hypothesis testing in a mixture case control model summary we consider a problem of testing mixture proportions using two sample data one from group one and the other from a mixture of groups one and two with unknown proportion for being in group two various statistical applications including microarray study infectious epidemiological studies case control studies with contaminated controls clinical trials allowing nonresponders genetic studies for gene mutation and fishery applications can be formulated in this setup under the assumption that the log ratio of probability density functions from the two groups is linear in the observations we propose a generalized score test statistic to test the mixture proportion under some regularity conditions it is shown that this statistic converges to a weighted chi squared random variable under the null hypothesis of 0 where the weight depends only on the sampling fraction of both groups the permutation method is used to provide more reliable finite sample approximation simulation results and two real data applications are presented

product integration in the presence of a singularity this paper considers the approximate evaluation of int a b k x f x dx where k is a fixed lebesgue integrable function by quadrature rules of the form sigma i 0 m n w in f x in normally rules of this kind are used only for smooth functions f and any singularities are incorporated into k here however we allow f to have a singularity either at an endpoint or in the interior general convergence properties are established for a wide class of product integration rules more detailed results are established for the class of rules which are exact if f belongs to a specified family of piecewise polynomials

on the convergence of an interpolatory product rule for evaluating cauchy principal value integrals etablissement des theoremes de convergence des regles de produits interpolatoires pour l evaluation d integrales singulieres de cauchy avec estimation asymptotique du reste amelioration et generalisation de resultats anterieurs

robustness in the presence of mixed parametric uncertainty and unmodeled dynamics continuing the development of the structured singular value approach to robust control design the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty the problem is shown to be equivalent to a smooth constrained finite dimensional optimization problem in view of the fact that the functional to be maximized may have several local extrema an upper bound on mu whose computation is numerically tractable is established this leads to a sufficient condition of robust stability and performance a historical perspective on the development of the mu theory is included

numerical integration in the presence of an interior singularity abstract the convergence of sequences of integration rules applied to an improperly integrable function with an interior singularity is investigated the rules studied include the gauss jacobi rules and the interpolatory integration rules based on the zeros of 1 x 2 m p n 2m x m 0 1 for certain values of and the results are then applied to the study of the convergence of hunter s method for cauchy principal value integrals

interpolary product integration in the presence of singularities l 2 theory we study interpolatory product integration rules based on gauss radau and lobatto points with respect to a generalized jacobi weight function as applied to the integration of functions with singularities we give sufficient conditions for convergence of such rules for both endpoint and interior singularities

generalized mixed μ bounds for real and complex multiple block uncertainty with internal matrix structure new absolute stability results that unify and extend existing structured singular value bounds for mixed uncertainty are developed using frequency domain arguments these bounds generalize prior upper bounds for mixed by permitting the treatment of non diagonal real uncertain blocks as well as accounting for internal matrix structure in the uncertainty several specializations to well known absolute stability criteria from the classical literature are given which allow for a systematic comparison of these results in addressing the problem of robust stability for real parameter uncertainty

asymptotics for the remainder of a class of positive quadratures for integrands with an interior singularity for functions with an interior singularity the errors of a class of positive quadrature formulae with high algebraic degree are reduced to those of the much simpler euler maclaurin type formulae applying this method to certain classes of functions such as for example f x h x x u where 1 with a sufficiently smooth functionh we obtain the main term of the error expansion for quadrature rules of ultraspherical type

efficient nonparametric estimation of causal effects in randomized trials with noncompliance causal approaches based on the potential outcome framework provide a useful tool for addressing noncompliance problems in randomized trials we propose a new estimator of causal treatment effects in randomized clinical trials with noncompliance we use the empirical likelihood approach to construct a profile random sieve likelihood and take into account the mixture structure in outcome distributions so that our estimator is robust to parametric distribution assumptions and provides substantial finite sample efficiency gains over the standard instrumental variable estimator our estimator is asymptotically equivalent to the standard instrumental variable estimator and it can be applied to outcome variables with a continuous ordinal or binary scale we apply our method to data from a randomized trial of an intervention to improve the treatment of depression among depressed elderly patients in primary care practices

properties of interpolatory product integration rules this paper is concerned with the approximate evaluation of int 1 1 k x f x d x where k is lebesgue integrable and f is at least riemann integrable and preferably smooth the integral is approximated by a rule of the form sum nolimits i 1 n w ni k f x ni where the points x ni are chosen in some prescribed way and the weights w ni k are such that the rule is exact if f is any polynomial of degree n for suitable choices of the points and suitable functions k the rule is shown to converge to the exact integral and the companion rule sum w ni k f x ni to int k x f x dx for all riemann integrable functions f the companion property ensures for example that the weights have the asymptotic positivity property if k is positive error bounds are given that guarantee rapid convergence of the rule if f is smooth finally by establishing a connection with mean convergence of lagrangian interpolation some negative theorems are proved in pa

rates of convergence of gaussian quadrature for singular integrands the authors obtain the rates of convergence or divergence of gaussian quadrature on functions with an algebraic or logarithmic singularity inside or at an endpoint of the interval of integration a typical result is the following for a bounded smooth weight function on 1 1 the error in n point gaussian quadrature of f x ix y1 1 is o n 2 28 if y 1 and o n if y e 1 1 provided we avoid the singularity if we ignore the singularityy the error is o n1 23 log n 8 loglog n 8t 1 for almost all choices of y these assertions are sharp with respect to order

efficient nonparametric estimation of mixture proportions summary by constructing a sequence of multinomial approximations and related maximum likelihood estimators we derive a cramer rao lower bound for nonparametric estimators of the mixture proportions and thereby characterize asymptotically optimal estimators for the case of the sampling model m2 of hosmer 1973 it is shown that the sequence of maximum likelihood estimators which can be obtained explicitly is asymptotically optimal in this sense the results hold true even when the multinomial approximations involve cells chosen adaptively from the data in a wellspecified way

numerical integration based on approximating splines abstract convergence results are proved for product integration rules based on approximating splines these results are both for bounded and unbounded integrands pointwise and uniform convergence results are proved for sequences of cauchy principal values of these approximating splines

on the evaluation of cauchy principal value integrals by rules based on quasi interpolating splines abstract in this paper we consider the numerical evaluation of one dimensional cauchy principal value integrals of the form 1 1 k x f x x d x 1 by rules obtained by subtracting out the singularity and then applying quadrature formulas based on quasi interpolating splines

on the convergence of closed interpolatory integration rules based on the zeros of gegenbauer polynomials abstract the gegenbauer polynomials c n x are orthogonal with respect to the weight function 1 x2 1 2 it is known that the interpolatory integration rules based on the zeros of 1 x2 c n 2 x converge for all riemann integrable functions for 1 2 4 this is shown to hold also for 1 2 1 2

numerical evaluation of cauchy principal value integrals with singular integrands convergence results are proved for sequences of interpolatory integration rules for cauchy principal value integrals of the form fk x f x x a dx 1 a 1 when f x is singular at a point a 7 i and the singularity is ignored or avoided

mean convergence of hermite fejer interpolation abstract weighted l p convergence of hermite fejer interpolation based on the zeros of generalized jacobi polynomials is investigated the main result of the paper gives necessary and sufficient conditions for such convergence for all continuous functions

a semiparametric two component compound mixture model and its application to estimating malaria attributable fractions summary malaria remains a major epidemiologic problem in many developing countries malaria is defined as the presence of parasites and symptoms usually fever due to the parasites in endemic areas an individual may have symptoms attributable either to malaria or to other causes from a clinical viewpoint it is important to correctly diagnose an individual who has developed symptoms so that the appropriate treatments can be given from an epidemiologic and economic viewpoint it is important to determine the proportion of malaria affected cases in individuals who have symptoms so that policies on intervention program can be developed once symptoms have developed in an individual the diagnosis of malaria can be based on the analysis of the parasite levels in blood samples however even a blood test is not conclusive as in endemic areas many healthy individuals can have parasites in their blood slides therefore data from this type of study can be viewed as coming from a mixture distribution with the components corresponding to malaria and nonmalaria cases a unique feature in this type of data however is the fact that a proportion of the nonmalaria cases have zero parasite levels therefore one of the component distributions is itself a mixture distribution in this article we propose a semiparametric likelihood approach for estimating the proportion of clinical malaria using parasite level data from a group of individuals with symptoms our approach assumes the density ratio for the parasite levels in clinical malaria and nonclinical malaria cases can be modeled using a logistic model we use empirical likelihood to combine the zero and nonzero data the maximum semiparametric likelihood estimate is more efficient than existing nonparametric estimates using only the frequencies of zero and nonzero data on the other hand it is more robust than a fully parametric maximum likelihood estimate that assumes a parametric model for the nonzero data simulation results show that the performance of the proposed method is satisfactory the proposed method is used to analyze data from a malaria survey carried out in tanzania

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

product integration of singular integrands based on cubic spline interpolation at equally spaced nodes in this paper the convergence of product integration rules based on cubic spline interpolation at equally spaced nodes with not a knot end condition is investigated for integrand functions with a interior or endpoint singularity in the integration interval

rates of convergence of gauss lobattom and radau integration rules for singular integrands rates of convergence or divergence are obtained in the application of gauss lobatto and radau integration rules to functions with an algebraic or logarithmic singularity inside or at an endpoint of the interval of integration a typical result is the following for a generalized jacobi weight function on 1 1 the error in applying an n point rule to f x ix vi 6 is o n 2 28 if y 1 and o n 1 6 if y e 1 1 providedwe avoid the singularity if we ignore the singularity y the error is o n 1 28 log n 6 log log n 1 e for almost all choices of y these assertions are sharp with respect to order 1 introduction this paper is a sequel to that of lubinsky and myself 2 on the rates of convergence of gauss integration rules for singular integrands in this paper we extend the results in 2 in several directions on the one hand we extend the results for gauss integration rules to results for lobatto and radau rules this extension follows easily from the generalized markov stieltjes inequality for lobatto and radau rules given in lemma 3 2 and from a representation of the coefficients or weights or cotes numbers in the lobatto and radau rules in terms of the coefficients in the gauss rule with respect to a related weight function a second extension is to the special case of gauss jacobi integration rules as well as lobatto jacobi and radau jacobi rules for certain values of the parameters defining the jacobi weight function

numerical evaluation of cauchy principal value integrals based on local spline approximation operators abstract in this paper a local spline approximation method defined for any function l 1 1 1 is applied to evaluate cauchy principal value integrals and convergence results with an error bound are established some comparisons with other methods and numerical examples are given

nonparametric estimation of component distributions in a multivariate mixture suppose k variate data are drawn from a mixture of two distributions each having independent components it is desired to estimate the univariate marginal distributions in each of the products as well as the mixing proportion this is the setting of two class fully parametrized latent models that has been proposed for estimating the distributions of medical test results when disease status is unavailable the problem is one of inference in a mixture of distributions without training data and until now it has been tackled only in a fully parametric setting we investigate the possibility of using nonparametric methods of course when k 1 the problem is not identifiable from a nonparametric viewpoint we show that the problem is almost identifiable when k 2 there the set of all possible representations can be expressed in terms of any one of those representations as a two parameter family furthermore it is proved that when k geq3 the problem is nonparametrically identifiable under particularly mild regularity conditions in this case we introduce root n consistent nonparametric estimators of the 2k univariate marginal distributions and the mixing proportion finite sample and asymptotic properties of the estimators are described

pairwise rank based likelihood for estimation and inference on the mixture proportion summary we consider the problem of estimation and inference on the mixture parameter in the two sample problem when sample data from the two distributions as well as from a third population consisting of a mixture of the two are used under a general nonparametric model where the relationship between the two populations is unspecified we develop a pairwise rank based likelihood simultaneous inference on the mixture proportion and a parameter representing the probability an observation from one population is greater than an observation from the other population is based on this likelihood under some regularity conditions it is shown that the maximum pairwise rank likelihood estimator is consistent and has an asymptotic normal distribution simulation results indicate that the performance of this statistic is satisfactory the methodology is demonstrated on a data set in prostate cancer

structured matrix norms for real and complex block structured uncertainty abstract new upper bounds for robust stability are developed for block structured real and complex uncertainty involving arbitrary spatial norms specifically norm bounded block structured uncertainty is considered wherein the defining norm is not necessarily the maximum singular value in the case where the defining norm on the uncertainty characterization is set equal to the spectral norm the resulting upper bounds generalize upper bounds for mixed by permitting the treatment of nondiagonal real uncertain blocks as well as accounting for internal matrix structure in the uncertainty the overall framework provides for a considerably simplified proof of the standard mixed bound

bounds on generalized structured singular values via the perron root of matrix majorants abstract the purpose of this paper is to present several bounds upon the structured singular value we first adopt a generalized notion of the structured singular value which is useful for problems where uncertainties are assumed to be bounded in an l p induced matrix norm two different type of bounds in terms of perron root and interaction parameters respectively are given for the new structured singular value and their relations are discussed these bounds are useful in that they are easy to compute and may be further analyzed to provide insights useful in design