topological structure of solution sets to asymptotic boundary value problems abstract topological structure is investigated for second order vector asymptotic boundary value problems because of indicated obstructions the r structure is firstly studied for problems on compact intervals and then by means of the inverse limit method on non compact intervals the information about the structure is furthermore employed by virtue of a fixed point index technique in frechet spaces developed by ourselves earlier for obtaining an existence result for nonlinear asymptotic problems some illustrating examples are supplied

assessing the performance of an allocation rule the problem of estimating the error rates of a sample based rule on the basis of the same sample used in its construction is considered the apparent error rate is an obvious nonparametric estimate of the conditional error rate of a sample rule but unfortunately it provides too optimistic an assessment attention is focussed on the formation of improved estimates mainly through appropriate bias correction of the apparent error rate in this respect the role of the bootstrap a computer based methodology is highlighted

estimation of error rate for linear discriminant functions by resampling non gaussian populations abstract this article presents simulation results comparing various resampling estimators of classification error rate for linear discriminant type classification algorithms three non gaussian multivariate populations are studied namely exponential cauchy and uniform simulations are conducted for small sample sizes two class and three class problems and 2 d 3 d and 5 d distributions estimation procedures and sample sizes are the same as in our previous study of gaussian populations again 200 bootstrap replications are used for each simulation trial for exponential and uniform distributions the 0 632 estimator generally performs best however for cauchy distributions the convex bootstrap and the e 0 often outperform the 0 632 estimator

efficient bootstrap simulation on donne deux idees pour renforcer une simulation la premiere est d equilibrer les echantillons simules la seconde est de faire un usage explicite des approximations qui ne demandent pas de simulation

the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

nonparametric confidence and tolerance regions in canonical variate analysis canonical variate analysis is a much used descriptive multivariate technique for grouped data in which the group means are represented by points in a low typically two dimensional space if inferential use is to be made of the technique some knowledge of the variability of these points under repeated sampling is necessary the traditional approach is to construct confidence or tolerance circles around each point by using approximate asymptotic results based on normal assumptions for the data jackknife and bootstrap resampling schemes are proposed in this paper as a basis for constructing nonparametric regions these regions are compared with the normal based circles on the well known fisher iris data and also by means of a small monte carlo study incorporating nonnormal as well as normal data it is suggested that the normal based circles may often be inappropriate and areas of further investigation are identified

a note on the modified likelihood for density estimation abstract let f be a kernel estimate with window width of the density f its performance is assessed by the kullback leibler information distance i f f f log f f log f this article establishes conditions for the asymptotic equivalence of the cross validation estimate and the jackknife estimate of the term f log f and provides the common limiting value this gives insight into the modified likelihood criterion for choosing introduced by habbema hermans and van den broek 1974 and duin 1976

foundations for an algebraic theory of conditioning abstract we investigate some mathematical results in ring theory in order to lay down an algebraic framework for a theory of measure free conditional objects

estimation of allocation rates in a cluster analysis context abstract a sample of multivariate observations is assumed to be drawn from a mixture of a given number of underlying populations the mixture likelihood approach to clustering is used to allocate each individual in the sample to its population of origin on the basis of the estimated posterior probabilities of population membership estimation of the correct allocation rate is considered for individual populations as well as for the overall mixture by averaging functions of the maximum of these posterior probabilities the estimates of the correct allocation rates provide a means of assessing the performance of the mixture approach to clustering the bootstrap technique is investigated for its effectiveness in reducing the bias of the estimates so obtained results are reported for three real data sets and a simulation study it is demonstrated that the proposed estimates generally provide useful information on the unobservable allocation rates of the mixture approach encouraging results are obtained for

optimal mean square error calibration of classifier error estimators under bayesian models a recently proposed bayesian modeling framework for classification facilitates both the analysis and optimization of error estimation performance the bayesian error estimator is then defined to have optimal mean square error performance but in many situations closed form representations are unavailable and approximations may not be feasible to address this we present a method to optimally calibrate arbitrary error estimators for minimum mean square error performance within a supposed bayesian framework assuming a fixed sample size classification rule and error estimation rule as well as a fixed bayesian model the calibration is done by first computing a calibration function that maps error estimates to their optimally calibrated values off line once found this calibration function may be easily applied to error estimates on the fly whenever the assumptions apply we demonstrate that calibrated error estimators offer significant improvement in performance relative to classical error estimators under bayesian models with both linear and non linear classification rules

an algebraic synthesis of the foundations of logic and probability abstract this paper develops a unified algebraic theory of logic and probability based upon an initial boolean algebra logic of propositions together with its extensionally associated boolean algebra of models probabilistic events that satisfy the axioms of the boolean algebra of propositions a proposition may be true in some models and false in others but each model must unambiguously assign true or false to each proposition thus the examples models in which different propositions are true may be all some or none according as those propositions are necessary possible or impossible a probability measure p on the collection of models events induces a probability measure on the propositions the crucial if then relation is defined so that p if p then q equals p q p the conditional probability of q given p this conditional probability is less than the probability of the material conditional q or not p unless p p 1 or p q p 1 at these two extremes the two propositional constructions coalesce thus an additional relation q p is needed in logic and probability namely q given p this is accomplished in the algebraic logic generated by the various sum ideals entailments of various propositions that may serve as the explicit condition of a conditional proposition q p the corresponding relation b a in the model realm defines the notion of a conditional event in this context of algebraic logic and model theory a new algebraic structure called the conditional closure l l of a boolean logic l is defined consisting of the ordered pairs q p of all propositions of l excluding those for which the condition p is equivalent to 0 that is impossible the conditional closure is not altogether a boolean algebra not altogether distributive for instance but it has many subalgebras that are boolean including l itself this paper contains partially assumed formulas for operating on the conditional propositions q p using and or not and of course if then this allows complex conditional expressions to be reduced to simple conditionals of boolean propositions which have a conditional probability an example calculation from the dice table illustrates how to simplify a compound conditional proposition and compute its probability the structure of l l is explored including the finite conditional closures of finite boolean logics formulas are given for the probabilistic difference between familiar conditional propositions that are equivalent in the two valued logic where all propositions are either necessary or impossible but not equivalent in the conditional closure of a logic with possible propositions the probability that a proposition p is a theorem true in all models is contrasted from the probability of the models in which the proposition is true the latter being p p while the former is p p 1 this leads to a natural connection with fuzzy set theory since p 1 can also be expressed as p 1 where 1 is the sum ideal filter of necessary propositions given the boolean axioms this paper ends with a representation of each conditional proposition q p as the p measurable two valued truth function of q restricted to those models that satisfy p

surrogates for numerical simulations optimization of eddy promoter heat exchangers abstract although the advent of fast and inexpensive parallel computers has rendered numerous previously intractable calculations feasible many numerical simulations remain too resource intensive to be directly inserted into engineering optimization efforts an attractive alternative to direct insertion considers models for computational systems the expensive simulation is evoked only to construct and validate a simplified input output model this simplified input output model then serves as a simulation surrogate in subsequent engineering optimization studies we present here a simple bayesian validated statistical framework for the construction validation and purposive application of static computer simulation surrogates as an example we consider dissipation transport optimization of laminar flow eddy promoter heat exchangers parallel spectral element navier stokes calculations serve to construct and validate surrogates for the flowrate and nusselt number these surrogates then represent the originating navier stokes equations in the ensuing design process a validation based a posteriori error framework serves to quantify the effect of surrogate for simulation substitution

error rate estimation by mixture decomposition abstract we show how error rate estimation may be viewed as a problem of mixture decomposition we apply the idea to the average conditional error rate approach to estimation and explore the effectiveness of one particular decomposition method by simulation

empirical functionals and efficient smoothing parameter selection a striking feature of curve estimation is that the smoothing parameter h 0 which minimizes the squared error of a kernel or smoothing spline estimator is very difficult to estimate this is manifest both in slow rates of convergence and in high variability of standard methods such as cross validation we quantify this difficulty by describing nonparametric information bounds and exhibit asymptotically efficient estimators of h 0 that attain the bounds the efficient estimators are substantially less variable than cross validation and other current procedures and simulations suggest that they may offer improvements at moderate sample sizes at least in terms of minimizing the squared error

on the asymptotic properties of smoothed estimators of the classifiction error rate abstract a smoothed correction of the classification error rate counting estimate is examined the consistency of the estimate under very general conditions is established for normal distributions asymptotic theory reveals that the relative deficiency of the counting estimate with respect to the smoothed one can tend to infinity as the sample size increases

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

topological structure of solution sets for impulsive differential inclusions in frechet spaces abstract in this paper we consider the existence of solutions as well as the topological and geometric structure of solution sets for first order impulsive differential inclusions in some frechet spaces both the initial and terminal problems are considered using ingredients from topology and homology the topological structures of solution sets closedness and compactness as well as some geometric properties contractibility acyclicity a r and r are investigated some of our existence results are obtained via the method of taking the inverse system limit on noncompact intervals

double bootstrap methods that use a single double bootstrap simulation we show that when the double bootstrap is used to improve performance of bootstrap methods for bias correction techniques based on using a single double bootstrap sample for each single bootstrap sample can be particularly effective in particular they produce third order accuracy for much less computational expense than is required by conventional double bootstrap methods however this improved level of performance is not available for the single double bootstrap methods that have been suggested to construct confidence intervals or distribution estimators

on the structure of fixed point sets of compact maps in b0 spaces with applications to integral and differential equations in unbounded domain abstract in this paper we present certain applications of the topological degree theory in locally convex linear topological spaces as a result we obtain two generalizations of the krasnosielski perov rabinovitz theorem on the characterization of sets of solutions of functional equations known for the banach space case onto the case of a locally convex space on the other hand we make use of such consequences of the degree theory as the borsuk theorem and browder domain invariance theorem in order to reach a certain generalization of the lasota opial covering theorem and a new theorem on the characterization of sets of solutions of functional equations at the end we apply the abstract topological theorems to study the structure of sets of solutions of a certain integral equation and the darboux problem for a hyperbolic equation in an unbounded domain a general assumption is that the kernel of the integral equation and the right hand side of the hyperbolic equation satisfy the caratheodory conditions

conditioning on mv algebras and additive measures i abstract we present a lattice ordered semigroup approach for the foundation of conditional events which covers the special situations where the underlying unconditional events are boolean or fuzzy respectively our proposal is quite different from other ring theoretical approaches the problem of extending additivity of uncertainty measures from unconditional to conditional events will be discussed

bias associated with the discriminant analysis approach to the estimation of mixing proportions abstract the estimation of the mixing proportion 1 in which the first of two multivariate normal groups occur is considered on the basis of a sample drawn from a mixture of them under the model studied there are also data of known origin available from each of the groups for the formation of a discriminant rule r the bias of the estimator of 1 based on the proportion of the mixture data assigned to the first group by r is investigated also the case where the data of known origin are taken to be correlated is considered

the bootstrap and identification of prognostic factors via cox s proportional hazards regression model this paper describes the use of the bootstrap 1 2 a new computer based statistical methodology to help validate a regression model resulting from the fitting of cox s proportional hazards model3 to a set of censored survival data as an example we define a prognostic model for outcome in childhood acute lymphocytic leukemia with the cox model and use of a training set of 224 patients to validate the accuracy of the model we use a bootstrap resampling technique to mimic the population under study in two stages first we select the important prognostic factors via a stepwise regression procedure with 100 bootstrap samples secondly we estimate the corresponding regression parameters for these important factors with 400 bootstrap samples the bootstrap result suggests that the model constructed from the training set is reasonable