bounding option prices by semidefinite programming a cutting plane algorithm in a recent article bertsimas and popescu showed that a tight upper bound on a europeantype call option price given the first n moments of the distribution of the underlying security price can be obtained by solving an associated semidefinite programming problem sdp the purpose of this paper is to improve and extend their results we will show that a tight lower bound can be calculated by solving another sdp also we will show that these problems can be solved very quickly by a newly developed cutting plane algorithm when n is less than six or seven

shift in the velocity of a front due to a cutoff we consider the effect of a small cutoff ensuremath varepsilon on the velocity of a traveling wave in one dimension simulations done over more than ten orders of magnitude as well as a simple theoretical argument indicate that the effect of the cutoff ensuremath varepsilon is to select a single velocity that converges when ensuremath varepsilon ensuremath rightarrow 0 to the one predicted by the marginal stability argument for small ensuremath varepsilon the shift in velocity has the form k mathrm ln ensuremath varepsilon ensuremath 2 and our prediction for the constant k agrees very well with the results of our simulations a very similar logarithmic shift appears in more complicated situations in particular in finite size effects of some microscopic stochastic systems our theoretical approach can also be extended to give a simple way of deriving the shift in position due to initial conditions in the fisher kolmogorov or similar equations

multinomial goodness of fit tests for logistic regression models we examine the properties of several tests for goodness of fit for multinomial logistic regression one test is based on a strategy of sorting the observations according to the complement of the estimated probability for the reference outcome category and then grouping the subjects into g equal sized groups a g x c contingency table where c is the number of values of the outcome variable is constructed the test statistic denoted as cg is obtained by calculating the pearson chi2 statistic where the estimated expected frequencies are the sum of the model based estimated logistic probabilities simulations compare the properties of cg with those of the ungrouped pearson chi2 test x2 and its normalized test z the null distribution of cg is well approximated by the chi2 distribution with g 2 x c 1 degrees of freedom the sampling distribution of x2 is compared with a chi2 distribution with n x c 1 degrees of freedom but shows erratic behavior with a few exceptions the sampling distribution of z adheres reasonably well to the standard normal distribution power simulations show that cg has low power for a sample of 100 observations but satisfactory power for a sample of 400 the tests are illustrated using data from a study of cytological criteria for the diagnosis of breast tumors

plant friendly input design convex relaxation and quality a common practice in a system identification exercise is to perturb the system of interest and use the resulting data to build a model the problem of interest in this contribution is to synthesize an input signal that is maximally informative for generating good quality models while being plant friendly i e least hostile to plant operation in this contribution limits on input move sizes are the plant friendly specifications the resulting optimization problem is nonlinear and nonconvex hence the original plant friendly input design problem is relaxed which results in a convex optimization problem we formulate a semidefinite programme using the theory of generalized tchebysheff inequalities to derive tight bounds on the quality of relaxation simulations show that the relaxation results in more plant friendly input signals

an inequality for nearly log concave distributions with applications to learning we prove that given a nearly log concave distribution in any partition of the space to two well separated sets the measure of the points that do not belong to these sets is large we apply this isoperimetric inequality to derive lower bounds on the generalization error in learning we further consider regression problems and show that if the inputs and outputs are sampled from a nearly log concave distribution the measure of points for which the prediction is wrong by more than epsi0 and less than epsi1 is roughly linear in epsi1 epsi0 as long as epsi0 is not too small and epsi1 not too large we also show that when the data are sampled from a nearly log concave distribution the margin cannot be large in a strong probabilistic sense

completely positive reformulations for polynomial optimization polynomial optimization encompasses a very rich class of problems in which both the objective and constraints can be written in terms of polynomials on the decision variables there is a well established body of research on quadratic polynomial optimization problems based on reformulations of the original problem as a conic program over the cone of completely positive matrices or its conic dual the cone of copositive matrices as a result of this reformulation approach novel solution schemes for quadratic polynomial optimization problems have been designed by drawing on conic programming tools and the extensively studied cones of completely positive and of copositive matrices in particular this approach has been applied to solve key combinatorial optimization problems along this line of research we consider polynomial optimization problems that are not necessarily quadratic for this purpose we use a natural extension of the cone of completely positive matrices namely the cone of completely positive tensors we provide a general characterization of the class of polynomial optimization problems that can be formulated as a conic program over the cone of completely positive tensors as a consequence of this characterization it follows that recent related results for quadratic problems can be further strengthened and generalized to higher order polynomial optimization problems also we show that the conditions underlying the characterization are conceptually the same regardless of the degree of the polynomials defining the problem to illustrate our results we discuss in further detail special and relevant instances of polynomial optimization problems

the use of cusums and other techniques in modelling continuous covariates in logistic regression the assessment of continuous covariates singly as possible predictors in a multivariable logistic regression model is an important first step in the analysis an approach to plotting which uses a cusum cumulative sum of the binary response variable is described extreme deviation statistics associated with the cusum may be used to detect monotonic and non monotonic trends probability plots of the covariate in the two groups defined by the response variable may help to determine the appropriate scale transformation of the covariate and to anticipate possible problems with the logistic fit the ratio of the variances in the response non response groups is informative about the need for a quadratic term in the logistic model smoothed scatterplots of the response are valuable in displaying the observed and fitted values the techniques are illustrated with two data sets

multiple objectives satisficing under uncertainty we propose a class of functions called multiple objective satisficing mos criteria for evaluating the level of compliance of a set of objectives in meeting their targets collectively under uncertainty the mos criteria include the joint targets achievement probability joint success probability criterion as a special case and also extend to situations when the probability distributions are not fully characterized we focus on a class of mos criteria that favors diversification which has the potential to mitigate severe shortfalls in scenarios when any objective fails to achieve its target naturally this class excludes joint success probability we further propose the shortfall aware mos criterion s mos which is inspired by the probability measure and is diversification favoring we also show how to build tractable approximations of the s mos criterion because the s mos criterion maximization is not a convex optimization problem we propose improvement algorithms via solving sequences of convex optimization problems we report encouraging computational results on a blending problem in meeting specification targets even in the absence of full probability distribution description

on reduced semidefinite programs for second order moment bounds with applications we show that the complexity of computing the second order moment bound on the expected optimal value of a mixed integer linear program with a random objective coefficient vector is closely related to the complexity of characterizing the convex hull of the points 1 atopwithdelims varvec x 1 atopwithdelims varvec x varvec x in mathcal x 1x1x x x where mathcal x x is the feasible region in fact we can replace the completely positive programming formulation for the moment bound on mathcal x x with an associated semidefinite program provided we have a linear or a semidefinite representation of this convex hull as an application of the result we identify a new polynomial time solvable semidefinite relaxation of the distributionally robust multi item newsvendor problem by exploiting results from the boolean quadric polytope for mathcal x x described explicitly by a finite set of points our formulation leads to a reduction in the size of the semidefinite program we illustrate the usefulness of the reduced semidefinite programming bounds in estimating the expected range of random variables with two applications arising in random walks and best worst choice models

goodness of fit testing in high dimensional generalized linear models we propose a family of tests to assess the goodness of fit of a high dimensional generalized linear model our framework is flexible and may be used to construct an omnibus test or directed against testing specific non linearities and interaction effects or for testing the significance of groups of variables the methodology is based on extracting left over signal in the residuals from an initial fit of a generalized linear model this can be achieved by predicting this signal from the residuals by using modern powerful regression or machine learning methods such as random forests or boosted trees under the null hypothesis that the generalized linear model is correct no signal is left in the residuals and our test statistic has a gaussian limiting distribution translating to asymptotic control of type i error under a local alternative we establish a guarantee on the power of the test we illustrate the effectiveness of the methodology on simulated and real data examples by testing goodness of fit in logistic regression models software implementing the methodology is available in the r package grptests

distributionally robust convex optimization distributionally robust optimization is a paradigm for decision making under uncertainty where the uncertain problem data are governed by a probability distribution that is itself subject to uncertainty the distribution is then assumed to belong to an ambiguity set comprising all distributions that are compatible with the decision maker s prior information in this paper we propose a unifying framework for modeling and solving distributionally robust optimization problems we introduce standardized ambiguity sets that contain all distributions with prescribed conic representable confidence sets and with mean values residing on an affine manifold these ambiguity sets are highly expressive and encompass many ambiguity sets from the recent literature as special cases they also allow us to characterize distributional families in terms of several classical and or robust statistical indicators that have not yet been studied in the context of robust optimization we determine conditions under which distributionally robust optimization problems based on our standardized ambiguity sets are computationally tractable we also provide tractable conservative approximations for problems that violate these conditions

probabilistic combinatorial optimization moments semidefinite programming and asymptotic bounds we address the problem of evaluating the expected optimal objective value of a 0 1 optimization problem under uncertainty in the objective coefficients the probabilistic model we consider prescribes limited marginal distribution information for the objective coefficients in the form of moments we show that for a fairly general class of marginal information a tight upper lower bound on the expected optimal objective value of a 0 1 maximization minimization problem can be computed in polynomial time if the corresponding deterministic problem is solvable in polynomial time we provide an efficiently solvable semidefinite programming formulation to compute this tight bound we also analyze the asymptotic behavior of a general class of combinatorial problems that includes the linear assignment spanning tree and traveling salesman problems under knowledge of complete marginal distributions with and without independence we calculate the limiting constants exactly

portfolio value at risk optimization for asymmetrically distributed asset returns we present a new approach to portfolio optimization by separating asset return distributions into positive and negative half spaces the approach minimizes a so called partitioned value at risk pvar measure by using the statistical information from the two half spaces respectively we show that the proposed pvar approach is a signiflcant improvement in several important aspects when compared to markowitz mean variance optimization approach first our approach which accommodates ambiguous asymmetric return distributions and captures portfolio risk in higher moments does not require asset distributions being elliptically symmetric or multivariate normal second using simulated and real data our approach generates better risk return tradeofis in the optimal portfolios the difierence between the two approaches increases in the degree of asymmetry in the underlying asset distributions third when given the support of asset returns our pvar measure becomes a coherent risk measure proposed by artzner et al 1999 whereas conventional risk measures such as variance and var fail to do so moreover our pvar measure is an asymmetric risk measure which is difierent from symmetric risk measures like variance and worst case mean covariance var wvar therefore our proposed pvar is a signiflcant addition to the existing portfolio risk measures we believe that the pvar approach can be very useful for better portfolio allocations than the mean variance or other symmetric risk metrics approach during market downturns when asset return distributions are often fat tailed or skewed

generalized logistic models abstract a class of models indexed by two shape parameters is introduced both to extend the scope of the standard logistic model to asymmetric probability curves and improve the fit in the noncentral probability regions one parameter subclasses can be used to examine symmetric or asymmetric deviations from the logistic model the delta algorithm is adapted to obtain maximum likelihood estimates of the parameters a review is made of other proposed generalizations the standard linear logistic model is widely used for modeling the dependence of binary data on explanatory variables its success is due to its broad applicability simplicity of form and ease of interpretation this model works well for many common applications however it assumes that the expected probability curve is skew symmetric about and that the shape of is the cumulative distribution function of the logistic distribution symmetric data with a shallower or steeper slope of ascent may not be fitted well by this model

partially finite convex programming part ii explicit lattice models in part i of this work we derived a duality theorem for partially finite convex programs problems for which the standard slater condition fails almost invariably our result depended on a constraint qualification involving the notion ofquasi relative interior the derivation of the primal solution from a dual solution depended on the differentiability of the dual objective function the differentiability of various convex functions in lattices was considered at the end of part i in part ii we shall apply our results to a number of more concrete problems including variants of semi infinite linear programming l1 approximation constrained approximation and interpolation spectral estimation semi infinite transportation problems and the generalized market area problem of lowe and hurter 1976 as in part i we shall use lattice notation extensively but as we illustrated there in concrete examples lattice theoretic ideas can be avoided if preferred by direct calculation

goodness of fit tests for random effect models with binary responses correlated binary responses are very common in longitudinal and repeated measures studies random effects models are often used to analyze such data and the h likelihood estimating procedure provides an inferential tool the objective of this study is to introduce goodness of fit score statistics for the fixed part of these models the proposed statistics are based on processes that partition the observations into mutually exclusive groups weighted versions of these statistics are also introduced and are based on the correlation between an appropriately adjusted candidate covariate for entrance into the model and the model residuals a simulation study indicates that the weighted statistics perform better than their unweighted counterparts whereas the statistics that are based on the partitioning of the covariate space seem to perform slightly better compared with those based on other grouping procedures the use of the proposed statistics is illustrated using a real data example

a hierarchy of near optimal policies for multistage adaptive optimization in this paper we propose a new tractable framework for dealing with linear dynamical systems affected by uncertainty applicable to multistage robust optimization and stochastic programming we introduce a hierarchy of near optimal polynomial disturbance feedback control policies and show how these can be computed by solving a single semidefinite programming problem the approach yields a hierarchy parameterized by a single variable the degree of the polynomial policies which controls the trade off between the optimality gap and the computational requirements we evaluate our framework in the context of three classical applications two in inventory management and one in robust regulation of an active suspension system in which very strong numerical performance is exhibited at relatively modest computational expense

effect of selection on ancestry an exactly soluble case and its phenomenological generalization we consider a family of models describing the evolution under selection of a population whose dynamics can be related to the propagation of noisy traveling waves for one particular model that we shall call the exponential model the properties of the traveling wave front can be calculated exactly as well as the statistics of the genealogy of the population one striking result is that for this particular model the genealogical trees have the same statistics as the trees of replicas in the parisi mean field theory of spin glasses we also find that in the exponential model the coalescence times along these trees grow like the logarithm of the population size a phenomenological picture of the propagation of wave fronts that we introduced in a previous work as well as our numerical data suggest that these statistics remain valid for a larger class of models while the coalescence times grow like the cube of the logarithm of the population size

a distributionally robust perspective on uncertainty quantification and chance constrained programming the objective of uncertainty quantification is to certify that a given physical engineering or economic system satisfies multiple safety conditions with high probability a more ambitious goal is to actively influence the system so as to guarantee and maintain its safety a scenario which can be modeled through a chance constrained program in this paper we assume that the parameters of the system are governed by an ambiguous distribution that is only known to belong to an ambiguity set characterized through generalized moment bounds and structural properties such as symmetry unimodality or independence patterns we delineate the watershed between tractability and intractability in ambiguity averse uncertainty quantification and chance constrained programming using tools from distributionally robust optimization we derive explicit conic reformulations for tractable problem classes and suggest efficiently computable conservative approximations for intractable ones

correlation polytopes their geometry and complexity a family of polytopes correlation polytopes which arise naturally in the theory of probability and propositional logic is defined these polytopes are tightly connected to combinatorial problems in the foundations of quantum mechanics and to the ising spin model correlation polytopes exhibit a great deal of symmetry exponential size symmetry groups which leave the polytope invariant and act transitively on its vertices are defined using the symmetries a large family of facets is determined a conjecture concerning the full facet structure of correlation polytopes is formulated the conjecture however implies that np co np various complexity results are proved it is shown that deciding membership in a correlation polytope is an np complete problem and deciding facets is probably not even in np the relations between the polytope symmetries and its complexity are indicated

computing bounds for stochastic programming problems by means of a generalized moment problem bounds on the expected value of a convex function are obtained by means of an approximating generalized moment problem numerical implementation is discussed in the context of stochastic programming problems

the value of stochastic modeling in two stage stochastic programs with cost uncertainty although stochastic programming is probably the most effective framework for handling decision problems that involve uncertain variables it is always a costly task to formulate the stochastic model that accurately embodies our knowledge of these variables in practice this might require one to collect a large amount of observations to consult with experts of the specialized field of practice or to make simplifying assumptions about the underlying system when none of these options seem feasible a common heuristic has been to simply seek the solution of a version of the problem where each uncertain variable takes on its expected value otherwise known as the solution of the mean value problem in this paper we show that when 1 the stochastic program takes the form of a two stage mixed integer stochastic linear programs and 2 the uncertainty is limited to the objective function the solution of the mean value problem is in fact robust with respect to the selection of a stochastic model we also propose tractable methods that will bound the actual value of stochastic modeling i e how much improvement can be achieved by investing more efforts in the resolution of the stochastic model our framework is applied to an airline fleet composition problem in the three cases that are considered our results indicate that resolving the stochastic model can not lead to more than a 7 improvement of expected profits thus providing arguments against the need to develop these more sophisticated models

equation free modelling of evolving diseases coarse grained computations with individual based models we demonstrate how direct simulation of stochastic individualbased models can be combined with continuum numericalanalysis techniques to study the dynamics of evolving diseases sidestepping the necessity of obtaining explicit populationlevel models the approach analyses the unavailable in closed form coarse macroscopic equations estimating the necessary quantities through appropriately initialized short bursts of individualbased dynamic simulation we illustrate this approach by analysing a stochastic and discrete model for the evolution of disease agents caused by point mutations within individual hosts building up from classical susceptibleinfectedrecovered and susceptibleinfectedrecoveredsusceptible models our example uses a onedimensional lattice for variant space and assumes a finite number of individuals macroscopic computational tasks enabled through this approach include stationarystate computation coarse projective integration parametric continuation and stability analysis

distributionally robust optimization with polynomial densities theory models and algorithms in distributionally robust optimization the probability distribution of the uncertain problem parameters is itself uncertain and a fictitious adversary e g nature chooses the worst distribution from within a known ambiguity set a common shortcoming of most existing distributionally robust optimization models is that their ambiguity sets contain pathological discrete distributions that give nature too much freedom to inflict damage we thus introduce a new class of ambiguity sets that contain only distributions with sum of squares sos polynomial density functions of known degrees we show that these ambiguity sets are highly expressive as they conveniently accommodate distributional information about higher order moments conditional probabilities conditional moments or marginal distributions exploiting the theoretical properties of a measure based hierarchy for polynomial optimization due to lasserre siam j optim 21 3 864 885 2011 we prove that certain worst case expectation constraints are polynomial time solvable under these new ambiguity sets we also show how sos densities can be used to approximately solve the general problem of moments we showcase the applicability of the proposed approach in the context of a stylized portfolio optimization problem and a risk aggregation problem of an insurance company

generalized chebychev inequalities theory and applications in decision analysis in many decision analysis problems we have only limited information about the relevant probability distributions in problems like these it is natural to ask what conclusions can be drawn on the basis of this limited information for example in the early stages of analysis of a complex problem we may have only limited fractile information for the distributions in the problem what can we say about the optimal strategy or certainty equivalents given these few fractiles this paper describes a very general framework for analyzing these kinds of problems where given certain moments of a distribution we can compute bounds on the expected value of an arbitrary objective function by suitable choice of moment and objective functions we can formulate and solve many practical decision analysis problems we describe the general framework and theoretical results discuss computational strategies and provide specific results for examples in dynamic programming decision analysis with incomplete information bayesian statistics and option pricing

the generalized moment based filter can we solve the filtering problem from the only knowledge of few moments of the noise terms in this technical note by exploiting set of distributions based filtering we solve this problem without introducing additional assumptions on the distributions of the noises e g gaussianity or on the final form of the estimator e g linear estimator given the moments e g mean and variance of random variable x it is possible to define the set of all distributions that are compatible with the moments information this set can be equivalently characterized by its extreme distributions a family of mixtures of dirac s deltas the lower and upper expectation of any function g of x are obtained in correspondence of these extremes and can be computed by solving a linear programming problem the filtering problem can then be solved by running iteratively this linear programming problem in this technical note we discuss theoretical properties of this filter we show the connection with set membership estimation and its practical applications

goodness of fit processes for logistic regression simulation results in this paper we use simulations to compare the performance of new goodness of fit tests based on weighted statistical processes to three currently available tests the hosmer lemeshow decile of risk test the pearson chi square and the unweighted sum of squares tests the simulations demonstrate that all tests have the correct size the power for all tests to detect lack of fit due to an omitted quadratic term with a sample of size 100 is close to or exceeds 50 per cent to detect moderate departures from linearity and is over 90 per cent for these same alternatives for sample size 500 all tests have low power with sample size 100 to detect lack of fit due to an omitted interaction between a dichotomous and continuous covariate while the power exceeds 80 per cent to detect extreme interaction with a sample size of 500 the power is low to detect any alternative link function with sample size 100 and for most alternative links for sample size 500 only in the case of sample size 500 and an extremely asymmetric link function is the power over 80 per cent the results from these simulations show that no single test new or current performs best in detecting lack of fit due to an omitted covariate or incorrect link function however one of the new weighted tests has power comparable to other tests in all settings simulated and had the highest power in the difficult case of an omitted interaction term we illustrate the tests within the context of a model for factors associated with abstinence from drug use in a randomized trial of residential treatment programmes we conclude the paper with a summary and specific recommendations for practice

a probabilistic model for minmax regret in combinatorial optimization in this paper we propose a new probabilistic model for minimizing the anticipated regret in combinatorial optimization problems with distributional uncertainty in the objective coefficients the interval uncertainty representation of data is supplemented with information on the marginal distributions as a decision criterion we minimize the worst case conditional value at risk of regret the proposed model includes the interval data minmax regret model as a special case for the class of combinatorial optimization problems with a compact convex hull representation a polynomial sized mixed integer linear program is formulated when a the range and mean are known and b the range mean and mean absolute deviation are known and a mixed integer second order cone program is formulated when c the range mean and standard deviation are known for the subset selection problem of choosing k elements of maximum total weight out of a set of n elements the probabilistic regret model is shown to be solvable in polynomial time in the instances a and b above this extends the current known polynomial complexity result for minmax regret subset selection with range information only

a lack of fit test for the mean function in a generalized linear model abstract a supremum type statistic based on partial sums of residuals is proposed to test the validity of the mean function of the response variable in a generalized linear model the proposed test does not need a partition of the space of covariates to handle the case of nonreplication the new test is consistent against the alternative under which the deterministic part of the model is not the one specified in the null hypothesis the asymptotic null distribution of the test statistic can be approximated through simulations this approximation is valid even if the variance of the response variable is misspecified in the model for practical sample sizes the adequacy of this large sample approximation to the null distribution of the test statistic is carefully examined power comparisons with other lack of fit tests are also performed to show the advantage of the test in particular we find that the new procedure is rather sensitive to detect a misspecified mean function if the assumed mean functio

goodness of fit tests in models for life history data based on cumulative hazard rates to check the validity of an assumed parametric model for survival data one may compare t the nonparametric nelson aalen plot of the cumulative hazard rate with a t the estimated parametric cumulative hazard rate being for example the maximum likelihood estimator convergence in distribution of n t a t and more general processes is studied employing the general framework of counting processes which allows for quite general models for life history data and for quite general censoring schemes the results are applied to the construction of 2 type statistics for goodness of fit cramer von mises and kolmogorov smirnov type tests are presented in the case where the unknown parameter is one dimensional

a class of single class minimax probability machines for novelty detection single class minimax probability machines mpms offer robust novelty detection with distribution free worst case bounds on the probability that a pattern will fall inside the normal region however in practice they are too cautious in labeling patterns as outlying and so have a high false negative rate fnr in this paper we propose a more aggressive version of the single class mpm that bounds the best case probability that a pattern will fall inside the normal region these two mpms can then be used together to delimit the solution space by using the hyperplane lying in the middle of this pair of mpms a better compromise between false positives fps and false negatives fns and between recall and precision can be obtained experiments on the real world data sets show encouraging results

on distributional robust probability functions and their computations abstract consider a random vector and assume that a set of its moments information is known among all possible distributions obeying the given moments constraints the envelope of the probability distribution functions is introduced in this paper as distributional robust probability function we show that such a function is computable in the bi variate case under some conditions connections to the existing results in the literature and its applications in risk management are discussed as well

evolution on a smooth landscape we study in detail a recently proposed simple discrete model for evolution on smooth landscapes an asymptotic solution of this model for long times is constructed we find that the dynamics of the population is governed by correlation functions that although being formally down by powers ofn the population size nonetheless control the evolution process after a very short transient the long time behavior can be found analytically since only one of these higher order correlators the two point function is relevant we compare and contrast the exact findings derived herein with a previously proposed phenomenological treatment employing mean field theory supplemented with a cutoff at small population density finally we relate our results to the recently studied case of mutation on a totally flat landscape

test the reliability of doubly robust estimation with missing response data in statistical inference one has to make sure that the underlying regression model is correctly specified otherwise the resulting estimation may be biased model checking is an important method to detect any departure of the regression model from the true one missing data is a ubiquitous problem in social and medical studies if the underlying regression model is correctly specified recent researches show great popularity of the doubly robust estimates method for handling missing data because of its robustness to the misspecification of either the missing data model or the conditional mean model i e the model for the conditional expectation of true regression model conditioning on the observed quantities however little work has been devoted to the goodness of fit test for doubly robust estimates method in this paper we propose a testing method to assess the reliability of the estimator derived from the doubly robust estimating equation with possibly missing response and always observed auxiliary variables numerical studies demonstrate that the proposed test can control type i errors well furthermore the proposed method can detect departures from model assumptions in the marginal mean model of interest powerfully a real dementia data set is used to illustrate the method for the diagnosis of model misspecification in the problem of missing response with an always observed auxiliary variable for cross sectional data

second order cone programming formulations for feature selection this paper addresses the issue of feature selection for linear classifiers given the moments of the class conditional densities the problem is posed as finding a minimal set of features such that the resulting classifier has a low misclassification error using a bound on the misclassification error involving the mean and covariance of class conditional densities and minimizing an l1 norm as an approximate criterion for feature selection a second order programming formulation is derived to handle errors in estimation of mean and covariances a tractable robust formulation is also discussed in a slightly different setting the fisher discriminant is derived feature selection for fisher discriminant is also discussed experimental results on synthetic data sets and on real life microarray data show that the proposed formulations are competitive with the state of the art linear programming formulation

probabilistic estimates for mixed model validation problems with cal h _ infty type uncertainties a mixed deterministic probabilistic model validation problem is investigated in this technical note which consists in an additive uncertain model with model uncertainty characterized by the h norm the data available for validation are time domain experimental data corrupted by a random noise sequence our aim is to compute the probability for such an uncertain model to be validated by the data and our main results are bounds on this probability that are computable based on the distribution of chi square random variables when the noise is a gaussian variable and solvable as an lmi problem when only statistical information such as the expectation and covariance of the noise are known

a robust minimax approach to classification when constructing a classifier the probability of correct classification of future data points should be maximized we consider a binary classification problem where the mean and covariance matrix of each class are assumed to be known no further assumptions are made with respect to the class conditional distributions misclassification probabilities are then controlled in a worst case setting that is under all possible choices of class conditional densities with given mean and covariance matrix we minimize the worst case maximum probability of misclassification of future data points for a linear decision boundary this desideratum is translated in a very direct way into a convex second order cone optimization problem with complexity similar to a support vector machine problem the minimax problem can be interpreted geometrically as minimizing the maximum of the mahalanobis distances to the two classes we address the issue of robustness with respect to estimation errors in the means and covariances of the classes via a simple modification of the input data we also show how to exploit mercer kernels in this setting to obtain nonlinear decision boundaries yielding a classifier which proves to be competitive with current methods including support vector machines an important feature of this method is that a worst case bound on the probability of misclassification of future data is always obtained explicitly

a minimax theorem with applications to machine learning signal processing and finance this paper concerns a fractional function of the form x ta sqrt x tbx where b is positive definite we consider the game of choosing x from a convex set to maximize the function and choosing a b from a convex set to minimize it we prove the existence of a saddle point and describe an efficient method based on convex optimization for computing it we describe applications in machine learning robust fisher linear discriminant analysis signal processing robust beamforming and robust matched filtering and finance robust portfolio selection in these applications x corresponds to some design variables to be chosen and the pair a b corresponds to the statistical model which is uncertain

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

analysis of evolution through competitive selection recent studies of in vitro evolution of dna via protein binding indicate that the evolution behavior is qualitatively different in different parameter regimes i here present a general theory that is valid for a wide range of parameters and which reproduces and extends previous results specifically the mean field theory of a general translation invariant model can be reduced to the basic diffusion equation with a dynamic boundary condition the simple analytical form yields both quantitatively accurate predictions and valuable insight into the principles involved

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

optimal inequalities in probability theory a convex optimization approach we propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for p x in s for a set s defined by polynomial inequalities and a random vector x defined on omega subseteq cal r n that has a given collection of up to kth order moments in the univariate case we provide optimal bounds on p x in s when the first k moments of x are given as the solution of a semidefinite optimization problem in k 1 dimensions in the multivariate case if the sets s and omega are given by polynomial inequalities we obtain an improving sequence of bounds by solving semidefinite optimization problems of polynomial size in n for fixed k we characterize the complexity of the problem of deriving tight moment inequalities we show that it is np hard to find tight bounds for k geq 4 and omega cal r n and for k geq 2 and omega cal r n when the data in the problem is rational for k 1 and omega cal r n we show that we can find tight upper bounds by solving n convex optimization problems when the set s is convex and we provide a polynomial time algorithm when s and omega are unions of convex sets over which linear functions can be optimized efficiently for the case k 2 and omega cal r n we present an efficient algorithm for finding tight bounds when s is a union of convex sets over which convex quadratic functions can be optimized efficiently

a branching random walk seen from the tip we show that all the time dependent statistical properties of the rightmost points of a branching brownian motion can be extracted from the traveling wave solutions of the fisher kpp equation the distribution of all the distances between the rightmost points has a long time limit which can be understood as the delay of the fisher kpp traveling waves when the initial condition is modified the limiting measure exhibits the surprising property of superposability the statistical properties of the distances between the rightmost points of the union of two realizations of the branching brownian motion shifted by arbitrary amounts are the same as those of a single realization we discuss the extension of our results to more general branching random walks

regret in the newsvendor model with partial information traditional stochastic inventory models assume full knowledge of the demand probability distribution however in practice it is often difficult to completely characterize the demand distribution especially in fast changing markets in this paper we study the newsvendor problem with partial information about the demand distribution e g mean variance symmetry unimodality in particular we derive the order quantities that minimize the newsvendor s maximum regret of not acting optimally most of our solutions are tractable which makes them attractive for practical application our analysis also generates insights into the choice of the demand distribution as an input to the newsvendor model in particular the distributions that maximize the entropy perform well under the regret criterion our approach can be extended to a variety of problems that require a robust but not conservative solution

normal goodness of fit tests for multinomial models with large degrees of freedom abstract goodness of fit tests for independent multinomials with parameters to be estimated are usually based on pearson s x 2 or the likelihood ratio g 2 both are included in the family of power divergence statistics sd for increasing sample sizes each sd has an asymptotic x 2 distribution provided that the number of cells remains fixed we are dealing with an increasing cells approach where the number j of independent multinomials increases while the number of classes for each multinomial and the number of parameters remain fixed extending results on x 2 and g 2 the asymptotic normality of any sd is obtained for increasing cells the corresponding normal goodness of fit tests discussed here apply for models with large degrees of freedom with no restrictions imposed on the sizes n j of each multinomial allowing large as well as small expectations sparse data within each cell the asymptotic expectation and variance of sd are easy to compute for pearson s x 2 and simplify considerably for

bounding probability of small deviation a fourth moment approach in this paper we study the problem of upper bounding the probability that a random variable is above its expected value by a small amount relative to the expected value by means of the second and the fourth central moments of the random variable in this particular context many classical inequalities yield only trivial bounds we obtain tight upper bounds by studying the primal dual moments generating conic optimization problems as an application we demonstrate that given the new probability bounds a substantial sharpening and simplification of a recent result and its analysis by feige see feige u 2006 on sums of independent random variables with unbounded variances and estimating the average degree in a graph siam j comput 35 964 984 can be obtained also these bounds lead to new properties of the distribution of the cut values for the max cut problem we expect the new probability bounds to be useful in many other applications

superreplication of financial derivatives via convex programming we give a method based on convex programming to calculate the optimal superreplicating and subreplicating prices and corresponding hedging portfolios of a financial derivative in terms of other financial derivatives in a discrete time setting our method produces a model that matches the superreplicating or subreplicating price within an arbitrary precision and is consistent with the other financial derivatives prices applications include robust replication in terms of call prices with various strikes and maturities of forward start options volatility and variance swaps and derivatives cliquet calls barrier options and lookback and asian options numerical examples show that in some cases the best superreplicating and or subreplicating prices are within 10 of the price obtained by a standard model but considerably differ from it in other cases our method can incorporate bid ask spreads interest rates and dividends and various limitations to the diffusion model the supplementary material is a

analytic approach to the evolutionary effects of genetic exchange we present an approximate analytic study of our previously introduced model of evolution including the effects of genetic exchange this model is motivated by the process of bacterial transformation we solve for the velocity the rate of increase of fitness as a function of the fixed population size n we find the velocity increases with ln n eventually saturating at an n which depends on the strength of the recombination process the analytical treatment is seen to agree well with direct numerical simulations of our model equations

a comparison of goodness of fit tests for the logistic regression model recent work has shown that there may be disadvantages in the use of the chi square like goodness of fit tests for the logistic regression model proposed by hosmer and lemeshow that use fixed groups of the estimated probabilities a particular concern with these grouping strategies based on estimated probabilities fitted values is that groups may contain subjects with widely different values of the covariates it is possible to demonstrate situations where one set of fixed groups shows the model fits while the test rejects fit using a different set of fixed groups we compare the performance by simulation of these tests to tests based on smoothed residuals proposed by le cessie and van houwelingen and royston a score test for an extended logistic regression model proposed by stukel the pearson chi square and the unweighted residual sum of squares these simulations demonstrate that all but one of royston s tests have the correct size an examination of the performance of the tests when the correct model has a quadratic term but a model containing only the linear term has been fit shows that the pearson chi square the unweighted sum of squares the hosmer lemeshow decile of risk the smoothed residual sum of squares and stukel s score test have power exceeding 50 per cent to detect moderate departures from linearity when the sample size is 100 and have power over 90 per cent for these same alternatives for samples of size 500 all tests had no power when the correct model had an interaction between a dichotomous and continuous covariate but only the continuous covariate model was fit power to detect an incorrectly specified link was poor for samples of size 100 for samples of size 500 stukel s score test had the best power but it only exceeded 50 per cent to detect an asymmetric link function the power of the unweighted sum of squares test to detect an incorrectly specified link function was slightly less than stukel s score test we illustrate the tests within the context of a model for factors associated with low birth weight

a cutting plane algorithm for semi definite programming problems with applications to failure discriminant analysis we will propose a new cutting plane algorithm for solving a class of semi definite programming problems sdp with a small number of variables and a large number of constraints problems of this type appear when we try to classify a large number of multi dimensional data into two groups by a hyper ellipsoidal surface among such examples are cancer diagnosis failure discrimination of enterprises also a certain class of option pricing problems can be formulated as this type of problem we will show that the cutting plane algorithm is much more efficient than the standard interior point algorithms for solving sdp

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

dual gauge programs with applications to quadratic programming and the minimum norm problem a gauge functionf is a nonnegative convex function that is positively homogeneous and satisfiesf 0 0 norms and pseudonorms are specific instances of a gauge function this paper presents a gauge duality theory for a gauge program which is the problem of minimizing the value of a gauge functionf over a convex set the gauge dual program is also a gauge program unlike the standard lagrange dual we present sufficient conditions onf that ensure the existence of optimal solutions to the gauge program and its dual with no duality gap these sufficient conditions are relatively weak and are easy to verify and are independent of any qualifications on the constraints the theory is applied to a class of convex quadratic programs and to the minimuml p norm problem the gauge dual program is shown to provide a smaller duality than the standard dual in a certain sense discussed in the text

a chance constrained programming framework to handle uncertainties in radiation therapy treatment planning abstract a stochastic programming framework is proposed for radiation therapy treatment planning this framework takes into account uncertainty in setting up or positioning patients identically day to day during the treatment period because uncertainties are unavoidable constraint violations are tolerated to some degree in practice under this assumption a chance constrained programming ccp framework is developed to handle setup uncertainties in treatment planning the proposed framework can be employed under different distributional assumptions the goal of the proposed approach is to maximize both the statistical confidence level of a treatment plan and the homogeneity of the dose distributions this novel perspective provides a user centric and personalized optimization model that allows a trade off between sufficient tumor coverage and sparing healthy tissues under uncertainty we describe testing of the performance of the proposed ccp models in terms of plan quality robustness and homogeneity and confidence level of the constraints using clinical data for a prostate cancer patient optimized ccp plans are also compared to plans developed using a deterministic approach that does not take uncertainties into account numerical experiments confirmed that the ccp is able to control setup uncertainties in target coverage and sparing of organs at risk

robust sample average approximation sample average approximation saa is a widely popular approach to data driven decision making under uncertainty under mild assumptions saa is both tractable and enjoys strong asymptotic performance guarantees similar guarantees however do not typically hold in finite samples in this paper we propose a modification of saa which we term robust saa which retains saa s tractability and asymptotic properties and additionally enjoys strong finite sample performance guarantees the key to our method is linking saa distributionally robust optimization and hypothesis testing of goodness of fit beyond robust saa this connection provides a unified perspective enabling us to characterize the finite sample and asymptotic guarantees of various other data driven procedures that are based upon distributionally robust optimization this analysis provides insight into the practical performance of these various methods in real applications we present examples from inventory management and portfolio allocation and demonstrate numerically that our approach outperforms other data driven approaches in these applications

a conic programming approach to generalized tchebycheff inequalities consider the problem of finding optimal bounds on the expected value of piecewise polynomials over all measures with a given set of moments this is a special class ofgeneralized tchebycheff inequalities in probability theory we study this problem within the framework of conic programming relying on a general approximation scheme for conic programming we show that these bounds can be numerically computed or approximated via semidefinite programming we also describe some applications of this class of generalized tchebycheff inequalities in probability finance and inventory theory

probability bounds with cherry trees a third order upper bound is presented on the probability of the union of a finite number of events by means of graphs called cherry trees these are graphs that we construct recursively in such a way that every time we pick a new vertex connect it with two already existing vertices if the latters are always adjacent we call the cherry tree a t cherry tree a cherry tree has a weight that provides us with the upper bound on the union any hunter worsley bound can be majorized by a t cherry bound constructed by the use of the hunter worsley tree a cherry tree bound can be identified as a feasible solution to the dual of the boolean probability bounding problem a t cherry tree bound can be identified as the objective function value of the dual vector corresponding to a dual feasible basis in the boolean problem this enables us to improve on the bound algorithmically if we use the dual method of linear programming

the speed of evolution in large asexual populations we consider an asexual biological population of constant size n evolving in discrete time under the influence of selection and mutation beneficial mutations appear at rate u and their selective effects s are drawn from a distribution g s after introducing the required models and concepts of mathematical population genetics we review different approaches to computing the speed of logarithmic fitness increase as a function of n u and g s we present an exact solution of the infinite population size limit and provide an estimate of the population size beyond which it is valid we then discuss approximate approaches to the finite population problem distinguishing between the case of a single selection coefficient g s s s b and a continuous distribution of selection coefficients analytic estimates for the speed are compared to numerical simulations up to population sizes of order 10300

a model with darwinian dynamics on a rugged landscape we discuss the population dynamics with selection and random diffusion keeping the total population constant in a fitness landscape associated with constraint satisfaction a paradigm for difficult optimization problems we obtain a phase diagram in terms of the size of the population and the diffusion rate with a glass phase inside which the dynamics keeps searching for better configurations and outside which deleterious mutations spoil the performance the phase diagram is analogous to that of dense active matter in terms of temperature and drive

the focused information criterion a variety of model selection criteria have been developed of general and specific types most of these aim at selecting a single model with good overall properties for example formulated via average prediction quality or shortest estimated overall distance to the true model the akaike the bayesian and the deviance information criteria along with many suitable variations are examples of such methods these methods are not concerned however with the actual use of the selected model which varies with context and application the present article takes the view that the model selector should instead focus on the parameter singled out for interest in particular a model that gives good precision for one estimand may be worse when used for inference for another estimand we develop a method that for a given focus parameter estimates the precision of any submodel based estimator the framework is that of large sample likelihood inference using an unbiased estimate of limiting risk we propose a foc

diffusion reproduction processes interesting physics emerges from studying a population of reproducing individuals each can be regarded as a random walker but it can either duplicate or die novel features of the collective behavior are quite surprising if individuals reproduce or die freely the life expectation is proportional to the size of the population and if it is kept constant the center of mass moves in space as if it were a single walker i e the diffusion constant is independent of population s size biology inspired interactions are also considered

positivity and optimization for semi algebraic functions we describe algebraic certificates of positivity for functions belonging to a finitely generated algebra of borel measurable functions with particular emphasis on algebras generated by semi algebraic functions in this case the standard global optimization problem with constraints given by elements of the same algebra is reduced via a natural change of variables to the better understood case of polynomial optimization a collection of simple examples and numerical experiments complement the theoretical parts of the article

goodness of fit tests for ordinal response regression models it is well documented that the commonly used pearson chi square and deviance statistics are not adequate for assessing goodness of fit in logistic regression models when continuous covariates are modelled in recent years several methods have been proposed which address this shortcoming in the binary logistic regression setting or assess model fit differently however these techniques have typically not been extended to the ordinal response setting and few techniques exist to assess model fit in that case we present the modified pearson chi square and deviance tests that are appropriate for assessing goodness of fit in ordinal response models when both categorical and continuous covariates are present the methods have good power to detect omitted interaction terms and reasonable power to detect failure of the proportional odds assumption or modelling the wrong functional form of a continuous covariate these tests also provide immediate information as to where a model may not fit well in addition the methods are simple to understand and implement and are non specific that is they do not require prespecification of a type of lack of fit to detect