local linear regression smoothers and their minimax efficiencies in this paper we introduce a smooth version of local linear regression estimators and address their advantages the mse and mise of the estimators are computed explicitly it turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency they are not only efficient in rates but also nearly efficient in constant factors in the nonparametric regression context the asymptotic minimax lower bound is developed via the heuristic of the hardest onedimensional subproblem of donoho and liu connections of the minimax risk with the modulus of continuity are made the lower bound is also applicable for estimating conditional mean regression and conditional quantiles for both fixed and random design regression problems

the beta model maximum likelihood cramer rao bounds and hypothesis testing we study the maximum likelihood estimator in a setting where the dependent variable is a random graph and covariates are available on a graph level the model generalizes the well known beta model for random graphs by replacing the constant model parameters with regression functions cramer rao bounds are derived for special cases of the undirected beta model the directed beta model and the covariate based beta model the corresponding maximum likelihood estimators are compared with the bounds by means of simulations moreover examples are given on how to use the presented maximum likelihood estimators to test for directionality and significance finally the applicability of the model is demonstrated using temporal social network data describing communication among healthcare workers

partially linear estimation with application to sparse signal recovery from measurement pairs we address the problem of estimating a random vector from two sets of measurements and such that the estimator is linear in we show that the partially linear minimum mean square error plmmse estimator does not require knowing the joint distribution of and in full but rather only its second order moments this renders it of potential interest in various applications we further show that the plmmse method is minimax optimal among all estimators that solely depend on the second order statistics of and we demonstrate our approach in the context of recovering a signal which is sparse in a unitary dictionary from noisy observations of it and of a filtered version we show that in this setting plmmse estimation has a clear computational advantage while its performance is comparable to state of the art algorithms we apply our approach both in static and in dynamic estimation applications in the former category we treat the problem of image enhancement from blurred noisy image pairs we show that plmmse estimation performs only slightly worse than state of the art algorithms while running an order of magnitude faster in the dynamic setting we provide a recursive implementation of the estimator and demonstrate its utility in tracking maneuvering targets from position and acceleration measurements

joint estimation of porosity and saturation using stochastic rock physics modeling sediment porosity and saturation affect bulk modulus shearmodulus anddensity consequently estimatinghydrocarbon saturation and reservoir porosity from seismic data is a joint estimation problem uncertainty in porosity will lead toerrorsinsaturationprediction andviceversa porosityand saturation can be jointly estimated using stochastic rockphysicsmodelingandformalbayesianestimationmethodology knowledge of shear impedance reduces the uncertainty inporosityandthusalsoreducesuncertaintyinsaturationestimation this study investigates joint estimation of porosity and saturation by using rock physics stochastic modeling and bayesian estimation theory to derive saturation and porosity maps of expected pay sands in the field example the uncertainty in porosity quantified by the standard deviation stdassociatedwiththeposteriorprobabilitydensityfunctionpdf derivedfrominversionofseismicdataismuchless than the uncertainty in the derived saturation for a typical case the std associated with saturation is 24 while porositystdisabout1 34porosityunitsgivenseismic derived inversionattributeswithreasonableaccuracy comparisonof these numbers with prior estimates showed that inversion of seismic data decreased the uncertainty in porosity to 15 of the prior uncertainty while saturation uncertainty was only reduced to 92 of the prior uncertainty although these resultsmayvaryfromonelocationtoanother themethodology isgeneralandcanbeappliedtootherlocations

limits of sensing temporal concentration changes by single cells berg and purcell biophys j 20 193 1977 calculated how the accuracy of concentration sensing by single celled organisms is limited by noise from the small number of counted molecules here we generalize their results to the sensing of concentration ramps which is often the biologically relevant situation e g during bacterial chemotaxis we calculate lower bounds on the uncertainty of ramp sensing by three measurement devices a single receptor an absorbing sphere and a monitoring sphere we contrast two strategies simple linear regression of the input signal versus maximum likelihood estimation and show that the latter can be twice as accurate as the former finally we consider biological implementations of these two strategies and identify possible signatures that maximum likelihood estimation is implemented by real biological systems

entropy and boundary conditions in random rhombus tilings the tilings of rhombi in two dimensions and of rhomboedra in three dimensions are studied when they are constrained by fixed boundary conditions we establish a link between those conditions and free or periodic boundary ones the entropy is written as a functional integral which is treated via a saddle point method we can exhibit the dominant states of the statistical ensemble of tilings and show that they can display a strong structural inhomogeneity caused by the boundary this inhomogeneity is responsible for a difference of entropy between the studied fixed boundary tilings and free boundary ones this method uses a representation of tilings by membranes embedded in a higher dimensional hypercubic lattice it is illustrated in the case of 60 degree rhombus tilings

analysis of repeated measurements using nonparametric smoothers and randomization tests the mixed model analysis of variance anova which is commonly applied to repeated measurements taken over time depends on specialized assumptions about the error distribution and fails to exploit information contained in the ordering of the data points over time this paper describes a procedure that overcomes these disadvantages while preserving familiar features of the mixed model anova group profiles are estimated by nonparametric smoothing of observed mean profiles group and time main effects and the group by time interaction effect are tested using randomization tests results of zerbe 1979 journal of the american statistical association 74 215 221 are used to construct f test approximations for the randomization tests of the group and group by time effects a new approximate f test for time effect is proposed a simulation study demonstrates that the approximations perform well and that smoothing increases the power of the tests for time main effect and group by time interaction the procedure is applied to data on hormone levels in cows

robust linear channel estimation methods for per subcarrier transmit antenna selection the linear minimum mean squared error mse channel estimator for systems employing per subcarrier transmit antenna selection is developed frequency domain correlations after the selection process are shown to be approximated well using a simple function which makes near optimal channel estimation practically possible however the resultant estimators are not robust to errors in the assumed model in terms of the antenna to subcarrier assignment used at transmission as we motivate both analytically and via simulations this is an issue when the channels for conveying this information to the receiver are severely constrained we present windowed channel estimations and new robust estimation algorithms either individually or in combination as solutions to counter this sensitivity the robust estimators are developed based on the principles of 1 minimising the worst case mse over the set of possible models and 2 minimising the expected mse over the set of possible models the latter estimator is preferred due to the lower implementation complexity and better mse performance we also prove theoretical asymptotic in signal to noise ratio performances of the two estimators used with the proposed correlation model simulation results illustrate the performance gains and improved robustness offered by the developed estimators

local polynomial regression analysis of clustered data this paper proposes a classical weighted least squares type of local polynomial smoothing for the analysis of clustered data with the key idea of using generalised inverses of correlation matrices the estimator has a simple closed form expression simplicity is achieved also for nonparametric generalised linear models with arbitrary link function via a transformation our approach can be characterised by local observations with local variances which yields intuitively correct results in the sense that correct incorrect specification of within cluster correlation has respective positive negative effects the approach is a natural extension of classical local polynomial smoothing consequently existing theory can be largely carried over and important issues such as bandwidth selection can be tackled in the classical fashion moreover the approach can handle various types of covariate such as cluster level subject level or partially cluster level numerical studies support the theoretical results the method is illustrated with a real example on luteinising hormone levels in cows

minimax mse ratio estimation with signal covariance uncertainties in continuation to an earlier work we further consider the problem of robust estimation of a random vector or signal with an uncertain covariance matrix that is observed through a known linear transformation and corrupted by additive noise with a known covariance matrix while in the earlier work we developed and proposed a competitive minimax approach of minimizing the worst case mean squared error mse difference regret criterion here we study in the same spirit the minimum worst case mse ratio regret criterion namely the worst case ratio rather than difference between the mse attainable using a linear estimator ignorant of the exact signal covariance and the minimum mse mmse attainable by optimum linear estimation with a known signal covariance we present the optimal linear estimator under this criterion in two ways the first is as a solution to a certain semidefinite programming sdp problem and the second is as an expression that is of closed form up to a single parameter whose value can be found by a simple line search procedure we then show that the linear minimax ratio regret estimator can also be interpreted as the mmse estimator that minimizes the mse for a certain choice of signal covariance that depends on the uncertainty region we demonstrate that in applications the proposed minimax mse ratio regret approach may outperform the well known minimax mse approach the minimax mse difference regret approach and the plug in approach where in the latter one uses the mmse estimator with an estimated covariance matrix replacing the true unknown covariance

local polynomial kernel regression for generalized linear models and quasi likelihood functions abstract we investigate the extension of the nonparametric regression technique of local polynomial fitting with a kernel weight to generalized linear models and quasi likelihood contexts in the ordinary regression case local polynomial fitting has been seen to have several appealing features in terms of intuitive and mathematical simplicity one noteworthy feature is the better performance near the boundaries compared to the traditional kernel regression estimators these properties are shown to carry over to generalized linear model and quasi likelihood settings we also derive the asymptotic distributions of the proposed class of estimators that allow for straightforward interpretation and extensions of state of the art bandwidth selection methods

nonparametric additive regression for repeatedly measured data summary we develop an easily computed smooth backfitting algorithm for additive model fitting in repeated measures problems our methodology easily copes with various settings such as when some covariates are the same over repeated response measurements we allow for a working covariance matrix for the regression errors showing that our method is most efficient when the correct covariance matrix is used the component functions achieve the known asymptotic variance lower bound for the scalar argument case smooth backfitting also leads directly to design independent biases in the local linear case simulations show our estimator has smaller variance than the usual kernel estimator this is also illustrated by an example from nutritional epidemiology

estimating the asymptotics of solid partitions we study the asymptotic behavior of solid partitions using transition matrix monte carlo simulations if p 3 n denotes the number of solid partitions of an integer n we show that lim n rightarrow infty n 3 4 log p 3 n sim 1 822 pm 0 001 this shows clear deviation from the value 1 7898 attained by macmahon numbers m 3 n that was conjectured to hold for solid partitions as well in addition we find estimates for other sub leading terms in log p 3 n in a pattern deviating from the asymptotics of line and plane partitions we need to add an oscillatory term in addition to the obvious sub leading terms the period of the oscillatory term is proportional to n 1 4 the natural scale in the problem this new oscillatory term might shed some insight into why partitions in dimensions greater than two do not admit a simple generating function

linear minimax regret estimation of deterministic parameters with bounded data uncertainties we develop a new linear estimator for estimating an unknown parameter vector x in a linear model in the presence of bounded data uncertainties the estimator is designed to minimize the worst case regret over all bounded data vectors namely the worst case difference between the mean squared error mse attainable using a linear estimator that does not know the true parameters x and the optimal mse attained using a linear estimator that knows x we demonstrate through several examples that the minimax regret estimator can significantly increase the performance over the conventional least squares estimator as well as several other least squares alternatives

arctic octahedron in three dimensional rhombus tilings and related integer solid partitions three dimensional integer partitions provide a convenient representation of codimension one three dimensional random rhombus tilings calculating the entropy for such a model is a notoriously difficult problem we apply transition matrix monte carlo simulations to evaluate their entropy with high precision we consider both free and fixed boundary tilings our results suggest that the ratio of free and fixed boundary entropies is free fixed 3 2 and can be interpreted as the ratio of the volumes of two simple nested polyhedra this finding supports a conjecture by linde moore and nordahl concerning the arctic octahedron phenomenon in three dimensional random tilings

recent developments in nonparametric density estimation advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where restrictions are imposed upon the class of densities or on the form of the likelihood function nonparametric density estimators that are data adaptive and lead to locally smoothed estimators are also discussed these include variable partition histograms estimators based on statistically equivalent blocks nearest neighbor estimators variable kernel estimators and adaptive kernel estimators for the multivariate case extensions of methods of univariate density estimation are usually straightforward but can be computationally expensive a method of multivariate density estimation that did not spring from a univariate generalization is described namely projection pursuit density estimation in which both dimensionality reduction and density estimation can be pursued at the same time finally some areas of related research are mentioned such as nonparametric estimation of functionals of a density robust parametric estimation semiparametric models and density estimation for censored and incomplete data directional and spherical data and density estimation for dependent sequences of observations

solution of the dimer problem on a hexagonal lattice with boundary the dimer problem is solved exactly for a hexagonal lattice with general boundary using a known generating function from the theory of partitions it is shown that the leading term in the entropy depends on the shape of the boundary

a competitive minimax approach to robust estimation of random parameters we consider the problem of estimating in the presence of model uncertainties a random vector x that is observed through a linear transformation h and corrupted by additive noise we first assume that both the covariance matrix of x and the transformation h are not completely specified and develop the linear estimator that minimizes the worst case mean squared error mse across all possible covariance matrices and transformations h in the region of uncertainty although the minimax approach has enjoyed widespread use in the design of robust methods we show that its performance is often unsatisfactory to improve the performance over the minimax mse estimator we develop a competitive minimax approach for the case where h is known but the covariance of x is subject to uncertainties and seek the linear estimator that minimizes the worst case regret namely the worst case difference between the mse attainable using a linear estimator ignorant of the signal covariance and the optimal mse attained using a linear estimator that knows the signal covariance the linear minimax regret estimator is shown to be equal to a minimum mse mmse estimator corresponding to a certain choice of signal covariance that depends explicitly on the uncertainty region we demonstrate through examples that the minimax regret approach can improve the performance over both the minimax mse approach and a plug in approach in which the estimator is chosen to be equal to the mmse estimator with an estimated covariance matrix replacing the true unknown covariance we then show that although the optimal minimax regret estimator in the case in which the signal and noise are jointly gaussian is nonlinear we often do not lose much by restricting attention to linear estimators

new local estimation procedure for a non parametric regression function for longitudinal data this paper develops a new estimation of nonparametric regression functions for clustered or longitudinal data we propose to use cholesky decomposition and profile least squares techniques to estimate the correlation structure and regression function simultaneously we further prove that the proposed estimator is as asymptotically efficient as if the covariance matrix were known a monte carlo simulation study is conducted to examine the finite sample performance of the proposed procedure and to compare the proposed procedure with the existing ones based on our empirical studies the newly proposed procedure works better than the naive local linear regression with working independence error structure and the efficiency gain can be achieved in moderate sized samples our numerical comparison also shows that the newly proposed procedure outperforms some existing ones a real data set application is also provided to illustrate the proposed estimation procedure

semiparametric models for longitudinal data with application to cd4 cell numbers in hiv seroconverters summary the paper describes a semiparametric model for longitudinal data which is illustrated by its application to data on the time evolution of cd4 cell numbers in hiv seroconverters the essential ingredients of the model are a parametric linear model for covariate adjustment a nonparametric estimation of a smooth time trend serial correlation between measurements on an individual subject and random measurement error a back fitting algorithm is used in conjunction with a cross validation prescription to fit the model a notable feature in the application is that the onset of hiv infection is associated with a sudden drop in cd4 cells followed by a longer term slower decay the model is also used to estimate an individual s curve by combining his data with the population curve shrinkage toward the population mean trajectory is controlled in a natural way by the estimated covariance structure of the data

image denoising using scale mixtures of gaussians in the wavelet domain we describe a method for removing noise from digital images based on a statistical model of the coefficients of an overcomplete multiscale oriented basis neighborhoods of coefficients at adjacent positions and scales are modeled as the product of two independent random variables a gaussian vector and a hidden positive scalar multiplier the latter modulates the local variance of the coefficients in the neighborhood and is thus able to account for the empirically observed correlation between the coefficient amplitudes under this model the bayesian least squares estimate of each coefficient reduces to a weighted average of the local linear estimates over all possible values of the hidden multiplier variable we demonstrate through simulations with images contaminated by additive white gaussian noise that the performance of this method substantially surpasses that of previously published methods both visually and in terms of mean squared error

marginal nonparametric kernel regression accounting for within subject correlation there has been substantial recent interest in non and semiparametric methods for longitudinal or clustered data with dependence within clusters it has been shown rather inexplicably that when standard kernel smoothing methods are used in a natural way higher efficiency is obtained by assuming independence than by using the true correlation structure it is shown here that this result is a natural consequence of how standard kernel methods incorporate the within subject correlation in the asymptotic setting considered where the cluster sizes are fixed and the cluster number increases in this paper an alternative kernel smoothing method is proposed unlike the standard methods the smallest variance of the new estimator is achieved when the true correlation is assumed asymptotically the variance of the proposed method is uniformly smaller than that of the most efficient working independence approach a small simulation study shows that significant improvement is obtained for finite samples

partial linear regression models for clustered data this article considers the analysis of clustered data via partial linear regression models adopting the idea of modeling the within cluster correlation from the method of generalized estimating equations a least squares type estimate of the slope parameter is obtained through piecewise local polynomial approximation of the nonparametric component this slope estimate has several advantages a it attains n1 2 consistency without undersmoothing b it is efficient when correct within cluster correlation is used assuming multivariate normality of the error c the preceding properties hold regardless of whether or not the nonparametric component is of cluster level and d this estimation method naturally extends to deal with generalized partial linear models simulation studies and a real example are presented in support of the theory

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

on the asymptotics of marginal regression splines with longitudinal data summary there have been studies on how the asymptotic efficiency of a nonparametric function estimator depends on the handling of the within cluster correlation when nonparametric regression models are used on longitudinal or cluster data in particular methods based on smoothing splines and local polynomial kernels exhibit different behaviour we show that the generalized estimation equations based on weighted least squares regression splines for the nonparametric function have an interesting property the asymptotic bias of the estimator does not depend on the working correlation matrix but the asymptotic variance and therefore the mean squared error is minimized when the true correlation structure is specified this property of the asymptotic bias distinguishes regression splines from smoothing splines

random tilings of high symmetry ii boundary conditions and numerical studies we perform numerical studies including monte carlo simulations of high rotational symmetry random tilings for computational convenience our tilings obey fixed boundary conditions in regular polygons such tilings are put in correspondence with algorithms for sorting lists in computer science we obtain statistics on path counting and vertex coordination which compare well with predictions of mean field theory and allow estimation of the configurational entropy which tends to the value 0 568 per vertex in the limit of continuous symmetry tilings with phason strain appear to share the same entropy as unstrained tilings as predicted by mean field theory we consider the thermodynamic limit and argue that the limiting fixed boundary entropy equals the limiting free boundary entropy although these differ for finite rotational symmetry

empirical bias bandwidths for local polynomial nonparametric regression and density estimation abstract a data based local bandwidth selector is proposed for nonparametric regression by local fitting of polynomials the estimator called the empirical bias bandwidth selector ebbs is rather simple and easily allows multivariate predictor variables and estimation of any order derivative of the regression function ebbs minimizes an estimate of mean squared error consisting of a squared bias term plus a variance term the variance term used is exact not asymptotic though it involves the conditional variance of the response given the predictors that must be estimated the bias term is estimated empirically not from an asymptotic expression thus ebbs is similar to the double smoothing approach of hardle hall and marron and a local bandwidth selector of schucany but is developed here for a far wider class of estimation problems than what those authors considered ebbs is tested on simulated data and its performance seems quite satisfactory local polynomial smoothing of a histogram is a high

on the asymptotics of higher dimensional partitions we conjecture that the asymptotic behavior of the numbers of solid three dimensional partitions is identical to the asymptotics of the three dimensional macmahon numbers evidence is provided by an exact enumeration of solid partitions of all integers 68 whose numbers are reproduced with surprising accuracy using the asymptotic formula with one free parameter and better accuracy on increasing the number of free parameters we also conjecture that similar behavior holds for higher dimensional partitions and provides some preliminary evidence for four and five dimensional partitions

flip dynamics in three dimensional random tilings we study single flip dynamics in sets of three dimensional rhombus tilings with fixed polyhedral boundaries this dynamics is likely to be slowed down by so called cycles such structures arise when tilings are encoded via the partition on tiling method and are susceptible to break connectivity by flips or at least ergodicity because they locally suppress a significant amount of flip degrees of freedom we first address the so far open question of the connectivity of tiling sets by elementary flips we prove exactly that sets of tilings of codimension 1 and 2 are connected for any dimension and tiling size for higher codimension tilings of dimension 3 the answer depends on the precise choice of the edge orientations which is a non trivial issue in most cases we can prove connectivity despite the existence of cycles in the few remaining cases among which one is the icosahedral symmetry the question remains open we also study numerically flip assisted diffusion to explore the possible effects of the previously mentioned cycles cycles do not seem to slow down significantly the dynamics at least as far as self diffusion is concerned

variable bandwidth and local linear regression smoothers in this paper we introduce an appealing nonparametric method for estimating the mean regression function the proposed method combines the ideas of local linear smoothers and variable bandwidth hence it also inherits the advantages of both approaches we give expressions for the conditional mse and mise of the estimator minimization of the mise leads to an explicit formula for an optimal choice of the variable bandwidth moreover the merits of considering a variable bandwidth are discussed in addition we show that the estimator does not have boundary effects and hence does not require modifications at the boundary the performance of a corresponding plug in estimator is investigated simulations illustrate the proposed estimation method

boundary effects in the dimer problem on a non bravais lattice the dimer problem on the hexagonal lattice is solved for various boundary shapes the number of arrangements of dimers in the asymptotic limit of large lattices proves to be very sensitive to the precise form of the lattice boundary if n is the number of lattice points and for large n wn 2 the number of dimer configurations then the molecular freedom w reduces to 1 in the case of standard boundaries parallelogram rectangle and there is in the thermodynamic limit no freedom in placing a dimer at all but in the case of a hexagonal lattice which has been given the shape of a macroscopic honeycomb and which thus exhibits the symmetry properties of the unit cell the molecular freedom is found to tend to w 3 4 3 1 299 for n investigating various other boundary forms the regular hexagon proves to be a convenient surface shape with a maximum molecular freedom which differs markedly from the known value w 1 381 for the hexagonal lattice wrapped on a torus

characterization of non stationary channels using mismatched wiener filtering a common simplification in the statistical treatment of linear time varying ltv wireless channels is the approximation of the channel as a stationary random process inside certain time frequency regions we develop a methodology for the determination of local quasi stationarity lqs regions i e local regions in which a channel can be treated as stationary contrary to previous results relying on to some extent heuristic measures and thresholds we consider a finite length wiener filter as realistic channel estimator and relate the size of lqs regions in time to the degradation of the mean square error mse of the estimate due to outdated and thus mismatched channel statistics we show that for certain power spectral densities psds of the channel a simplified but approximate evaluation of the matched mse based on the assumption of an infinite filtering length yields a lower bound on the actual matched mse moreover for such psds the actual mse degradation is upper bounded and the size of the actual lqs regions is lower bounded by the approximate evaluation using channel measurements we compare the evolution of the lqs regions based on the actual and the approximate mse they show strong similarities

semiparametric estimation in general repeated measures problems summary the paper considers a wide class of semiparametric problems with a parametric part for some covariate effects and repeated evaluations of a nonparametric function special cases in our approach include marginal models for longitudinal or clustered data conditional logistic regression for matched case control studies multivariate measurement error models generalized linear mixed models with a semiparametric component and many others we propose profile kernel and backfitting estimation methods for these problems derive their asymptotic distributions and show that in likelihood problems the methods are semiparametric efficient although generally not true it transpires that with our methods profiling and backfitting are asymptotically equivalent we also consider pseudolikelihood methods where some nuisance parameters are estimated from a different algorithm the methods proposed are evaluated by using simulation studies and applied to the kenya haemoglobin data

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

quasi likelihood estimation in semiparametric models abstract suppose the expected value of a response variable y may be written h x t where x and t are covariates each of which may be vector valued is an unknown parameter vector is an unknown smooth function and h is a known function in this article we outline a method for estimating the parameter of this type of semiparametric model using a quasi likelihood function algorithms for computing the estimates are given and the asymptotic distribution theory for the estimators is developed the generalization of this approach to the case in which y is a multivariate response is also considered the methodology is illustrated on two data sets and the results of a small monte carlo study are presented

a transfer matrix monte carlo study of random penrose tilings the authors investigate the entropic properties of a simple two dimensional random quasicrystal model a random tiling by the 36 and 72 rhombi applying the transfer matrix monte carlo tmmc method to random tilings for the first time they have calculated the entropy as a function of phason strain they confirm earlier results that the state of zero phason strain i e ten fold symmetry has the largest entropy the entropy is 0 4810 5 per tile in addition by fitting the dependence of the entropy on the phason strain they determined the three stiffness constants in the phason elasticity one of which is not measurable by previous approaches they compare the efficacy of the tmmc method with that of other methods

a minimum squared error framework for generalized sampling we treat the problem of reconstructing a signal from its nonideal samples where the sampling and reconstruction spaces as well as the class of input signals can be arbitrary subspaces of a hilbert space our formulation is general and includes as special cases reconstruction from finitely many samples as well as uniform sampling of continuous time signals which are not necessarily bandlimited to obtain a good approximation of the signal in the reconstruction space from its samples we suggest two design strategies that attempt to minimize the squared norm error between the signal and its reconstruction the approaches we propose differ in their assumptions on the input signal if the signal is known to lie in an appropriately chosen subspace then we propose a method that achieves the minimal squared error on the other hand when the signal is not restricted we show that the minimal norm reconstruction cannot generally be obtained instead we suggest minimizing the worst case squared error between the reconstructed signal and the best possible but usually unattainable approximation of the signal within the reconstruction space we demonstrate both theoretically and through simulations that the suggested methods can outperform the consistent reconstruction approach previously proposed for this problem

covariate adjusted correlation analysis with application to fmr1 premutation female carrier data motivated by molecular data on female premutation carriers of the fragile x mental retardation 1 fmr1 gene we present a new method of covariate adjusted correlation analysis to examine the association of messenger rna mrna and number of cgg repeat expansion in the fmr1 gene the association between the molecular variables in female carriers needs to adjust for activation ratio actratio a measure which accounts for the protective effects of one normal x chromosome in females carriers however there are inherent uncertainties in the exact effects of actratio on the molecular measures of interest to account for these uncertainties we develop a flexible adjustment that accommodates both additive and multiplicative effects of actratio nonparametrically the proposed adjusted correlation uses local conditional correlations which are local method of moments estimators to estimate the pearson correlation between two variables adjusted for a third observable covariate the local method of moments estimators are averaged to arrive at the final covariate adjusted correlation estimator which is shown to be consistent we also develop a test to check the nonparametric joint additive and multiplicative adjustment form simulation studies illustrate the efficacy of the proposed method application to fmr1 premutation data on 165 female carriers indicates that the association between mrna and cgg repeat after adjusting for actratio is stronger finally the results provide independent support for a specific jointly additive and multiplicative adjustment form for actratio previously proposed in the literature

fundamental limits of linear equalizers diversity capacity and complexity linear equalizers les have been widely adopted for practical systems due to their low computational complexity however it is also well known that les provide inferior performance relative to a maximum likelihood equalizer mle or other near mles because les usually cannot collect the diversity order enabled by the transmitter and at the same time they lose mutual information more importantly unlike mle or near mles the performance of les has not been well quantified this hinders more general applications of les in wireless systems in this paper we reveal a fundamental link between a channel parameter orthogonality deficiency od of the channel matrix and the diversity and capacity of les we identify that when the od of channel matrix has an upper bound strictly less than 1 the same diversity order as that of mles is collected by les and the outage capacity loss relative to mles is also a constant over signal to noise ratio snr these results can be applied to designing a framework for hybrid equalizers furthermore by studying the statistical properties of the od and comparing the complexity of different equalizers we show that hybrid equalizers can trade off between performance and complexity by tuning the channel matrix od the theoretical analysis is corroborated by computer simulations

numerical estimation of the asymptotic behaviour of solid partitions of an integer the number of solid partitions of a positive integer is an unsolved problem in combinatorial number theory in this paper solid partitions are studied numerically by the method of exact enumeration for integers up to 50 and by monte carlo simulations using wang landau sampling method for integers up to 8000 it is shown that limn ln p3 n n3 4 1 79 0 01 where p3 n is the number of solid partitions of the integer n this result strongly suggests that the macmahon conjecture for solid partitions though not exact could still give the correct leading asymptotic behaviour

on posterior distributions for signals in gaussian noise with unknown covariance matrix a bayesian approach to estimate parameters of signals embedded in complex gaussian noise with unknown color is presented the study specifically focuses on a bayesian treatment of the unknown noise covariance matrix making up a nuisance parameter in such problems by integrating out uncertainties regarding the noise color an enhanced ability to estimate both the signal parameters as well as properties of the error is exploited several noninformative priors for the covariance matrix such as the reference prior the jeffreys prior and modifications to this are considered some of the priors result in analytical solutions whereas others demand numerical approximations in the linear signal model connections are made between the standard adaptive maximum likelihood aml estimate and a bayesian solution using the jeffreys prior with adjustments to the jeffreys prior correspondence to the regularized solution is also established this in turn enables a formal treatment of the regularization parameter simulations indicate that significant improvements compared to the aml estimator can be obtained by considering both the derived regularized solutions as well as the one obtained using the reference prior the simulations also indicate the possibility of enhancing the predictions of properties of the error as uncertainties in the noise color are acknowledged

random tilings of high symmetry i mean field theory we study random tiling models in the limit of high rotational symmetry in this limit a mean field theory yields reasonable predictions for the configurational entropy of free boundary rhombus tilings in two dimensions we base our mean field theory on an iterative tiling construction inspired by the work of de bruijn in addition to the entropy we consider correlation functions phason elasticity and the thermodynamic limit tilings of dimension other than two are considered briefly

fixed boundary octagonal random tilings a combinatorial approach some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied a geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems thus generalizing previous results on the more restricted class of codimension one tilings in particular this method gives access to counting formulas which are directly related to questions of entropy in these statistical systems methods and tools from the field of enumerative combinatorics are used

nonparametric function estimation for clustered data when the predictor is measured without with error abstract we consider local polynomial kernel regression with a single covariate for clustered data using estimating equations we assume that at most m observations are available on each cluster in the case of random regressors with no measurement error in the predictor we show that it is generally the best strategy to ignore entirely the correlation structure within each cluster and instead pretend that all observations are independent in the further special case of longitudinal data on individuals with fixed common observation times we show that equivalent to the pooled data approach is the strategy of fitting separate nonparametric regressions at each observation time and constructing an optimal weighted average we also consider what happens when the predictor is measured with error using the simex approach to correct for measurement error we construct an asymptotic theory for both the pooled and the weighted average estimators surprisingly for the same amount of smoothing the weighted a

configurational entropy of codimension one tilings and directed membranes the calculation of random tiling configurational entropy amounts to an enumeration of partitions a geometrical description of the configuration space is given in terms of integral points in a high dimensional space and the entropy is deduced from the integral volume of a convex polytope in some cases the latter volume can be expressed in a compact multiplicative formula and in all cases in terms of binomial series the origin of which is given a geometrical meaning our results mainly concern codimension one tilings but can also be extended to higher codimension tilings we also discuss the link between freeboundary and fixed boundary condition problems

hybrid fm polynomial phase signal modeling parameter estimation and cramer rao bounds parameter estimation for a class of nonstationary signal models is addressed the class contains combination of a polynomial phase signal pps and a frequency modulated fm component of the sinusoidal or hyperbolic type such signals appear in radar and sonar applications involving moving targets with vibrating or rotating components a novel approach is proposed that allows us to decouple estimation of the fm parameters from those of the pps relying on properties of the multilag high order ambiguity function ml haf the accuracy achievable by any unbiased estimator of the hybrid fm pps parameters is investigated by means of the cramer rao lower bounds crlbs both exact and large sample approximate expressions of the bounds are derived and compared with the performance of the proposed methods based on monte carlo simulations

quasi ml period estimation from incomplete timing data given a noisy sequence of possibly shifted integer multiples of a certain period it is often of interest to accurately estimate the period with known integer regressors the problem is classical linear regression in many applications however the regressors are unknown integers and only loose bounds on the period are available examples include hop period and timing estimation wherein hops may be missed at the output of the frequency discriminator or the emitter may hop out of band pulse repetition interval pri analysis and passive rotating beam radio scanning we study several pertinent period estimators our emphasis is on a quasi maximum likelihood approach developed herein and an earlier method based on the fourier transform of a dirac delta train representation of the data surprisingly both are capable of attaining the clairvoyant crame spl acute r rao bound at moderate signal to noise ratios snrs even for short e g 10 samples we carefully address parameter identifiability issues and corroborate our findings with extensive simulations

hidden relationships bayesian estimation with partial knowledge we address the problem of bayesian estimation where the statistical relation between the signal and measurements is only partially known we propose modeling partial bayesian knowledge by using an auxiliary random vector called instrument the statistical relations between the instrument and the signal and between the instrument and the measurements are known however the joint probability function of the signal and measurements is unknown two types of statistical relations are considered corresponding to second order moment and complete distribution function knowledge we propose two approaches for estimation in partial knowledge scenarios the first is based on replacing the orthogonality principle by an oblique counterpart and is proven to coincide with the method of instrumental variables from statistics although developed in a different context the second is based on a worst case design strategy and is shown to be advantageous in many aspects we provide a thorough analysis showing in which situations each of the methods is preferable and propose a nonparametric method for approximating the estimators from a set of examples finally we demonstrate our approach in the context of enhancement of facial images that have undergone unknown degradation and image zooming

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures