pseudounitary symmetry and the gaussian pseudounitary ensemble of random matrices employing the currently discussed notion of pseudo hermiticity we define a pseudounitary group further we develop a random matrix theory that is invariant under such a group and call this ensemble of pseudo hermitian random matrices the pseudounitary ensemble we obtain exact results for the nearest neighbor level spacing distribution for 2 x 2 pt invariant hamiltonian matrices that have forms approximately sln 1 s near zero spacing for three independent elements and approximately s for four independent elements this shows a level repulsion in a marked distinction with an algebraic form s beta in the wigner surmise we believe that this paves the way for a description of varied phenomena in two dimensional statistical mechanics quantum chromodynamics and so on

pseudo hermiticity versus pt symmetry iii equivalence of pseudo hermiticity and the presence of antilinear symmetries we show that a diagonalizable non hermitian hamiltonian h is pseudo hermitian if and only if it has an antilinear symmetry i e a symmetry generated by an invertible antilinear operator this implies that the eigenvalues of h are real or come in complex conjugate pairs if and only if h possesses such a symmetry in particular the reality of the spectrum of h implies the presence of an antilinear symmetry we further show that the spectrum of h is real if and only if there is a positive definite inner product on the hilbert space with respect to which h is hermitian or alternatively there is a pseudo canonical transformation of the hilbert space that maps h into a hermitian operator

on the pseudo hermitian nondiagonalizable hamiltonians we consider a class of possibly nondiagonalizable pseudo hermitian operators with discrete spectrum and we establish for such a class the equivalence between the pseudo hermiticity property and the existence of an antilinear involutory symmetry moreover we prove that this class actually coincides with the one of possibly nondiagonalizable weak pseudo hermitian operators and that in no case unless they are diagonalizable and have a real spectrum they are hermitian with respect to a definite inner product finally we show that a typical degeneracy of the real eigenvalues which reduces to the well known kramers degeneracy in the hermitian case occurs whenever a fermionic possibly nondiagonalizable pseudo hermitian hamiltonian admits an antilinear symmetry like the time reversal operator t some consequences and applications are briefly discussed

alternative descriptions and bipartite compound quantum systems we analyse some features of alternative hermitian and quasi hermitian quantum descriptions of simple and bipartite compound systems we show that alternative descriptions of two interacting subsystems are possible if and only if the metric operator of the compound system can be obtained as a tensor product of positive operators on component spaces some examples also show that such property could be strictly connected with symmetry properties of the non hermitian hamiltonian

pseudo hermiticity and generalized pt and cpt symmetries we study certain linear and antilinear symmetry generators and involution operators associated with pseudo hermitian hamiltonians and show that the theory of pseudo hermitian operators provides a simple explanation for the recent results of bender brody and jones quant ph 0208076 on the cpt symmetry of a class of pt symmetric non hermitian hamiltonians we present a natural extension of these results to the class of diagonalizable pseudo hermitian hamiltonians h with a discrete spectrum in particular we introduce generalized parity p time reversal t and charge conjugation c operators and establish the pt and cpt invariance of h

on fitting cox s proportional hazards models to survey data summary binder 1992 proposed a method of fitting cox s proportional hazards models to survey data with complex sampling designs he defined the regression parameter of interest as the solution to the partial likelihood score equation based on all the data values of the survey population under study and developed heuristically a procedure to estimate the regression parameter and the corresponding variance in this paper we provide a formal justification of binder s method furthermore we present an alternative approach which regards the survey population as a random sample from an infinite universe and accounts for this randomness in the statistical inference under the alternative approach the regression parameter retains its original interpretation as the log hazard ratio and the statistical conclusion applies to other populations the related problem of survival function estimation is also studied

power calculation for case cohort studies with nonrare events summary case cohort design has been advocated in many epidemiologic studies when studying rare diseases or events in this design with a rare event all the events are selected for risk factor assessment when the event is not rare it is desirable to consider a generalized case cohort design where only a fraction of events are sampled we provide a valid test statistic to compare hazards functions between two samples for this generalized design and give a method for calculating power our result generalizes the result in cai and zeng 2004 biometrics60 1015 1024 and it shows numerically that efficiency loss due to sampling only part of the events is very low under nonrare events situation

more efficient estimators for case cohort studies the case cohort study design used to reduce costs in large cohort studies is a random sample of the entire cohort named the subcohort augmented with subjects having the disease of interest but not in the subcohort sample when several diseases are of interest several case cohort studies may be conducted using the same subcohort with each disease analyzed separately ignoring the additional exposure measurements collected on subjects with the other diseases this is not an efficient use of the data and in this paper we propose more efficient estimators we consider both joint and separate analyses for the multiple diseases we propose an estimating equation approach with a new weight function and we establish the consistency and asymptotic normality of the resulting estimator simulation studies show that the proposed methods using all available information gain efficiency we apply our proposed method to the data from the busselton health study

a case cohort design for epidemiologic cohort studies and disease prevention trials summary suppose that a cohort of individuals is to be followed in order to relate failure rates to preceding covariate histories a design is proposed which involves covariate data only for cases experiencing failure and for members of a randomly selected subcohort odds ratio and relative risk estimation procedures are presented for such a case cohort design a small simulation study compares case cohort relative risk estimation procedures to full cohort and synthetic case control analyses relevance to epidemiologic cohort studies and disease prevention trials is discussed

alternative descriptions in quaternionic quantum mechanics we characterize the quasianti hermitian quaternionic operators in quaternionic quantum mechanics by means of their spectra moreover we state a necessary and sufficient condition for a set of quasianti hermitian quaternionic operators to be anti hermitian with respect to a uniquely defined positive scalar product in a infinite dimensional right quaternionic hilbert space according to such results we obtain two alternative descriptions of a quantum optical physical system in the realm of quaternionic quantum mechanics while no alternative can exist in complex quantum mechanics and we discuss some differences between them

weighted estimating equations for semiparametric transformation models with censored data from a case cohort design summary in a case cohort design introduced by prentice 1986 covariates are assembled only for a subcohort randomly selected from the entire cohort and any additional cases outside the subcohort semiparametric transformation models are considered here for failure time data from the case cohort design weighted estimating equations are proposed for esti mation of the regression parameters the estimation procedure of survival probability at given covariate levels is also provided asymptotic properties are derived for the estimators using finite population sampling theory u statistics theory and martingale convergence results the finite sample properties of the proposed estimators as well as the efficiency relative to the full cohort estimators are assessed via simulation studies a case cohort dataset from the atherosclerosis risk in communities study is used to illustrate the estimating procedure

semiparametric transformation models for the case cohort study a general class of semiparametric transformation models is studied for analysing survival data from the case cohort design which was introduced by prentice 1986 weighted estimating equations are proposed for simultaneous estimation of the regression parameters and the transformation function it is shown that the resulting regression estimators are asymptotically normal with variance covariance matrix that has a closed form and can be consistently estimated by the usual plug in method simulation studies show that the proposed approach is appropriate for practical use an application to a case cohort dataset from the atherosclerosis risk in communities study is also given to illustrate the methodology

exact pt symmetry is equivalent to hermiticity we show that a quantum system possessing an exact antilinear symmetry in particular pt symmetry is equivalent to a quantum system having a hermitian hamiltonian we construct the unitary operator relating an arbitrary non hermitian hamiltonian with exact pt symmetry to a hermitian hamiltonian we apply our general results to pt symmetry in finite dimensions and give the explicit form of the above mentioned unitary operator and hermitian hamiltonian in two dimensions our findings lead to the conjecture that non hermitian cpt symmetric field theories are equivalent to certain nonlocal hermitian field theories

pseudounitary operators and pseudounitary quantum dynamics we consider pseudounitary quantum systems and discuss various properties of pseudounitary operators in particular we prove a characterization theorem for block diagonalizable pseudounitary operators with finite dimensional diagonal blocks furthermore we show that every pseudounitary matrix is the exponential of i 1 times a pseudo hermitian matrix and determine the structure of the lie groups consisting of pseudounitary matrices in particular we present a thorough treatment of 2 2 pseudounitary matrices and discuss an example of a quantum system with a 2 2 pseudounitary dynamical group as other applications of our general results we give a proof of the spectral theorem for symplectic transformations of classical mechanics demonstrate the coincidence of the symplectic group sp 2n with the real subgroup of a matrix group that is isomorphic to the pseudounitary group u n n and elaborate on an approach to second quantization that makes use of the underlying pseudounitary dynamical groups

finite dimensional pt symmetric hamiltonians this paper investigates finite dimensional pt symmetric hamiltonians it is shown here that there are two ways to extend real symmetric hamiltonians into the complex domain i the usual approach is to generalize such hamiltonians to include complex hermitian hamiltonians ii alternatively one can generalize real symmetric hamiltonians to include complex pt symmetric hamiltonians in the first approach the spectrum remains real while in the second approach the spectrum remains real if the pt symmetry is not broken both generalizations give a consistent theory of quantum mechanics but if d 2 a d dimensional hermitian matrix hamiltonian has more arbitrary parameters than a d dimensional pt symmetric matrix hamiltonian

a littelmann type formula for duistermaat heckman measures given a compact hamiltonian t manifold m some component of whose moment map is a palais smale morse function we give a formula for the duistermaat heckman measure the fourier transform of the equivariant symplectic volume as a sum of volumes of simplices unlike previous formulae for this measure this sum has only positive terms in the case of m a flag manifold our results are an asymptotic version of littelmann s multiplicity result in representation theory 9

case cohort studies with interval censored failure time data the case cohort design has been widely used as a means of cost reduction in assembling or measuring expensive covariates in large cohort studies the existing literature on the case cohort design is mainly focused on right censored data in practice however the failure time is often subject to interval censoring it is known only to fall within some random time interval in this paper we consider the case cohort study design for interval censored failure time and develop a sieve semiparametric likelihood approach for analyzing data from this design under the proportional hazards model we construct the likelihood function using inverse probability weighting and build the sieves with bernstein polynomials the consistency and asymptotic normality of the resulting regression parameter estimator are established and a weighted bootstrap procedure is considered for variance estimation simulations show that the proposed method works well for practical situations and an application to real data is provided

marginal hazards model for case cohort studies with multiple disease outcomes case cohort study designs are widely used to reduce the cost of large cohort studies while achieving the same goals especially when the disease rate is low a key advantage of the case cohort study design is its capacity to use the same subcohort for several diseases or for several subtypes of disease in order to compare the effect of a risk factor on different types of diseases times to different events need to be modelled simultaneously valid statistical methods that take the correlations among the outcomes from the same subject into account need to be developed to this end we consider marginal proportional hazards regression models for case cohort studies with multiple disease outcomes we also consider generalized case cohort designs that do not require sampling all the cases which is more realistic for multiple disease outcomes we propose an estimating equation approach for parameter estimation with two different types of weights consistency and asymptotic normality of the proposed estimators are established large sample approximation works well in small samples in simulation studies the proposed methods are applied to the busselton health study

selecting an efficient design for assessing exposure disease relationships in an assembled cohort summary we develop approximate methods to compare the efficiencies and to compute the power of alternative potential designs for sampling from a cohort before beginning to collect exposure data our methods require only that the cohort be assembled meaning that the numbers of individuals nkj at risk at pairs of event times tk and ti tk are available to compute nkj one needs to know the entry follow up censoring and event history but not the exposure for each individual our methods apply to any unbiased control sampling design in which cases are compared to a random sample of noncases at risk at the time of an event we apply our methods to approximate the efficiencies of the nested case control design the case cohort design and an augmented case cohort design compared to the full cohort design in an assembled cohort of 17 633 members of an insurance cooperative who were followed for mortality from prostatic cancer the assumptions underlying the approximation are that exposure is unrelated both to the hazard of an event and to the hazard for censoring the approximations performed well in simulations when both assumptions held and when the exposure was moderately related to censoring

pseudo hermitian hamiltonians indefinite inner product spaces and their symmetries we extend the definition of generalized parity p charge conjugation c and time reversal t operators to nondiagonalizable pseudo hermitian hamiltonians and use these generalized operators to describe the full set of symmetries of a pseudo hermitian hamiltonian according to a fourfold classification in particular we show that t p and ct p are the generators of the p antiunitary symmetries moreover a necessary and sufficient condition is provided for a pseudo hermitian hamiltonian h to admit a p reflecting symmetry which generates the p pseudounitary and the p pseudoantiunitary symmetries finally a physical example is considered and some hints on the p unitary evolution of a physical system are also given

generalized case cohort sampling a class of cohort sampling designs including nested case control case cohort and classical case control designs involving survival data is studied through a unified approach using cox s proportional hazards model by finding an optimal sample reuse method via local averaging a closed form estimating function is obtained leading directly to the estimators of the regression parameters that are relatively easy to compute and are more efficient than some commonly used estimators in case cohort and nested case control studies a semiparametric efficient estimator can also be found with some further computation in addition the class of sampling designs in this study provides a variety of sampling options and relaxes the restrictions of sampling schemes that are currently available

robust variance estimation for the case cohort design large cohort studies of rare outcomes require extensive data collection often for many relatively uninformative subjects sampling schemes have been proposed that oversample certain groups for example the case cohort design of prentice 1986 biometrika 73 1 11 provides an efficient method of analysis of failure time data however the variance estimate must explicitly correct for correlated score contributions a simple robust variance estimator is proposed that allows for more complicated sampling mechanisms the variance estimate uses a jackknife estimate of the variance of the individual influence function and is shown to be equivalent to a robust variance estimator proposed by lin and wei 1989 journal of the american statistical association 84 1074 1078 for the standard cox model simulation results indicate excellent agreement with corrected asymptotic estimates and appropriate test size the technique is illustrated with data evaluating the efficacy of mammography screening in reducing breast cancer mortality

case cohort and case control analysis with cox s model prentice 1986 proposed the case cohort design and studied a pseudolikelihood estimator of regression parameters in cox s model we derive a class of estimating equations for case cohort sampling each depending on a different estimator of the population distribution which lead naturally to simple estimators that improve on prentice s pseudolikelihood estimator we also discuss an equivalence between case control and case cohort sampling in terms of the estimation of regression parameters in cox s model

weighted semiparametric likelihood method for fitting a proportional odds regression model to data from the case cohort design the problem of fitting a proportional odds regression model to data from the case cohort design proposed by prentice is considered a weighted semiparametric likelihood method is proposed under the proportional odds model the maximum weighted semiparametric likelihood estimators of both the regression parameter and the transformation function are shown to be consistent and normally distributed the applicability of the weighted semiparametric likelihood method to the semiparametric transformation regression models is also discussed in particular when the proportional hazards regression model is fitted estimators proposed by chen and lo can be generated by the weighted semiparametric likelihood method under different weighting schemes a simulation study suggests that the case cohort design is also useful under the proportional odds regression model and the proposed method performs well with practical finite sample sizes

a canonical form for matrices under consimilarity abstract square complex matrices a b are said to be consimilar if a sb s 1 for some nonsingular matrix s consimilarity is an equivalence relation that is a natural matrix generalization of rotation of scalars in the complex plane we survey the known forms to which a given complex matrix may be reduced by unitary consimilarity and describe a canonical form to which it may be reduced by a general consimilarity we derive a useful criterion for two matrices to be consimilar and show that every matrix is consimilar to its own conjugate transpose and adjoint to a real matrix and to a hermitian matrix

on solutions of the matrix equations x axb c and x axb c this paper studies the solutions of complex matrix equations x axb c and x axb c and obtains explicit solutions of the equations by the method of characteristic polynomial and a method of real representation of a complex matrix respectively

pseudo hermiticity for a class of nondiagonalizable hamiltonians we give two characterization theorems for pseudo hermitian possibly nondiagonalizable hamiltonians with a discrete spectrum that admit a block diagonalization with finite dimensional diagonal blocks in particular we prove that for such an operator h the following statements are equivalent 1 h is pseudo hermitian 2 the spectrum of h consists of real and or complex conjugate pairs of eigenvalues and the geometric multiplicity and the dimension of the diagonal blocks for the complex conjugate eigenvalues are identical 3 h is hermitian with respect to a positive semidefinite inner product we further discuss the relevance of our findings for the merging of a complex conjugate pair of eigenvalues of diagonalizable pseudo hermitian hamiltonians in general and the pt symmetric hamiltonians and the effective hamiltonian for a certain closed frw minisuperspace quantum cosmological model in particular

quasistationary quaternionic hamiltonians and complex stochastic maps we show that the complex projections of time dependent quasianti hermitian quaternionic hamiltonian dynamics are complex stochastic dynamics in the space of complex quasi hermitian density matrices if and only if a quasistationarity condition is fulfilled i e if and only if is an hermitian positive time independent complex operator an example is also discussed

weak pseudo hermiticity and antilinear commutant we inquire into some properties of diagonalizable pseudo hermitian operators showing that their definition can be relaxed and that the pseudo hermiticity property is strictly connected with the existence of an antilinear symmetry this result is then illustrated by considering the particular case of the complex morse potential

marginal regression models for multivariate failure time data abstract in this article we propose a general cox type regression model to formulate the marginal distributions of multivariate failure time data this model has a nested structure in that it allows different baseline hazard functions among distinct failure types and imposes a common baseline hazard function on the failure times of the same type we prove that the maximum quasi partial likelihood estimator for the vector of regression parameters under the independence working assumption is consistent and asymptotically normal with a covariance matrix for which a consistent estimator is provided furthermore we establish the uniform consistency and joint weak convergence of the aalen breslow type estimators for the cumulative baseline hazard functions and develop a resampling technique to approximate the joint distribution of these processes which enables one to make simultaneous inference about the survival functions over the time axis and across failure types finally we assess the small sample pro

competing risks analysis of the case cohort design the case cohort design has been put forward as an alternative design in epidemiological follow up studies which is less expensive than a full scale cohort study in this design all cases are assembled together with a randomly selected sub cohort which is then used as a comparison group one advantage being that the same sub cohort may serve as a comparison group for a number of different case series however the fact that all case groups are compared with the same sub cohort creates a correlation between estimated effects of exposures on the different outcomes this asymptotic correlation is derived using martingale central limit theory by studying competing risks data sampled using a case cohort design in a monte carlo simulation study we study effects on the correlation of factors such as number of cases and size of sub cohort

additive hazards regression for case cohort studies summary the case cohort design is a common means of reducing cost in large epidemiological cohort studies under this design covariates are measured only on the cases and a subcohort randomly selected from the entire cohort in this paper we demonstrate how to use the case cohort data to estimate the regression parameter of the additive hazards model which specifies that the conditional hazard function given a set of covariates is the sum of an arbitrary baseline hazard function and a regression function of the covariates the proposed estimator is shown to be consistent and asymptotically normal with an easily estimated variance the subcohort may be selected by independent bernoulli sampling with arbitrary selection probabilities or by stratified simple random sampling the efficiencies of various sampling schemes are investigated both analytically and by simulation a real example is provided