optimal eigen expansions and uniform bounds let x k k in mathbb z in mathbb l 2 mathcal t be a stationary process with associated lag operators varvec mathcal c h uniform asymptotic expansions of the corresponding empirical eigenvalues and eigenfunctions are established under almost optimal conditions on the lag operators in terms of the eigenvalues spectral gap in addition the underlying dependence assumptions are optimal in a certain sense including both short and long memory processes this allows us to study the relative maximum deviation of the empirical eigenvalues under very general conditions among other things convergence to an extreme value distribution is shown we also discuss how the asymptotic expansions transfer to the long run covariance operator varvec mathcal g in a general framework

minimax renyi redundancy the redundancy for universal lossless compression of discrete memoryless sources in campbell s setting is characterized as a minimax renyi divergence which is shown to be equal to the maximal alpha mutual information via a generalized redundancy capacity theorem special attention is placed on the analysis of the asymptotics of minimax renyi divergence which is determined up to a term vanishing in blocklength

unexpected properties of bandwidth choice when smoothing discrete data for constructing a functional data classifier the data functions that are studied in the course of functional data analysis are assembled from discrete data and the level of smoothing that is used is generally that which is appropriate for accurate approximation of the conceptually smooth functions that were not actually observed existing literature shows that this approach is effective and even optimal when using functional data methods for prediction or hypothesis testing however in the present paper we show that this approach is not effective in classification problems there a useful rule of thumb is that undersmoothing is often desirable but there are several surprising qualifications to that approach first the effect of smoothing the training data can be more significant than that of smoothing the new data set to be classified second undersmoothing is not always the right approach and in fact in some cases using a relatively large bandwidth can be more effective and third these perverse results are the consequence of very unusual properties of error rates expressed as functions of smoothing parameters for example the orders of magnitude of optimal smoothing parameter choices depend on the signs and sizes of terms in an expansion of error rate and those signs and sizes can vary dramatically from one setting to another even for the same classifier

3 d shape estimation of dna molecules from stereo cryo electron micro graphs using a projection steerable snake we introduce a three dimensional 3 d parametric active contour algorithm for the shape estimation of dna molecules from stereo cryo electron micrographs we estimate the shape by matching the projections of a 3 d global shape model with the micrographs we choose the global model as a 3 d filament with a b spline skeleton and a specified radial profile the active contour algorithm iteratively updates the b spline coefficients which requires us to evaluate the projections and match them with the micrographs at every iteration since the evaluation of the projections of the global model is computationally expensive we propose a fast algorithm based on locally approximating it by elongated blob like templates we introduce the concept of projection steerability and derive a projection steerable elongated template since the two dimensional projections of such a blob at any 3 d orientation can be expressed as a linear combination of a few basis functions matching the projections of such a 3 d template involves evaluating a weighted sum of inner products between the basis functions and the micrographs the weights are simple functions of the 3 d orientation and the inner products are evaluated efficiently by separable filtering we choose an internal energy term that penalizes the average curvature magnitude since the exact length of the dna molecule is known a priori we introduce a constraint energy term that forces the curve to have this specified length the sum of these energies along with the image energy derived from the matching process is minimized using the conjugate gradients algorithm we validate the algorithm using real as well as simulated data and show that it performs well

guessing under source uncertainty this paper considers the problem of guessing the realization of a finite alphabet source when some side information is provided the only knowledge the guesser has about the source and the correlated side information is that the joint source is one among a family a notion of redundancy is first defined and a new divergence quantity that measures this redundancy is identified this divergence quantity shares the pythagorean property with the kullback leibler divergence good guessing strategies that minimize the supremum redundancy over the family are then identified the min sup value measures the richness of the uncertainty set the min sup redundancies for two examples the families of discrete memoryless sources and finite state arbitrarily varying sources are then determined

improved bounds on lossless source coding and guessing moments via renyi measures this paper provides upper and lower bounds on the optimal guessing moments of a random variable taking values on a finite set when side information may be available these moments quantify the number of guesses required for correctly identifying the unknown object and similarly to arikan s bounds they are expressed in terms of the arimoto renyi conditional entropy although arikan s bounds are asymptotically tight the improvement of the bounds in this paper is significant in the non asymptotic regime relationships between moments of the optimal guessing function and the map error probability are also established characterizing the exact locus of their attainable values the bounds on optimal guessing moments serve to improve non asymptotic bounds on the cumulant generating function of the codeword lengths for fixed to variable optimal lossless source coding without prefix constraints non asymptotic bounds on the reliability function of discrete memoryless sources are derived as well relying on these techniques lower bounds on the cumulant generating function of the codeword lengths are derived by means of the smooth renyi entropy for source codes that allow decoding errors

on properties of functional principal components analysis summary functional data analysis is intrinsically infinite dimensional functional principal component analysis reduces dimension to a finite level and points to the most significant components of the data however although this technique is often discussed its properties are not as well understood as they might be we show how the properties of functional principal component analysis can be elucidated through stochastic expansions and related results our approach quantifies the errors that arise through statistical approximation in successive terms of orders n 1 2 n 1 n 3 2 where n denotes sample size the expansions show how spacings among eigenvalues impact on statistical performance the term of size n 1 2 illustrates first order properties and leads directly to limit theory which describes the dominant effect of spacings thus for example spacings are seen to have an immediate first order effect on properties of eigenfunction estimators but only a second order effect on eigenvalue estimators our results can be used to explore properties of existing methods and also to suggest new techniques in particular we suggest bootstrap methods for constructing simultaneous confidence regions for an infinite number of eigenvalues and also for individual eigenvalues and eigenvectors

arimoto renyi conditional entropy and bayesian m ary hypothesis testing this paper gives upper and lower bounds on the minimum error probability of bayesian m ary hypothesis testing in terms of the arimoto renyi conditional entropy of an arbitrary order alpha the improved tightness of these bounds over their specialized versions with the shannon conditional entropy alpha 1 is demonstrated in particular in the case where m is finite we show how to generalize fano s inequality under both the conventional and list decision settings as a counterpart to the generalized fano s inequality allowing m to be infinite a lower bound on the arimoto renyi conditional entropy is derived as a function of the minimum error probability explicit upper and lower bounds on the minimum error probability are obtained as a function of the arimoto renyi conditional entropy for both positive and negative alpha furthermore we give upper bounds on the minimum error probability as functions of the renyi divergence in the setup of discrete memoryless channels we analyze the exponentially vanishing decay of the arimoto renyi conditional entropy of the transmitted codeword given the channel output when averaged over a random coding ensemble

hierarchical universal coding in an earlier paper we proved a strong version of the redundancy capacity converse theorem of universal coding stating that for most sources in a given class the universal coding redundancy is essentially lower bounded by the capacity of the channel induced by this class since this result holds for general classes of sources it extends rissanen s 1986 strong converse theorem for parametric families while our earlier result has established strong optimality only for mixture codes weighted by the capacity achieving prior our first result herein extends this finding to a general prior for some cases our technique also leads to a simplified proof of the above mentioned strong converse theorem the major interest in this paper however is in extending the theory of universal coding to hierarchical structures of classes where each class may have a different capacity in this setting one wishes to incur redundancy essentially as small as that corresponding to the active class and not the union of classes our main result is that the redundancy of a code based on a two stage mixture first within each class and then over the classes is no worse than that of any other code for most sources of most classes if in addition the classes can be efficiently distinguished by a certain decision rule then the best attainable redundancy is given explicitly by the capacity of the active class plus the normalized negative logarithm of the prior probability assigned to this class these results suggest some interesting guidelines as for the choice of the prior we also discuss some examples with a natural hierarchical partition into classes

dispersion operators and resistant second order functional data analysis inferences related to the second order properties of functional data as expressed by covariance structure can become unreliable when the data are non gaussian or contain unusual observations in the functional setting it is often difficult to identify atypical observations as their distinguishing characteristics can be manifold but subtle in this paper we introduce the notion of a dispersion operator investigate its use in probing the second order structure of functional data and develop a test for comparing the second order characteristics of two functional samples that is resistant to atypical observations and departures from normality the proposed test is a regularized m test based on a spectrally truncated version of the hilbert schmidt norm of a score operator defined via the dispersion operator we derive the asymptotic distribution of the test statistic investigate the behaviour of the test in a simulation study and illustrate the method on a structural biology dataset

detecting and localizing differences in functional time series dynamics a case study in molecular biophysics abstractmotivated by the problem of inferring the molecular dynamics of dna in solution and linking them with its base pair composition we consider the problem of comparing the dynamics of functional time series fts and of localizing any inferred differences in frequency and along curvelength the approach we take is one of fourier analysis where the complete second order structure of the fts is encoded by its spectral density operator indexed by frequency and curvelength the comparison is broken down to a hierarchy of stages at a global level we compare the spectral density operators of the two fts across frequencies and curvelength based on a hilbert schmidt criterion then we localize any differences to specific frequencies and finally we further localize any differences along the length of the random curves that is in physical space a hierarchical multiple testing approach guarantees control of the averaged false discovery rate over the selected frequencies in this sense we are ab

a strong version of the redundancy capacity theorem of universal coding the capacity of the channel induced by a given class of sources is well known to be an attainable lower bound on the redundancy of universal codes with respect to this class both in the minimax sense and in the bayesian maximin sense we show that this capacity is essentially a lower bound also in a stronger sense that is for most sources in the class this result extends rissanen s 1984 1986 lower bound for parametric families we demonstrate the applicability of this result in several examples e g parametric families with growing dimensionality piecewise fixed sources arbitrarily varying sources and noisy samples of learnable functions finally we discuss implications of our results to statistical inference

data driven version of neyman s smooth test of fit abstract neyman s smooth test for testing uniformity is a recognized goodness of fit procedure as stated by lariccia the test can be viewed as a compromise between omnibus test procedures with generally low power in all directions and procedures whose power is focused in the direction of a specific alternative the basic idea behind this test is to embed the null density into say a k dimensional exponential family and then to construct an asymptotically optimal test for the parametric testing problem the resulting procedure is neyman s test with k components the most difficult problem related with using this test is the choice of k recommendations in statistical literature are sometimes confusing some authors advocate a small number of components whereas others show that in some situations a larger number of components is profitable all existing suggestions concerning how to select k exploit in fact some preliminary knowledge about a possible alternative in this article a new data driven met

fourier analysis of stationary time series in function space we develop the basic building blocks of a frequency domain framework for drawing statistical inferences on the second order structure of a stationary sequence of functional data the key element in such a context is the spectral density operator which generalises the notion of a spectral density matrix to the functional setting and characterises the second order dynamics of the process our main tool is the functional discrete fourier transform fdft we derive an asymptotic gaussian representation of the fdft thus allowing the transformation of the original collection of dependent random functions into a collection of approximately independent complex valued gaussian random functions our results are then employed in order to construct estimators of the spectral density operator based on smoothed versions of the periodogram kernel the functional generalisation of the periodogram matrix the consistency and asymptotic law of these estimators are studied in detail as immediate consequences we obtain central limit theorems for the mean and the long run covariance operator of a stationary functional time series our results do not depend on structural modelling assumptions but only functional versions of classical cumulant mixing conditions and are shown to be stable under discrete observation of the individual curves

new tests for equality of several covariance functions for functional data in this article we propose two new tests for the equality of the covariance functions of several functional populations namely a quasi gpf test and a quasi fmax test whose test statistics are o

a wilcoxon mann whitney type test for infinite dimensional data the wilcoxon mann whitney test is a robust competitor of the t test in the univariate setting for finite dimensional multivariate non gaussian data several extensions of the wilcoxon mann whitney test have been shown to outperform hotelling s t 2 test in this paper we study a wilcoxon mann whitney type test based on spatial ranks in infinite dimensional spaces we investigate its asymptotic properties and compare it with several existing tests the proposed test is shown to be robust with respect to outliers and to have better power than some competitors for certain distributions with heavy tails we study its performance using real and simulated data

functional data analysis for sparse longitudinal data we propose a nonparametric method to perform functional principal components analysis for the case of sparse longitudinal data the method aims at irregularly spaced longitudinal data where the number of repeated measurements available per subject is small in contrast classical functional data analysis requires a large number of regularly spaced measurements per subject we assume that the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random subject specific trajectory plus measurement errors basic elements of our approach are the parsimonious estimation of the covariance structure and mean function of the trajectories and the estimation of the variance of the measurement errors the eigenfunction basis is estimated from the data and functional principal components score estimates are obtained by a conditioning step this conditional estimation method is conceptually simple and straightforward to implement

guessing subject to distortion we investigate the problem of guessing a random vector x within distortion level d our aim is to characterize the best attainable performance in the sense of minimizing in some probabilistic sense the number of required guesses g x until the error falls below d the underlying motivation is that g x is the number of candidate codewords to be examined by a rate distortion block encoder until a satisfactory codeword is found in particular for memoryless sources we provide a single letter characterization of the least achievable exponential growth rate of the spl rho th moment of g x as the dimension of the random vector x grows without bound in this context we propose an asymptotically optimal guessing scheme that is universal both with respect to the information source and the value of spl rho we then study some properties of the exponent function e d spl rho along with its relation to the source coding exponents finally we provide extensions of our main results to the gaussian case guessing with side information and sources with memory

distances and inference for covariance operators a framework is developed for inference concerning the covariance operator of a functional random process where the covariance operator itself is an object of interest for statistical analysis distances for comparing positive definite covariance matrices are either extended or shown to be inapplicable to functional data in particular an infinite dimensional analogue of the procrustes size and shape distance is developed convergence of finite dimensional approximations to the infinite dimensional distance metrics is also shown for inference a frechet estimator of both the covariance operator itself and the average covariance operator is introduced a permutation procedure to test the equality of the covariance operators between two groups is also considered additionally the use of such distances for extrapolation to make predictions is explored as an example of the proposed methodology the use of covariance operators has been suggested in a philological study of cross linguistic dependence as a way to incorporate quantitative phonetic information it is shown that distances between languages derived from phonetic covariance functions can provide insight into the relationships between the romance languages

bootstrap based testing of equality of mean functions or equality of covariance operators for functional data we investigate the properties of a simple bootstrap method for testing the equality of mean functions or of covariance operators in functional data theoretical size and power results are derived for certain test statistics whose limiting distributions depend on unknown infinite dimensional parameters simulations demonstrate good size and power of the bootstrap based functional tests

entropy analysis and new constructions of biometric key generation systems in this paper a new soft two level approach for the generation of multiple and revocable biometric keys adapted to the analog nature of biometric signals is proposed it consists of a novel randomized soft code offset construction for the euclidean metric at the first level and a code redundancy construction for the hamming metric possibly based on a reed solomon code at the second level the shannon entropy analysis shows that the new construction achieves optimal security with respect to biometric template protection whereas the previously proposed constructions for the euclidean metric are shown to be insecure in the multiple key setting in addition a general code offset construction for the hamming metric is analyzed in terms of the shannon entropy instead of the usual min entropy this allows a nice characterization of the codes that provide biometric template protection in the multiple key scenario as well as a further differentiation among these codes with respect to the achievable security level

asymptotics and non asymptotics for universal fixed to variable source coding universal fixed to variable lossless source coding for memoryless sources is studied in the finite blocklength and higher order asymptotic regimes optimal three term fixed error asymptotic expressions are derived for general fixed to variable codes and for prefix codes it is shown that the non prefix type size code in which codeword lengths are chosen in ascending order of type class size achieves the optimal third order term and outperforms classical two stage codes converse results are proved making use of a result on the distribution of the empirical entropy and laplace s approximation finally the fixed to variable coding problem without a prefix constraint is shown to be essentially the same as the universal guessing problem

estimation of the mean of functional time series and a two sample problem summary the paper is concerned with inference based on the mean function of a functional time series we develop a normal approximation for the functional sample mean and then focus on the estimation of the asymptotic variance kernel using these results we develop and asymptotically justify testing procedures for the equality of means in two functional samples exhibiting temporal dependence evaluated by means of a simulation study and application to a real data set these two sample procedures enjoy good size and power in finite samples

reducing guesswork via an unreliable oracle alice holds a random variable x and bob is trying to guess its value by asking questions of the form is x x alice answers truthfully and the game terminates once bob guesses correctly before the game begins bob is allowed to reach out to an oracle carole and ask her any yes no question i e a question of the form is x in a carole is known to lie with a given probability p what should bob ask carole if he would like to minimize his expected guessing time when carole is always truthful p 0 it is not difficult to check that bob should order the symbol probabilities in descending order and ask carole whether the index of x with respect to this order is even or odd we show that this strategy is almost optimal for any lying probability p up to a small additive constant upper bounded by 1 4 we discuss a connection to the cutoff rate of the bsc with feedback

functional linear regression analysis for longitudinal data we propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data where both the predictor and response are functions of a covariate such as time predictor and response processes have smooth random trajectories and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects in longitudinal studies the number of repeated measurements per subject is often small and may be modeled as a discrete random number and accordingly only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit we propose a functional regression approach for this situation using functional principal component analysis where we estimate the functional principal component scores through conditional expectations this allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory the resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories asymptotic properties for a sample of n subjects are investigated under mild conditions as n oo and we obtain consistent estimation for the regression function besides convergence results for the components of functional linear regression such as the regression parameter function we construct asymptotic pointwise confidence bands for the predicted trajectories a functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced extending the standard r 2 to the functional case the proposed methods are illustrated with a simulation study longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index

searching for structure in curve samples abstract the shape of a regression curve can to a large extent be characterized by the succession of structural features like extrema inflection points and so on when analyzing a sample of regression curves it is often important to know at an early stage of data analysis which structural features are occurring consistently in each curve of the sample such a definition is usually not easy due to substantial interindividual variation both in the x and the y axis and due to the influence of noise a method is proposed for identifying typical features without relying on an a priori specified functional model for the curves the approach is based on the frequencies of occurrence of structural features as for example maxima in the curve sample along the x axis important tools are nonparametric regression and differentiation and kernel density estimation apart from a theoretical foundation the usefulness of the method is documented by application to two interesting biomedical areas growth and develop

minimum guesswork with an unreliable oracle we study a guessing game where alice holds a discrete random variable x and bob tries to sequentially guess its value before the game begins bob can obtain side information about x by asking an oracle carole any binary question of his choosing carole s answer is however unreliable and is incorrect with probability epsilon we show that bob should always ask carole whether the index of x is odd or even with respect to a descending order of probabilities this question simultaneously minimizes all the guessing moments for any value of epsilon in particular this result settles a conjecture of burin and shayevitz we further consider a more general setup where bob can ask a multiple choice m ary question and then observe carole s answer through a noisy channel when the channel is completely symmetric i e when carole decides whether to lie regardless of bob s question and has no preference when she lies a similar question about the ordered index of x modulo m is optimal interestingly however the problem of testing whether a given question is optimal appears to be generally difficult in other symmetric channels we provide supporting evidence for this difficulty by showing that a core property required in our proofs becomes np hard to test in the general m ary case we establish this hardness result via a reduction from the problem of testing whether a system of modular difference disequations has a solution which we prove to be np hard for m geq 3

noisy guesses we consider the problem of guessing a random finite alphabet secret n vector where the guesses are transmitted via a noisy channel we provide a single letter formula for the best achievable exponential growth rate of the rho th moment of the number of guesses as a function of n this formula exhibits a fairly clear insight concerning the penalty due to the noise we describe two different randomized schemes that achieve the optimal guessing exponent one of them is fully universal in the sense of being independent of source that governs the vector to be guessed the channel that corrupts the guesses and the moment power rho interestingly it turns out that in general the optimal guessing exponent function exhibits a phase transition when it is examined either as a function of the channel parameters or as a function of rho as long as the channel is not too distant in a certain sense to be defined precisely from the identity channel i e the clean channel or equivalently as long rho is larger than a certain critical value rho text c there is no penalty at all in the guessing exponent compared to the case of noiseless guessing

tests for error correlation in the functional linear model in this chapter we consider two tests for error correlation in the fully functional linear model which we call methods i and ii they complement the tools described in section 8 6 and the graphical goodness of fit checks used in chapter 9 to construct the test statistics finite dimensional residuals are computed in two different ways and then their autocorrelations are suitably defined from these autocorrelation matrices two quadratic forms are constructed whose limiting distribution are chi squared with known numbers of degrees of freedom different for the two forms the test statistics can be relatively easily computed using the r package fda

detecting changes in the mean of functional observations principal component analysis has become a fundamental tool of functional data analysis it represents tne functional data as x i t t 1 i l v l t wnere is the common mean v l are the eigenfunctions of the covariance operator and the i l are the scores inferential procedures assume that the mean function t is the same for all values of i if in fact the observations do not come from one population but rather their mean changes at some point s the results of principal component analysis are confounded by the change s it is therefore important to develop a methodology to test the assumption of a common functional mean we develop such a test using quantities which can be readily computed in the r package fda the null distribution of the test statistic is asymptotically pivotal with a well known asymptotic distribution the asymptotic test has excellent finite sample performance its application is illustrated on temperature data from england

the renyi redundancy of generalized huffman codes huffman s algorithm gives optimal codes as measured by average codeword length and the redundancy can be measured as the difference between the average codeword length and shannon s entropy if the objective function is replaced by an exponentially weighted average then a simple modification of huffman s algorithm gives optimal codes the redundancy can now be measured as the difference between this new average and a renyi s 1961 generalization of shannon s entropy by decreasing some of the codeword lengths in a shannon code the upper bound on the redundancy given in the standard proof of the noiseless source coding theorem is improved the lower bound is improved by randomizing between codeword lengths allowing linear programming techniques to be used on an integer programming problem these bounds are shown to be asymptotically equal the results are generalized to the renyi case and are related to r g gallager s 1978 bound on the redundancy of huffman codes

robust functional estimation using the median and spherical principal components summary we present robust estimators for the mean and the principal components of a stochastic process in l2 r robustness and asymptotic properties of the estimators are studied theoretically by simulation and by example it is shown that the proposed estimators are generally more robust to outliers than the commonly used sample mean and principal components although their properties depend on the spacings of the eigenvalues of the covariance function

second order comparison of gaussian random functions and the geometry of dna minicircles given two samples of continuous zero mean iid gaussian processes on 0 1 we consider the problem of testing whether they share the same covariance structure our study is motivated by the problem of determining whether the mechanical properties of short strands of dna are significantly affected by their base pair sequence though expected to be true had so far not been observed in three dimensional electron microscopy data the testing problem is seen to involve aspects of ill posed inverse problems and a test based on a karhunen loeve approximation of the hilbert schmidt distance of the empirical covariance operators is proposed and investigated when applied to a dataset of dna minicircles obtained through the electron microscope our test seems to suggest potential sequence effects on dna shape supplemental material available online

pairwise curve synchronization for functional data summary data collected by scientists are increasingly in the form of trajectories or curves often these can be viewed as realizations of a composite process driven by both amplitude and time variation we consider the situation in which functional variation is dominated by time variation and develop a curve synchronization method that uses every trajectory in the sample as a reference to obtain pairwise warping functions in the first step these initial pairwise warping functions are then used to create improved estimators of the underlying individual warping functions in the second step a truncated averaging process is used to obtain robust estimation of individual warping functions the method compares well with other available time synchronization approaches and is illustrated with berkeley growth data and gene expression data for multiple sclerosis

guessing revisited a large deviations approach the problem of guessing a random string is revisited a close relation between guessing and compression is first established then it is shown that if the sequence of distributions of the information spectrum satisfies the large deviation property with a certain rate function then the limiting guessing exponent exists and is a scalar multiple of the legendre fenchel dual of the rate function other sufficient conditions related to certain continuity properties of the information spectrum are briefly discussed this approach highlights the importance of the information spectrum in determining the limiting guessing exponent all known prior results are then re derived as example applications of our unifying approach

totally stably tame variables abstract we improve the constructions of ann polon math 76 2001 67 to obtain new automorphisms and variables of a x y we introduce the concept of totally stably tame variables this allows us to prove that all our variables are stably tame comparing different notions of length we show that some of our variables are not in berson s class cf j pure appl algebra 170 2002 131 for definition

portmanteau test of independence for functional observations in various fields observations are curves over some natural time interval these curves often arise from finely spaced measurements e g in physical sciences and finance or from smoothing unequally spaced sparse measurements recent years have seen the development of tools for analyzing such data in the growing field of functional data analysis to validate the assumptions underlying these tools it is important to verify that the functional observations form a simple random sample if the curves form a functional time series then model validation typically relies on checking whether model residuals are independent and identically distributed we propose a test for independence and identical distribution of functional observations to reduce dimension curves are projected on the most important functional principal components then a test statistic based on lagged cross covariances of the resulting vectors is constructed we show that this dimension reduction step introduces asymptotically negligib

self modelling warping functions the paper introduces a semiparametric model for functional data the warping functions are assumed to be linear combinations ofqcommon components which are estimated from the data hence the name self modelling even small values ofqprovide remarkable model flexibility comparable with nonparametric methods at the same time this approach avoids overfitting because the common components are estimated combining data across individuals as a convenient by product component scores are often interpretable and can be used for statistical inference an example of classification based on scores is given abstract from author

common functional principal components functional principal component analysis fpca based on the karhunen loeve decomposition has been successfully applied in many applications mainly for one sample problems in this paper we con sider common functional principal components for two sample prob lems our research is motivated not only by the theoretical challenge of this data situation but also by the actual question of dynamics of implied volatility iv functions for different maturities the log returns of ivs are samples of smooth random functions and the methods proposed here study the similarities of their stochastic be havior first we present a new method for estimation of functional principal components from discrete noisy data next we present the two sample inference for fpca and develop the two sample theory we propose bootstrap tests for testing the equality of eigenvalues eigenfunctions and mean functions of two functional samples illus trate the test properties by simulation study and apply the method to the iv analysis

components and completion of partially observed functional data functional data are traditionally assumed to be observed on the same domain motivated by a data set of heart rate temporal profiles we develop methodology for the analysis of incomplete functional samples where each curve may be observed on a subset of the domain and unobserved elsewhere we formalize this observation regime and develop the fundamental procedures of functional data analysis for this framework estimation of parameters mean and covariance operator and principal component analysis principal scores of a partially observed function cannot be computed directly and we solve this challenging issue by estimating their best predictions as linear functionals of the observed part of the trajectory next we propose a functional completion procedure that recovers the missing part by using the observed part of the curve we construct prediction intervals for principal scores and bands for missing parts of trajectories the prediction problems are seen to be ill posed inverse problems regularization techniques are used to obtain a stable solution a simulation study shows the good performance of our methods we illustrate the methods on the heart rate data and provide practical computational algorithms and theoretical arguments and proofs of all results

comments on an inequality on guessing and its application to sequential decoding in the above paper by e arikan see ibid vol 42 no 1 p 99 105 1996 an asymptotically tight upper bound on the spl rho th moment spl rho spl ges 0 of the minimal number of guesses required to determine the value of a random variable was derived we show that we can tighten this bound for the case of positive integer moments when spl rho 1 the bound is improved by a factor of 2 and that the new bound also applies to a class of nonminimal guessing sequences