gamow states as continuous linear functionals over analytical test functions the space of analytical test functions rapidly decreasing on the real axis i e schwartz test functions of the type s on the real axis is used to construct the rigged hilbert space rhs h gamow states gs can be defined in rhs starting from dirac s formula it is shown that the expectation value of a self adjoint operator acting on a gs is real we have computed exactly the probability of finding a system in a gs and found that it is finite the validity of recently proposed approximations to calculate the expectation value of self adjoint operators in a gs is discussed

likelihood ration test wald test and kuhn tucker test in linear models with inequality constraints on the regression parameters likelihood ratio test wald test and kuhn tucker test in linear models with inequality constraints on the regression parameters this paper considers the problem of testing statistical hypotheses in linear regression models with inequality constraints on the regression coefficients the kuhn tucker multiplier test statistic is defined and its relationships with the likelihood ratio test and the wald test are examined it is shown in particular that these relationships are the same as in the equality constrained case it is emphasized however that their common asymptotic distribution is a mixture of chi square distributions under the null hypothesis

on the rhs description of resonances and virtual states we discuss the role of the m ller wave operators in a rigged hilbert space rhs model for resonances we also construct a new rhs of analytic functions valid for the description of resonances and virtual states these become generalized state vectors in a general physically realizable representation

nonparametric maximum likelihood estimation of survival functions with a general stochastic ordering and its dual summary this paper discusses estimation of survival functions under an arbitrary partial stochas tic ordering of the underlying populations this technique is especially useful when data are not easily acquired from one or more of the populations but data from the other populations are available estimation of the ordered survival functions entails use of a fenchel duality theorem that results in an algorithm to identify and resolve violations of the underlying ordering the algorithm produces estimates which converge to the maximum likelihood estimates of the survival functions given the underlying ordering the method is illustrated by an example in life testing of components and in clinical trials estimation of a survival function p t is often of prime interest if the functional form of the survival function is unknown nonparametric methods such as the product limit estimator kaplan meier 1958 are needed johansen 1978 showed that the product limit estimator is a maximum likelihood estimator in the class of all distributions under a generalized maximum likelihood framework kiefer wolfowitz 1956 since the generalized maximum likelihood esti mates for continuous survival functions place probability only on those time points at which observations occur suppose however that one desires to estimate a number of survival functions when a partial stochastic ordering exists among them and under censoring for example in a clinical trial of two treatments for a rare debilitating disease let pi p2 and p3 be the survival functions respectively for the disease free group and for the disease groups under the two treatments the well documented survival function p1 can be used to generate better estimates for p2 and p3 by using the order restrictions pi t p2 t and pi t p3 t brunk et al 1966 developed the maximum likelihood estimators for two stochastically ordered distributions for uncensored independent random samples dykstra

bounds on mixtures of distributions arising in order restricted inference in testing hypotheses involving order restrictions on a collection of parameters distributions arise which are mixtures of standard distributions since tractable expressions for the mixing proportions generally do not exist even for parameter collections of moderate size the implementation of these tests may be difficult stochastic upper and lower bounds are obtained for such test statistics in a variety of these kinds of problems these bounds are also shown to be tight the tightness results point out some situations in which the bounds could be used to obtain approximate methods these results can also be applied to obtain the least favorable configutiation when testing the equality of two multinomial populations versus a stochastic ordering alternative

maximum likelihood estimation of the survival functions of stochastically ordered random variables abstract many times aspects of populations exist that must satisfy a stochastic ordering requirement nevertheless estimates may not bear out this stochastic ordering because of the inherent variability of the observations this article will consider the problem of finding maximum likelihood estimates of stochastically ordered survival functions for the cases a one survival function being fixed in advance and b estimating two survival functions when the data include censored observations a numerical example is discussed in detail to illustrate the solution to this problem

cone contraction and reference point methods for multi criteria mixed integer optimization abstract interactive approaches employing cone contraction for multi criteria mixed integer optimization are introduced in each iteration the decision maker dm is asked to give a reference point new aspiration levels the subsequent pareto optimal point is the reference point projected on the set of admissible objective vectors using a suitable scalarizing function thereby the procedures solve a sequence of optimization problems with integer variables in such a process the dm provides additional preference information via pair wise comparisons of pareto optimal points identified using such preference information and assuming a quasiconcave and non decreasing value function of the dm we restrict the set of admissible objective vectors by excluding subsets which cannot improve over the solutions already found the procedures terminate if all pareto optimal solutions have been either generated or excluded in this case the best pareto point found is an optimal solution such convergence is expected in the special case of pure integer optimization indeed numerical simulation tests with multi criteria facility location models and knapsack problems indicate reasonably fast convergence in particular under a linear value function we also propose a procedure to test whether or not a solution is a supported pareto point optimal under some linear value function

the rigged hilbert space approach to the lippmann schwinger equation ii the analytic continuation of the lippmann schwinger bras and kets the analytic continuation of the lippmann schwinger bras and kets is obtained and characterized it is shown that the natural mathematical setting for the analytic continuation of the solutions of the lippmann schwinger equation is the rigged hilbert space rather than just the hilbert space it is also argued that this analytic continuation entails the imposition of a time asymmetric boundary condition upon the group time evolution resulting in a semigroup time evolution physically the semigroup time evolution is simply a retarded or advanced propagator

reply to comment on on the inconsistency of the bohm gadella theory with quantum mechanics in this reply we show that when we apply standard distribution theory to the lippmann schwinger equation the resulting spaces of test functions would comply with the hardy axiom only if classic results of paley and wiener of gelfand and shilov and of the theory of ultradistributions were wrong also we point out several differences between the standard method of constructing rigged hilbert spaces in quantum mechanics and the method used in time asymmetric quantum theory

solving the discrete multiple criteria problem using convex cones an interactive method employing pairwise comparisons of attainable solutions is developed for solving the discrete deterministic multiple criteria problem assuming a single decision maker who has an implicit quasi concave increasing utility or value function the method chooses an arbitrary set of positive multipliers to generate a proxy composite linear objective function which is then maximized over the set of solutions the maximizing solution is compared with several solutions using pairwise judgments asked of the decision maker responses are used to eliminate alternatives using convex cones based on expressed preferences and then a new set of weights is found that satisfies the indicated preferences the requisite theory and proofs as well as a detailed numerical example are included in addition the results of some computational experiments to test the effectiveness of the method are described

on the inconsistency of the bohm gadella theory with quantum mechanics the bohm gadella theory sometimes referred to as the time asymmetric quantum theory of scattering and decay is based on the hardy axiom the hardy axiom asserts that the solutions of the lippmann schwinger equation are functionals over spaces of hardy functions the preparation registration arrow of time provides the physical justification for the hardy axiom in this paper it is shown that the hardy axiom is incorrect because the solutions of the lippmann schwinger equation do not act on spaces of hardy functions it is also shown that the derivation of the preparation registration arrow of time is flawed thus hardy functions neither appear when we solve the lippmann schwinger equation nor should they appear it is also shown that the bohm gadella theory does not rest on the same physical principles as quantum mechanics and that it does not solve any problem that quantum mechanics cannot solve the bohm gadella theory must therefore be abandoned

a likelihood ratio test against stochastic ordering in several populations abstract the likelihood ratio test is often used to test hypotheses involving a stochastic ordering distribution theory for the likelihood ratio test has been developed only for two stochastically ordered distributions for testing equality of distributions against a stochastic ordering in several populations this paper derives the null asymptotic distribution of the likelihood ratio test statistic which is characterized by minimization problems and has no closed form a monte carlo simulation is conducted to study the limiting distribution because the limiting distribution depends on the specific values of the unknown distributions under the null hypothesis asymptotic and bootstrap approaches are proposed to overcome practical difficulties and implement tests based on the likelihood principle asymptotic validities for these tests are established and simulations are carried out to check their performances for finite sample sizes the tests are applied to an example involving data for survival time f

the rigged hilbert space approach to the lippmann schwinger equation part i we exemplify the way the rigged hilbert space deals with the lippmann schwinger equation by way of the spherical shell potential we explicitly construct the lippmann schwinger bras and kets along with their energy representation their time evolution and the rigged hilbert spaces to which they belong it will be concluded that the natural setting for the solutions of the lippmann schwinger equation and therefore for scattering theory is the rigged hilbert space rather than just the hilbert space

a review of interactive methods for multiobjective integer and mixed integer programming this paper makes a review of interactive methods devoted to multiobjective integer and mixed integer programming moip momip problems the basic concepts concerning the characterization of the non dominated solution set are first introduced followed by a remark about non interactive methods vs interactive methods then we focus on interactive moip momip methods including their characterization according to the type of preference information required from the decision maker the computing process used to determine non dominated solutions and the interactive protocol used to communicate with the decision maker we try to draw out some contrasts and similarities of the different types of methods

wald criteria for jointly testing equality and inequality restrictions on montre comment le test de wald s applique a des ensembles de restrictions d egalite et d inegalite et on presente la distribution pour de grands echantillons du test sous l hypothese nulle

the local nature of hypothesis tests involving inequality constraints in nonlinear models this paper derives several properties unique to nonlinear model hypothesis testing problems involving linear or nonlinear inequality constraints in the null or alternative hypothesis the paper is organized around a lemma which characterizes the set containing the least favorable parameter value for a nonlinear model inequality constraints hypothesis test we then present two examples which illustrate several implications of this lemma we also discuss the impact of these properties on the empirical implementation and interpretation of these test procedures

nonparametric estimation of convex models via mixtures we present a general approach to estimating probability measures constrained to lie in a convex set we represent constrained measures as mixtures of simple known extreme measures and so the problem of estimating a constrained measure becomes one of estimating an unconstrained mixing measure convex constraints arise in many modeling situations such as estimation of the mean and estimation under stochastic ordering constraints we describe mixture representation techniques for these and other situations and discuss applications to maximum likelihood and bayesian estimation

a unified approach to likelihood inference on stochastic orderings in a nonparametric context abstract for data in a two way contingency table with ordered margins we consider various hypotheses of stochastic orders among the conditional distributions considered by rows and show that each is equivalent to requiring that an invertible transformation of the vectors of conditional row probabilities satisfies an appropriate set of linear inequalities this leads to the construction of a general algorithm for maximum likelihood estimation under multinomial sampling and provides a simple framework for deriving the asymptotic distribution of log likelihood ratio tests the usual stochastic ordering and the so called uniform and likelihood ratio orderings are considered as special cases in particular for each of these three orderings we determine the transformation required to apply the estimation algorithm we then consider testing the hypothesis that the rows are identically distributed against the alternative that they are stochastically ordered as well as testing each stochastic order against an u

using cutting planes in an interactive reference point approach for multiobjective integer linear programming problems we propose an interactive approach for multiple objective integer linear programming moilp problems that combines the use of the tchebycheff metric with cutting plane techniques at each interaction the method computes the nondominated solution for the moilp problem that is closest to a reference point according to the tchebycheff metric the information provided by the decision maker in each dialogue phase is used to adjust the next reference point through a sensitivity analysis stage cutting plane techniques enable the method to take advantage of computations performed at previous iterations to solve the next scalarizing integer program we address both theoretical issues and the computational implementation

comment on on the inconsistency of the bohm gadella theory with quantum mechanics rigged hilbert spaces of hardy functions lead to a consistent theory of resonance scattering and decay contrary to the claims of a recent article 8 the theory holds for a wide range of potentials and rigorously describes the asymmetric time evolution of resonances

an algorithm for restricted least squares regression abstract a commonly occurring problem in statistics is that of minimizing a least squares expression subject to side constraints here a simple iterative algorithm is presented and shown to converge to the desired solution several examples are presented including finding the closest concave convex function to a set of points and other general quadratic programming problems the dual problem to the basic problem is also discussed and a solution for it is given in terms of the algorithm finally extensions to expressions other than least squares are given

asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints summary the analysis of moment structural models has become an important tool of investigation in behavioural educational and economic studies the chi squared largesample test is routinely employed to assess the goodness of fit of the model considered however in order to invoke the standard asymptotic distribution theory certain regularity conditions have to be met here we consider the case where the population value of the parameter vector is a boundary point of the feasible region we show that in this case the asymptotic distribution of test statistic is a mixture of chi squared distributions the problem of finding the corresponding weights is discussed

statistical inference for uniform stochastic ordering in several populations tion for k population problems under uniform stochastic ordering restrictions we derive closed form estimates even with right censored data by a reparameterization which reduces the problem to a well known isotonic regression problem we also derive the asymptotic distribution of the likelihood ratio statistic for testing equality of the k populations against the uniform stochastic ordering restriction this asymptotic distribution is of the chi bar square type as discussed by robertson wright and dykstra these distributional results are obtained by appealing to elegant results from empirical process theory and showing that the proposed test is asymptotically distribution free recurrence formulas are derived for the weights of the chi bar square distribution for particular cases the theory developed in this paper is illustrated by an example involving data for survival times for carcinoma of the oropharynx

the rigged hilbert space approach to the gamow states we use the resonances of the spherical shell potential to present a thorough description of the gamow quasinormal states within the rigged hilbert space it will be concluded that the natural setting for the gamow states is a rigged hilbert space whose test functions fall off at infinity faster than gaussians

on computing the trace of the kernel of the homogeneous fredholm s equation a method for computing the trace of the kernel of the homogeneous fredholm s equation for resonant states arising from nonlocal potentials is proposed we show that this integral formulation is convergent

the lippmann schwinger equations in the rigged hilbert space we give a rigorous meaning to the lippmann schwinger equations relating generalized eigenvectors of the free and perturbed hamiltonians in a scattering process some related formulae are also presented a discussion is made on the context of rigged hilbert space formulation of quantum mechanics

inference for likelihood ratio ordering in the two sample problem abstract we obtain the maximum likelihood estimators of two multinomial probability vectors under the constraint that they are likelihood ratio ordered we extend this estimation approach to the case of two univariate distributions and show strong consistency of the estimators we also derive and study the asymptotic distribution of the likelihood ratio statistic for testing the equality of two discrete probability distributions against the alternative that one distribution is greater than the other in the likelihood ratio ordering sense finally we examine a data set pertaining to average daily insulin dose from the boston collaborative drug surveillance program and compare our testing procedure to testing procedures for other stochastic orderings