on conditional and intrinsic autoregressions summary gaussian conditional autoregressions have been widely used in spatial statistics and bayesian image analysis where they are intended to describe interactions between random variables at fixed sites in euclidean space the main appeal of these distributions is in the markovian interpretation of their full conditionals intrinsic autoregressions are limiting forms that retain the markov property despite being improper they can have advantages over the standard autoregressions both conceptually and in practice for example they often avoid difficulties in parameter estimation without apparent loss or exhibit appealing invariances as in texture analysis however on small arrays and in nonlattice applications both forms of autoregression can lead to undesirable second order characteristics either in the variables themselves or in contrasts among them this paper discusses standard and intrinsic autoregressions and describes how the problems that arise can be alleviated using dempster s 1972 algorithm or an appropriate modification the approach represents a partial synthesis of standard geostatistical and gaussian markov random field formulations some nonspatial applications are also mentioned

a formula for the kronecker products of schur functions of hook shapes abstract the kronecker product of two homogeneous symmetric polynomials p 1 and p 2 is defined by means of the frobenius map by the formula p 1 p 2 f f 1 p 1 f 1 p 2 when p 1 and p 2 are schur functions s i and s j respectively then the resulting product s i s j is the frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the diagrams i and j taking the scaler product of s i s j with a third schur function s k gives the so called kronecker coefficient c ijk s i s j s k which gives the multiplicity of the representation corresponding to k in the tensor product in this paper we give an explicit formula for c ijk where i and j are hooks and k is arbitrary

brief paper uniqueness of prediction error estimates of multivariable moving average models in prediction error pe identification of the parameter estimates is given by the global minimum of a scalar valued function of the innovation sample covariance matrix it may happen that the loss function has multiple local minimum points so that a numerical search routine can fail to find the global minimum such a situation usually referred to as lack of uniqueness of the estimates was experienced in practice and also theoretically examined for various model structures a unique minimum of the criterion is also crucial for convergence of recursive pe algorithms in this paper multivariable moving average ma models are considered it is proved that for such models any reasonable pe criterion has asymptotically a unique stationary point furthermore it is shown that this stationary point is a global minimum which corresponds to the true parameter vector this extends the result known for univariate ma models to the multivariate case

bayesian computation via the gibbs sampler and related markov chain monte carlo methods the use of the gibbs sampler for bayesian computation is reviewed and illustrated in the context of some canonical examples other markov chain monte carlo simulation methods are also briefly described and comments are made on the advantages of sample based approaches for bayesian inference summaries

multi modal clique graph matching for view based 3d model retrieval multi view matching is an important but a challenging task in view based 3d model retrieval to address this challenge we propose an original multi modal clique graph mcg matching method in this paper we systematically present a method for mcg generation that is composed of cliques which consist of neighbor nodes in multi modal feature space and hyper edges that link pairwise cliques moreover we propose an image set based clique edgewise similarity measure to address the issue of the set to set distance measure which is the core problem in mcg matching the proposed mcg provides the following benefits 1 preserves the local and global attributes of a graph with the designed structure 2 eliminates redundant and noisy information by strengthening inliers while suppressing outliers and 3 avoids the difficulty of defining high order attributes and solving hyper graph matching we validate the mcg based 3d model retrieval using three popular single modal data sets and one novel multi modal data set extensive experiments show the superiority of the proposed method through comparisons moreover we contribute a novel real world 3d object data set the multi view rgb d object data set to the best of our knowledge it is the largest real world 3d object data set containing multi modal and multi view information

extension of the blahut arimoto algorithm for maximizing directed information in this paper we extend the blahut arimoto algorithm for maximizing massey s directed information the algorithm can be used for estimating the capacity of channels with delayed feedback where the feedback is a deterministic function of the output in order to maximize the directed information we apply the ideas from the regular blahut arimoto algorithm i e the alternating maximization procedure to our new problem we provide both upper and lower bound sequences that converge to the optimum global value our main insight in this paper is that in order to find the maximum of the directed information over a causal conditioning probability mass function one can use a backward index time maximization combined with the alternating maximization procedure we give a detailed description of the algorithm showing its complexity and the memory needed and present several numerical examples

var forecasting under misspecification the paper considers multi step forecasting of a stationary vector process under a quadratic loss function with a collection of finite order vector autoregressions var under severe misspecification it is preferable to use the multi step loss function also for parameter estimation we propose a modification to shibata s ann statist 8 1980 147 final prediction error criterion to jointly choose the var lag order and one of two predictors the maximum likelihood estimator plug in predictor or the loss function estimator plug in predictor a monte carlo experiment illustrates the theoretical results and documents the empirical performance of the selection criterion

statistical analysis of field experiments using neighbouring plots the paper describes and provides examples of four different applications of the use of neighbouring plot values in the analysis of agricultural field experiments the topics include the use of check plots to control environmental variation in large unreplicated variety trials spatial models based on first differences to accommodate fertility effects in trials that have some degree of replication adjustment for interplot competition using a simultaneous equations formulation and the analysis of trials in which plot values are affected by the particular treatments on neighbouring plots

asymptotically optimal estimation in misspecified time series models a concept of asymptotically efficient estimation is presented whena misspecified parametric time series model is fitted to a stationary process efficiency of several minimum distance estimates is proved and the behavior ofthe gaussian maximum likelihood estimate is studied furthermore the behaviorof estimates that minimize the h step prediction error is discussed briefly the paper answers to some extent the question what happens when a misspecifiedmodel is fitted to time series data and one acts as if the model were true

mean squared error of estimation or prediction under a general linear model abstract the problem considered is that of predicting a linear combination of the fixed and random effects of a mixed effects linear model more generally the problem considered is that of predicting an unobservable random variable from a set of observable random variables the best linear unbiased predictor depends on parameters which generally are unknown various exact or approximate expressions are given for the mean squared error mse of the predictor obtained by replacing the unknown parameters with estimates several estimators of the mse are investigated

uniform convergence of sample second moments of families of time series arrays we consider abstractly defined time series arrays yt t 1 le t le t requiring only that their sample lagged second moments converge and that their end values y1 j t and yt j t be of order less than t for each j ge 0 we show that under quite general assumptions various types of arrays that arise naturally in time series analysis have these properties including regression residuals from a time series regression seasonal adjustments and infinite variance processes rescaled by their sample standard deviation we establish a useful uniform convergence result namely that these properties are preserved in a uniform way when relatively compact sets of absolutely summable filters are applied to the arrays this result serves as the foundation for the proof in a companion paper by findley potscher and wei of the consistency of parameter estimates specified to minimize the sample mean squared multistep ahead forecast error when invertible short memory models are fit to short or long memory time series or time series arrays

inconsistent signal feasibility problems least squares solutions in a product space presents parallel projection methods to find least squares solutions to inconsistent convex set theoretic signal synthesis problems the problem of finding a signal that minimizes a weighted average of the squares of the distances to constraint sets is reformulated in a product space where it is equivalent to that of finding a point that lies in a particular subspace and at minimum distance from the cartesian product of the original sets a solution is obtained in the product space via methods of alternating projections which naturally lead to methods of parallel projections in the original space the convergence properties of the proposed methods are analyzed and signal synthesis applications are demonstrated

diagnostic testing of univariate time series models summary the portmanteau statistic for testing the adequacy of an autoregressive moving average model is based on the first m autocorrelations of the residuals from the fitted model this paper examines the properties of this test for various choices of m a modification which allows the use of small values of m is shown to result in a more powerful test the lagrange multiplier statistic godfrey 1979 and a test statistic examined by newbold 1980 are also discussed

brief paper some properties of the output error method the output error identification method is studied in various respects the stationary points of the associated loss function are investigated sufficient conditions for a unique local minimum are given the loss functions can be minimized using a quasilinearization algorithm such an algorithm will give good local convergence it is however shown that global convergence does not always occur the output error method is also compared with some other estimation methods from the accuracy point of view it is proved that a prediction error method will give better accuracy an instrumental variable technique may give better or worse accuracy depending on the actual noise correlation

maximization of worst case weighted sum rate for miso downlink systems with imperfect channel knowledge the problem of robust weighted sum rate maximization wsrmax in multicell downlink multi input single output systems is considered we assume that channel state information csi of all users is imperfectly known at the base stations the problem is known to be np hard even in the case of perfect csi we propose optimal and suboptimal but fast converging algorithms for wsrmax problem with csi errors assuming bounded ellipsoidal model for the csi errors we optimize the worst case weighted sum rate the proposed optimal algorithm is based on branch and bound bb technique and it globally solves the worst case wsrmax problem with an optimality certificate as the convergence speed of the bb method can be slow for large networks we also provide a fast but possibly suboptimal algorithm based on alternating optimization technique and sequential convex programming the optimal bb based algorithm can be used to provide performance benchmarks for any suboptimal algorithm numerical results show that the convergence speed of the suboptimal algorithm is fast and it finds a close to optimal solution in only a few iterations

on the asymptotic accuracy of pseudo linear regression algorithms the accuracy properties of a general pseudo linear regression plr method are examined both off line and on line algorithms are considered assuming the parameter estimates converge they will under weak conditions be asymptotically gaussian distributed expressions for the corresponding covariance matrices are given it is shown that the asymptotic covariance matrix of the off line plr algorithm is bounded from above by the matrix corresponding to the on line plr algorithm and from below by that corresponding to the prediction error method some simple numerical illustrations of the theoretical results are also included

stability of coefficients in the kronecker product of a hook and a rectangle we use recent work of jonah blasiak 2012 arxiv 1209 2018 to prove a stability result for the coefficients in the kronecker product of two schur functions one indexed by a hook partition and one indexed by a rectangle partition we also give nearly sharp bounds for the size of the partition starting with which the kronecker coefficients are stable moreover we show that once the bound is reached no new schur functions appear in the decomposition of kronecker product we call this property superstability thus one can recover the schur decomposition of the kronecker product from the smallest case in which the superstability holds the bound for superstability is sharp our study of this particular case of the kronecker product is motivated by its usefulness for the understanding of the quantum hall effect scharf t et al 1994 j phys a math gen 27 4211 9

adaptive alternating minimization algorithms the classical alternating minimization or projection algorithm has been successful in the context of solving optimization problems over two variables the iterative nature and simplicity of the algorithm has led to its application in many areas such as signal processing information theory control and finance a general set of sufficient conditions for the convergence and correctness of the algorithm are known when the underlying problem parameters are fixed in many practical situations however the underlying problem parameters are changing over time and the use of an adaptive algorithm is more appropriate in this paper we study such an adaptive version of the alternating minimization algorithm more precisely we consider the impact of having a slowly time varying domain over which the minimization takes place as a main result of this paper we provide a general set of sufficient conditions for the convergence and correctness of the adaptive algorithm perhaps somewhat surprisingly these conditions seem to be the minimal ones one would expect in such an adaptive setting we present applications of our results to adaptive decomposition of mixtures adaptive log optimal portfolio selection and adaptive filter design

an algorithm for maximizing expected log investment return let the random stock market vector x geq 0 be drawn according to a known distribution function f x x in r m a log optimal portfolio b ast is any portfolio b achieving maximal expected log return w ast sup b e ln b t x where the supremum is over the simplex b geq 0 sum i 1 m b i 1 an algorithm is presented for finding b ast the algorithm consists of replacing the portfolio b by the expected portfolio b b i e b i x i b t x corresponding to the expected proportion of holdings in each stock after one market period the improvement in w b after each iteration is lower bounded by the kullback leibler information number d b b between the current and updated portfolios thus the algorithm monotonically improves the return w an upper bound on w ast is given in terms of the current portfolio and the gradient and the convergence of the algorithm is established

a random field approach to the analysis of field plot experiments and other spatial experiments several nearest neighbor methods for the analysis of data from spatial experiments e g agricultural field experiments have recently been proposed these methods attempt to account for the effect of spatial heterogeneity on the estimation of treatment contrasts typically this is accomplished indirectly by differencing or by using residuals from neighboring plots to construct covariates we examine an alternative approach in which the spatial heterogeneity is modeled directly the model underlying our approach is similar to the model underlying a geostatistical kriging analysis and as in the latter model the observations are regarded collectively as a partial realization of a random field a randomization study of uniformity trial data suggests that the random field approach often provides more accurate estimates of treatment contrasts than nearest neighbor approaches in addition the random field approach is devoid of ambiguities as to the handling of border plots and is generally more flexible than nearest neighbor approaches

the exact multi period mean square forecast error for the first order autoregressive model the finite sample behaviour of the multi period least squares forecast is considered in the simple normal autoregressive model yt yt 1 ut where 1 necessary and sufficient conditions are established for the existence of the forecast bias and the mean square forecast error msfe and an exact expression for the msfe is given exact numerical results are obtained for both the stationary and the fixed start up case our main conclusions are that for small values of the msfe is a decreasing function of the number of forecast periods and that the behaviour of the msfe in the stationary and the fixed start up case is very similar except for values of close to 1

the exact multi period mean square forecast error for the first order autoregressive model with an intercept abstract we study the exact finite sample behaviour of the mean square forecast error msfe of multi period least squares forecasts in the normal autoregressive model y t y t 1 u t we obtain necessary and sufficient conditions for the existence of the msfe and give an exact expression which we use to obtain numerical results for both the stationary and the fixed start up model we conclude inter alia that the behaviour of the msfe in the model with intercept can be very different from that in the model without intercept especially when is close to unity

intrinsic autoregressions and related models on the two dimensional lattice summary stationary autoregressions on a two dimensional lattice are generalized to intrinsic models where only increments are assumed to be stationary prediction formulae and the asymptotic behaviour of the semivariogram are derived for parameter estimation we propose an approximate maximum likelihood estimator a generalization of whittle s estimator it is derived also for general intrinsic models

robustness of maximum likelihood estimates for multi step predictions the exponential smoothing case summary we extend the argument initiated by cox 1961 that the exponential smoothing formula can be made more robust for multi step forecasts if the smoothing parameter is adjusted as a function of the forecast horizon 1 the consistency property of the estimator which minimizes the sum of squares of the sample 1 step ahead forecast errors makes the robustness result useful in practice we also generalize the consistency result to some other parsimonious nonstationary models which have been popular in use the asymptotic distribution of the estimated smoothing parameter adjusted for forecast horizon l leads to the development of diagnostic tools which are based on i step forecasts

powers of the vandermonde determinant and the quantum hall effect the expansion of the laughlin ansatz for describing the ground state wavefunction for the fractional quantum hall effect as a linear combination of slater determinantal wavefunctions for n particles is discussed in terms of the corresponding expansion of even powers of the vandermonde alternant into schur functions two new algorithms for computing the coefficients of the complete expansion are given they appear to be substantially more efficient than other methods and avoid any use of symmetric group characters a number of examples are given and the results obtained for n 7 8 and 9 reviewed the separate calculation of individual coefficients is also discussed

optimal pulse shape design for digital communication systems by projections onto convex sets recent developments in dsp hardware has made it possible to use almost any pulse shape for digital data trans mission this resulted in a search for algorithms capable of constructing pulse shapes matching to the properties of a given channel the projection onto convex sets technique is suitable for the solution of this problem particularly because of its ffexibility in modeling a variety of constraints the use of the technique for the optimal pulse shape design problem is demonstrated through a detailed example from power line communications i introduction ince nyquist s invention of the intersymbol interference isi the optimal pulse shaping problem for digital data transmission over channels with various properties has been investigated by a number of researchers 1 7 the main differences between various digital nyquist pulse design al gorithms published in the literature are the type of the filter fir or iir they choose to generate the pulse shape and the formulation and solution of the optimization problem some of these approaches use general optimization techniques and they can handle additional constraints if needed however addition of each new constraint requires reformulation of the problem and the solution algorithm the projections onto convex sets pocs technique is an optimization method introduced in the 1960 s 8 a number of successful applications of the pocs technique in the signal processing area have been reported in the recent years 9 ll the pocs technique is attractive for optimum pulse shape design particularly because of its modular way of formulating problems with many constraints in this technique the addition of new constraints or removal of some of the existing ones does not require reformulation it is also possible to handle certain inconsistent requirements in the pocs formulation 12 if there are nonintersecting convex sets it is possible to find the vector which is as close as possible to one set of constraints while satisfying the rest

multi step estimation and forecasting in dynamic models abstract we consider the situation in which a researcher uses a misspecified model for forecasting if the interest is in multi step forecasting and the accuracy of the forecasts is measured through the sum of squared multi step forecast errors then in large samples it is preferable to minimize corresponding sum of squared in sample multi step forecast errors we derive the asymptotic properties of the resulting estimator to asses the behavior of the estimator in small samples we perform a monte carlo experiment in most of the cases considered ols outperforms the estimator we conjecture that this occurs because the estimator is still defined on the basis of squared errors

fitting time series models by minimizing multistep ahead errors a frequency domain approach this paper brings together two topics in the estimation of time series forecasting models the use of the multistep ahead error sum of squares as a criterion to be minimized and frequency domain methods for carrying out this minimization the methods are developed for the wide class of time series models having a spectrum which is linear in unknown coefficients this includes the ima 1 1 model for which the common exponentially weigh ted moving average predictor is optimal besides more general structural models for series exhibiting trends and seasonality the method is extended to include the box jenkins air line model the value of the multistep criterion is that it provides protection against using an incorrectly specified model the value of frequency domain estimation is that the iteratively reweighted least squares scheme for fitting generalized linear models is readily extended to construct the parameter estimates and their standard errors it also yields insight into the loss of efficiency when the model is correct and the robustness of the criterion against an incorrect model a simple example is used to illustrate the method and a real example demonstrates the extension to seasonal models the discussion considers a diagnostic test statistic for indicating an incorrect model

uniqueness of estimated k step prediction models of arma processes the direct estimation of the k step prediction models of arma processes is discussed the emphasis is on the uniqueness properties of the parameter estimates of such models obtained by using either a prediction error method pem or a pseudo linear regression plr algorithm the main result is that both pem and plr when applied to such models have a certain uniqueness property more specifically it is shown that all the limit models corresponding to either method behave precisely as the true optimal predictor furthermore if a minimal parameterization is used then the true predictor is the unique limit model both for pem and for plr

uniqueness of the maximum likelihood estimates of arma model parameters an elementary proof under mild conditions the likelihood function associated with an arma model asymptotically has a unique stationary point which is a global maximum corresponding to the true parameter values this note gives a new proof of this fact the proof is considerably simpler than that previously published

poisson gamma random field models for spatial statistics summary doubly stochastic bayesian hierarchical models are introduced to account for uncertainty and spatial variation in the underlying intensity measure for point process models inhomogeneous gamma process random fields and more generally markov random fields with infinitely divisible distributions are used to construct positively autocorrelated intensity measures for spatial poisson point processes these in turn are used to model the number and location of individual events a data augmentation scheme and markov chain monte carlo numerical methods are employed to generate samples from bayesian posterior and predictive distributions the methods are developed in both continuous and discrete settings and are applied to a problem in forest ecology

on a system of two dimensional recurrence equations summary the solution of a system of two dimensional recurrence equations is discussed particularly in the context of the auto correlation structure of a stationary first order autonormal scheme on the doubly infinite rectangular lattice a simple approximation in the region of primary interest is described

mathematical framework for pseudo spectra of linear stochastic difference equations although spectral analysis of stationary stochastic processes has solid mathematical foundations this is not always so for some non stationary cases here we establish a rigorous mathematical extension of the classic fourier spectrum to the case in which there are ar roots in the unit circle i e the transfer function of the linear time invariant filter has poles on the unit circle to achieve it we embed the classical problem in a wider framework extend the discrete time fourier transform and defined a new extended fourier transform pair pseudo covariance function pseudo spectrum our approach is a proper extension of the classical spectral analysis within which the fourier transform pair auto covariance function spectrum is a particular case consequently spectrum and pseudo spectrum coincide when the first one is defined

bayesian analysis of agricultural field experiments the paper describes bayesian analysis for agricultural field experiments a topic that has received very little previous attention despite a vast frequentist literature adoption of the bayesian paradigm simplifies the interpretation of the results especially in ranking and selection also complex formulations can be analysed with comparative ease by using markov chain monte carlo methods a key ingredient in the approach is the need for spatial representations of the unobserved fertility patterns this is discussed in detail problems caused by outliers and by jumps in fertility are tackled via hierarchical f formulations that may find use in other contexts the paper includes three analyses of variety trials for yield and one example involving binary data none is entirely straightforward some numerical comparisons with frequentist analyses are made

analysis of field experiments by least squares smoothing summary assuming a smooth trend plus independent error model for the environmental effects in the yields of a field plot experiment least squares smoothing methods are developed to estimate both the treatment effects and the unknown trend treatment estimates are closely related to those resulting from a generalized least squares analysis in which the covariance structure for the environmental effects has a particular form however the main emphases are on the accuracy of treatment estimates under a fixed smooth trend plus error model and the exploratory power of the basic method to isolate trend effects of unknown form although the detailed development is for the one dimensional case generalizations of the smoothness concept and extensions to two dimensions are also discussed application of the basic method is illustrated on three data sets and the results compared with other analyses

modeling of time series arrays by multistep prediction or likelihood methods abstract an estimation theory is provided for the fitting of possibly incorrect invertible short memory models to short or long memory time series or time series arrays by multistep prediction error minimization or gaussian likelihood maximization by array we mean data y t t 1 t t that depend on the number of observations t such as regression or other estimated model residuals or the outputs of time varying filters for example seasonal adjustments our theory only requires the modeled array to have basic properties for a s i p convergence of parameter estimates the array s sample lagged second moments must converge a s i p and its end values y 1 j t and y t j t must be of order less than t 1 2 or an appropriately differenced version of the observed array must have these properties in findley et al ann statist 29 2001 815 broad classes of arrays were shown to have these properties even for the special case of autoregressive moving average models fit to stationary gaussian time series data our result on the convergence of parameter estimates minimizing p step ahead prediction error sums of squares is new

on the kronecker product of sn characters abstract let be irreducible s n characters and assume appears in the kronecker product with maximal first part 1 then 1 min i i a similar result holds for the maximal first column we also give a recursive formula for as an application we show that if n 1 1 1 then s n 2 3 2 3 2 3 s n 1 where denotes the outer tensor product these results are applied to study the character where runs through the partitions with no more then k parts this character is closely related to the polynomial identities of the algebra of k k matrices

iterative reconstruction in x ray fluorescence tomography based on radon inversion we describe a new approach for the inversion of the generalized attenuated radon transform in x ray fluorescence computed tomography xfct the approach consists of using the radon inverse as an approximation for the actual one followed by an iterative refinement also we analyze the problem of retrieving the attenuation map directly from the emission data giving rise to a novel alternating method for the solution we applied our approach to real and simulated xfct data and compared its performance to previous inversion algorithms for the problem showing its main advantages better images than those obtained by other analytic methods and much faster than iterative methods in the discrete setting

robust reduced rank filtering for erratic seismic noise attenuation abstractsingular spectrum analysis ssa or cadzow reduced rank filtering is an efficient method for random noise attenuation ssa starts by embedding the seismic data into a hankel matrix rank reduction of this hankel matrix followed by antidiagonal averaging is utilized to estimate an enhanced seismic signal rank reduction is often implemented via the singular value decomposition svd the svd is a nonrobust matrix factorization technique that leads to suboptimal results when the seismic data are contaminated by erratic noise the term erratic noise designates non gaussian noise that consists of large isolated events with known or unknown distribution we adopted a robust low rank factorization that permitted use of the ssa filter in situations in which the data were contaminated by erratic noise in our robust ssa method we replaced the quadratic error criterion function that yielded the truncated svd solution by a bisquare function the hankel matrix was then approximated by the product of two low

approximating long memory dna sequences by short memory process abstract this paper analyzes the approximation of a general long memory arfima p d q process by a short memory arma 1 1 process to validate this approximation a mean square error forecast criterion is considered and the calculation of the mean square error between the observation x t l of an arfima process and the l step ahead forecast of the arma 1 1 process is presented the performance of the arma 1 1 approximation to an arfima model is illustrated by using an application to a dna sequence of orf virus this paper gives some theoretical justification on the mean square error forecast criterion based on a selected arma 1 1 model when compared to the general arfima p d q process the paper also provides a dna sequence of orf virus analysis when the time series is originated from the tangent values of the chaos game representation coordinates for each nucleotide the conclusions of this paper work well because the estimator value of d is small d 0 16 the paper also gives the other parameter estimate of the fitted arfima 0 d 1 model and one step predictions using arma 1 1 model

constrained monte carlo maximum likelihood for dependent data maximum likelihood estimates mles in autologistic models and other exponential family models for dependent data can be calculated with markov chain monte carlo methods the metropolis algorithm or the gibbs sampler which simulate ergodic markov chains having equilibrium distributions in the model from one realization of such a markov chain a monte carlo approximant to the whole likelihood function can be constructed the parameter value if any maximizing this function approximates the mle

selecting optimal multistep predictors for autoregressive processes of unknown order we consider the problem of choosing the optimal in the sense of mean squared prediction error multistep predictor for an autoregressive ar process of finite but unknown order if a working ar model which is possibly misspecified is adopted for multistep predictions then two competing types of multistep predictors i e plug in and direct predictors can be obtained from this model we provide some interesting examples to show that when both plug in and direct predictors are considered the optimal multistep prediction results cannot be guaranteed by correctly identifying the underlying model s order this finding challenges the traditional model order selection criteria which usually aim to choose the order of the true model a new prediction selection criterion which attempts to seek the best combination of the prediction order and the prediction method is proposed to rectify this difficulty when the underlying model is stationary the validity of the proposed criterion is justified theoretically to obtain this result asymptotic properties of accumulated squares of multistep prediction errors are investigated in addition to overcoming the above difficulty some other advantages of the proposed criterion are also mentioned