a class of logarithmically completely monotonic functions and the best bounds in the second kershaw s double inequality in the article the sufficient and necessary conditions such that a class of functions which involve the psi function j and the ratio c x t c x s are logarithmically completely monotonic are established the best bounds for the ratio c x t c x s are given and some comparisons with known results are carried out where s and t are two real numbers and x min s t

on gamma function inequalities on montre que certaines fonctions comportant les quotients de fonctions gamma sont completement monotones on en deduit des inegalites on etablit aussi la divisibilite infinie de plusieurs distributions de probabilite dont les transformees de laplace comportent des quotients de fonctions gamma

common structure of techniques for choosing smoothing parameters in regression problems on tente de relier deux methodes de choix du parametre de lissage l une basee sur la mesure du risque et l autre sur le degre d ajustement de la regression aux donnees

noise estimation in signal restoration using regularization several estimators of variance are compared in the context of problems where smoothing is incorporated in the estimation of regression functions many of the estimators are scaled quadratic forms in the data and some incorporate steps that try to accommodate substantial nonconstancy in the response function the estimators are described and an account is provided of an empirical study based on four one dimensional source functions of varying degrees of irregularity

bayesian inference for semiparametric regression using a fourier representation summary this paper presents the bayesian analysis of a semiparametric regression model that consists of parametric and nonparametric components the nonparametric component is represented with a fourier series where the fourier coefficients are assumed a priori to have zero means and to decay to 0 in probability at either algebraic or geometric rates the rate of decay controls the smoothness of the response function the posterior analysis automatically selects the amount of smoothing that is coherent with the model and data posterior probabilities of the parametric and semiparametric models provide a method for testing the parametric model against a non specific alternative the bayes estimator s mean integrated squared error compares favourably with the theoretically optimal estimator for kernel regression

parameter selection for total variation based image restoration using discrepancy principle there are two key issues in successfully solving the image restoration problem 1 estimation of the regularization parameter that balances data fidelity with the regularity of the solution and 2 development of efficient numerical techniques for computing the solution in this paper we derive a fast algorithm that simultaneously estimates the regularization parameter and restores the image the new approach is based on the total variation tv regularized strategy and morozov s discrepancy principle the tv norm is represented by the dual formulation that changes the minimization problem into a minimax problem a proximal point method is developed to compute the saddle point of the minimax problem by adjusting the regularization parameter adaptively in each iteration the solution is guaranteed to satisfy the discrepancy principle we will give the convergence proof of our algorithm and numerically show that it is better than some state of the art methods in terms of both speed and accuracy

influence diagnostics for the cross validated smoothing parameter in spline smoothing abstract this article addresses the problem of influence in estimating the smoothing parameter when fitting a univariate smoothing spline diagnostics are presented that can identify observations that locally influence the choice of smoothing parameter by generalized cross validation either through their case weights or observed responses generalized cross validation and the diagnostic methods are given a frequency interpretation and illustrated with examples

regularized fast recursive least squares algorithms for adaptive filtering fast recursive least squares frls algorithms are developed by using factorization techniques which represent an alternative way to the geometrical projections approach or the matrix partitioning based derivations the estimation problem is formulated within a regularization approach and priors are used to achieve a regularized solution which presents better numerical stability properties than the conventional least squares one the numerical complexity of the presented algorithms is explicitly related to the displacement rank of the a priori covariance matrix of the solution it then varies between o 5m and that of the slow rls algorithms to update the kalman gain vector m being the order of the solution an important advantage of the algorithms is that they admit a unified formulation such that the same equations may equally treat the prewindowed and the covariance cases independently from the used priors the difference lies only in the involved numerical complexity which is modified through a change of the dimensions of the intervening variables simulation results are given to illustrate the performances of these algorithms

brief fast spline smoothing via spectral factorization concepts when tuning the smoothness parameter of nonparametric regression splines the evaluation of the so called degrees of freedom is one of the most computer intensive tasks in the paper a closed form expression of the degrees of freedom is obtained for the case of cubic splines and equally spaced data when the number of data tends to infinity state space methods kalman filtering and spectral factorization techniques are used to prove that the asymptotic degrees of freedom are equal to the variance of a suitably defined stationary process the closed form expression opens the way to fast spline smoothing algorithms whose computational complexity is about one half of standard methods or even one fourth under further approximations

a new lower bound in the second kershaw s double inequality in the paper a new and elegant lower bound in the second kershaw s double inequality is established some alternative simple and polished proofs are given several deduced functions involving the gamma and psi functions are proved to be decreasingly monotonic and logarithmically completely monotonic and some remarks and comparisons are stated

non parametric tests for right censored data with biased sampling testing the equality of two survival distributions can be difficult in a prevalent cohort study when non random sampling of subjects is involved due to the biased sampling scheme independent censoring assumption is often violated although the issues about biased inference caused by length biased sampling have been widely recognized in statistical epidemiological and economical literature there is no satisfactory solution for efficient two sample testing we propose an asymptotic most efficient nonparametric test by properly adjusting for length biased sampling the test statistic is derived from a full likelihood function and can be generalized from the two sample test to a k sample test the asymptotic properties of the test statistic under the null hypothesis are derived using its asymptotic independent and identically distributed representation we conduct extensive monte carlo simulations to evaluate the performance of the proposed test statistics and compare them with the conditional test and the standard logrank test for different biased sampling schemes and right censoring mechanisms for length biased data empirical studies demonstrated that the proposed test is substantially more powerful than the existing methods for general left truncated data the proposed test is robust still maintains accurate control of type i error rate and is also more powerful than the existing methods if the truncation patterns and right censoring patterns are the same between the groups we illustrate the methods using two real data examples

some properties of the divided difference of psi and polygamma functions abstract let n 1 n 1 n for n 0 where n stands for the psi and polygamma functions for p q r and min p q let d x p x q n 1 n x be the divided difference of the functions n 1 for x in this paper we establish the necessary and sufficient conditions for the function n x n 1 x 2 n x n 2 x to be completely monotonic on in particular we find that the function n 1 2 n n 2 is strictly decreasing from 0 onto n n 1 n 1 n 2 these not only generalize and strengthen some known results but also yield many new and interesting ones

a comparison of variance estimators in nonparametric regression summary we compare two estimators of error variance both based on quadratic forms in the residuals about smoothing spline fits to data the estimators are compared over the whole range of values of the smoothing parameter as well as for data based choices of the smoothing parameter we show that the commonly used estimator of variance has the serious drawback of underestimating the error variance for small choices of the smoothing parameter this drawback is not shared by a simple but more computationally intensive alternative

estimating the distribution of a renewal process from times at which events from an independent process are detected the analysis of length biased data has been mostly limited to the interarrival interval of a renewal process covering a specific time point motivated by a surveillance problem we consider a more general situation where this time point is random and related to a specific event for example status change or onset of a disease we also consider the problem when additional information is available on whether the event intervals interarrival intervals covering the random event end within or after a random time period which we call a window period following the random event under the assumptions that the occurrence rate of the random event is low and the renewal process is independent of the random event we provide formulae for the estimation of the distribution of interarrival times based on the observed event intervals procedures for testing the required assumptions are also furnished we apply our results to human immunodeficiency virus hiv test data from public test sites in seattle washington where the random event is hiv infection and the window period is from the onset of hiv infection to the time at which a less sensitive hiv test becomes positive results show that the estimator of the intertest interval length distribution from event intervals ending within the window period is less biased than the estimator from all event intervals the latter estimator is affected by right truncation finally we discuss possible applications to estimating hiv incidence and analyzing length biased samples with right or left truncated data

a study of methods of choosing the smoothing parameter in image restoration by regularization the method of regularization is portrayed as providing a compromise between fidelity to the data and smoothness with the tradeoff being determined by a scalar smoothing parameter various ways of choosing this parameter are discussed in the case of quadratic regularization criteria they are compared algebraically and their statistical properties are comparatively assessed from the results of all extensive simulation study based on simple images

wendel s and gautschi s inequalities refinements extensions and a class of logarithmically completely monotonic functions abstract in the article sufficient and necessary conditions that a class of functions involving ratio of euler s gamma functions and originating from wendel s and gautschi s inequalities are logarithmically completely monotonic are presented from this wendel s gautschi s kershaw s laforgia s bustoz ismail s merkle s and elezovic giordano pecaric s inequalities are refined extended and sharpened and a double inequality on the divided differences of the psi and polygamma functions is deduced straightforwardly

alignment using genetic programming with causal trees for identification of protein functions a hybrid evolutionary model is used to propose a hierarchical homology of protein sequences to identify protein functions systematically the proposed model offers considerable potentials considering the inconsistency of existing methods for predicting novel proteins because some novel proteins might align without meaningful conserved domains maximizing the score of sequence alignment is not the best criterion for predicting protein functions this work presents a decision model that can minimize the cost of making a decision for predicting protein functions using the hierarchical homologies particularly the model has three characteristics i it is a hybrid evolutionary model with multiple fitness functions that uses genetic programming to predict protein functions on a distantly related protein family ii it incorporates modified robust point matching to accurately compare all feature points using the moment invariant and thin plate spline theorems and iii the hierarchical homologies holding up a novel protein sequence in the form of a causal tree can effectively demonstrate the relationship between proteins this work describes the comparisons of nucleocapsid proteins from the putative polyprotein sars virus and other coronaviruses in other hosts using the model

a new upper bound in the second kershaw s double inequality and its generalizations in the paper a new upper bound in the second kershaw s double inequality involving ratio of gamma functions is established and as generalizations of the second kershaw s double inequality the divided differences of the psi and polygamma functions are bounded

bayesian confidence intervals for the cross validated smoothing spline summary we consider the model y t g ti ei i 17 2 n where g t t 0 1 is a smooth function and the ei are independent n 0 a2 errors with g2 unknown the cross validated smoothing spline can be used to estimate g non parametrically from

spline smoothing and optimal rates of convergence in nonparametric regression models linear estimation is considered in nonparametric regression models of the form yi f xi ei xi e a b where the zero mean errors are uncorrelated with common variance a2 and the response function f is assumed only to have a bounded square integrable qth derivative the linear estimator which minimizes the maximum mean squared error summed over the observation points is derived and the exact minimax rate of convergence is obtained for practical problems where bounds on 11 f2q 11 and a2 may be unknown generalized cross validation is shown to give an adaptive estimator which achieves the minimax optimal rate under the additional assumption of normality 1 the model consider the nonparametric regression model

ml parameter estimation for markov random fields with applications to bayesian tomography markov random fields mrfs have been widely used to model images in bayesian frameworks for image reconstruction and restoration typically these mrf models have parameters that allow the prior model to be adjusted for best performance however optimal estimation of these parameters sometimes referred to as hyperparameters is difficult in practice for two reasons i direct parameter estimation for mrfs is known to be mathematically and numerically challenging ii parameters can not be directly estimated because the true image cross section is unavailable we propose a computationally efficient scheme to address both these difficulties for a general class of mrf models and we derive specific methods of parameter estimation for the mrf model known as generalized gaussian mrf ggmrf we derive methods of direct estimation of scale and shape parameters for a general continuously valued mrf for the ggmrf case we show that the ml estimate of the scale parameter spl sigma has a simple closed form solution and we present an efficient scheme for computing the ml estimate of the shape parameter p by an off line numerical computation of the dependence of the partition function on p we present a fast algorithm for computing ml parameter estimates when the true image is unavailable to do this we use the expectation maximization em algorithm we develop a fast simulation method to replace the e step and a method to improve the parameter estimates when the simulations are terminated prior to convergence experimental results indicate that our fast algorithms substantially reduce the computation and result in good scale estimates for real tomographic data sets

nonparametric input estimation in physiological systems problems methods and case studies input estimation from output data is an important problem in the analysis of physiological systems because many signals of interest are not directly accessible to measurement when the system is time invariant this problem is often referred to as deconvolution three representative physiological problems regarding hormone secretion insulin dynamics and hepatic glucose production are used to illustrate the major challenges ill conditioning confidence intervals assessment infrequent and nonuniform sampling nonnegativity constraints and computational efficiency the paper provides a critical overview of the existing techniques focusing on regularization theory and bayesian estimation in order to overcome some inadequacies of the existing methods some new results are derived in particular the connection between the maximum likelihood estimate of the regularization parameter and the notion of equivalent degree of freedom is studied moreover a fast svd based numerical algorithm is developed that includes the optimization of the regularization parameter and the computation of confidence intervals the proposed techniques are validated on a benchmark problem and are shown to provide effective solutions to the three physiological case studies

a comparison theorem for two divided differences and applications to special functions abstract in this paper we present a general comparison theorem for two divided differences of a three times differentiable function this gives a unified treatment for logarithmically complete monotonicity monotonicity and inequalities involving some special functions including gamma psi and polygamma functions as their consequences we not only refine and generalize some important results but also present simple and interesting alternative proofs of certain earlier results

bandwidth choice for nonparametric regression this paper is concerned with the problem of choosing a bandwidth parameter for nonparametric regression we analyze a tapered fourier series estimate and discuss the relationship of this estimate to a kernel estimate we first consider a method based on an unbiased estimate of mean square error and show that the bandwidth thus chosen is asymptotically optimal other methods are examined as well and are shown to be asymptotically equivalent a small simulation shows however that for small or moderate sample size the methods perform quite differently

methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation the application of regularization to ill conditioned problems necessitates the choice of a regularization parameter which trades fidelity to the data with smoothness of the solution the value of the regularization parameter depends on the variance of the noise in the data the problem of choosing the regularization parameter and estimating the noise variance in image restoration is examined an error analysis based on an objective mean square error mse criterion is used to motivate regularization two approaches for choosing the regularization parameter and estimating the noise variance are proposed the proposed and existing methods are compared and their relationship to linear minimum mean square error filtering is examined experiments are presented that verify the theoretical results

asymptotic optimality of the fast randomized versions of gcv and c_l in ridge regression and regularization ridge regression is a well known technique to estimate the coefficients of a linear model the method of regularization is a similar approach commonly used to solve underdetermined linear equations with discrete noisy data when applying such a technique the choice of the smoothing or regularization parameter h is crucial generalized cross validation gcv and mallows c l are two popular methods for estimating a good value for h from the data their asymptotic properties such as consistency and asymptotic optimality have been largely studied craven and wahba 1979 golub heath and wahba 1979 speckman 1985 very interesting convergence results for the actual random parameter given by gcv and c l have been shown by li 1985 1986 recently girard 1987 1989 has proposed fast randomized versions of gcv and c l the purpose of this paper is to show that the above convergence results also hold for these new methods

piecewise and local image models for regularized image restoration using cross validation we describe two broad classes of useful and physically meaningful image models that can be used to construct novel smoothing constraints for use in the regularized image restoration problem the two classes termed piecewise image models pims and focal image models lims respectively capture unique image properties that can be adapted to the image and that reflect structurally significant surface characteristics members of the pim and lim classes are easily formed into regularization operators that replace differential type constraints we also develop an adaptive strategy for selecting the best pim or lim for a given problem from among the defined class and we explain the construction of the corresponding regularization operators considerable attention is also given to determining the regularization parameter via a cross validation technique and also to the selection of an optimization strategy for solving the problem several results are provided that illustrate the processes of model selection parameter selection and image restoration the overall approach provides a new viewpoint on the restoration problem through the use of new image models that capture salient image features that are not well represented through traditional approaches

bayesian interpretation of periodograms the usual nonparametric approach to spectral analysis is revisited within the regularization framework both usual and windowed periodograms are obtained as the squared modulus of the minimizer of regularized least squares criteria then particular attention is paid to their interpretation within the bayesian statistical framework finally the question of unsupervised hyperparameter and window selection is addressed it is shown that maximum likelihood solution is both formally achievable and practically useful

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

a complete monotonicity property of the gamma function abstract a logarithmically completely monotonic function is completely monotonic the function 1 ln x 1 x ln x 1 is strictly completely monotonic on 0 the function x 1 x x is strictly logarithmically completely monotonic on 0

the average posterior variance of a smoothing spline and a consistent estimate of the average squared error a smoothing spline estimator can be interpreted in two ways either as the solution to a variational problem or as the posterior mean when a particular gaussian prior is placed on the unknown regression function the estimate of the average posterior variance of a spline estimate proposed by wahba is shown to converge in probability to a quantity proportional to the expected average squared error this result is established by relating this statistic to a consistent risk estimate based on generalized cross validation

a class of logarithmically completely monotonic functions and application to the best bounds in the second gautschi kershaw s inequality in this article the logarithmically complete monotonicity of the function c x b c x a 1 a b exp j x c are discussed where a b c are real numbers and c is the classical euler s gamma function from this the best upper and lower bounds for walls ratio c x 1 c x s are established which refine the second gautschi kershaw s inequality

bayesian confidence intervals for smoothing splines abstract the frequency properties of wahba s bayesian confidence intervals for smoothing splines are investigated by a large sample approximation and by a simulation study when the coverage probabilities for these pointwise confidence intervals are averaged across the observation points the average coverage probability acp should be close to the nominal level from a frequency point of view this agreement occurs because the average posterior variance for the spline is similar to a consistent estimate of the average squared error and because the average squared bias is a modest fraction of the total average squared error these properties are independent of the bayesian assumptions used to derive this confidence procedure and they explain why the acp is accurate for functions that are much smoother than the sample paths prescribed by the prior this analysis accounts for the choice of the smoothing parameter bandwidth using cross validation in the case of natural splines an adaptive method for avo

some extensions of w gautschi s inequalities for the gamma function it has been shown by w gautschi that if 0 i xi s f x exp i s x 1 the following closer bounds are proved exp i s 4 x 12 f exp i s x s i f x s 2 and x 2 t x 2 s 4 these are compared with each other and with inequalities given by t erber and j d keckic and p m vasic

some inequalities and monotonicity properties associated with the gamma and psi functions abstract in this paper we present new completely monotonic functions and bernstein functions relating to the gamma and psi functions as consequences we establish new upper and lower bounds for the gamma function in terms of function also we present some estimates for stirling s formula remainder

unsupervised frequency tracking beyond the nyquist frequency using markov chains this paper deals with the estimation of a sequence of frequencies from a corresponding sequence of signals this problem arises in fields such as doppler imaging where its specificity is twofold first only short noisy data records are available typically four sample long and experimental constraints may cause spectral aliasing so that measurements provide unreliable ambiguous information second the frequency sequence is smooth here this information is accounted for by a markov model and application of the bayes rule yields the a posteriori density the maximum a posteriori is computed by a combination of viterbi and descent procedures one of the major features of the method is that it is entirely unsupervised adjusting the hyperparameters that balance data based and prior based information is done automatically by maximum likelihood ml using an expectation maximization em based gradient algorithm we compared the proposed estimate to a reference one and found that it performed better variance was greatly reduced and tracking was correct even beyond the nyquist frequency

testing for elevated disease rates using smoothed estimates a method of testing for elevated disease rates using smoothed estimates is introduced for counts aggregated to a specific administrative level a test statistic is developed based on an approximation to the smoothed estimate from a penalized multinomial likelihood and an approximate critical value is derived empirical investigation reveals that smoothing can increase the ability to detect elevated rates the methods are applied to data on gastric cancer in nova scotia canada