sigma delta modulation with i i d gaussian inputs the response of a single loop sigma delta modulator to an independent identically distributed i i d gaussian input signal is analyzed a continuous time stochastic model is developed and the connection of the model to the system is described a condition is given so that the difference between the behavior of the model and that of the true system can be made arbitrarily small theories from renewal and wiener processes are applied to show the convergence and mixing properties of the output sequence also derived is the power spectrum of the quantization noise compared to the spectrum when the input is dc the i i d gaussian random process smears the discrete spectrum into band structures

on simple oversampled a d conversion in shift invariant spaces it has been found that the quantization error e for a conventional oversampled analog to digital a d conversion behaves like spl par e spl par sup 2 o spl tau sup 2 with respect to the sampling rate spl tau recently conventional a d conversion has been extended to a d conversion based on shift invariant spaces as consequences of such extension it offers rich choices to build a nonideal a d conversion system of high accuracy and low computational complexity as well as reduces the noise sensitivity and computational complexity in digital to analog d a conversion therefore it is necessary to establish the estimate of quantization error for the extended a d conversion based on shift invariant spaces in this paper we introduce a constructive method to establish an estimate of the quantization error as e sup 2 o spl tau sup 2 for oversampled a d conversion in shift invariant spaces meanwhile we demonstrate that the bit rate required to encode the converted digital signal in such a d conversion scheme only increases as the logarithm of the sampling ratio therefore the quantization error is an exponentially decaying function of the bit rate in order to establish such an estimate we need the nonuniform sampling theorem for shift invariant spaces which as the necessary preparation is studied prior to introducing the constructive method

estimating treatment effects in a two arm parallel trial of a continuous outcome for a continuous outcome in a two arm trial that satisfies normal distribution assumptions we can transform the standardized mean difference with the use of the cumulative distribution function to be the effect size measure p x

double interpolation for digital to analog conversion interpolative digital to analog converters generate an output that has only a few analog levels they provide fine resolution by oscillating rapidly between these levels in such a manner that the average output represents the value of the applied code here we describe an improved method of interpolating that results in reduced noise in the signal band a theory of the interpolation confirmed by experiments demonstrates that switching between only two levels at 1 3 mhz could provide 16 bit resolution for telephone signals

on constrained randomized quantization randomized dithered quantization is a method capable of achieving white reconstruction error independent of the source dithered quantizers have traditionally been considered within their natural setting of uniform quantization in this paper we extend conventional dithered quantization to nonuniform quantization via a subterfage dithering is performed in the companded domain closed form necessary conditions for optimality of the compressor and expander mappings are derived for both fixed and variable rate randomized quantization numerically mappings are optimized by iteratively imposing these necessary conditions the framework is extended to include an explicit constraint that deterministic or randomized quantizers yield reconstruction error that is uncorrelated with the source surprising theoretical results show direct and simple connection between the optimal constrained quantizers and their unconstrained counterparts numerical results for the gaussian source provide strong evidence that the proposed constrained randomized quantizer outperforms the conventional dithered quantizer as well as the constrained deterministic quantizer moreover the proposed constrained quantizer renders the reconstruction error nearly white in the second part of the paper we investigate whether uncorrelated reconstruction error requires random coding to achieve asymptotic optimality we show that for a gaussian source the optimal vector quantizer of asymptotically high dimension whose quantization error is uncorrelated with the source is indeed random thus random encoding in this setting of rate distortion theory is not merely a tool to characterize performance bounds but a required property of quantizers that approach such bounds

opportunities and challenges of combined effect measures based on prioritized outcomes many authors have proposed different approaches to combine multiple endpoints in a univariate outcome measure in the literature in case of binary or time to event variables composite endpoints which combine several event types within a single event or time to first event analysis are often used to assess the overall treatment effect a main drawback of this approach is that the interpretation of the composite effect can be difficult as a negative effect in one component can be masked by a positive effect in another recently some authors proposed more general approaches based on a priority ranking of outcomes which moreover allow to combine outcome variables of different scale levels these new combined effect measures assign a higher impact to more important endpoints which is meant to simplify the interpretation of results whereas statistical tests and models for binary and time to event variables are well understood the latter methods have not been investigated in detail so far in this paper we will investigate the statistical properties of prioritized combined outcome measures we will perform a systematical comparison to standard composite measures such as the all cause hazard ratio in case of time to event variables or the absolute rate difference in case of binary variables to derive recommendations for different clinical trial scenarios we will discuss extensions and modifications of the new effect measures which simplify the clinical interpretation moreover we propose a new method on how to combine the classical composite approach with a priority ranking of outcomes using a multiple testing strategy based on the closed test procedure

chain codes and their linear reconstruction filters a differential chain code implementation of the basic eight directional and n ring chain code for the encoding of line drawings is considered several distortion and rate criteria are considered and they are evaluated for the special case of encoding infinite straight lines these results indicate that the differential code outperforms ring codes and the ring delta code using a linear decoding filter for reconstructing a smoothed approximation to the original line is also considered optimal reconstruction filters for infinite straight lines at various angles are derived an angle invariant reconstruction filter that minimizes the expected square distortion taken over all angles is derived the overall performance using this filter is close to that of the optimum filter furthermore the impulse response of the angle invariant filter is approximately rectangular which can lead to simple hardware implementations

sample size estimation for the van elteren test a stratified wilcoxon mann whitney test the van elteren test is a type of stratified wilcoxon mann whitney test for comparing two treatments accounting for strata in this paper we study sample size estimation methods for the asymptotic version of the van elteren test assuming that the stratum fractions ratios of each stratum size to the total sample size and the treatment fractions ratios of each treatment size to the stratum size are known in the study design in particular we develop three large sample sample size estimation methods and present a real data example to illustrate the necessary information in the study design phase in order to apply the methods simulation studies are conducted to compare the performance of the methods and recommendations are made for method choice finally sample size estimation for the van elteren test when the stratum fractions are unknown is also discussed

adjusting o brien s test to control type i error for the generalized nonparametric behrens fisher problem summaryo brien 1984 biometrics40 1079 1087 introduced a simple nonparametric test procedure for testing whether multiple outcomes in one treatment group have consistently larger values than outcomes in the other treatment group we first explore the theoretical properties of o brien s test we then extend it to the general nonparametric behrens fisher hypothesis problem when no assumption is made regarding the shape of the distributions we provide conditions when o brien s test controls its error probability asymptotically and when it fails we also provide adjusted tests when the conditions do not hold throughout this article we do not assume that all outcomes are continuous simulations are performed to compare the adjusted tests to o brien s test the difference is also illustrated using data from a parkinson s disease clinical trial

quantization noise in single loop sigma delta modulation with sinusoidal inputs an exact nonlinear difference equation is derived and solved for a simple sigma delta modulator consisting of a discrete time integrator and a binary quantizer inside a single feedback loop and an arbitrary input signal it is shown that the system can be represented as an affine operation discrete time integration of a biased input followed by a memoryless nonlinearity an extension of the transform method for the analysis of nonlinear systems is applied to obtain formulas for first and second order time average moments of the binary quantization noise including the sample mean energy and autocorrelation the results are applied to the special case of a sinusoidal input signal to evaluate these time averages and the power spectrum in the limit of large oversampling ratios the marginal moments behave as if the quantization noise had a uniform distribution the spectrum is neither white nor continuous however even in the limit of large oversampling ratios

generalized treatment effects for clinical trials practice in the analysis of clinical trials with continuously measured endpoints is to focus on the difference or percentage change in mean or median response however treatments may have effects on the distribution of responses other than on the average response we sought an approach to such generalized treatment effects that i targets a parameter that is easily understood by our clinical colleagues and ii employs confidence intervals as the basis for inference we consider one such approach based on work in reliability theory namely setting pr y x as the target parameter and compare this approach to an earlier one due to o brien the two approaches have similar properties when they both seek to reject the null hypothesis of no effect due to different variances but differ when the larger variance corresponds to the larger mean in that case our approach views that the larger variance attenuates the effect of the larger mean out suggested approach applies easily to positive control clinical equivalence trials copyright 2000 john wiley sons ltd

reachability problems for a class of integro differential equations abstract we study control problems for some integro differential equations using the hilbert uniqueness method to do that we follow a harmonic analysis approach our results can be applied to concrete examples in viscoelasticity theory

probabilistic index models summary we present a semiparametric statistical model for the probabilistic index which can be defined as p y y a where y and y a are independent random response variables associated with covariate patterns x and x a respectively a link function defines the relationship between the probabilistic index and a linear predictor asymptotic normality of the estimators and consistency of the covariance matrix estimator are established through semiparametric theory the model is illustrated with several examples and the estimation theory is validated in a simulation study

a theory of nonsubtractive dither a detailed mathematical investigation of multibit quantizing systems using nonsubtractive dither is presented it is shown that by the use of dither having a suitably chosen probability density function moments of the total error can be made independent of the system input signal but that statistical independence of the error and the input signals is not achievable similarly it is demonstrated that values of the total error signal cannot generally be rendered statistically independent of one another but that their joint moments can be controlled and that in particular the error sequence can be rendered spectrally white the properties of some practical dither signals are explored and recommendations are made for dithering in audio video and measurement applications the paper collects all of the important results on the subject of nonsubtractive dithering and introduces important new ones with the goal of alleviating persistent and widespread misunderstandings regarding the technique

eigenvalue and eigenfunction asymptotics for regular sturm liouville problems abstract in this paper we obtain asymptotic formulas for eigenvalues eigenfunctions and the reciprocals of the eigenfunction norms for eigenvalue problems associated with the general sturm liouville equation pu r q u 0 having regular endpoints the method is based on a conversion to liouville normal form and an iterative procedure of solving the associated volterra integral equation producing an asymptotic expansion of the solution in higher powers of 1 1 2 as

some windows with very good sidelobe behavior some of the windows presented by harris 1 are not correct in terms of their reported peak sidelobes and optimal behavior we present corrected plots of harris windows and also derive additional windows with very good sidelobes and optimal behavior under several different constraints the temporal weightings are characterized as a sum of weighted cosines over a finite duration the plots enable the reader to select a window to suit his requirements in terms of bias due to nearby sidelobes and bias due to distant sidelobes

a saturation analysis of error spectrum shaping quantizers error spectrum shaping ess quantizers are analyzed in the context of realistic fir reconstruction filters and nonzero probability of saturation conditions for stable operation are derived and a suboptimal but effective and computationally efficient design method is proposed excellent agreement is obtained between the analysis and simulations even under these more stringent constraints ess quantizers are shown to achieve significant signal to quantization noise ratio sqnr gains when quantizing oversampled gaussian sources

combining dependent tests with incomplete repeated measurements summary in comparing the effectiveness of two treatments repeated measurements of the same characteristic are often taken under two or more distinct conditions for each experimental subject a locally best linear combination of individual test statistics obtained under different conditions is proposed to test the equality of the two treatments the test procedure allows different patterns of missing observations for the two groups to be compared special cases such as combining wilcoxon tests t tests and 2 x 2 tables are discussed in detail with real examples

probabilistic index an intuitive non parametric approach to measuring the size of treatment effects effect sizes es tell the magnitude of the difference between treatments and ideally should tell clinicians how likely their patients will benefit from the treatment currently used es are expressed in statistical rather than in clinically useful terms and may not give clinicians the appropriate information we restrict our discussion to studies with two groups one with n patients receiving a new treatment and the other with m patients receiving the usual or no treatment the standardized mean difference e g cohen s d is a well known index for continuous outcomes there is some intuitive value to d but measuring improvement in standard deviations sd is a statistical concept that may not help a clinician how much improvement is a half sd a more intuitive and simple to calculate es is the probability that the response of a patient given the new treatment x is better than the one for a randomly chosen patient given the old or no treatment y i e p x y larger values meaning better outcomes this probability has an immediate identity with the area under the curve auc measure in procedures for receiver operator characteristic roc curve comparing responses to two treatments it also can be easily calculated from the mann whitney u wilcoxon or kendall statistics we describe the characteristics of an ideal es we propose p x y as an alternative index summarize its correspondence with well known non parametric statistics compare it to the standardized mean difference index and illustrate with clinical data copyright 2005 john wiley sons ltd

analysis of failure time data with dependent interval censoring summary this article develops a method for the analysis of screening data for which the chance of being screened is dependent on the event of interest informative censoring because not all subjects make all screening visits the data on the failure of interest is interval censored we propose a model that will properly adjust for the dependence to obtain an unbiased estimate of the nonparametric failure time function and we provide an extension for applying the method for estimation of the regression parameters from a discrete time proportional hazards regression model the method is applied on a data set from an observational study of cytomegalovirus shedding in a population of hiv infected subjects who participated in a trial conducted by the aids clinical trials group

statistics of the binary quantizer error in single loop sigma delta modulation with white gaussian input representations and statistical properties of the process e defined by e n 1 e n n are given here u u b sign u m and n n 0 is gaussian white noise the process e represents the binary quantizer error in a model for single loop sigma delta modulation the innovations variables are found and the existence and uniqueness of an invariant probability measure ergodicity properties as well as the existence of the exponential moment with respect to the invariant probability are proved using markov process theory we consider also e as a random perturbation for small values of the variance of n of the orbits of sn 1 sn here sn has the uniform invariant distribution on the interval m h m b analytical approximations to the structure of the power spectrum of e are obtained using a linear prediction in terms of the innovations variables and the perturbation approach

fir filters with sigma delta modulation encoding two methods are presented for building fir finite impulse response digital filters where either the impulse response or the input signal is encoded using sigma delta modulation for each method both single loop and two stage sigma delta modulation are used for encoding these systems are operated at a sampling rate higher than the nyquist rate of the signal and their performance depends on the oversampling ratio except in the decoding circuit these filters require no multiplication and hence can be built using simple hardware an analysis of the systems is provided and examples for filters realized using these approaches are given

boundary controllability and source reconstruction in a viscoelastic string under external traction abstract treatises on vibrations devote large space to study the dynamical behavior of an elastic system subject to known external tractions since usually a system is part of a chain of mechanisms which disturb the system for example due to the periodic rotation of shafts these disturbances affect the horizontal component of traction in a time dependent way in this paper we shall study control and source identification problems for a viscoelastic string subject to external traction using moment theory

multiloop sigma delta quantization it is shown that for a multiloop sigma delta modulator driven by a dc input the statistical behavior described by long term time averages of the error process from an inner multibit quantizer is consistent with that of a signal independent white uniform noise the same result also holds for sinusoidal inputs provided that the modulator has three or more loops this behavior is exploited to obtain trade offs between asymptotic system performances and oversampling ratios for several sigma delta modulator decoding filter pairs for an l loop modulator with a simple sinc sup l 1 decoding filter it is shown that doubling the oversampling ratio can result in an improvement of 3 01 2l 1 db in the signal to quantization noise ratio

on controllability of a rotating string we consider the problem of controllability of a rotating string we say that a set of initial data of the string is controllable if for any initial data of this set by suitable manipulation of the exterior force the string goes to rest the main result of the paper is a description of controllable sets of initial data the equation is not strongly hyperbolic to get exact controllability of such equation in the sharp time interval we use control functions from sobolev spaces with noninteger indices to prove our results we apply the method of moments that has been widely used in control theory of distributed parameter systems since the classical papers of h o fattorini and d l russell we use recent results about exponential bases in sobolev spaces with noninteger indices

the structure of the limit cycles in sigma delta modulation it is shown that when the input to a sigma delta modulator is a dc level that can be expressed as a rational number b a when normalized with respect to the quantizer step the output bit string is periodic with a period that is multiple of the denominator a on the basis of number theory the structure of these cycles for single loop modulators is determined and the noise contribution is computed around such levels the noise has two peaks for which the maximum value and the width are proportional to the relative signal bandwidth and to the inverse of the period of the cycle respectively the effect of the limit cycles on the performance of the a d and d a converters using sigma delta modulation is discussed a comparison between single loop and double loop modulators from the point of view of this phenomena is made

granular quantization noise in the first order delta sigma modulator a unified approach to analyzing the granular quantization error of the first order delta sigma modulator is presented the approach handles many of the previously analyzed input sequences in addition to a large class of new input sequences by averaging over the arbitrarily small amount of circuit noise assumed to be present at the analog input to the delta sigma modulator a simple expression for the autocorrelation of the quantization error is derived each term in the expression is formally equal to the quantization error of a nonoverloaded uniform quantizer operating upon a finite partial sum of consecutive input sequence samples hence existing results concerning uniform quantizers are directly applicable in evaluating the autocorrelation expression for specific input sequences the theory is also applicable to deterministic input sequences and has been applied to obtain a new closed form result for sinusoidal input sequences ergodic results which assert that under mild conditions the autocorrelation equals the time average autocorrelation in probability are presented

on controllability of a non homogeneous elastic string with memory abstract we are motivated by the problem of control for a non homogeneous elastic string with memory we reduce the problem of controllability to a non standard moment problem the solution of the latter problem is based on an auxiliary riesz basis property result for a family of functions quadratically close to the nonharmonic exponentials this result requires the detailed analysis of an integro differential equation and is of interest in itself for function theory controllability of the string implies observability of a dual system

power and sample size estimation for the clustered wilcoxon test the wilcoxon rank sum test is widely used for two group comparisons of nonnormal data an assumption of this test is independence of sampling units both within and between groups which will be violated in the clustered data setting such as in ophthalmological clinical trials where the unit of randomization is the subject but the unit of analysis is the individual eye for this purpose we have proposed the clustered wilcoxon test to account for clustering among multiple subunits within the same cluster rosner glynn and lee 2003 biometrics 59 1089 1098 2006 biometrics 62 1251 1259 however power estimation is needed to plan studies that use this analytic approach we have recently published methods for estimating power and sample size for the ordinary wilcoxon rank sum test rosner and glynn 2009 biometrics 65 188 197 in this article we present extensions of this approach to estimate power for the clustered wilcoxon test simulation studies show a good agreement between estimated and empirical power these methods are illustrated with examples from randomized trials in ophthalmology enhanced power is achieved with use of the subunit as the unit of analysis instead of the cluster using the ordinary wilcoxon rank sum test

exponential bases in sobolev spaces in control and observation problems for the wave equation the fourier method in control systems reduces the study of controllability observability to the study of related exponential families in this paper we present examples of such systems specifically those for which we can prove that the related exponential families form a riesz basis in corresponding appropriately defined sobolev spaces this makes it possible to choose natural pairs of spaces the state space observability space and the control space state space depending on whether an observation or a control problem is studied respectively so that the observation and control operators are isomorphisms

two stage sigma delta modulation an exact analysis of the nonlinear difference equation describing a discrete time two stage sigma delta quantizer with dc and sinusoidal inputs is presented it is shown that for dc inputs the binary quantizer error of the second quantizer which appears at the output of the modulator as a second order difference is asymptotically white uniformly distributed and uncorrelated with the input for sinusoidal inputs this result holds with one exception the autocorrelation which is a sum of weighted bessel functions does not correspond to that of a white noise process at large oversampling ratios the autocorrelation is approximately white and hence the white additive noise model is a good approximation for evaluating the first and second order moments it is found that for both a sinc sup 3 decoder filter and an ideal low pass filter decoder the average quantization noise power is inversely proportional to the fifth power of the oversampling ratio r for both types of inputs this represents an improvement over the single loop sigma delta quantizer by a factor of r sup 2 simulation results that agree with the theoretical analysis are presented

probabilistic index an intuitive non parametric approach to measuring the size of the treatment effects by l acion j j peterson s temple and s arndt statistics in medicine 2006 25 4 591 602

a digital requantizer with shaped requantization noise that remains well behaved after nonlinear distortion a major problem in oversampling digital to analog converters and fractional n frequency synthesizers which are ubiquitous in modern communication systems is that the noise they introduce contains spurious tones the spurious tones are the result of digitally generated quantized signals passing through nonlinear analog components this paper presents a new method of digital requantization called successive requantization special cases of which avoids the spurious tone generation problem sufficient conditions are derived that ensure certain statistical properties of the quantization noise including the absence of spurious tones after nonlinear distortion a practical example is presented and shown to satisfy these conditions

power and sample size estimation for the wilcoxon rank sum test with application to comparisons of c statistics from alternative prediction models the wilcoxon mann whitney wmw u test is commonly used in nonparametric two group comparisons when the normality of the underlying distribution is questionable there has been some previous work on estimating power based on this procedure lehmann 1998 nonparametrics in this article we present an approach for estimating type ii error which is applicable to any continuous distribution and also extend the approach to handle grouped continuous data allowing for ties we apply these results to obtaining standard errors of the area under the receiver operating characteristic curve auroc for risk prediction rules under h 1 and for comparing auroc between competing risk prediction rules applied to the same data set these results are based on sas callable functions to evaluate the bivariate normal integral and are thus easily implemented with standard software

combining several ordinal measures in clinical studies in medical research it is rare that a single variable is sufficient to represent all relevant aspects of epidemiological risk genomic activity adverse events or clinical response since biological systems tend to be neither linear nor hierarchical in nature the assumptions of traditional multivariate statistical methods based on the linear model can often not be justified on theoretical grounds establishing concept validity through empirical validation is not only problematic but also time consuming this paper proposes the use of u statistics for scoring multivariate ordinal data and a family of simple non parametric tests for analysis the scoring method is demonstrated to be applicable to scoring clinical response profiles in the treatment of psoriasis and then to identifying genomic pathways that best correlate with these profiles

multistage sigma delta modulation a theoretical basis is provided for multistage sigma delta modulation msm which is a cascade realization of several single loop sigma delta modulators with a linear combinatorial network equations are derived describing the output and the quantization noise of msm for an arbitrary input signal and the noise shaping characteristic of msm is investigated the spectral characteristics of an m stage sigma delta modulator with both dc and sinusoidal inputs are developed for both types of inputs the binary quantizer noise of the mth m or 3 quantizer which appears at the output as an mth order difference is asymptotically white uniformly distributed and uncorrelated with the input level it is also found that for an m stage sigma delta quantizer with either an ideal low pass filter or a sinc sup m 1 filter decoder the average quantization noise of the system is inversely proportional to the 2m 1 th power of the oversampling ratio this implies that the high order systems are favourable in terms of the trade off between the quantization noise and oversampling ratio simulation results are presented to support the theoretical analysis

a hybrid procedure for detecting global treatment effects in multivariate clinical trials theory and applications to fmri studies in multivariate clinical trials a key research endpoint is ascertaining whether a candidate treatment is more efficacious than an established alternative this global endpoint is clearly of high practical value for studies such as those arising from neuroimaging where the outcome dimensions are not only numerous but they are also highly correlated and the available sample sizes are typically small in this paper we develop a two stage procedure testing the null hypothesis of global equivalence between treatments effects and demonstrate its application to analysing phase ii neuroimaging trials prior information such as suitable statistics of historical data or suitably elicited expert clinical opinions are combined with data collected from the first stage of the trial to learn a set of optimal weights we apply these weights to the outcome dimensions of the second stage responses to form the linear combination z and t tests statistics while controlling the test s false positive rate we show that the proposed tests hold desirable asymptotic properties and characterise their power functions under wide conditions in particular by comparing the power of the proposed tests with that of hotelling s t2 we demonstrate their advantages when sample sizes are close to the dimension of the multivariate outcome we apply our methods to fmri studies where we find that for sufficiently precise first stage estimates of the treatment effect standard single stage testing procedures are outperformed copyright 2011 john wiley sons ltd

generalized pairwise comparisons of prioritized outcomes in the two sample problem this paper extends the idea behind the u statistic of the wilcoxon mann whitney test to perform generalized pairwise comparisons between two groups of observations the observations are outcomes captured by a single variable possibly repeatedly measured or by several variables of any type e g discrete continuous time to event when several outcomes are considered they must be prioritized we show that generalized pairwise comparisons extend well known non parametric tests and illustrate their interest using data from two randomized clinical trials we also show that they lead to a general measure of the difference between the groups called the proportion in favor of treatment denoted which is related to traditional measures of treatment effect for a single variable

performance in waveform quantization the quantization noise and dynamic range of commonly known analog digital converters or modulators are computed from the equations derived by analyzing the generalized quantizer which deploys integer coefficient transversal filters the results show that signal to noise ratio s n curves for high sampling frequency do not have a constant slope of one at a low signal level the s n curve has a slope of 1 5 from the results equations to approximate s n curves in the linear region are derived they can be used to plot a s n curve for any sampling frequency and any number of quantization steps

a class of proportional win fractions regression models for composite outcomes the win ratio is gaining traction as a simple and intuitive approach to analysis of prioritized composite endpoints in clinical trials to extend it from two sample comparison to regression we propose a novel class of semiparametric models that includes as special cases both the two sample win ratio and the traditional cox proportional hazards model on time to the first event under the assumption that the covariate specific win and loss fractions are proportional over time the regression coefficient is unrelated to the censoring distribution and can be interpreted as the log win ratio resulting from one unit increase in the covariate u statistic estimating functions in the form of an arbitrary covariate specific weight process integrated by a pairwise residual process are constructed to obtain consistent estimators for the regression parameter the asymptotic properties of the estimators are derived using uniform weak convergence theory for u processes visual inspection of a score process provides useful clues as to the plausibility of the proportionality assumption extensive numerical studies using both simulated and real data from a major cardiovascular trial show that the regression methods provide valid inference on covariate effects and outperform the two sample win ratio in both efficiency and robustness the proposed methodology is implemented in the r package wr publicly available from the comprehensive r archive network cran

power and sample size calculations for the wilcoxon mann whitney test in the presence of death censored observations we consider a clinical trial of a potentially lethal disease in which patients are randomly assigned to two treatment groups and are followed for a fixed period of time a continuous endpoint is measured at the end of follow up for some patients however death or severe disease progression may preclude measurement of the endpoint a statistical analysis that includes only patients with endpoint measurements may be biased an alternative analysis includes all randomized patients with rank scores assigned to the patients who are available for the endpoint measurement on the basis of the magnitude of their responses and with worst rank scores assigned to those patients whose death precluded the measurement of the continuous endpoint the worst rank scores are worse than all observed rank scores the treatment effect is then evaluated using the wilcoxon mann whitney test in this paper we derive closed form formulae for the power and sample size of the wilcoxon mann whitney test when missing measurements of the continuous endpoints because of death are replaced by worst rank scores we distinguish two approaches for assigning the worst rank scores in the tied worst rank approach all deaths are weighted equally and the worst rank scores are set to a single value that is worse than all measured responses in the untied worst rank approach the worst rank scores further rank patients according to their time of death so that an earlier death is considered worse than a later death which in turn is worse than all measured responses in addition we propose four methods for the implementation of the sample size formulae for a trial with expected early death we conduct monte carlo simulation studies to evaluate the accuracy of our power and sample size formulae and to compare the four sample size estimation methods

discrete sequential boundaries for clinical trials summary pocock 1977 o brien fleming 1979 and slud wei 1982 have proposed different methods to construct discrete sequential boundaries for clinical trials these methods require that the total number of decision times be specified in advance in the present paper we propose a more flexible way to construct discrete sequential boundaries the method is based on the choice of a function a t which characterizes the rate at which the error level ac is spent the boundary at a decision time is determined by a t and by past and current decision times but does not depend on the future decision times or the total number of decision times

simultaneous control problems for systems of elastic strings and beams abstract controllability problems for n strings or beams controlled from a common endpoint are studied we give necessary and sufficient conditions for simultaneous spectral and approximate controllability and describe the space of simultaneously reachable states as a function of the position of the endpoint for each type of controllability result the sharp controllability time is obtained the proofs are based on new results in nonharmonic fourier series which describe riesz bases of exponential divided differences

the analysis of multiple endpoints in clinical trials treatment comparisons in randomized clinical trials usually involve several endpoints such that conventional significance testing can seriously inflate the overall type i error rate one option is to select a single primary endpoint for formal statistical inference but this is not always feasible another approach is to apply bonferroni correction i e multiply each p value by the total number of endpoints its conservatism for correlated endpoints is examined for multivariate normal data a third approach is to derive an appropriate global test statistic and this paper explores one such test applicable to any set of asymptotically normal test statistics quantitative binary and survival endpoints are all considered within this general framework two examples are presented and the relative merits of the proposed strategies are discussed

oversampled sigma delta modulation oversampled sigma delta modulation has been proposed as a practical implementation for high rate analog to digital conversion because of its simplicity and its robustness against circuit imperfections to date mathematical developments of the basic properties of such systems have been based either on simplified continuous time approximate models or on linearized discrete time models where the quantizer is replaced by an additive white uniform noise source in this paper we rigorously derive several basic properties of a simple discrete time single integrator loop sigma delta modulator with an accumulate and dump demodulator the derivation does not require any assumptions on the correlation or distribution of the quantizer error and hence involves no linearization of the nonlinear system but it does show that when the input is constant the state sequence of the integrator in the encoder loop can be modeled exactly as a linear system in an appropriate space two basic properties are developed 1 the behavior of the sigma delta quantizer when driven by a constant input and its relation to uniform quantization and 2 the rate distortion tradeoffs between the oversampling ratio and the average mean squared quantization error

deterministic analysis of oversampled a d conversion and decoding improvement based on consistent estimates discusses the deterministic analysis of oversampled a d conversion adc the properties derivable from such an analysis and the consequences on reconstruction using nonlinear decoding given a band limited input x producing a quantized version c the authors consider the set of all input signals that are band limited and produce c they call any element of this set a consistent estimate of x regardless of the type of encoder simple predictive or noise shaping they show that this set is convex and as a consequence any nonconsistent estimate can be improved they also show that the classical linear decoding estimates are not necessarily consistent numerical tests performed on simple adc single loop and multiloop spl sigma spl delta modulation show that consistent estimates yield a mean square error mse that decreases asymptotically with the oversampling ratio faster than the linear decoding mse by approximately 3 db octave this implies an asymptotic mse of the order of spl oscr r sup spl minus 2n 2 instead of spl oscr r sup spl minus 2n 1 in linear decoding where r is the oversampling ratio and n the order of the modulator methods of improvement of nonconsistent estimates based on the deterministic knowledge of the quantized signal are proposed for simple adc predictive adc single loop and multiloop spl sigma spl delta modulation

weighted analysis of composite endpoints with simultaneous inference for flexible weight constraints composite endpoints are widely used as primary endpoints of randomized controlled trials across clinical disciplines a common critique of the conventional analysis of composite endpoints is that all disease events are weighted equally whereas their clinical relevance may differ substantially we address this by introducing a framework for the weighted analysis of composite endpoints and interpretable test statistics which are applicable to both binary and time to event data to cope with the difficulty of selecting an exact set of weights we propose a method for constructing simultaneous confidence intervals and tests that asymptotically preserve the family wise type i error in the strong sense across families of weights satisfying flexible inequality or order constraints based on the theory of 2 distributions we show that the method achieves the nominal simultaneous coverage rate with substantial efficiency gains over scheffe s procedure in a simulation study and apply it to trials in cardiovascular disease and enteric fever 2016 the authors statistics in medicine published by john wiley sons ltd

spectral analysis of quantization noise in a single loop sigma delta modulator with dc input an exact discrete time analysis of the moments and spectra of the quantization noise of a discrete time single loop sigma delta modulator with a dc input is presented an exact difference equation for the discrete time nonlinear system is used to evaluate the first and second order moments and power spectrum of the binary quantizer noise and the binary quantizer output for a single loop sigma delta encoder with a dc input it is shown that the sample mean and power of the binary quantization noise are consistent with the common uniform distribution assumption but that the autocorrelation and power spectrum are not consistent with the white noise assumption the results are used to evaluate the overall sample average mean squared quantization error as a function of the decimation filter used

dimension reduction for multivariate response data this article concerns the analysis of multivariate response data with multivariate regressors methods for reducing the dimensionality of response variables are developed with the goal of preserving as much regression information as possible we note parallels between this goal and the goal of sliced inverse regression which intends to reduce the regressor dimension in a univariate regression while preserving as much regression information as possible a detailed discussion is given for the case where the response is a curve measured at fixed points the problem in this setting is to select basis functions for fitting an aggregate of curves we propose that instead of focusing on goodness of fit attention should be shifted to the problem of explaining the variation of the curves in terms of the regressor variables a data adaptive basis searching method based on dimension reduction theory is proposed simulation results and an application to a climatology problem are given

testing for individual and population equivalence based on the proportion of similar responses this paper focuses on the classical problem of comparison of treatment effects we show that we can base a simple and intuitive approach to comparison of two treatments on the proportion of similar responses this approach is equivalent to the standard comparison of the treatment means in the normal case with equal known variances but is quite different in other cases our approach applies under two different settings testing a null hypothesis of no treatment difference against an alternative hypothesis of a difference and testing the null hypothesis of at least a specific difference against an alternative hypothesis of equivalence we develop our approach both for parallel groups independent samples and cross over paired samples studies the two situations give rise to the known concepts of population and individual equivalence we present a graphical procedure to supplement the method

on the stability of sigma delta modulators a framework for stability analysis of sigma delta modulators is presented it is argued that limit cycles for constant inputs are natural objects to investigate in this context a number of analytical and approximate techniques to aid the stability analysis of the double loop and interpolative modulators are discussed and lead to ways of improving the design that explicitly take stability into account

properties of composite time to first event versus joint marginal analyses of multiple outcomes many clinical studies eg cardiovascular outcome trials investigate the effect of an intervention on multiple event time outcomes the most common method of analysis is a so called composite analysis of a composite outcome defined as the time to the first component event other approaches have been proposed including the win ratio or win difference for ordered outcomes and the application of the wei lachin test herein we assess the influence of the marginal and joint distributions of the component events and their correlation structures on the operating characteristics of these methods for the analysis of multiple events the operating characteristics are investigated using a bivariate exponential model with a shared frailty under which these properties are obtained in closed form while the composite time to first event analysis provides an unbiased test of the hypothesis of equality of the distribution of the time to first event we show that it can provide a biased test of the joint null hypothesis of equal marginal hazards when the correlation of event times differs between groups the same applies to the win ratio however the operating characteristics of the wei lachin or other tests of the joint equality of the marginal hazards are unaffected furthermore when the correlations are equal the wei lachin test is more powerful to detect a difference in marginal hazards than the composite analysis test careful consideration of the properties of the various methods for analysis of composite outcome measures are in order before adopting one as primary analysis in a clinical study

analysis of dc input response for a class of one bit feedback encoders a class of one bit feedback encoders that includes sigma delta modulation encoders data modulation encoders and some dpcm encoders is defined for any encoder from the class it is shown that the encoder output sequence in response to a dc input sequence is a typical sequence for a certain strict sense stationary ergodic process whose power spectral density is determined the result is applied to give the performance of an analog to digital conversion system operated at high sampling rates

exact controllability and stabilization of a vibrating string with an interior point mass in this article we examine the problems of boundary control and stabilization for a one dimensional wave equation with interior point masses we show that singularities in waves are smoothed one order as they cross a point mass thus in the case of one interior point mass with e g l 2 dirichlet control at the left end the most general reachable space from 0 that one can expect is l 2 times h 1 to the left of the mass and h 1 times l 2 to the right of the mass we show that this is in fact the optimal result modulo certain compatibility conditions several related results for both control and stabilization of such systems are also given

non parametric estimates of overlap kernel densities provide accurate non parametric estimates of the overlapping coefficient or the proportion of similar responses psr in two populations non parametric estimates avoid strong assumptions on the shape of the populations such as normality or equal variance and possess sampling variation approaching that of parametric estimates we obtain accurate standard error estimates by bootstrap resampling we illustrate the practical use of these methods in two examples and use simulations to explore the properties of the estimators under various sampling situations

controllability and observability of a heat equation with hyperbolic memory kernel abstract the exact controllability and observability for a heat equation with hyperbolic memory kernel in anisotropic and nonhomogeneous media are considered due to the appearance of such a kind of memory the speed of propagation for solutions to the heat equation is finite and the corresponding controllability property has a certain nature similar to hyperbolic equations and is significantly different from that of the usual parabolic equations by means of carleman estimate we establish a positive controllability and observability result under some geometric condition on the other hand by a careful construction of highly concentrated approximate solutions to hyperbolic equations with memory we present a negative controllability and observability result when the geometric condition is not satisfied

sample size determination for some common nonparametric tests abstract the article discusses the problem of determining the number of observations required by some common nonparametric tests so that the tests have power at least 1 against alternatives that differ sufficiently from the hypothesis being tested it is shown that the number of observations depends on certain simple probabilities a method is suggested for fixing the value of the appropriate probability when determining sample size

boundary controllability and observability of a viscoelastic string in this paper we consider an integrodifferential system which governs the vibration of a viscoelastic one dimensional object we assume that we can act on the system at the boundary and we prove that it is possible to control both the position and the velocity at every point of the body and at a certain time t large enough we shall prove this result using moment theory and we shall prove that the solution of this problem leads to identification of a riesz sequence which solves controllability and observability the results presented here are constructive and can lead to simple numerical algorithms

friedman type statistics and consistent multiple comparisons for unbalanced designs with missing data abstract a generalization of the friedman test using the marginal likelihood principle kalbfleisch and prentice 1973 is presented and its asymptotic power given it allows use of a variety of score functions and handles ties and unbalanced designs some well known statistics including the tests of kruskal and wallis 1952 prentice 1979 and rinaman 1983 are proven to be special cases whereas others e g klotz 1980 groggel and skillings 1986 rai 1987 skillings and mack 1981 are shown to be less appropriate multiple comparisons are considered under both the global hypothesis and alternatives evaluating non centrality parameters under local shift alternatives the procedures of klotz 1980 and for a special case skillings and mack 1981 can be either anticonservative showing differences that do not exist or insensitive ignoring differences that do exist depending on the distribution of missing data a new scheffe type procedure for arbitrarily missing data is presented and recomme

on the connection between exponential bases and certain related sequences in l2 π π a sequence of vectors fn in a separable hilbert space h is a frame if there are positive constants a b such that a f 2 n f fn 2 b f 2 for all f h and fn is a riesz sequence if it is a riesz basis in the closure of the space spanned by the vectors fn the latter of the following two questions has been raised by khrushchev nikolskii and pavlov can every frame of complex exponentials ei nx in l2 be made into a riesz basis by removing from ei nx a suitable collection of the functions ei nx can every riesz sequence ei nx in l2 be made into a riesz basis by adjoining to ei nx a suitable collection of exponentials ei nx ei nx we show that this is indeed so for all frames and riesz sequences ei nx studied so far and we prove that we can always solve both problems in a certain weak sense however our main conclusion is that the answer to both questions is no

double loop sigma delta modulation with dc input a discrete time model having a two bit 2 b quantizer is analyzed exactly and an analytical expression for the quantizer noise sequence is found rigorous answers are then provided to two fundamental questions for a double loop sigma delta modulation system with dc input 1 what is the long term statistical behavior of the internal quantizer noise 2 how does the asymptotic mean square sigma delta quantization error vary as a function of the oversampling ratio

analysis of a clinical trial involving a combined mortality and adherence dependent interval censored endpoint clinical trials often involve a variety of clinical and laboratory measures that are used as endpoints and sometimes two of these measures are combined in one endpoint when the individual components of such a combined endpoint are time to event measurements the analysis is straightforward if each of the components is measured frequently and regularly over time however the analysis of the combined endpoint is more difficult when one component of the endpoint is right censored and the other is interval censored this paper describes a statistic based on a rank ordering of events for such a combined measure the power of the test statistic is explored

procedures for comparing samples with multiple endpoints summary five procedures are considered for the comparison of two or more multivariate samples these procedures include a newly proposed nonparametric rank sum test and a generalized least squares test also considered are the following tests ordinary least squares hotelling s p and a bonferroni per experiment error rate approach applications are envisaged in which each variable represents a qualitatively different measure of response to treatment the null hypothesis of no treatment difference is tested with power directed towards alternatives in which at least one treatment is uniformly better than the others in all simulations the nonparametric procedure provided relatively good power and accurate control over the size of the test and is recommended for general use alternatively the generalized least squares procedure may also be useful with normally distributed data in moderate or large samples a convenient expression for this procedure is obtained and its asymptotic relative efficiency with respect to the ordinary least squares test is evaluated clinical trials are often conducted for the purpose of evaluating the relative efficacy of two or more modes of therapy when the therapies are administered to independent groups and efficacy is measured on the basis of a single response variable appropriate parametric and nonparametric one way analysis of variance anova procedures and their properties are well known often however efficacy is measured by more than one variable although univariate methods for assessing each characteristic individually are useful in this setting there is often the additional need for a single overall objective probability statement that addresses the question of whether or not the experimental therapy is efficacious this need is particularly acute when the medical question is controversial and the sample size is small although a large literature exists on the comparison of multivariate samples the procedures tend to be infrequently used in practice as pointed out by meier 1975 p 523 the standard analysis for the comparison of two multivariate samples which is based on hotelling s t2 statistic addresses somewhat the wrong question and consequently has very poor power for the alternatives of primary interest specifically the t2 procedure addresses the question are one or more of the treatments different making no distinction between variables that change favorably and variables that change unfavorably a second approach would be to assign per experiment error rates to each of the univariate tests by using bonferroni s inequality i e by multiplying each univariate p value by the number of variables studied the desired overall probability statement is then given by the minimum of the per experiment univariate p values although this approach may be useful in some settings it may lack power for alternatives in which most or all measures of efficacy are improved this will be of particular concern when the number k of endpoints studied is large relative to sample size in fact statistical significance will not be possible in some instances this problem will be exacerbated when the measures of efficacy are highly

global rank tests for multiple possibly censored outcomes clinical trials often collect multiple outcomes on each patient as the treatment may be expected to affect the patient on many dimensions for example a treatment for a neurological disease such as als is intended to impact several dimensions of neurological function as well as survival the assessment of treatment on the basis of multiple outcomes is challenging both in terms of selecting a test and interpreting the results several global tests have been proposed and we provide a general approach to selecting and executing a global test the tests require minimal parametric assumptions are flexible about weighting of the various outcomes and are appropriate even when some or all of the outcomes are censored the test we propose is based on a simple scoring mechanism applied to each pair of subjects for each endpoint the pairwise scores are then reduced to a summary score and a rank sum test is applied to the summary scores this can be seen as a generalization of previously proposed nonparametric global tests e g o brien 1984 we discuss the choice of optimal weighting schemes based on power and relative importance of the outcomes as the optimal weights are generally unknown in practice we also propose an adaptive weighting scheme and evaluate its performance in simulations we apply the methods to analyze the impact of a treatment on neurological function and death in an als trial

decimation for sigma delta modulation decimation is an important component of oversampled analog to digital conversion it transforms the digitally modulated signal from short words occurring at high sampling rate to longer words at the nyquist rate here we are concerned with the initial stage of decimation where the word rate decreases to about four times the nyquist rate we show that digital filters comprising cascades of integrate and dump functions can match the structure of the noise from sigma delta modulation to provide decimation with negligible loss of signal to noise ratio explicit formulas evaluate particular tradeoffs between modulation rate signal to noise ratio length of digital words and complexity of the modulating and decimating functions

a weighted combined effect measure for the analysis of a composite time to first event endpoint with components of different clinical relevance composite endpoints combine several events within a single variable which increases the number of expected events and is thereby meant to increase the power however the interpretation of results can be difficult as the observed effect for the composite does not necessarily reflect the effects for the components which may be of different magnitude or even point in adverse directions moreover in clinical applications the event types are often of different clinical relevance which also complicates the interpretation of the composite effect the common effect measure for composite endpoints is the all cause hazard ratio which gives equal weight to all events irrespective of their type and clinical relevance thereby the all cause hazard within each group is given by the sum of the cause specific hazards corresponding to the individual components a natural extension of the standard all cause hazard ratio can be defined by a weighted all cause hazard ratio where the individual hazards for each component are multiplied with predefined relevance weighting factors for the special case of equal weights across the components the weighted all cause hazard ratio then corresponds to the standard all cause hazard ratio to identify the cause specific hazard of the individual components any parametric survival model might be applied the new weighted effect measure can be tested for deviations from the null hypothesis by means of a permutation test in this work we systematically compare the new weighted approach to the standard all cause hazard ratio by theoretical considerations monte carlo simulations and by means of a real clinical trial example

noise aided processing revisiting ditheringin a spl sigma spl delta quantizer in this paper we show that a particular spl sigma spl delta quantizer which is a strongly nonlinear dynamical system can provide noise enhanced processing effects in some sense in particular we will show that quantizing a sine which means adding a small amount of noise in the input can improve the quantization process in particular the output local signal to noise ratio snr exhibits a maximum when plotted against the noise amplitude

a use of double integration in sigma delta modulation sigma delta modulation is viewed as a technique that employs integration and feedback to move quantization noise out of baseband this technique may be iterated by placing feedback loop around feedback loop but when three or more loops are used the circuit can latch into undesirable overloading modes in the desired mode a simple linear theory gives a good description of the modulation even when the quantization has only two levels a modulator that employs double integration and two level quantization is easy to implement and is tolerant of parameter variation at sampling rates of 1 mhz it provides resolution equivalent to 16 bit pcm for voiceband signals digital filters that are suitable for converting the modulation to pcm are also described

modulo sigma delta modulation a modulo sigma delta modulator is introduced and the behavior of the quantization error is derived the system consists of a modulo limiter followed by a sigma delta modulator the limiter confines the input to the no overload region of the sigma delta modulator and the modulo arithmetic performed by the limiter is amenable to recently developed techniques for the exact analysis of quantizer error behavior in sigma modulators with bounded inputs the quantization error behavior is derived for a modulator driven by a quasi stationary random process the limit distribution and the power of the quantization error are found except for some singular cases the normalized quantization error is uniformly distributed in 1 2 1 2 the power spectrum and the autocorrelation function of the quantization error with a causal arma p q process input are also derived it is shown that the quantization noise is white when the input is a random process with stationary independent increments simulation results support the theoretical analysis

dithering and its effects on sigma delta and multistage sigma delta modulation the spectrum of the quantization error in a dithered sigma delta modulator is derived under the constraint that the dithering signal does not cause overload the results apply to dc sinusoidal and more general quasi stationary signals it is shown in the case of a simple sigma delta modulation that no overload dithering can smooth the error spectrum and can make the quantization error asymptotically uncorrelated with the input it does not however make the quantization error white in the case of multistage sigma delta modulation with the appropriate dithering the quantization error becomes white even for a system with only two stages the signal to quantization noise ratio sqnr is derived for sigma delta and multistage sigma delta oversampled analog to digital conversion with additive dithering simulation results are presented to support the theoretical analysis

reconstruction of oversampled band limited signals from spl sigma spl delta encoded binary sequences considers the application of spl sigma spl delta modulators to analog to digital conversion the authors have previously shown that for constant input signals optimal nonlinear decoding can achieve large gains in signal to noise ratio snr over linear decoding the present paper shows a similar result for band limited input signals the new nonlinear decoding algorithm is based on projections onto convex sets pocs and alternates between a time domain operation and a band limitation to find a signal invariant under both the time domain operation results in a quadratic programming problem the band limitation can be based on singular value decomposition of a certain matrix the authors show simulation results for the snr performance of a pocs based decoder and a linear decoder for the single loop double loop and two stage spl sigma spl delta modulators and for a specific fourth order interpolative modulator depending on the modulator and the oversampling ratio improvements in snr of up to 10 20 db can be achieved

on the robustness of single loop sigma delta modulation sigma delta modulation a widely used method of analog to digital a d signal conversion is known to be robust to hardware imperfections i e bit streams generated by slightly imprecise hardware components can be decoded comparably well we formulate a model for robustness and give a rigorous analysis for single loop sigma delta modulation applied to constant signals dc inputs for n time cycles with an arbitrary small enough initial condition u sub o and a quantizer that may contain an offset error the mean square error mse of any decoding scheme for this quantizer with u sub o and the offset error known is bounded below by 1 96n sup 3 we also determine the asymptotically best possible mse as n spl rarr spl infin for perfect decoding when u sub o 0 and u sub o 1 2 the robustness result is the upper bound that a triangular linear filter decoder with both u sub o and the offset error unknown achieves an mse of 40 3n sup 3 these results establish the known result that the o 1 n sup 3 decay of the mse with n is optimal in the single loop case under weaker assumptions than previous analyses and show that a suitable linear decoder is robust against offset error these results are obtained using methods from number theory and fourier analysis

quantization noise spectra several results describing the behavior of quantization noise in a unified and simplified manner are discussed exact formulas for quantizer noise spectra are developed they are applied to a variety of systems and inputs including scalar quantization pcm dithered pcm sigma delta modulation dithered sigma delta modulation two stage sigma delta modulation and second order sigma delta modulation

versions of the sign test in the presence of ties summary the question of how to treat tied observations is often a problem in nonparametric tests in this article the many different procedures that have been suggested for dealing with ties in the sign test are reviewed and compared we discuss a new version of the sign test that is especially well suited for situations in which the probability of an observation equalling the hypothesized median is known to be very high this test is based on the test statistic formed by adding the number of positive observations and two thirds of the number of zeroes studies of the power and significance level of many tests that have been proposed for this problem indicate that the asymptotic uniformly most powerful ump test performs very well for almost all settings considered even when the sample size is small this asymptotic ump test which has been little used to date is recommended for use in practice a particular application of the use of the sign test in the study of fluctuating asymmetry of meristic traits in ornamental goldfish is presented