converges to a normal distribution with the zero mean and the variance equal to the semiparametric efficiency bound

automatic bayesian curve fitting summary a method of estimating a variety of curves by a sequence of piecewise polynomials is proposed motivated by a bayesian model and an appropriate summary of the resulting posterior distribution a joint distribution is set up over both the number and the position of the knots defining the piecewise polynomials throughout we use reversible jump markov chain monte carlo methods to compute the posteriors the methodology has been successful in giving good estimates for smooth functions i e continuous and differentiable as well as functions which are not differentiable and perhaps not even continuous at a finite number of points the methodology is extended to deal with generalized additive models

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

on the approximation rate of hierarchical mixtures of experts for generalized linear models we investigate a class of hierarchical mixtures of experts hme models where generalized linear models with nonlinear mean functions of the form xt are mixed here is the inverse link function it is shown that mixtures of such mean functions can approximate a class of smooth functions of the form h x where h e w 2 k a sobolev class over 0 1 s as the number of experts m in the network increases an upper bound of the approximation rate is given as o m 2 s in lp norm this rate can be achieved within the family of hme structures with no more than s layers where s is the dimension of the predictor x

adaptspec adaptive spectral estimation for nonstationary time series we propose a method for analyzing possibly nonstationary time series by adaptively dividing the time series into an unknown but finite number of segments and estimating the corresponding local spectra by smoothing splines the model is formulated in a bayesian framework and the estimation relies on reversible jump markov chain monte carlo rjmcmc methods for a given segmentation of the time series the likelihood function is approximated via a product of local whittle likelihoods thus no parametric assumption is made about the process underlying the time series the number and lengths of the segments are assumed unknown and may change from one mcmc iteration to another the frequentist properties of the method are investigated by simulation and applications to electroencephalogram and the el nino southern oscillation phenomenon are described in detail

conditional spectral analysis of replicated multiple time series with application to nocturnal physiology abstractthis article considers the problem of analyzing associations between power spectra of multiple time series and cross sectional outcomes when data are observed from multiple subjects the motivating application comes from sleep medicine where researchers are able to noninvasively record physiological time series signals during sleep the frequency patterns of these signals which can be quantified through the power spectrum contain interpretable information about biological processes an important problem in sleep research is drawing connections between power spectra of time series signals and clinical characteristics these connections are key to understanding biological pathways through which sleep affects and can be treated to improve health such analyses are challenging as they must overcome the complicated structure of a power spectrum from multiple time series as a complex positive definite matrix valued function this article proposes a new approach to such analyses based on a tensor pr

bayesian semiparametric joint models for functional predictors motivated by the need to understand and predict early pregnancy loss using hormonal indicators of pregnancy health this article proposes a semiparametric bayesian approach for assessing the relationship between functional predictors and a response a multivariate adaptive spline model is used to describe the functional predictors and a generalized linear model with a random intercept describes the response through specifying the random intercept to follow a dirichlet process jointly with the random spline coefficients we obtain a procedure that clusters trajectories according to shape and according to the parameters of the response model for each cluster this very flexible method allows for the incorporation of covariates in the models for both the response and the trajectory we apply the method to postovulatory progesterone data from the early pregnancy study and find that the model successfully predicts early pregnancy loss

a computationally practical simulation estimator for panel data in this paper i develop a practical extension of mcfadden s method of simulated moments estimator for limited dependent variable models to the panel data case the method is based on a factorization of the msm first order condition into transition probabilities along with the development of a new highly accurate method for similating these transition probabilities a series of monte carlo tests show that this msm estimator performs quite well relative to quadrature based ml estimators even when large numbers of quadrature points are employed the estimator also performs well relative to simulated ml even when a highly accurate method is used to simulate the choice probabilities in terms of computational speed complex panel data models involving random effects and arma errors may be estimated via msm in times similar to those necessary for estimation of simple random effects models via ml quadrature

limiting exit location distributions in the stochastic exit problem consider a two dimensional continuous time dynamical system with an attracting fixed point s if the deterministic dynamics are perturbed by white noise random perturbations of strength epsilon the system state will eventually leave the domain of attraction omega of s we analyze the case when as epsilon to0 the exit location on the boundary partial omega is increasingly concentrated near a saddle point h of the deterministic dynamics we show using formal methods that the asymptotic form of the exit location distribution on partial omega is generically non gaussian and asymmetric and classify the possible limiting distributions a key role is played by a parameter mu equal to the ratio lambda s h lambda u h of the stable and unstable eigenvalues of the linearized deterministic flow at h if mu 1 then the exit location distribution is generically asymptotic as epsilon to0 to a weibull distribution with shape parameter 2 mu on the cal o epsilon mu 2

mixtures of experts of autoregressive time series asymptotic normality and model specification we consider a class of nonlinear models based on mixtures of local autoregressive time series at any given time point we have a certain number of linear models denoted as experts where the vector of covariates may include lags of the dependent variable additionally we assume the existence of a latent multinomial variable whose distribution depends on the same covariates as the experts that determines which linear process is observed this structure denoted as mixture of experts me is considerably flexible in modeling the conditional mean function as shown by jiang and tanner we present a formal treatment of conditions to guarantee the asymptotic normality of the maximum likelihood estimator mle under stationarity and nonstationarity and under correct model specification and model misspecification the performance of common model selection criteria in selecting the number of experts is explored via monte carlo simulations finally we present applications to simulated and real data sets to illustrate the ability of the proposed structure to model not only the conditional mean but also the whole conditional density