uncertainty in model based seasonal adjustment procedures and construction of minimax filters abstract unobserved component autoregressive integrated moving average models are often the cornerstone of model based seasonal adjustment procedures unfortunately these models are inherently underidentified and ad hoc assumptions must be made prior to the analysis this article investigates the effect of seasonal adjustment filters on a class of observationally equivalent models bounds on the mean squared error mse associated with arbitrary linear filters are derived the article also derives robust seasonal adjustment filters the filters are robust in the sense that they minimize the maximum mse from the set of observationally equivalent models the article shows that the minimax and minimum extraction filters are equivalent for a certain class of models empirical results for a number of economic time series are presented

equivariant stable homotopy and segal s burnside ring conjecture analogous results were proved later for ko in the generality of compact lie groups by atiyah and segal 8 and for kfq the algebraic k theory spectrum associated to the finite field fq by rector 29 using quillen s 26 computation of it kfq in each case the answer involves an appropriate completed representation ring of g and the cohomology theory in question is constructed from the permutative category of finite dimensional vector spaces over a field see 31 if one considers cohomology theories constructed from other permutative categories one expects to find analogous computations in terms of a completed representation ring of g in the given category appropriately defined in particular stable cohomotopy ts is constructed from the category of finite sets 31 and for this category the analogue of the representation ring is a well known object the burnside ring a g 13 a g is a commutative ring with augmentation so one may speak of a g the completed burnside ring moreover there is

estimation prediction and interpolation for nonstationary series with the kalman filter abstract we show how our definition of the likelihood of an autoregressive integrated moving average arima model with missing observations alternative to that of kohn and ansley and based on the usual assumptions made in estimation of and forecasting with arima models permits a direct and standard state space representation of the nonstationary original data so that the ordinary kalman filter and fixed point smoother can be efficiently used for estimation forecasting and interpolation in this way the problem of estimating missing values in nonstationary series is considerably simplified the results are extended to regression models with arima errors and a computer program is available from the authors

a smoothness priors state space modeling of time series with trend and seasonality abstract a smoothness priors modeling of time series with trends and seasonalities is shown an observed time series is decomposed into local polynomial trend seasonal globally stationary autoregressive and observation error components each component is characterized by an unknown variance white noise perturbed difference equation constraint the constraints or bayesian smoothness priors are expressed in state space model form trading day factors are also incorporated in the model a kalman predictor yields the likelihood for the unknown variances hyperparameters likelihoods are computed for different constraint order models in different subsets of constraint equation model classes akaike s minimum aic procedure is used to select the best model fitted to the data within and between the alternative model classes smoothing is achieved by using a fixed interval smoother algorithm examples are shown

filtering and smoothing algorithms for state space models abstract the paper reviews and generalizes recent filtering and smoothing algorithms for observations generated by a state model in particular the paper discusses the modified kalman filter derived by ansley and kohn 1985 and kohn and ansley 1986 to deal with state space models having partially diffuse initial conditions and shows how to compute the limiting normalized likelihood of the observations for such models the paper also discusses and generalizes the new smoothing algorithm presented by kohn and ansley 1987c 1989 and extends it to state space models with partially diffuse initial conditions

nonuniqueness of the positive dirichlet problem for parabolic equations in cylinders abstract we investigate uniqueness of nonnegative solutions to parabolic equations in cylinders of r n 1 with zero dirichlet boundary condition where bases of cylinders are unbounded domains we give necessary and or sufficient conditions for the uniqueness in terms of geometric properties of the bases

missing observations in arima models skipping approach versus additive outlier approach optimal estimation of missing values in arma models is typically performed by using the kalman filter for likelihood evaluation skipping in the computations the missing observations obtaining the maximum likelihood ml estimators of the model parameters and using some smoothing algorithm the same type of procedure has been extended to nonstationary arima models in go mez and maravall 1994 an alternative procedure suggests filling in the holes in the series with arbitrary values and then performing ml estimation of the arima model with additive outliers ao when the model parameters are not known the two methods di er since the ao likelihood is a ected by the arbitrary values we develop the proper likelihood for the ao approach in the general non stationary case and show the equivalence of this and the skipping method finally the two methods are compared through simulation and their relative advantages assessed the comparison also includes the ao method with the uncorrected likelihood 1999 elsevier science s a all rights reserved

the effect of seasonal adjustment filters on tests for a unit root abstract we consider the effect of seasonal adjustment filters in univariate dynamic models we concentrate our analysis on the behavior of the least squares estimator of the sum of the autoregressive coefficients in a regression we show the existence of a limiting upward bias with the x 11 filter when the process does not contain a unit root we quantify the extent of this bias for a range of models and filtering procedures the asymptotic bias has interesting implications with respect to the power of tests for a unit root in order to assess the importance of this effect we present an extensive simulation study of both the size and power of the usual dickey fuller 1979 and phillips perron 1988 statistics we show that in many cases there is considerable reduction in power compared to the benchmark cases where the data is unfiltered finally some practical implications of our study are addressed with respect to tests for unit roots with seasonally adjusted data

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

trend estimation and de trending via rational square wave filters this paper gives an account of some techniques of linear filtering which can be used for extracting trends from economic time series of limited duration and for generating de trended series a family of rational square wave filters is described which enable designated frequency ranges to be selected or rejected their use is advocated in preference to other filters which are commonly used in quantitative economic analysis

an arima model based approach to seasonal adjustment abstract this article proposes a model based procedure to decompose a time series uniquely into mutually independent additive seasonal trend and irregular noise components the series is assumed to follow the gaussian arima model properties of the procedure are discussed and an actual example is given

estimation of median income of four person families a bayesian time series approach abstract this article develops a general methodology for small domain estimation based on data from repeated surveys the results are directly applied to the estimation of median income of four person families for the 50 states and the district of columbia these estimates are needed by the u s department of health and human services hhs to formulate its energy assistance program for low income families the u s bureau of the census by an informal agreement has provided such estimates to hhs through a linear regression methodology since the latter part of the 1970s the current method is an empirical bayes method eb that uses the current population survey cps estimates as well as the most recent decennial census estimates updated by the per capita income estimates of the bureau of economic analysis however with the existing methodology standard errors associated with these estimates are not easy to obtain the eb estimates when used naively can lead to underestimation of standard errors mo

an intrinsic metric approach to uniqueness of the positive cauchy neumann problem for parabolic equations abstract we study uniqueness of nonnegative solutions of the cauchy neumann problem for parabolic equations in unbounded domains and give a sufficient condition for the uniqueness of nonnegative solutions to hold we also give a parabolic harnack inequality with neumann boundary conditions

estimation prediction and interpolation for arima models with missing data abstract we show how to define and then compute efficiently the marginal likelihood of an arima model with missing observations the computation is carried out by using the univariate version of the modified kalman filter introduced by ansley and kohn 1985a which allows a partially diffuse initial state vector we also show how to predict and interpolate missing observations and obtain the mean squared error of the estimate

sensitivity analysis of seasonal adjustments empirical case studies abstract three detailed case studies illustrating the seasonal analysis of economic time series are presented using component models for seasonal and nonseasonal behavior analyses are performed within a semi bayesian framework where inferences for target quantities of interest such as seasonally adjusted values are obtained as posterior distributions conditional on observed data and fitted parameter values such an approach is similar to previous model based methods of seasonal analysis but new models and algorithms are used and more important a sensitivity analysis is performed to determine the extent to which conclusions vary across a range of plausible fitted models it is found that sensitivity to variation across plausible models is not unusual in practice the logical conclusion of the investigation is that a fully bayesian analysis is required that averages conditional posteriors over a posterior distribution for the model parameters such an analysis is necessarily sensitive to the choice of

errors in variables and seasonal adjustment procedures abstract seasonal adjustment procedures attempt to estimate the sample realizations of an unobservable economic time series in the presence of both seasonal and irregular factors in this article we consider a factor that has not been considered explicitly in previous treatments of seasonal adjustment measurement error because of the sample design used in the current population survey measurement error will not be a white noise process but instead it will be characterized by serial correlation of a known form we first consider what effect the serially correlated measurement error has on estimation of the nonseasonal component in seasonal adjustment models we also consider the effect of measurement error on the widely used seasonal adjustment process x 11 estimated unobserved components models are used to estimate the precision root mean squared error of the official and optimal seasonally adjusted data

signal extraction for finite nonstationary time series summary for a signal plus noise model where both signal and noise are generated by nonstation ary arima autoregressive integrated moving average models we use the transformation approach of ansley kohn 1985 to construct an estimate of the signal and obtain its mean squared error given a finite number of observations a method for efficient computa tion of the signal estimate and its mean squared error is also presented consider observations z t generated by the signal plus noise nmodel z t s t n t 1s1 where both signal s t and noise n t follow nonstationary arima processes we give a solution to the finite sample signal extraction problem by defining an estimate of the signal s t given observations z 1 z n and finding its mean squared error the major difficulty in handling the model with nonstationary components is that the distribution of the observations is not defined unless assumptions are made about the initial values of the series moreover it is usually very difficult to formulate reasonable assumptions about the initial values because they affect the correlation structure of the whole series see for example the discussion of bell 1984 p 652 kohn ansley 1986 use a transformation approach to overcome the initial value problem in defining the likelihood and obtaining predittors and interpolators for an arima model which may have missing values this approach does not require any assumptions about the initial values and generalizes the usual differencing approach of box jenkins 1976 for the case where there are no missing observations in the present paper we apply the transformation approach of kohn ansley 1986 and ansley kohn 1985 to solve the signal extraction problem we define a linear estimator a t of s t so that s t sa t is invariant to the initial values and furthermore is unbiased in the sense that e s t s t 0 moreover it has minimum mean squared error amongst all such estimators our approach allows missing observations and extends to signal extraction for several additive signals when each is an arima process we also show how to compute the signal estimate and its mean squared error efficiently by writing the model in state space form placing a diffuse prior on the initial values of the process and then using the modified kalman filter and the modified smoothing algorithm of ansley kohn 1985 and kohn ansley 1986 gersch kitagawa 1983 kitagawa gersch 1984 and harvey todd 1983 use

smoothing and interpolation with the state space model abstract a new result dealing with smoothing and interpolation in the state space model is developed and explored the result simplifies the derivation of existing smoothing algorithms and provides alternate forms that have analytic and practical computing advantages the connection to signal extraction and interpolation is explored and diffuse specifications are considered

on the peterson hit problem let pk f2 x1 x2 xk pk f2 x1 x2 xk be the polynomial algebra over the prime field of two elements f2f2 in k variables x1 x2 xkx1 x2 xk each of degree 1 we study the hit problem set up by f peterson of finding a minimal set of generators for pkpk as a module over the mod 2 steenrod algebra aa in this paper we study a minimal set of generators for aa module pkpk in some so called generic degrees and apply these results to explicitly determine the hit problem for k 4k 4

on stably decomposing products of classifying spaces john martino1 stewart priddy2 jason douma3 1 department of mathematics and statistics western michigan university kalamazoo mi 49008 usa e mail martino math stat wmich edu 2 department of mathematics northwestern university evanston il 60208 usa e mail priddy math nwu edu 3 department of mathematics university of sioux falls sioux falls sd 57105 usa e mail jason douma thecoo edu

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

blow up and global existence for the porous medium equation with reaction on a class of cartan hadamard manifolds abstract we consider the porous medium equation with power type reaction terms u p on negatively curved riemannian manifolds and solutions corresponding to bounded nonnegative and compactly supported data if p m small data give rise to global in time solutions while solutions associated to large data blow up in finite time if p m large data blow up at worst in infinite time and under the stronger restriction p 1 1 m 2 all data give rise to solutions existing globally in time whereas solutions corresponding to large data blow up in infinite time the results are in several aspects significantly different from the euclidean ones as has to be expected since negative curvature is known to give rise to faster diffusion properties of the porous medium equation

stochastic linear trends models and estimators abstract the paper considers stochastic linear trends in series with a higher than annual frequency of observation using an approach based on arima models some of the trend models or the model interpretation of trend estimation filters most often found in statistics and econometrics are analysed and compared the properties of the trend optimal estimator are derived and the analysis is extended to seasonally adjusted and or detrended series it is seen that under fairly general conditions the estimator of the unobserved component is noninvertible and will not accept a convergent autoregressive representation this has implications concerning unit root testing and vad model fitting

the hit problem for the modular invariants of linear groups abstract let the mod 2 steenrod algebra a and the general linear group glk gl k f 2 act on pk f 2 x1 xk with deg xi 1 in the usual manner we prove that for a family of some rather small subgroups g of glk every element of positive degree in the invariant algebra pkg is hit by a in pk in other words pkg a pk where pkg and a denote respectively the submodules of pkg and a consisting of all elements of positive degree this family contains most of the parabolic subgroups of glk it should be noted that the smaller the group g is the harder the problem turns out to be remarkably when g is the smallest group of the family the invariant algebra pkg is a polynomial algebra in k variables whose degrees are 8 and fixed while k increases it has been shown by hu ng trans amer math soc 349 1997 3893 3910 that for g glk the inclusion pkglk a pk is equivalent to a weak algebraic version of the long standing conjecture stating that the only spherical classes in q0s0 are the elements of hopf invariant 1 and those of kervaire invariant 1

estimation filtering and smoothing in state space models with incompletely specified initial conditions on definit la vraisemblance pour un modele d espace d etats avec des conditions initiales incompletement specifiees par transformation des donnees pour eliminer la dependance sur des conditions non specifiees on etend cette approche pour obtenir des estimations sur les vecteurs d etat les predicteurs et les interpolateurs pour des observations manquantes

the relative burnside module and the stable maps between classifying spaces of compact lie groups tom dieck s burnside ring of compact lie groups is generalized to the relative case for any g d n a compact lie group and its normal subgroup a g n is defined to be an appropriate set of the equivalence classes of compact g enr s with free n action in such a way that i a g d n 7g n so b n g where b n g is the classifying space of principal n g bundle under the product situation i e g f x k n k a f x k d k is also denoted by a f k as it turns out to be the usual a f k when both f and k are finite then a couple of applications are given to the study of stable maps between classifying spaces of compact lie groups a conceptual proof of feshbach s double coset formula and a density theorem on the map ap a l h a a bl bh p for any compact lie groups l k when p is odd some restriction is applied to l when p 2 this latter result may be regarded as the pushout of feshbach s density theorem and the theorem of may snaith zelewski over the celebrated carlsson solution of segal s burnside ring conjecture 0 introduction even before the arrival of carlsson s celebrated paper c the segal conjecture for finite groups was known to imply the complete calculation of the stable maps between classifying spaces of finite groups agm lmm more recently snaith s2 got interested in the stable maps between classifying spaces of compact lie groups while studying his explicit brauer induction problem see s2 which is a nice reference on this topic snaith s research led to the further generalization of the segal conjecture by msz theorem a msz let f and j be finite groups and let k be a compact lie group then a induces the isomorphism a a f x j k a j fxj bfj bfk f where a f x j k is the grothendieck a f x j module of principal f x j k bundles over finite f x j sets and aj fxj is the completion of received by the editors june 23 1993 1991 mathematics subject classification primary 55p42 55r35 57s15 secondary 55q91 55r9 1

the porous medium equation on riemannian manifolds with negative curvature the large time behaviour abstract we consider nonnegative solutions of the porous medium equation pme on cartan hadamard manifolds whose negative curvature can be unbounded we take compactly supported initial data because we are also interested in free boundaries we classify the geometrical cases we study into quasi hyperbolic quasi euclidean and critical cases depending on the growth rate of the curvature at infinity we prove sharp upper and lower bounds on the long time behaviour of the solutions in terms of corresponding bounds on the curvature in particular we estimate the location of the free boundary a global harnack principle follows we also present a change of variables that allows to transform radially symmetric solutions of the pme on model manifolds into radially symmetric solutions of a corresponding weighted pme on euclidean space and back this equivalence turns out to be an important tool of the theory

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

exact likelihood of vector autoregressive moving average process with missing or aggregated data summary this note points out that by using the kalman filter with nonconstant coefficients we can compute the exact likelihood of an autoregressive moving average process observed with noise when some of our observations are either missing or aggregated

automorphisms of p groups given as cyclic by elementary abelian central extensions abstract we determine the structure of aut g and its relation to symplectic groups when g is a non abelian p group given by a central extension of the form 1 z p g z p z p 1

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

the negative answer to kameko s conjecture on the hit problem abstract the goal of this paper is to give a negative answer to kameko s conjecture on the hit problem stating that the cardinal of a minimal set of generators for the polynomial algebra p k considered as a module over the steenrod algebra a is dominated by an explicit quantity depending on the number of the polynomial algebra s variables k the conjecture was shown by kameko himself for k 3 in his phd thesis in the johns hopkins university in 1990 and recently proved by us and kameko for k 4 however we claim that it turns out to be wrong for any k 4 in order to deny kameko s conjecture we study a minimal set of generators for a module p k in some so call generic degrees what we mean by generic degrees is a bit different from that of other authors in the fields such as crabb hubbuck nam repka selick wood we prove an inductive formula for the cardinal of the minimal set of generators in these generic degrees when the number of the variables k increases as an immediate consequence of this inductive formula we recognize that kameko s conjecture is no longer true for any k 4

the weak conjecture on spherical classes let mathcal a be the mod 2 steenrod algebra we construct a chain level representation of the dual of singer s algebraic transfer which maps singer s invariant theoretic model of the dual of the lambda algebra gamma k wedge to bf f 2 x 1 pm 1 ldots x k pm 1 and is the inclusion of the dickson algebra d k subset gamma k wedge into bf f 2 x 1 ldots x k this chain level representation allows us to confirm the weak conjecture on spherical classes see 9 assuming the truth of 1 either the conjecture that the dickson invariants of at least k 3 variables are homologically zero in tor k mathcal a bf f 2 bf f 2 2 or a conjecture on mathcal a decomposability of the dickson algebra in gamma k wedge we prove the conjecture in item 1 for k 3 and also show a weak form of the conjecture in item 2

blow up and global existence for solutions to the porous medium equation with reaction and slowly decaying density abstract we study existence of global solutions and finite time blow up of solutions to the cauchy problem for the porous medium equation with a variable density x and a power like reaction term x u p with p 1 this is a mathematical model of a thermal evolution of a heated plasma see 29 the density decays slowly at infinity in the sense that x x q as x with q 0 2 we show that for large enough initial data solutions blow up in finite time for any p 1 on the other hand if the initial datum is small enough and p p for a suitable p depending on m n then global solutions exist in addition if p p for a suitable p p depending on m n then the solution blows up in finite time for any nontrivial initial datum we need the extra hypothesis that q 0 for 0 small enough when m p p observe that p p if x is a multiple of x q for x large enough such results are in agreement with those established in 48 where x 1 and are related to some results in 32 33 the case of fast decaying density at infinity i e q 2 is examined in 36

the porous medium equation on riemannian manifolds with negative curvature the superquadratic case we study the long time behaviour of nonnegative solutions of the porous medium equation posed on cartan hadamard manifolds having very large negative curvature more precisely when the sectional or ricci curvatures diverge at infinity more than quadratically in terms of the geodesic distance to the pole we find an unexpected separate variable behaviour that reminds one of dirichlet problems on bounded euclidean domains as a crucial step we prove existence of solutions to a related sublinear elliptic problem a result of independent interest uniqueness of solutions vanishing at infinity is also shown along with comparison principles both in the parabolic and in the elliptic case our results complete previous analyses of the porous medium equation flow on negatively curved riemannian manifolds which were carried out first for the hyperbolic space and then for general cartan hadamard manifolds with a negative curvature having at most quadratic growth we point out that no similar analysis seems to exist for the linear heat flow we also translate such results into some weighted porous medium equations in the euclidean space having special weights

estimation error and the specification of unobserved component models abstract the paper deals with the problem of identifying stochastic unobserved two component models as in seasonal adjustment or trend cycle decompositions solutions based on the properties of the unobserved component estimation error are considered and analytical expressions for the variance of the errors in the final preliminary and concurrent estimators are obtained these expressions are straightforwardly derived from the arima model for the observed series the estimation error variance is always minimized at a canonical decomposition i e at a decomposition with one of the components noninvertible and a simple procedure to determine that decomposition is presented on occasion however the most precise final estimator may be obtained at a canonical decomposition different from the one that yields the most precise preliminary estimator two examples are presented first a simple trend plus cycle type model is used to illustrate the derivations the second example presents results for a class of models often encountered in actual time series

the cohomology of the steenrod algebra and representations of the general linear groups let tr k be the algebraic transfer that maps from the coinvariants of certain gl k representations to the cohomology of the steenrod algebra this transfer was defined by w singer as an algebraic version of the geometrical transfer tr k s bv k s s 0 it has been shown that the algebraic transfer is highly nontrivial more precisely that tr k is an isomorphism for k 1 2 3 and that tr k tr k is a homomorphism of algebras in this paper we first recognize the phenomenon that if we start from any degree d and apply sq repeatedly at most k 2 times then we get into the region in which all the iterated squaring operations are isomorphisms on the coinvariants of the gl k representations as a consequence every finite sq 0 family in the coinvariants has at most k 2 nonzero elements two applications are exploited the first main theorem is that tr k is not a n isomorphism for k 5 furthermore for every k 5 there are infinitely many degrees in which tr k is not an isomorphism we also show that if tr detects a nonzero element in certain degrees of ker sq 0 then it is not a monomorphism and further for each k tr k is not a monomorphism in infinitely many degrees the second main theorem is that the elements of any sq 0 family in the cohomology of the steenrod algebra except at most its first k 2 elements are either all detected or all not detected by tr k for every k applications of this study to the cases k 4 and 5 show that tr 4 does not detect the three families g d 3 and p and that tr 5 does not detect the family h n 1g n n 1