a one factor multivariate time series model of metropolitan wage rates abstract the paper formulates and estimates a single factor multivariate time series model the model is a dynamic generalization of the multiple indicator or factor analysis model it is shown to be a special case of the general state space model and can be estimated by maximum likelihood methods using the kalman filter algorithm the model is used to obtain estimates of the unobserved metropolitan wage rate for los angeles based on observations of sectoral wages within the standard metropolitan statistical area hypothesis tests model diagnostics and out of sample forecasts are used to evaluate the model

correlation coefficients for random variables on a unit sphere or hypersphere summary a coefficient to measure association between two random variables each distributed on the surface of the unit sphere in p dimensional space is proposed analogous to an existing correlation coefficient for circular data the sample estimate is studied in detail for the case p 3 and comparisons are made with other statistics in current use some applications are given to studying remanent magnetization in rock samples

generalised linear model selection by the predictive least quasi deviance criterion we consider the problem of selecting a model with the best predictive ability in a class of generalised linear models a predictive least quasi deviance criterion is proposed to measure the predictive ability of a model this criterion is obtained by applying the idea of the predictive minimum description length principle and the theory of quasi likelihood functions the resulting predictive quasi deviance function is an extension of the predictive stochastic complexity of the model under rather weak conditions the predictive least quasi deviance method is shown to be consistent in the sense that the probability of selecting the right model converges to one as the number of observations goes to infinity also we show that the selected model converges to the optimal model in expectation the method is then modified for finite sample applications examples and simulation results are presented

a drifting markov process on the circle with physical applications a discrete time markov process for directions in the plane is introduced the random direction at any time is influenced both by the direction at the preceding time and by a target direction that depends on the current time the deviations of the directions from their targets have a limiting distribution as time tends to infinity and asymptotic approximations to the limiting probability density function are given both for the case of weak dependence and for the case of strong dependence tests of independence and of independence of increments are presented various types of behaviour of the process are illustrated by simulations some physical applications notably concerning the commensurate versus incommensurate nature of solid materials are discussed the application of the process is illustrated also by the analysis of some data on semi diurnal wind directions

regression models for an angular response summary this paper considers the problem of regressing an angular response variate on a set of linear explanatory variables a general class of models is proposed in which the mean direction and dispersion of a von mises variate are related to the explanatory variables by general link functions appropriate regression diagnostics estimation and testing procedures are developed for fitting the models the meaning of correlation between an angular and a linear variable is clarified and leads to a general notion of multiple correlation associated with the regression model the methods are applied to a set of data arising from a study of movements of intertidal gastropods

estimation and inference with weak semi strong and strong identification this paper analyzes the properties of standard estimators tests and confidence sets cs s for parameters that are unidentified or weakly identified in some parts of the parameter space the paper also introduces methods to make the tests and cs s robust to such identification problems the results apply to a class of extremum estimators and corresponding tests and cs s that are based on criterion functions that satisfy certain asymptotic stochastic quadratic expansions and that depend on the parameter that determines the strength of identification this covers a class of models estimated using maximum likelihood ml least squares ls quantile generalized method of moments generalized empirical likelihood minimum distance and semi parametric estimators the consistency lack of consistency and asymptotic distributions of the estimators are established under a full range of drifting sequences of true distributions the asymptotic sizes in a uniform sense of standard and identification robust tests and cs s are established the results are applied to the arma 1 1 time series model estimated by ml and to the nonlinear regression model estimated by ls in companion papers the results are applied to a number of other models

invariance nonlinear models and asymptotic tests the invariance properties of some well known asymptotic tests are studied three types of invariance are considered invariance to the representation of the null hypothesis invariance to one to one transformations of the parameter space reparameterizations and invariance to one to one transformations of the model variables such as changes in measurement units tests that are not invariant include the wald test and generalized versions of it a widely used variant of the lagrange multiplier test neyman s c a test and a generalized version of the latter for all these tests we show that simply changing measurement units can lead to vastly different answers even when equivalent null hypotheses are tested this problem is illustrated by considering regression models with box cox transformations on the variables we observe in particular that various consistent estimators of the information matrix lead to test procedures with different invariance properties general sufficient conditions are then established under which the generalized c a test becomes invariant to reformulations of the null hypothesis and or to one to one transformations of the parameter space as well as to transformations of the variables in many practical cases where wald type tests lack invariance we find that special formulations of the generalized c a test are invariant and hardly more costly to compute than wald tests this computational simplicity stands in contrast with other invariant tests such as the likelihood ratio test we conclude that noninvariant asymptotic tests should be avoided or used with great care further in many situations the suggested implementation of the generalized c a test often yields an attractive substitute to the wald test which is not invariant and to other invariant tests which are more costly to perform

a universal prior for integers and estimation by minimum description length of the number of bits required to write down the observed data has been reformulated to extend the classical maximum likelihood principle the principle permits estimation of the number of the parameters in statistical models in addition to their values and even of the way the parameters appear in the models i e of the model structures the principle rests on a new way to interpret and construct a universal prior distribution for the integers which makes sense even when the parameter is an individual object truncated realvalued parameters are converted to integers by dividing them by their precision and their prior is determined from the universal prior for the integers by optimizing the precision 1 introduction in this paper we study estimation based upon the principle of minimizing the total number of binary digits required to rewrite the observed data when each observation is given with some precision instead of attempting at an absolutely shortest description which would be futile we look for the optimum relative to a class of parametrically given distributions this minimum description length mdl principle which we introduced in a less comprehensive form in 25 turns out to degenerate to the more familiar maximum likelihood ml principle in case the number of parameters in the models is fixed so that the description length of the parameters themselves can be ignored in another extreme case where the parameters determine the data it similarly degenerates to jaynes s principle of maximum entropy 14 but the main power of the new criterion is that it permits estimates of the entire model its parameters their number and even the way the parameters appear in the model i e the model structure hence there will be no need to supplement the estimated parameters with a separate hypothesis test to decide whether a model is adequately parameterized or perhaps over parameterized

how many variables should be entered in a regression equation abstract the optimal number of regressors is determined to minimize mean squared prediction error and is shown to be a small fraction of the number of data points as the number of regressors grows large the sp criterion provides an asymptotically optimal rule for the number of variables to enter

data compression and histograms in this paper the relationship between code length and the selection of the number of bins for a histogram density is considered for a sequence of iid observations on 0 1 first we use a shortest code length criterion to select the number of bins for a histogram a uniform almost sure asymptotic expansion for the code length is given and it is used to prove the asymptotic optimality of the selection rule in addition the selection rule is consistent if the true density is uniform 0 1 secondly we deal with the problem what is the best achievable average code length with underlying density functionf minimax lower bounds are derived for the average code length over certain smooth classes of underlying densitiesf for the smooth class with bounded first derivatives the rate in the lower bound is shown to be achieved by a code based on a sequence of histograms whose number of bins is changed predictively moreover this best code can be modified to ensure that the almost sure version of the code length has asymptotically the same behavior as its expected value i e the average code length

model selection and the principle of minimum description length this article reviews the principle of minimum description length mdl for problems of model selection by viewing statistical modeling as a means of generating descriptions of observed data the mdl framework discriminates between competing models based on the complexity of each description this approach began with kolmogorov s theory of algorithmic complexity matured in the literature on information theory and has recently received renewed attention within the statistics community here we review both the practical and the theoretical aspects of mdl as a tool for model selection emphasizing the rich connections between information theory and statistics at the boundary between these two disciplines we find many interesting interpretations of popular frequentist and bayesian procedures as we show mdl provides an objective umbrella under which rather disparate approaches to statistical modeling can coexist and be compared we illustrate the mdl principle by considering problems in regression nonpar

properties of sufficiency and statistical tests 1 in a previous paper dealing with the importance of properties of sufficiency in the statistical theory of small samples attention was mainly confined to the theory of estimation in the present paper the structure of small sample tests whether these are related to problems of estimation and fiducial distributions or are of the nature of tests of goodness of fit is considered further

a direct derivation of the exact fisher information matrix of gaussian vector state space models this paper deals with a direct derivation of fisher s information matrix of vector state space models for the general case by which is meant the establishment of the matrix as a whole and not element by element the method to be used is matrix differentiation see 4 we assume the model to be gaussian and use the negative logarithm of the likelihood function as used in the definition of fisher s information in a related paper klein et al 3 establish the information matrix by assembling its elements as derived in the literature 2 5 6 and for an approximation of the hessian of the log likelihood function one can refer to 7 8

the comparison of time series recorders the problem of comparing the outputs of several time series recorders when each receives the same signal together with stationary additive noise is considered when the signal is stationary or is the sum of stationary and periodic components methods of estimating some parameters using frequency domain methods are given the estimates are obtained by maximizing an approximate gaussian likelihood but do not require gaussian data the estimation is particularly simple if the spectra of the different error processes are equal asymptotic properties of the estimators are presented here with an application of the methods a small simulation study shows that the asymptotic theory is a reasonable approximation for this application

density estimation by stochastic complexity the results by p hall and e j hannan 1988 on optimization of histogram density estimators with equal bin widths by minimization of the stochastic complexity are extended and sharpened in two separate ways as the first contribution two generalized histogram estimators are constructed the first has unequal bin widths which together with the number of the bins are determined by minimization of the stochastic complexity using dynamic programming the other estimator consists of a mixture of equal bin width estimators each of which is defined by the associated stochastic complexity as the main contribution in the present work two theorems are proved which together extend the universal coding theorems to a large class of data generating densities the first gives an asymptotic upper bound for the code redundancy in the order of magnitude achieved with a special predictive type of histogram estimator which sharpens a related bound the second theorem states that this bound cannot be improved upon by any code whatsoever

prepivoting test statistics a bootstrap view of asymptotic refinements abstract approximate tests for a composite null hypothesis about a parameter may be obtained by referring a test statistic to an estimated critical value either asymptotic theory or bootstrap methods can be used to estimate the desired quantile the simple asymptotic test a refers the test statistic to a quantile of its asymptotic null distribution after unknown parameters have been estimated the bootstrap approach used here is based on the concept of prepivoting prepivoting is the transformation of a test statistic by the cdf of its bootstrap null distribution the simple bootstrap test b refers the prepivoted test statistic to a quantile of the uniform 0 1 distribution under regularity conditions the bootstrap test b has a smaller asymptotic order of error in level than does the asymptotic test a provided that the asymptotic null distribution of the test statistic does not depend on unknown parameters in the contrary case both a and b have the same order of level error certain class

approximation dans les espaces metriques et theorie de l estimation we investigate the relations between the speed of estimation and the metric structure of the parameter space especially in the case when its metric dimension is infinite given some distance d on generally hellinger distance in the case of n i i d variables we consider the minimax risk for n observations r n q mathop inf sup limits t n theta in theta text ie theta d q theta t n t n being any estimate of we shall look for functions r such that for positive constants c 1 q and c 2 q c 1 r q n r n q c2 r q n r n is the speed of estimation and we shall show under fairly general conditions including i i d variables and regular cases of markov chains and stationnary gaussian processes that r n is determined up to multiplicative constants by the metric structure of we shall also give a construction for some sort of universal estimates the risk of which is bounded by c 2 r q n in all cases where the preceding theory applies

stochastic complexity and modeling on demontre un theoreme fondamental qui donne une borne inferieure pour la longueur de code et donc pour les erreurs de prediction on definit les notions d information a priori et d information utile dans les donnees

a correlation coefficient for circular data summary a coefficient to measure association between angular variables is discussed its asymptotic distribution found and its properties developed comparisons with other statistics in current use are made and some examples given

construction of the exact fisher information matrix of gaussian time series models by means of matrix differential rules the fisher information matrix is of fundamental importance for the analysis of parameter estimation of time series models in this paper the exact information matrix of a multivariate gaussian time series model expressed in state space form is derived a computationally efficient procedure is used by applying matrix differential rules for the derivatives of a matrix function j j with respect to its vector argument an algorithm is given it is sketched for the general state space structure without specifying a parametrization it is then detailed for the vector autoregressive moving average varma model with a given parametrization where explicit recurrent relations are developed

lower bounds for the minimax risk using f divergences and applications lower bounds involving f divergences between the underlying probability measures are proved for the minimax risk in estimation problems our proofs just use simple convexity facts special cases and straightforward corollaries of our bounds include well known inequalities for establishing minimax lower bounds such as fano s inequality pinsker s inequality and inequalities based on global entropy conditions two applications are provided a new minimax lower bound for the reconstruction of convex bodies from noisy support function measurements and a different proof of a recent minimax lower bound for the estimation of a covariance matrix

comparing dynamic equilibrium models to data a bayesian approach abstract this paper studies the properties of the bayesian approach to estimation and comparison of dynamic equilibrium economies both tasks can be performed even if the models are nonnested misspecified and nonlinear first we show that bayesian methods have a classical interpretation asymptotically the parameter point estimates converge to their pseudotrue values and the best model under the kullback leibler distance will have the highest posterior probability second we illustrate the strong small sample behavior of the approach using a well known application the u s cattle cycle bayesian estimates outperform maximum likelihood results and the proposed model is easily compared with a set of bvars

on strong consistency of model selection in classification this paper considers model selection in classification in many applications such as pattern recognition probabilistic inference using a bayesian network prediction of the next in a sequence based on a markov chain the conditional probability p y y x x of class yisiny given attribute value xisinx is utilized by model we mean the equivalence relation in x for x x isinxx x harrp y y x x p y y x x forall yisiny by classification we mean the number of such equivalence classes is finite we estimate the model from n samples zn xi yi i 1 n isin xtimesy n using information criteria in the form empirical entropy h plus penalty term k 2 dn the model such that h k 2 dn is minimized is the estimated model where k is the number of independent parameters in the model and dn n 1 infin is a real nonnegative sequence such that lim supndn n 0 for autoregressive processes although the definitions of h and k are different it is known that the estimated model almost surely coincides with the true model as nrarrinfin if dn n 1 infin 2loglogn n 1 infin and that it does not if dn n 1 infin 2loglogn n 1 infin hannan and quinn the problem whether the same property is true for classification was open this paper solves the problem in the affirmative

asymptotics for likelihood ratio tests under loss of identifiability this paper describes the large sample properties of the likelihood ratio test statistic lrts when the parameters characterizing the true null distribution are not unique it is well known that the classical asymptotic theory for the likelihood ratio test does not apply to such problems and the lrts may not have the typical chi squared type limiting distribution this paper establishes a general quadratic approximation of the log likelihood ratio function in a hellinger neighborhood of the true density which is valid with or without loss of identifiability of the true distribution under suitable conditions the asymptotic null distribution of the lrts under loss of identifiability can be obtained by maximizing the quadratic form these results extend the work of chernoff and le cam in particular applications to testing the number of mixture components in finite mixture models are discussed

on the level error after bartlett adjustment of the likelihood ratio statistic summary it is well known that bartlett adjustment reduces level error of the likelihood ratio statistic from order n l to order n3 2 in the present note we show that level error of the adjusted statistic is actually order n2

estimation of a regression function by maxima of minima of linear functions in this paper estimation of a regression function from independent and identically distributed random variables is considered estimates are defined by minimization of the empirical l2 risk over a class of functions which are defined as maxima of minima of linear functions results concerning the rate of convergence of the estimates are derived in particular it is shown that for smooth regression functions satisfying the assumption of single index models the estimate is able to achieve up to some logarithmic factor the corresponding optimal one dimensional rate of convergence hence under these assumptions the estimate is able to circumvent the so called curse of dimensionality the small sample behavior of the estimates is illustrated by applying them to simulated data

minimax risk over hyperrectangles and implications consider estimating the mean of a standard gaussian shift when that mean is known to lie in an orthosymmetric quadratically convex set in l 2 such sets include ellipsoids hyperrectangles and l p bodies with p 2 the minimax risk among linear estimates is within 25 of the minimax risk among all estimates the minimax risk among truncated series estimates is within a factor 4 44 of the minimax risk this implies that the difficulty of estimation a statistical quantity is measured fairly precisely by the n widths a geometric quantity if the set is not quadratically convex as in the case of l p bodies with p 2 things change appreciably minimax linear estimators may be out performed arbitrarily by nonlinear estimates the ordinary kolmogorov n widths still determine the difficulty of linear estimation but the difficulty of nonlinear estimation is tied to the inner bernstein n widths which can be far smaller essential use is made of a new heuristic that the difficulty of the hardest rectangular subproblem is equal to the difficulty of the full problem

probability inequalities for likelihood ratios and convergence rates of sieve mles let y 1 y n be independent identically distributed with density p 0 and let f be a space of densities we show that the supremum of the likelihood ratios i 1 n p y i p 0 y i where the supremum is over p f with p l 2 p 0 1 2 2 e is exponentially small with probability exponentially close to 1 the exponent is proportional to ne 2 the only condition required for this to hold is that e exceeds a value determined by the bracketing hellinger entropy of f a similar inequality also holds if we replace f by f n and p 0 by q n where q n is an approximation to p 0 in a suitable sense these results are applied to establish rates of convergence of sieve mles furthermore weak conditions are given under which the optimal rate e n defined by h e n f ne n 2 where h f is the hellinger entropy of f is nearly achievable by sieve estimators

time series analysis of circular data summary the paper considers the problem of modelling and analysing a time series of measurements which take the form of unit two dimensional vectors several general classes of existing and new models are studied in terms of feasibility and flexibility an approach is recommended which uses two of these classes and which takes advantage of several standard time series algorithms that are available in modern software packages an application is given to the analysis of a series of data on wind directions

on stochastic complexity and nonparametric density estimation summary we use the concepts of stochastic complexity description length and model selection to develop data based methods for choosing smoothing parameters in nonparametric density estimation in the case of histogram estimators we derive a simple exact formula for stochastic complexity when the prior distribution of cell probabilities is uniform over the class of all possible choices the formula depends only on the data and the smoothing parameter which is readily chosen according to the criterion of minimum stochastic complexity approaches based on stochastic complexity and description length are shown to be asymptotically equivalent in certain circumstances they produce a degree of smoothing which is almost optimal from the viewpoint of minimizing l or supremum distance but which smooths a little more than is optimal in the sense of minimizing l distance for any finite value of r

on maximum likelihood estimation in infinite dimensional parameter spaces an approximate maximum likelihood estimate is known to be consistent under some compactness and integrability conditions in this paper we study its convergence rate and its asymptotic efficiency in estimating smooth functionals of the parameter we provide conditions under which the rate of convergence can be established this rate is essentially governed by the size of the space of score functions as measured by an entropy index we also show that for a large class of smooth functionals the plug in maximum likelihood estimate is asymptotically efficient that is it achieves the minimal fisher information bound the theory is illustrated by several nonparametric or semiparametric examples

on the histogram as a density estimator l 2 theory let f be a probability density on an interval i finite or infinite i includes its finite endpoints if any and f vanishes outside of i let x1 x k be independent random variables with common density f the empirical histogram for the x s is often used to estimate f to define this object choose a reference point xosi and a cell width h let nj be the number of x s falling in the j th class interval

lower bounds on expected redundancy for nonparametric classes the article focuses on lower bound results on expected redundancy for universal coding of independent and identically distributed data on 0 1 from parametric and nonparametric families after reviewing existing lower bounds we provide a new proof for minimax lower bounds on expected redundancy over nonparametric density classes this new proof is based on the calculation of a mutual information quantity or it utilizes the relationship between redundancy and shannon capacity it therefore unifies the minimax redundancy lower bound proofs in the parametric and nonparametric cases

likelihood ratio testing in linear state space models an application to dynamic stochastic general equilibrium models abstract this paper considers the problem of hypothesis testing in linear gaussian state space models we consider two hypotheses of interest a simple null and a hypothesis of explicit parameter restrictions we derive the asymptotic distributions of the corresponding likelihood ratio test statistics and compute the bartlett adjustments the results are non trivial because the unrestricted state space model is not even locally identified we apply our analysis to test the validity of the dynamic stochastic general equilibrium dsge models a monte carlo exercise illustrates our findings and confirms the importance of bartlett corrections at sample sizes typically encountered in macroeconomics

minimum complexity density estimation the authors introduce an index of resolvability that is proved to bound the rate of convergence of minimum complexity density estimators as well as the information theoretic redundancy of the corresponding total description length the results on the index of resolvability demonstrate the statistical effectiveness of the minimum description length principle as a method of inference the minimum complexity estimator converges to true density nearly as fast as an estimator based on prior knowledge of the true subclass of densities interpretations and basic properties of minimum complexity estimators are discussed some regression and classification problems that can be examined from the minimum description length framework are considered

some notes on rissanen s stochastic complexity a new version of stochastic complexity for a parametric statistical model is derived based on a class of two part codes we show that choosing the quantization in the first step according to the fisher information is optimal and we compare our approach to a recent result of rissanen 1983 an application to robust regression model selection is presented

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

invariants and likelihood ratio statistics because the likelihood ratio statistic is invariant under reparameterization it is possible to make a large sample expansion of the statistic itself and of its expectation in terms of invariants in particular the bartlett adjustment factor can be expressed in terms of invariant combinations of cumulants of the first two log likelihood derivatives such expansions are given first for a scalar parameter and then for vector parameters geometrical interpretation is given where possible and some special cases discussed

a rate of convergence result for a universal d semifaithful code the problem of optimal rate universal coding is considered in the context of rate distortion theory a d semifaithful universal coding scheme for discrete memoryless sources is given the main result is a refined covering lemma based on the random coding argument and the method of types the average codelength of the code is shown to approach its lower bound the rate distortion function at a rate o n sup 1 log n and this is conjectured to be optimal based on a result of a j pilc 1968 issues of constructiveness and universality are addressed

the minimum description length principle in coding and modeling we review the principles of minimum description length and stochastic complexity as used in data compression and statistical modeling stochastic complexity is formulated as the solution to optimum universal coding problems extending shannon s basic source coding theorem the normalized maximized likelihood mixture and predictive codings are each shown to achieve the stochastic complexity to within asymptotically vanishing terms we assess the performance of the minimum description length criterion both from the vantage point of quality of data compression and accuracy of statistical inference context tree modeling density estimation and model selection in gaussian linear regression serve as examples

metalearners for estimating heterogeneous treatment effects using machine learning there is growing interest in estimating and analyzing heterogeneous treatment effects in experimental and observational studies we describe a number of metaalgorithms that can take advantage of any supervised learning or regression method in machine learning and statistics to estimate the conditional average treatment effect cate function metaalgorithms build on base algorithms such as random forests rfs bayesian additive regression trees barts or neural networks to estimate the cate a function that the base algorithms are not designed to estimate directly we introduce a metaalgorithm the x learner that is provably efficient when the number of units in one treatment group is much larger than in the other and can exploit structural properties of the cate function for example if the cate function is linear and the response functions in treatment and control are lipschitz continuous the x learner can still achieve the parametric rate under regularity conditions we then introduce versions of the x learner that use rf and bart as base learners in extensive simulation studies the x learner performs favorably although none of the metalearners is uniformly the best in two persuasion field experiments from political science we demonstrate how our x learner can be used to target treatment regimes and to shed light on underlying mechanisms a software package is provided that implements our methods

dynamic identification of dynamic stochastic general equilibrium models this paper studies dynamic identification of parameters of a dynamic stochastic general equilibrium model from the first and second moments of the data classical results for dynamic simultaneous equations do not apply because the state space solution of the model does not constitute a standard reduced form full rank of the jacobian matrix of derivatives of the solution parameters with respect to the parameters of interest is necessary but not sufficient for identification we use restrictions implied by observational equivalence to obtain two sets of rank and order conditions one for stochastically singular models and another for nonsingular models measurement errors mean long run and a priori restrictions can be accommodated an example is considered to illustrate the results

covering numbers for convex functions in this paper we study the covering numbers of the space of convex and uniformly bounded functions in multidimension we find optimal upper and lower bounds for the epsilon covering number of cal c a b d b in the l p metric 1 leq p in terms of the relevant constants where d geq 1 a b 0 and cal c a b d b denotes the set of all convex functions on a b d that are uniformly bounded by b we summarize previously known results on covering numbers for convex functions and also provide alternate proofs of some known results our results have direct implications in the study of rates of convergence of empirical minimization procedures as well as optimal convergence rates in the numerous convexity constrained function estimation problems

minimax noiseless universal coding for markov sources upper and lower bounds are presented on the minimax redundancy for markov source noiseless block to variable universal coding the number of states block size and markov order are arbitrary but finite unlike earlier results the upper and lower bounds are absolute that is not merely asymptotic the upper bound is established by combinatorial bounds the lower bound is established by a bound on the average redundancy for an arbitrary distribution on the transition probabilities the bound can be optimized across the choice of distribution

a markov process for circular data summary we propose a discrete time markov process which takes values on the unit circle some properties of the process including the limiting behaviour and ergodicity are investigated many computations associated with this process are shown to be greatly simplified if the variables and parameters of the model are represented in terms of complex numbers a further discussion is given on some submodels in particular on the stationary process the proposed model is compared with some existing markov processes for circular data some statistical issues of the model such as statistical inference model selection and diagnostic checks are considered finally an application of the model to wind direction data is provided