optimal sample allocation for normal discrimination and logistic regression under stratified sampling abstract for two multivariate normal populations with a common covariance matrix and stratified sampling we consider two methods of estimation fisher s linear discriminant function and logistic regression intuition suggests that taking half of the observations from each population is a reasonable design choice based on minimizing the expected error regret asymptotic optimal sample allocations are found the results indicate that the differences in the expected error regret for optimal versus balanced allocation are generally quite small it is recommended that equal sample sizes for the two populations be used for these problems

the segal bargmann coherent state transform for compact lie groups abstract let k be an arbitrary compact connected lie group we describe on k an analog of the segal bargmann coherent state transform and we prove theorem 1 that this generalized coherent state transform maps l 2 k isometrically onto the space of holomorphic functions in l 2 g where g is the complexification of k and where is an appropriate heat kernel measure on g the generalized coherent state transform is defined in terms of the heat kernel on the compact group k and its analytic continuation to the complex group g we also define a k averaged version of the coherent state transform and we prove a similar result for it theorem 2 in the appendix we describe the coherent states which motivate the definitions of these transforms in section 9 we obtain inversion formulas for both the coherent state transform and the k averaged coherent state transform theorems 3 4 as a consequence we obtain formulas for certain bergman kernels on the complex group g theorems 5 6 in section 10 we derive a general paley wiener theorem for k and we exhibit a family of convolution transforms each of which is an isometric isomorphism of l 2 k onto a certain l 2 space of holomorphic functions on g these transforms include the k averaged coherent state transform as a special case we also discuss the analogous results for r 1 the results of section 10 are theorems 7 10 finally in section 11 we discuss the case of quotient spaces we prove theorem 11 that both the coherent state transform and the k averaged coherent state transform pass in a natural way from k to k h where h is a closed connected subgroup of k

forecasting technological substitutions with concurrent short time series abstract it has been shown in the literature that the data of technological substitutions exhibit a strong correlation across different time periods that is a strong serial correlation significant improvement in predicting such substitutions has been achieved by incorporating serial correlation and power transformation parameters into four growth curve models the modified models or data based transformed models however break down when the number of time points is small this article proposes a generalized growth curve model for forecasting technological substitutions with concurrent short time series the model combines the concepts of power transformations and repeated measurements with a common serial covariance structure concurrent time series for several cases provide the repeated measurement requirement of the model improvement in forecasting accuracy by using this model is demonstrated with a set of telephone switching data

sample reuse procedures for prediction of the unobserved portion of a partially observed vector summary predictive sample reuse methods are applied in several ways to the problem of supplying values for the unobserved portion of a vector when n other similar but fully observed vectors are in hand one method is to use a mixing matrix to combine two independently calculated vector predictors into a single predictor the other is to use regression techniques with varying constraints on the regression matrix the methods are applied to two data sets and for one of them compared with previous prediction analyses

analysis of drug dissolution data drug absorption in the human body depends on the dissolution rate of the drug suitable dissolution characteristics are important to ensure that the drug will achieve the desired therapeutic effects to assess the similarity of dissolution rates of several drug lots we apply a general growth curve model with different covariance structures the box cox power transformation and the naive log transformation are applied to a function of the dissolution rate the predictive sample reuse or cross validation method is employed in selecting an appropriate model with best predictive accuracy a testing procedure for examining the similarity among the drug lots is also conducted a partially bayesian approach is used for the assessment of dissolution equivalence

the segal bargmann transform for the heat equation associated with root systems we study the heat equation associated to a multiplicity function on a root system where the corresponding laplace operator has been defined by heckman and opdam in particular we describe the image of the associated segal bargmann transform as a space of holomorphic functions in the case where the multiplicity function corresponds to a riemannian symmetric space g k of non compact type we obtain a description of the image of the space of k invariant l2 function on g k under the segal bargmann transform associated to the heat equation on g k thus generalizing and reproving a result of b hall and j mitchell for spaces of complex type

segal bargmann transforms associated with finite coxeter groups using a polarization of a suitable restriction map and heat kernel analysis we construct a generalized segal bargmann transform associated with every finite coxeter group g on n we find the integral representation of this transform and we prove its unitarity to define the segal bargmann transform we introduce a hilbert space open image in new window of holomorphic functions on open image in new window with reproducing kernel equal to the dunkl kernel the definition and properties of open image in new window extend naturally those of the well known classical fock space the generalized segal bargmann transform allows to exhibit some relationships between the dunkl theory in the schrodinger model and in the fock model further we prove a branching decomposition of open image in new window as a unitary open image in new window module and a general version of hecke s formula for the dunkl transform

tests and model selection for the general growth curve model the model considered here is a generalized multivariate analysis of variance model useful especially for many types of growth curve problems including biological growth and technology substitution it is defined as yp x n xp x m tau m x r ar x n epsilon p x n where tau is unknown and x and a are known design matrices of ranks m less than p and r less than n respectively furthermore the columns of epsilon are independent p variate normal with mean vector 0 and common covariance matrix sigma in general p is the number of time or spatial points observed on each of the n cases m 1 is the degree of polynomial in time and r is the number of groups the main focus of this paper is the selection of models for the general growth curve model with regard to the covariance matrix sigma likelihood ratio tests and selection procedures based on sample reuse and predictions are proposed special emphasis is on the serial covariance structure for sigma which has been shown to be quite important in the prediction of biological data and technology substitution data one population and k population problems are considered some of the results are illustrated with two sets of biological data

large amplitude free vibrations of a mass grounded by linear and nonlinear springs in series this communication is concerned with the large amplitude free vibration of a mass grounded by linear and nonlinear springs in series based on a single equation of motion in terms of relative displacement variable a qualitative analysis is completed and some new and interesting dynamic behaviors are discovered these behaviors include oscillations in asymmetric single potential wells and existence of asymptotes in phase plane for the case of softening springs the ranges of oscillations are determined and expressions of exact periods for symmetric and asymmetric oscillations are established furthermore the construction of analytical approximations to period and periodic solution is briefly described

estimation of the area under the roc curve the area under the receiver operating characteristic curve is frequently used as a measure for the effectiveness of diagnostic markers in this paper we discuss and compare estimation procedures for this area these are based on i the mann whitney statistic ii kernel smoothing iii normal assumptions iv empirical transformations to normality these are compared in terms of bias and root mean square error in a large variety of situations by means of an extensive simulation study overall we find that transforming to normality usually is to be preferred except for bimodal cases where kernel methods can be effective

linear combinations of multiple diagnostic markers abstract the receiver operating characteristic roc curve is a simple and meaningful measure to assess the usefulness of diagnostic markers to use the information carried by multiple markers we note that fisher s linear discriminant function provides a linear combination of markers to maximize the sensitivity over the entire specificity range uniformly under the multivariate normal distribution model with proportional covariance matrices with no restriction on covariance matrices we also provide a solution of the best linear combination of markers in the sense that the area under the roc curve of this combination is maximized among all possible linear combinations we illustrate both situations discussed in the article with a cancer clinical trial data

on segal bargmann analysis for finite coxeter groups and its heat kernel we prove identities involving the integral kernels of three versions two being introduced here of the segal bargmann transform associated to a finite coxeter group acting on a finite dimensional real euclidean space the first version essentially having been introduced around the same time by ben said and orsted and independently by soltani and the dunkl heat kernel due to rosler of the dunkl laplacian associated with the same coxeter group all but one of our relations are originally due to hall in the context of standard segal bargmann analysis on euclidean space hall s results trivial dunkl structure and arbitrary finite dimension as well as our own results in deformed quantum mechanics non trivial dunkl structure dimension one are particular cases of the results proved here so we can understand all of these versions of the segal bargmann transform associated to a coxeter group as hall type transforms in particular we define an analogue of hall s version c generalized segal bargmann transform which is then shown to be dunkl convolution with the dunkl heat kernel followed by analytic continuation in the context of version c we also introduce a new segal bargmann space and a new transform associated to the dunkl theory also we have what appears to be a new relation in this context between the segal bargmann kernels for versions a and c

free vibrations of a mass grounded by linear and nonlinear springs in series abstract in many technical applications spring like flexible elements or real springs connected in series are used the operation range of these components determines whether the system behaviour has a linear or nonlinear characteristic in the relevant literature there exists the knowledge how the equivalent spring is obtained for linear springs connected serially however some cases occur in which one linear and one nonlinear spring arranged in series are used in such cases it is not possible to define an equivalent spring rate this study is concerned with such a system that consists of a mass grounded two springs one of them linear and the other nonlinear two methods are developed to analyze the dynamic behaviour of system one method makes use of a set of differential algebraic equations dae in short the other is based on getting a single equation of motion using relative displacement variables for the second method analytical solutions are also obtained by means of the lindstedt and the harmonic balance techniques it is observed that numerical and analytical solutions found for both methods are in very good agreement when v 0 1 and 0 1 10 where v 0 is the initial deflection of nonlinear spring and is the ratio of linear portion coefficient of the nonlinear spring to that of the linear spring

prediction and estimation of growth curves with special covariance structures abstract the model considered here is a generalized multivariate analysis of variance model useful especially for many types of growth curve problems including biological growth and technology substitutions it is defined as yp n xp m m rar n p n where is unknown and x and a are known design matrices of ranks m p and r n respectively further the columns of are independent p variate normal with mean vector 0 and common covariance matrix in general p is the number of time or spatial points observed on each of the n cases m 1 is the degree of polynomial and r is the number of groups this article focuses mainly on predicting future observations including partially observed vectors and future values on a given number of cases using their past observations these prediction problems as well as the estimation of parameters are considered for two parsimonious covariance structures that are very important in practice the results are illustrated with three real data sets som

bayesian multivariate hierarchical transformation models for roc analysis a bayesian multivariate hierarchical transformation model bmhtm is developed for receiver operating characteristic roc curve analysis based on clustered continuous diagnostic outcome data with covariates two special features of this model are that it incorporates non linear monotone transformations of the outcomes and that multiple correlated outcomes may be analysed the mean variance and transformation components are all modelled parametrically enabling a wide range of inferences the general framework is illustrated by focusing on two problems 1 analysis of the diagnostic accuracy of a covariate dependent univariate test outcome requiring a box cox transformation within each cluster to map the test outcomes to a common family of distributions 2 development of an optimal composite diagnostic test using multivariate clustered outcome data in the second problem the composite test is estimated using discriminant function analysis and compared to the test derived from logistic regression analysis where the gold standard is a binary outcome the proposed methodology is illustrated on prostate cancer biopsy data from a multi centre clinical trial

assessment and propagation of model uncertainty in most examples of inference and prediction the expression of uncertainty about unknown quantities y on the basis of known quantities x is based on a model m that formalizes assumptions about how x and y are related m will typically have two parts structural assumptions s such as the form of the link function and the choice of error distribution in a generalized linear model and parameters whose meaning is specific to a given choice of s it is common in statistical theory and practice to acknowledge parametric uncertainty about given a particular assumed structure s it is less common to acknowledge structural uncertainty about s itself a widely used approach involves enlisting the aid of x to specify a plausible single best choice s for s and then proceeding as if s were known to be correct in general this approach fails to assess and propagate structural uncertainty fully and may lead to miscalibrated uncertainty assessments about y given x when miscalibration occurs it will often result in understatement of inferential or predictive uncertainty about y leading to inaccurate scientific summaries and overconfident decisions that do not incorporate sufficient hedging against uncertainty in this paper i discuss a bayesian approach to solving this problem that has long been available in principle but is only now becoming routinely feasible by virtue of recent computational advances and examine its implementation in examples that involve forecasting the price of oil and estimating the chance of catastrophic failure of the us space shuttle

generalized hermite polynomials and the heat equation for dunkl operators based on the theory of dunkl operators this paper presents a general concept of multivariable hermite polynomials and hermite functions which are associated with finite reflection groups on n the definition and properties of these generalized hermite systems extend naturally those of their classical counterparts partial derivatives and the usual exponential kernel are here replaced by dunkl operators and the generalized exponential kernel k of the dunkl transform in the case of the symmetric group s n our setting includes the polynomial eigenfunctions of certain calogero sutherland type operators the second part of this paper is devoted to the heat equation associated with dunkl s laplacian as in the classical case the corresponding cauchy problem is governed by a positive one parameter semigroup this is assured by a maximum principle for the generalized laplacian the explicit solution to the cauchy problem involves again the kernel k which is on the way proven to be nonnegative for real arguments

an interpretation for the roc curve and inference using glm procedures the accuracy of a medical diagnostic test is often summarized in a receiver operating characteristic roc curve this paper puts forth an interpretation for each point on the roc curve as being a conditional probability of a test result from a random diseased subject exceeding that from a random nondiseased subject this interpretation gives rise to new methods for making inference about roc curves it is shown that inference can be achieved with binary regression techniques applied to indicator variables constructed from pairs of test results one component of the pair being from a diseased subject and the other from a nondiseased subject within the generalized linear model glm binary regression framework roc curves can be estimated and we highlight a new semiparametric estimator covariate effects can also be evaluated with the glm models the methodology is applied to a pancreatic cancer dataset where we use the regression framework to compare two different serum biomarkers asymptotic distribution theory is developed to facilitate inference and to provide insight into factors influencing variability of estimated model parameters

some new examples of markov processes which enjoy the time inversion property in this paper we give a sufficient condition on the semi group densities of an homogeneous markov process taking values in n which ensures that it enjoys the time inversion property our condition covers all previously known examples of markov processes satisfying this property as new examples we present a class of markov processes with jumps the dunkl processes and their radial parts

accurate approximate analytical solutions for nonlinear free vibration of systems with serial linear and nonlinear stiffness this paper deals with free vibration of a nonlinear system having combined linear and nonlinear springs in series the conservative oscillation system is formulated as a nonlinear ordinary differential equation having linear and nonlinear stiffness components the governing equation is linearized and associated with the harmonic balance method to establish new and accurate higher order analytical approximate solutions unlike the perturbation method which is restricted to nonlinear conservative systems with a small perturbed parameter and also unlike the classical harmonic balance method which results in a complicated set of algebraic equations the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation some examples are solved and compared with numerical integration solutions and published results new solutions to the nonlinear systems are also presented and discussed

nonparametric and semiparametric estimation of the receiver operating characteristic curve curve odc given by f g t first we consider nonparametric estimators based on empirical distribution functions and derive asymptotic properties next we consider the so called semiparametric binormal model in which it is assumed that the distributions f and g are normal after some unknown monotonic transformation of the measurement scale for this model we propose a generalized least squares procedure and compare it with the estimation algorithm of dorfman and alf which is based on grouped data asymptotic results are obtained small sample properties are examined via a simulation study finally we describe a minimum distance estimator for the roc curve which does not require grouping the data