central limit theorem for linear eigenvalue statistics of random matrices with independent entries we consider n x n real symmetric and hermitian wigner random matrices n 1 2 w with independent modulo symmetry condition entries and the null sample covariance matrices n 1 x x with independent entries of m x n matrix x assuming first that the 4th cumulant excess 4 of entries of w and x is zero and that their 4th moments satisfy a lindeberg type condition we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem clt as n m m n c 0 oo with the same variance as for gaussian matrices if the test functions of statistics are smooth enough essentially of the class c 5 this is done by using a simple interpolation trick from the known results for the gaussian matrices and the integration by parts presented in the form of certain differentiation formulas then by using a more elaborated version of the techniques we prove the clt in the case of nonzero excess of entries again for essentially 5 test function here the variance of statistics contains an additional term proportional to 4 the proofs of all limit theorems follow essentially the same scheme

bulk universality for generalized wigner matrices consider n n hermitian or symmetric random matrices h where the distribution of the i j matrix element is given by a probability measure ij with a subexponential decay let sigma ij 2 be the variance for the probability measure ij with the normalization property that sum i sigma 2 ij 1 for all j under essentially the only condition that c le n sigma ij 2 le c 1 for some constant c 0 we prove that in the limit n the eigenvalue spacing statistics of h in the bulk of the spectrum coincide with those of the gaussian unitary or orthogonal ensemble gue or goe we also show that for band matrices with bandwidth m the local semicircle law holds to the energy scale m 1

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

envelope constrained h filtering with fading measurements and randomly occurring nonlinearities in this paper the envelope constrained h filtering problem is investigated for a class of discrete time varying stochastic systems over a finite horizon the system under consideration involves fading measurements randomly occurring nonlinearities rons and mixed multiplicative and additive noises a novel envelope constrained performance criterion is proposed to better quantify the transient dynamics of the filtering error process over the finite horizon the purpose of the problem addressed is to design a time varying filter such that both the h performance and the desired envelope constraints are achieved at each time step by utilizing the stochastic analysis techniques combined with the ellipsoid description on the estimation errors sufficient conditions are established in the form of recursive matrix inequalities rmis reflecting both the envelope information and the desired h performance index the filter gain matrix is characterized by means of the solvability of the deduced rmis finally a simulation example is provided to show the effectiveness of the proposed filtering design scheme

analysis of factorial survival experiments several new methodological issues that arise within two way factorial designs for survival experiments are discussed within the framework of asymptotic theory for the proportional hazards model with two binary treatment covariates these issues include the proper formulation of null hypotheses and alternatives the choice among log rank and adjusted or stratified log rank statistics the asymptotic correlation between test statistics for the separate main effects the asymptotic power under the various possible methods of analysis of tests to detect main effects and interactions the comparison of power to detect main effects within a 2 x 2 factorial design with power in a three group trial where no patients are randomized simultaneously to both treatments and the problems of analysis arising when accrual or exposure to one of the treatments is terminated early for ethical reasons

sequential linear rank tests for two sample censored survival data under extremely general patterns of patient arrival allocation to treatment and loss to follow up in randomized clinical trial settings the sequentially computed logrank statistic mantel 1966 is shown under the null hypothesis of identically distributed lifetimes to have exactly uncorrelated increments and is shown via rebolledo s 1980 martingale invariance principle to satisfy a functional central limit theorem justifying sequential logrank tests of jones and whitehead 1979 generalizations are made to other twosample rank tests for censored survival data and practical applicability to real randomized clinical trials is discussed

large sample behaviour of the product limit estimator on the whole line weak convergence results are proved for the product limit estimator on the whole line applications are given to confidence band construction estimation of mean lifetime and to the theory of q functions the results are obtained using stochastic calculus and in probability linear bounds for empirical processes

estimation of a secondary parameter in a group sequential clinical trial summary in this article we investigate estimation of a secondary parameter in group sequential tests we study the model in which the secondary parameter is the mean of the normal distribution in a subgroup of the subjects the bias of the naive secondary parameter estimator is studied it is shown that the sampling proportions of the subgroup have a crucial effect on the bias as the sampling proportion of the subgroup at or just before the stopping time increases the bias of the naive subgroup parameter estimator increases as well an unbiased estimator for the subgroup parameter and an unbiased estimator for its variance are derived using simulations we compare the mean squared error of the unbiased estimator to that of the naive estimator and we show that the differences are negligible as an example the methods of estimation are applied to an actual group sequential clinical trial the beta blocker heart attack trial

inference for cumulative incidence quantiles via parametric and nonparametric approaches in survival analysis a point estimate and confidence interval for median survival time have been frequently used to summarize the survival curve however such quantile analyses on competing risks data have not been widely investigated in this paper we propose parametric inferences for quantiles from the cumulative incidence function and develop parametric confidence intervals for quantiles in addition we study a simplified method of inference for the nonparametric approach we compare the parametric and nonparametric inferences in empirical studies simulation studies show that the procedures perform well with parametric analyses yielding smaller mean square error when the model is not too badly misspecified we illustrate the methods with data from a breast cancer clinical trial copyright 2011 john wiley sons ltd

asymptotic expansions based on smooth functions in the central limit theorem stein s method is used to derive asymptotic expansions for expectations of smooth functions of sums of independent random variables together with lyapounov estimates of the error in the approximation

dyson brownian motion for general beta and potential at the edge in this paper we compare the solutions of the dyson brownian motion for general beta and potential v and the associated mckean vlasov equation near the edge under suitable conditions on the initial data and the potential v we obtain optimal rigidity estimates of particle locations near the edge after a short time t mathrm o 1 our argument uses the method of characteristics along with a careful estimate involving an equation of the edge with the rigidity estimates as an input we prove a central limit theorem for mesoscopic statistics near the edge which as far as we know has been done for the first time in this paper additionally combining our results with landon and yau edge statistics of dyson brownian motion arxiv 1712 03881 2017 we give a proof of the local ergodicity of the dyson brownian motion for general beta and potential at the edge i e we show the distribution of extreme particles converges to the tracy widom beta distribution in a short time

fixed energy universality for generalized wigner matrices we prove the wigner dyson mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and hermitian wigner matrices previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for hermitian matrices if the energy parameter is fixed we develop a homogenization theory of the dyson brownian motion and show that microscopic universality follows from mesoscopic statistics 2015 wiley periodicals inc

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

symmetric tests and confidence intervals for survival probabilities and quantiles of censored survival data we describe existing tests and introduce two new tests concerning the value of a survival function these tests may be used to construct a confidence interval for the survival probability at a given time or for a quantile of the survival distribution simulation studies show that error rates can differ substantially from their nominal values particularly at survival probabilities close to zero or one we recommend our new constrained bootstrap test for its good overall performance

group sequential tests for delayed responses with discussion summary group sequential methods are used routinely to monitor clinical trials and to provide early stopping when there is evidence of a treatment effect a lack of an effect or concerns about patient safety in many studies the response of clinical interest is measured some time after the start of treatment and there are subjects at each interim analysis who have been treated but are yet to respond we formulate a new form of group sequential test which gives a proper treatment of these pipeline subjects these tests can be applied even when the continued accrual of data after the decision to stop the trial is unexpected we illustrate our methods through a series of examples we define error spending versions of these new designs which handle unpredictable group sizes and provide an information monitoring framework that can accommodate nuisance parameters such as an unknown response variance by studying optimal versions of our new designs we show how the benefits of lower expected sample size that are normally achieved by a group sequential test are reduced when there is a delay in response the loss of efficiency for larger delays can be ameliorated by incorporating data on a correlated short term end point fitting a joint model for the two end points but still making inferences on the original longer term end point we derive p values and confidence intervals on termination of our new tests

optimal pulse shape design for digital communication systems by projections onto convex sets recent developments in dsp hardware has made it possible to use almost any pulse shape for digital data trans mission this resulted in a search for algorithms capable of constructing pulse shapes matching to the properties of a given channel the projection onto convex sets technique is suitable for the solution of this problem particularly because of its ffexibility in modeling a variety of constraints the use of the technique for the optimal pulse shape design problem is demonstrated through a detailed example from power line communications i introduction ince nyquist s invention of the intersymbol interference isi the optimal pulse shaping problem for digital data transmission over channels with various properties has been investigated by a number of researchers 1 7 the main differences between various digital nyquist pulse design al gorithms published in the literature are the type of the filter fir or iir they choose to generate the pulse shape and the formulation and solution of the optimization problem some of these approaches use general optimization techniques and they can handle additional constraints if needed however addition of each new constraint requires reformulation of the problem and the solution algorithm the projections onto convex sets pocs technique is an optimization method introduced in the 1960 s 8 a number of successful applications of the pocs technique in the signal processing area have been reported in the recent years 9 ll the pocs technique is attractive for optimum pulse shape design particularly because of its modular way of formulating problems with many constraints in this technique the addition of new constraints or removal of some of the existing ones does not require reformulation it is also possible to handle certain inconsistent requirements in the pocs formulation 12 if there are nonintersecting convex sets it is possible to find the vector which is as close as possible to one set of constraints while satisfying the rest

extrapolating a band limited function from its samples taken in a finite interval a band limited signal of finite energy can be reconstructed from its samples taken at the nyquist rate moreover the reconstruction is stable a feature crucial for implementation a small error in the sample values generates only a correspondingly small error in the resulting signal the nyquist sample values are mutually independent so that knowledge of them in a given interval t leq t leq 0 provides hardly any information about the behavior of the signal outside the interval however when the samples are taken at a greater rate a case referred to as over sampling they are interrelated and this redundancy can be exploited in various ways to improve the behavior of the reconstruction procedure a natural question is whether it can also be used to form accurate estimates of the signal outside the interval of observation this problem is relevant as well to prediction theory with oversampling when t infty so that the samples are known on the entire half line t they determine the signal everywhere although the reconstruction is now no longer stable here we examine the case of finite t of course a finite amount of data can yield only limited accuracy we prove that the samples can be used to form an approximation to the signal outside the sampling interval with an error which asymptotically as t rightarrow infty decreases exponentially in t over a range which grows linearly with t however as in the limiting case these approximations are not useful in practice since they require the sample values to be known exactly in the presence of measurement error the nature of the results changes good approximations are now available for only a bounded distance outside the interval of observation regardless of its length and their accuracy and range of validity can be increased only by improving the precision of sample reading since physical measurements are never perfect it is this conclusion which counts for applications the same results hold for the extrapolation of bounded band limited signals

a note on confidence limits for quartiles with right censored data two new methods are proposed for constructing the confidence limits for quartiles the small sample behaviour of these two methods is compared with the jennison and turnbull modified brookmeyer crowley method at three quartiles the simulation study indicates that one of the new methods the smoothed modified reflected method is preferred over the other methods when the censoring rate is less than 40 per cent while the jennison and turnbull method is preferred for higher censoring the results for the upper quartile are similar to the median for the lower quartile with small sample and high censoring semi infinite intervals may often occur the correct practice is to permit the semi infinite intervals without modifying them since it is more informative and gives closer to nominal level coverage

spectral statistics of erdős renyi graphs i local semicircle law we consider the ensemble of adjacency matrices of erd s renyi random graphs that is graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one we prove that as long as pn with a speed at least logarithmic in n the density of eigenvalues of the erd s renyi ensemble is given by the wigner semicircle law for spectral windows of length larger than n 1 up to logarithmic corrections as a consequence all eigenvectors are proved to be completely delocalized in the sense that the l norms of the l2 normalized eigenvectors are at most of order n 1 2 with a very high probability the estimates in this paper will be used in the companion paper spectral statistics of erd s renyi graphs ii eigenvalue spacing and the extreme eigenvalues 2011 preprint to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pn n2 3

interim analyses based on median survival times summary for a typical two arm survival study analyses are usually performed periodically to evaluate for example the treatment difference in this paper we propose to use differences or ratios of estimated median survival times repeatedly during a study to evaluate the relative merit of two treatment groups this new procedure provides quantitative assessments of the treatment difference and is completely model free our method is illustrated with a recent aids clinical study which evaluates the efficacy of a reduced dose of azt

mean square convergence of adaptive envelope constrained filtering a new type of adaptive filtering scheme for solving an envelope constrained filter design problem in a stochastic environment has been presented the steady state stochastic analysis of the adaptive scheme is established in the sense of the mean square it is shown that if a given initial point is in a neighborhood of the origin the adaptive scheme with a fixed step size produces a filter that converges in the mean square sense to within an upper boundary of the noise free optimum filter numerical examples involving pulse compression and channel equalization are presented to illustrate the convergence characteristics of the adaptive scheme operating in a stochastic environment

a new interpolation theorem with application to pulse transmission an interpolation theorem is determined for the case when there are a finite number of arbitrarily placed sampling instants and the interpolation function is the output of a known filter they are also the interpolation functions with the specified properties that have minimum energy the theorem is used to determine the input to a communications channel given a finite number of samples of its output this provides a generalization of matched filters and a perspective on the benefits of fractionally spaced equalization the theorem is also used to construct masks of a family of pulses that are specified by the range of pulse voltages at a finite number of sampling instants the theory determines how the pulse masks thus constructed is transformed when the pulse family is transmitted through a filter such as a length of transmission line

biometrika centenary survival analysis although papers relating to survival analysis have appeared in biometrika since the very first issue the current emphasis on the topic in this journal is fairly new especially as it relates to applications in biostatistics this survey traces the development of survival analysis throughout the twentieth century as reflected in the pages of biometrika focusing primarily on work published since 1980

isotropic self consistent equations for mean field random matrices we present a simple and versatile method for deriving an isotropic local laws for general random matrices constructed from independent random variables our method is applicable to mean field random matrices where all independent variables have comparable variances it is entirely insensitive to the expectation of the matrix in this paper we focus on the probabilistic part of the proof the derivation of the self consistent equations as a concrete application we settle in complete generality the local law for wigner matrices with arbitrary expectation

mesoscopic eigenvalue density correlations of wigner matrices we investigate to what extent the microscopic wigner gaudin mehta dyson wgmd or sine kernel statistics of random matrix theory remain valid on mesoscopic scales to that end we compute the connected two point spectral correlation function of a wigner matrix at two mesoscopically separated points in the mesoscopic regime density correlations are much weaker than in the microscopic regime our result is an explicit formula for the two point function this formula implies that the wgmd statistics are valid to leading order on all mesoscopic scales that in the real symmetric case there are subleading corrections matching precisely the wgmd statistics while in the complex hermitian case these subleading corrections are absent we also uncover non universal subleading correlations which dominate over the universal ones beyond a certain intermediate mesoscopic scale the proof is based on a hierarchy of schwinger dyson equations for a sufficiently large class of polynomials in the entries of the green function the hierarchy is indexed by a tree whose depth is controlled using stopping rules a key ingredient in the derivation of the stopping rules is a new estimate on the density of states which we prove to have bounded derivatives of all order on all mesoscopic scales

a confidence interval for the median survival time a nonparametric asymptotic confidence interval for the median survival time is developed for the case where data are subject to arbitrary right censoring this is accomplished by inverting a generalization of the sign test for censored data a simulation study shows that this nonparametric confidence interval performs well for a variety of underlying survival functions the procedure is applied to data from a clinical trial that compared f our dosage regimens of 5 uorouracil

repeated confidence intervals for the median survival time summary we describe methods for the construction of a confidence interval for median survival time based on right censored data these methods are extended to the construction of repeated confidence intervals for the median based on accumulating data here the overall probability that all intervals contain the true median is guaranteed at a fixed level the use of repeated confidence intervals for median survival time in post marketing surveillance is discussed a confidence interval for median survival time provides a useful summary of the survival experience of a group of patients if confidence intervals are calculated repeatedly as data accumulates the probability that at least one interval fails to contain the median may be much higher than the error rate for a single interval and if these confidence intervals are used in a decision making process the probability of an incorrect decision increases accordingly jennison turnbull 1984 have proposed methods for calculating repeated confidence intervals appropriate to such situations similar ideas have also been discussed by lai 1984 in 2 we propose a new form of single sample nonparametric confidence interval this interval has asymptotically correct coverage probability and monte carlo simulations suggest it is superior to its competitors for small sample sizes repeated confidence intervals for the median are presented in 3 and their stnall sample size performance is assessed by monte carlo simulation an example of their use is given in 4 all the methods considered can easily be modified to give confidence intervals for other quantiles or for the survival probability at a fixed time

accurate confidence limits for quantiles under random censoring in survival analysis estimates of median survival times in homogeneous samples are often based on the kaplan meier estimator of the survivor function confidence intervals for quantiles such as median survival are typically constructed via large sample theory or the bootstrap the former has suspect accuracy for small sample sizes under moderate censoring and the latter is computationally intensive in this paper improvements on so called test based intervals and reflected intervals cf slud byar and green 1984 biometrics 40 587 600 are sought using the edgeworth expansion for the distribution of the studentized nelson aalen estimator derived in strawderman and wells 1997 journal of the american statistical association 92 we propose a method for producing more accurate confidence intervals for quantiles with randomly censored data the intervals are very simple to compute and numerical results using simulated data show that our new test based interval outperforms commonly used methods for computing confidence intervals for small sample sizes and or heavy censoring especially with regard to maintaining specified coverage

a comparison of reflected versus test based confidence intervals for the median survival time based on censored data the small sample performance of some recently proposed nonparametric methods of constructing confidence intervals for the median survival time based on randomly right censored data is compared with that of two new methods most of these methods are equivalent for large samples all proposed intervals are either test based or reflected intervals in the sense defined in the paper coverage probabilities for the interval estimates were obtained by exact calculation for uncensored data and by stimulation for three life distributions and four censoring patterns in the range of situations studied test based methods often have less than nominal coverage while the coverage of the new reflected confidence intervals is closer to nominal although somewhat conservative and these intervals are easy to compute

a penalty approach to iterative algorithms for envelope constrained filter design the discrete time envelope constrained ec filtering problem can be formulated as a quadratic programming qp problem with affine inequality constraints this qp problem is approximated by an unconstrained minimization problem with two parameters descent direction based algorithms are applied to solve the unconstrained problem iteratively it is shown that these algorithms converge

gaussian fluctuations of eigenvalues in wigner random matrices we study the fluctuations of eigenvalues from a class of wigner random matrices that generalize the gaussian orthogonal ensemble we begin by considering an n n matrix from the gaussian orthogonal ensemble goe or gaussian symplectic ensemble gse and let xk denote eigenvalue number k under the condition that both k and n k tend to infinity as n w e show thatxk is normally distributed in the limit we also consider the joint limit distribution of eigenvalues xk1 x km from the goe or gse where k1 n km and ki 1 ki 1 i m 1 tend to infinity with n the result in each case is an m dimensional normal distribution using a recent universality result by tao and vu we extend our results to a class of wigner real symmetric matrices with non gaussian entries that have an exponentially de caying distribution and whose first four moments match the gaussian moments

estimates and confidence intervals for median and mean life in the proportional hazard model summary estimation procedures and confidence intervals are given for the median and mean survival time in the proportional hazard regression model for median survival the methods apply to censored data the procedures are based on cox s partial likelihood estimates of the linear model parameters in the log linear proportional hazard model and on breslow s estimate of the baseline hazard function the asymptotic properties of these semiparametric estimates are developed and they are compared with the optimal parametric estimates for the weibull regression model for the parameter values considered the more generally valid semiparametric estimate of mean survival loses little efficiency relative to the optimal parametric estimates in this model unless the weibull shape parameter is close to zero the efficiency loss for the semiparametric estimate of median survival is greater but not severe unless censoring is heavy we also compare the optimal parametric estimates of mean and median survival with the weibull shape parameter known and unknown and find that the efficiency loss for the unknown shape estimate is small

spectral statistics of erdős renyi graphs ii eigenvalue spacing and the extreme eigenvalues we consider the ensemble of adjacency matrices of erd s renyi random graphs i e graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one under the assumption p n gg n 2 3 we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum more precisely we prove 1 that the eigenvalue spacing of the erd s renyi graph in the bulk of the spectrum has the same distribution as that of the gaussian orthogonal ensemble and 2 that the second largest eigenvalue of the erd s renyi graph has the same distribution as the largest eigenvalue of the gaussian orthogonal ensemble as an application of our method we prove the bulk universality of generalized wigner matrices under the assumption that the matrix entries have at least 4 e moments

bootstrap type confidence intervals for quantiles of the survival distribution in this paper we outline and illustrate an easy to program method for analytically calculating both parametric and non parametric bootstrap type confidence intervals for quantiles of the survival distribution based on right censored data this new approach allows for the incorporation of covariates within the framework of parametric models the procedure is based upon the notion of fractional order statistics and is carried forth using a simple beta transformation of the estimated survival function parametric or non parametric it is the only direct method currently available in the sense that all other methods are based on inverting test statistics or employing confidence intervals for other survival quantities we illustrate that the new method has favourable coverage probabilities for median confidence intervals as compared to six other competing methods

a hereditarily indecomposable mathcal l _ infty space that solves the scalar plus compact problem we construct a hereditarily indecomposable banach space with dual space isomorphic to l1 every bounded linear operator on this space is expressible as i k with a scalar and k compact

envelope constrained filters adaptive algorithms we consider an envelope constrained ec optimal filter design problem involving a quadratic cost function and a number of linear inequality constraints using the duality theory and the space transformation function the optimal solution of the dual problem can be computed by finding the limiting point of an ordinary differential equation given in terms of the gradient flow an iterative algorithm is developed via discretizing the differential equation from the primal dual relationship the corresponding sequence of approximate solutions in the original ec filtering problem is obtained based on these results an adaptive algorithm is constructed for solving the stochastic ec filtering problem in which the input signal is corrupted by an additive random noise for illustration a practical example is solved for both noise free and noisy cases

group sequential analysis incorporating covariate information abstract in this article we survey existing results concerning the joint distribution of the sequence of estimates of the parameter vector when a model is fitted to accumulating data and provide a unified theory that explains the independent increments structure commonly seen in group sequential test statistics our theory covers normal linear models including the case of correlated observations and asymptotic results extend to generalized linear models and the proportional hazards regression model for survival data the asymptotic results are derived using standard methods for the nonsequential case and they hold as long as these nonsequential techniques are applicable at each individual analysis in all cases the joint distribution of the sequence of parameter estimates has the same form exactly or asymptotically as that of the sequence of means of an increasing number of independent identically distributed normal variables thus our results provide the formal basis for extending the scope of s

two sample inference for median survival times based on one sample procedures for censored survival data abstract confidence intervals for median survival times are derived for censored survival data the intervals are obtained by using the quantiles of the kaplan meier product limit estimator and have the same pitman efficiency as the intervals found by inverting the sign test two sample tests and confidence intervals for the difference in median survival times are then developed based on the comparison of the one sample confidence intervals several methods for choosing the confidence coefficients of the corresponding one sample confidence intervals are developed the pitman efficiencies of these two sample tests are the same as that of the median test proposed by brookmeyer and crowley 1982a the procedures can also be used for the accelerated failure time model proportional hazard model and the behrens fisher problem nonparametric two sample inference procedures are useful in comparing the responses of treatment and control groups in medical follow up studies the data are usually subjected to cens

on the 1 n corrections to the green functions of random matrices with independent entries we propose a general approach to the construction of 1 n corrections to the green function gn z of the ensembles of random real symmetric and hermitian n n matrices with independent entries hk l by this approach we study the correlation function cn z1 z2 of the normalized trace n 1trgn assuming that the average of mod hk l5 is bounded we found that to leading order cn z1 z2 n 2f z1 z2 where f z1 2 only depends on the second and the fourth moments of hk l for the correlation function of the density of energy levels we obtain an expression which in the scaling limit only depends on the second moment of hk l this can be viewed as supporting the universality conjecture of random matrix theory

functional laws of the iterated logarithm for the product limit estimator of a distribution function under random censorship or truncation functional laws of the iterated logarithm are established for a modified version of the classical product limit estimator of a distribution function when the data are subject to random censorship or truncation making use of stochastic integral representations and empirical process theory strong approximations involving i i d continuous parameter martingales are developed for the product limit estimator and these strong approximations are then applied to derive the functional laws

asymptotic properties of large random matrices with independent entries we study the normalized trace gn z n 1 tr h zi 1 of the resolvent of n n real symmetric matrices h 1 jk wjk n j k 1n assuming that their entries are independent but not necessarily identically distributed random variables we develop a rigorous method of asymptotic analysis of moments of gn z for iz 0 where 0 is determined by the second moment of wjk by using this method we find the asymptotic form of the expectation e gn z and of the connected correlator e gn z1 gn z2 e gn z1 e gn z2 we also prove that the centralized trace ngn z e ngn z has the gaussian distribution in the limit n based on these results we present heuristic arguments supporting the universality property of the local eigenvalue statistics for this class of random matrix ensembles

comparison of median survival times with adjustment for covariates brookmeyer and crowley derived a non parametric confidence interval for the median survival time of a homogeneous population by inverting a generalization of the sign test for censored data the 1 alpha confidence interval for the median is essentially the set of all values t such that the kaplan meier estimate of the survival curve at time t does not differ significantly from one half at the two sided alpha level su and wei extended this approach to the two sample problem and derived a confidence interval for the difference in median survival times based on the kaplan meier estimates of the individual survival curves and a minimum dispersion test statistic here i incorporate covariates into the analysis by assuming a proportional hazards model for the covariate effects while leaving the two underlying survival curves virtually unconstrained i generate a simultaneous confidence region for the two median survival times adjusted to any selected value z of the covariate vector using a test based approach analogous to brookmeyer and crowley s for the one sample case this region is in turn used to derive a confidence interval for the difference in median survival times between the two treatment groups at the selected value of z employment of a procedure suggested by aitchison sets the level of the simultaneous region to a value that should yield at least approximately the desired confidence coefficient for the difference in medians simulation studies indicate that the method provides reasonably accurate coverage probabilities