sparse linear discriminant analysis by thresholding for high dimensional data in many social economical biological and medical studies one objective is to classify a subject into one of several classes based on a set of variables observed from the subject because the probability distribution of the variables is usually unknown the rule of classification is constructed using a training sample the well known linear discriminant analysis lda works well for the situation where the number of variables used for classification is much smaller than the training sample size because of the advance in technologies modern statistical studies often face classification problems with the number of variables much larger than the sample size and the lda may perform poorly we explore when and why the lda has poor performance and propose a sparse lda that is asymptotically optimal under some sparsity conditions on the unknown parameters for illustration of application we discuss an example of classifying human cancer into two classes of leukemia based on a set of 7 129 genes and a training sample of size 72 a simulation is also conducted to check the performance of the proposed method

partial estimation of covariance matrices a classical approach to accurately estimating the covariance matrix of a p variate normal distribution is to draw a sample of size n p and form a sample covariance matrix however many modern applications operate with much smaller sample sizes thus calling for estimation guarantees in the regime n ll p we show that a sample of size n o m log6p is sufficient to accurately estimate in operator norm an arbitrary symmetric part of consisting of m n nonzero entries per row this follows from a general result on estimating hadamard products m where m is an arbitrary symmetric matrix

phantom maps and purity in modular representation theory iii abstract in the third installment of this paper we exhibit two phantom maps between kg modules whose composite is not projective whenever k is an uncountable field of characteristic p and g is a finite group of p rank at least 2

estimating sparse precision matrix optimal rates of convergence and adaptive estimation precision matrix is of significant importance in a wide range of applications in multivariate analysis this paper considers adaptive minimax estimation of sparse precision matrices in the high dimensional setting optimal rates of convergence are established for a range of matrix norm losses a fully data driven estimator based on adaptive constrained l1 minimization is proposed and its rate of convergence is obtained over a collection of parameter spaces the estimator called aclime is easy to implement and performs well numerically a major step in establishing the minimax rate of convergence is the derivation of a rate sharp lower bound a two directional lower bound technique is applied to obtain the minimax lower bound the upper and lower bounds together yield the optimal rates of convergence for sparse precision matrix estimation and show that the aclime estimator is adaptively minimax rate optimal for a collection of parameter spaces and a range of matrix norm losses simultaneously

toward a curse of dimensionality appropriate coda asymptotic theory for semi parametric models we argue that due to the curse of dimensionality there are major difficulties with any pure or smoothed likelihood based method of inference in designed studies with randomly missing data when missingness depends on a high dimensional vector of variables we study in detail a semi parametric superpopulation version of continuously stratified random sampling we show that all estimators of the population mean that are uniformly consistent or that achieve an algebraic rate of convergence no matter how slow require the use of the selection randomization probabilities we argue that in contrast to likelihood methods which ignore these probabilities inverse selection probability weighted estimators continue to perform well achieving uniform n 1 2 rates of convergence we propose a curse of dimensionality appropriate coda asymptotic theory for inference in non and semi parametric models in an attempt to formalize our arguments we discuss whether our results constitute a fatal blow to the likelihood principle and study the attitude toward these that a committed subjective bayesian would adopt finally we apply our coda theory to analyse the effect of the curse of dimensionality in several interesting semi parametric models including a model for a two armed randomized trial with randomization probabilities depending on a vector of continuous pretreatment covariates x we provide substantive settings under which a subjective bayesian would ignore the randomization probabilities in analysing the trial data we then show that any statistician who ignores the randomization probabilities is unable to construct nominal 95 per cent confidence intervals for the true treatment effect that have both i an expected length which goes to zero with increasing sample size and ii a guaranteed expected actual coverage rate of at least 95 per cent over the ensemble of trials analysed by the statistician during his or her lifetime however we derive a new interval estimator depending on the randomization probabilities that satisfies i and ii

some comments on computer implementation of random number generators abstract several workers in simulation studies are concerned with the correct and portable computer implementation of random number generators since approximate calculations may lead to accumulated errors in various machines all implementations must be exact in a previous work gentle this journal 1990 presented several implementations for multiplicative congruential generators the aim of this paper is to describe some more methods especially the implementation using the decomposition of the modulus is very efficient

a direct approach to sparse discriminant analysis in ultra high dimensions summary sparse discriminant methods based on independence rules such as the nearest shrunken centroids classifier tibshirani et al 2002 and features annealed independence rules fan fan 2008 have been proposed as computationally attractive tools for feature selection and classification with high dimensional data a fundamental drawback of these rules is that they ignore correlations among features and thus could produce misleading feature selection and inferior classification we propose a new procedure for sparse discriminant analysis motivated by the least squares formulation of linear discriminant analysis to demonstrate our proposal we study the numerical and theoretical properties of discriminant analysis constructed via lasso penalized least squares our theory shows that the method proposed can consistently identify the subset of discriminative features contributing to the bayes rule and at the same time consistently estimate the bayes classification direction even when the dimension can grow faster than any polynomial order of the sample size the theory allows for general dependence among features simulated and real data examples show that lassoed discriminant analysis compares favourably with other popular sparse discriminant proposals

adaptive covariance matrix estimation through block thresholding estimation of large covariance matrices has drawn considerable recent attention and the theoretical focus so far has mainly been on developing a minimax theory over a fixed parameter space in this paper we consider adaptive covariance matrix estimation where the goal is to construct a single procedure which is minimax rate optimal simultaneously over each parameter space in a large collection a fully data driven block thresholding estimator is proposed the estimator is constructed by carefully dividing the sample covariance matrix into blocks and then simultaneously estimating the entries in a block by thresholding the estimator is shown to be optimally rate adaptive over a wide range of bandable covariance matrices a simulation study is carried out and shows that the block thresholding estimator performs well numerically some of the technical tools developed in this paper can also be of independent interest

simulated division with approximate factoring for the multiple recursive generator with both unrestricted multiplier and non mersenne prime modulus abstract this paper focuses on devising a general and efficient way of generating random numbers for the multiple recursive generator with both unrestricted multiplier and non mersenne prime modulus we propose a new algorithm that embeds the technique of approximate factoring into the simulated division method the proposed new algorithm improves the decomposition method in terms of both the suitability for various word sizes of the computers and the efficiency characteristics such as the number of arithmetic operations required and the computational time empirical simulations are conducted to compare and evaluate the computational time of this algorithm with the decomposition method for various computers

phantom maps in the stable module category abstract we study phantom maps in the stable module category stmod kg wherekis a field andgis a finite group in this article we almost exclusively deal with maps out of countably generated modules we show that the space ph kg m n of stabilized phantom maps m n has an expression as a lim 1 space which allows some control on its vanishing then we present a situation where all maps are phantom and need not be trivial we provide explicit details for such a particular situation finally we construct a universal phantom map we use it to show that the composite of two phantom maps is trivial and to characterize the modules with no nontrivial outbound phantom maps

transfinite adams representability abstract we consider the following problems in a well generated triangulated category t let be a regular cardinal and t t the full subcategory of compact objects is every functor h t op ab that preserves products of h t x t for some x in t is every natural transformation t x t t y t of the form t f t for some f x y in t if the answer to both questions is positive we say that t satisfies adams representability a classical result going back to brown and adams shows that the stable homotopy category satisfies 0 adams representability the case 0 is well understood thanks to the work of christensen keller and neeman in this paper we develop an obstruction theory to decide whether t satisfies adams representability we derive necessary and sufficient conditions of homological nature and we compute several examples in particular we show that there are rings satisfying adams representability for all 0 and rings which do not satisfy adams representability for any 0 moreover we exhibit rings for which the answer to both questions is no for all 2

an exhaustive analysis of two term multiple recursive random number generators with efficient multipliers this paper presents results of an exhaustive computer search for the maximum spectral value in full period two term kth order multiple recursive generators mrgs with efficient multipliers for orders k 2 3 7 we provide a table of kth order mrgs with largest spectral values these mrgs not only have the best spectral values found so far but also can be implemented efficiently by applying the approximate factoring method

an analysis of linear congruential random number generators when multiplier restrictions exist abstract this research conducts an exhaustive search for the best spectral test performance in a full period linear congruential generator lcg with the largest prime modulus smaller than 2b for bit sizes b 8 9 31 three types of restrictions on multipliers are presented and some full period lcgs are presented for use in three different spectral test dimensions according to the exhaustive searches we perform the results indicate differences exist among the numbers of possible multipliers for the three types of multiplier restrictions we demonstrate that these differences can affect the performance of spectral tests for different dimensions

high dimensional classification using features annealed independence rules classification using high dimensional features arises frequently in many contemporary statistical studies such as tumor classification using microarray or other high throughput data the impact of dimensionality on classifications is largely poorly understood in a seminal paper bickel and levina 2004 show that the fisher discriminant performs poorly due to diverging spectra and they propose to use the independence rule to overcome the problem we first demonstrate that even for the independence classification rule classification using all the features can be as bad as the random guessing due to noise accumulation in estimating population centroids in high dimensional feature space in fact we demonstrate further that almost all linear discriminants can perform as bad as the random guessing thus it is paramountly important to select a subset of important features for high dimensional classification resulting in features annealed independence rules fair the conditions under which all the important features can be selected by the two sample t statistic are established the choice of the optimal number of features or equivalently the threshold value of the test statistics are proposed based on an upper bound of the classification error simulation studies and real data analysis support our theoretical results and demonstrate convincingly the advantage of our new classification procedure

combined multiple recursive random number generators we analyze the random number generators obtained by combining two or more multiple recursive generators we study the lattice structure of such combined generators and argue that combination is a good way of obtaining robust generators based on a recurrence with many nonzero coefficients and which also possess a fast implementation

theoretical analyses of forward and backward heuristics of multiple recursive random number generators this article focuses on heuristic algorithms that are capable of finding multiple recursive generators mrgs with maximum spectral value criterion in a short period of time two promising algorithms provided by kao and tang are the forward and backward heuristics here we provide a deep understanding of the forward and backward heuristics for an mrg the forward heuristic finds a sequence of partial sets of multipliers that lead toward a better spectral value for the second order mrgs the performance of the backward heuristic theoretically outperforms that of the forward heuristic the effectiveness of the backward heuristic is numerically confirmed by the experiments we present

effective and efficient restriction on producing the multipliers for the multiple recursive random number generator abstract the most efficient and easiest way for implementing a multiple recursive random number generator is the approximate factoring af method in the literature the af method can be applied in the three types of restriction on multiplier this paper considers the effectiveness and efficiency of these restrictions from the theoretical and empirical points of view the most effective and efficient restriction is presented the number of arithmetic operations required by the af method is also discussed

analysis of semi parametric regression models with non ignorable non response in this article we consider the problem of making inferences about the parameter 0 indexing the conditional mean of an outcome given a vector of regressors when a subset of the variables outcome or covariates are missing for some study subjects and the probability of non response depends upon both observed and unobserved data values that is non response is non ignorable we propose a new class of inverse probability of censoring weighted estimators that are consistent and asymptotically normal can for estimating 0 when the non response probabilities can be parametrically modelled and a can estimator exists the proposed estimators do not require full specification of the likelihood and their computation does not require numerical integration we show that the asymptotic variance of the optimal estimator in our class attains the semi parametric variance bound for the model in some models no can estimator of 0 exists we provide a general algorithm for determining when can estimators of 0 exist our results follow after specializing a general representation described in the article for the efficient score and the influence function of regular asymptotically linear estimators in an arbitrary semi parametric model with non ignorable non response in which the probability of observing complete data is bounded away from zero and the non response probabilities can be parametrically modelled 1997 by john wiley sons ltd

good parameters and implementations for combined multiple recursive random number generators combining parallel multiple recursive sequences provides an efficient way of implementing random number generators with long periods and good structural properties such generators are statistically more robust than simple linear congruential generators that fit into a computer word we made extensive computer searches for good parameter sets with respect to the spectral test for combined multiple recursive generators of different sizes we also compare different implementations and give a specific code in c that is faster than previous implementations of similar generators

high dimensional linear discriminant analysis optimality adaptive algorithm and missing data the paper develops optimality theory for linear discriminant analysis in the high dimensional setting a data driven and tuning free classification rule which is based on an adaptive constrained l1 minimization approach is proposed and analysed minimax lower bounds are obtained and this classification rule is shown to be simultaneously rate optimal over a collection of parameter spaces in addition we consider classification with incomplete data under the missingness completely at random model an adaptive classifier with theoretical guarantees is introduced and the optimal rate of convergence for high dimensional linear discriminant analysis under the missingness completely at random model is established the technical analysis for the case of missing data is much more challenging than that for complete data we establish a large deviation result for the generalized sample covariance matrix which serves as a key technical tool and can be of independent interest an application to lung cancer and leukaemia studies is also discussed

covariance regularized regression and classification for high dimensional problems in recent years many methods have been developed for regression in high dimensional settings we propose covariance regularized regression a family of methods that use a shrunken estimate of the inverse covariance matrix of the features in order to achieve superior prediction an estimate of the inverse covariance matrix is obtained by maximizing its log likelihood under a multivariate normal model subject to a constraint on its elements this estimate is then used to estimate coefficients for the regression of the response onto the features we show that ridge regression the lasso and the elastic net are special cases of covariance regularized regression and we demonstrate that certain previously unexplored forms of covariance regularized regression can outperform existing methods in a range of situations the covariance regularized regression framework is extended to generalized linear models and linear discriminant analysis and is used to analyze gene expression data sets with multiple class and survival outcomes

powers of the phantom ideal the mono epi me exact structure on the morphisms of an exact category a e is introduced and used to prove ideal versions of salce s lemma christensen s ghost lemma and wakamatsu s lemma for an exact category salce s lemma establishes a bijective correspondence i i between the class of special precovering ideals of a e and that of its special preenveloping ideals me extensions of morphisms are used to define an extension i j of ideals christensen s lemma asserts that the class of special precovering respectively special preenveloping ideals is closed under products and extensions and that the bijective correspondence of salce s lemma satisfies ij j i and i j j i wakamatsu s lemma asserts that if a covering ideal i is closed under me extensions then it is a special precovering ideal as an application it is proved that if g is a finite group and is the ideal of phantom morphisms in the category k g mod then n 1 is the object ideal generated by projective modules where n is the nilpotency index of the jacobson radical j if r is a semiprimary ring with jn 0 then n is generated by projective modules for a right coherent ring r over which every cotorsion left r module has a coresolution of length n by pure injective modules n 1 is generated by flat modules

a general instrumental variable framework for regression analysis with outcome missing not at random summary the instrumental variable iv design is a well known approach for unbiased evaluation of causal effects in the presence of unobserved confounding in this article we study the iv approach to account for selection bias in regression analysis with outcome missing not at random in such a setting a valid iv is a variable which i predicts the nonresponse process and ii is independent of the outcome in the underlying population we show that under the additional assumption iii that the iv is independent of the magnitude of selection bias due to nonresponse the population regression in view is nonparametrically identified for point estimation under i iii we propose a simple complete case analysis which modifies the regression of primary interest by carefully incorporating the iv to account for selection bias the approach is developed for the identity log and logit link functions for inferences about the marginal mean of a binary outcome assuming i and ii only we describe novel and approximately sharp bounds which unlike robins manski bounds are smooth in model parameters therefore allowing for a straightforward approach to account for uncertainty due to sampling variability these bounds provide a more honest account of uncertainty and allows one to assess the extent to which a violation of the key identifying condition iii might affect inferences for illustration the methods are used to account for selection bias induced by hiv testing nonparticipation in the evaluation of hiv prevalence in the zambian demographic and health surveys

identifiability of normal and normal mixture models with nonignorable missing data abstractmissing data problems arise in many applied research studies they may jeopardize statistical inference of the model of interest if the missing mechanism is nonignorable that is the missing mechanism depends on the missing values themselves even conditional on the observed data with a nonignorable missing mechanism the model of interest is often not identifiable without imposing further assumptions we find that even if the missing mechanism has a known parametric form the model is not identifiable without specifying a parametric outcome distribution although it is fundamental for valid statistical inference identifiability under nonignorable missing mechanisms is not established for many commonly used models in this article we first demonstrate identifiability of the normal distribution under monotone missing mechanisms we then extend it to the normal mixture and t mixture models with nonmonotone missing mechanisms we discover that models under the logistic missing mechanism are less

a direct estimation approach to sparse linear discriminant analysis this article considers sparse linear discriminant analysis of high dimensional data in contrast to the existing methods which are based on separate estimation of the precision matrix and the difference of the mean vectors we introduce a simple and effective classifier by estimating the product directly through constrained l1 minimization the estimator can be implemented efficiently using linear programming and the resulting classifier is called the linear programming discriminant lpd rule the lpd rule is shown to have desirable theoretical and numerical properties it exploits the approximate sparsity of and as a consequence allows cases where it can still perform well even when and or cannot be estimated consistently asymptotic properties of the lpd rule are investigated and consistency and rate of convergence results are given the lpd classifier has superior finite sample performance and significant computational advantages over the existing methods that require separate estimation

penalized classification using fisher s linear discriminant we consider the supervised classification setting in which the data consist of p features measured on n observations each of which belongs to one of k classes linear discriminant analysis lda is a classical method for this problem however in the high dimensional setting where p n lda is not appropriate for two reasons first the standard estimate for the within class covariance matrix is singular and so the usual discriminant rule cannot be applied second when p is large it is difficult to interpret the classification rule obtained from lda since it involves all p features we propose penalized lda a general approach for penalizing the discriminant vectors in fisher s discriminant problem in a way that leads to greater interpretability the discriminant problem is not convex so we use a minorization maximization approach in order to efficiently optimize it when convex penalties are applied to the discriminant vectors in particular we consider the use of l 1 and fused lasso penalties our proposal is equivalent to recasting fisher s discriminant problem as a biconvex problem we evaluate the performances of the resulting methods on a simulation study and on three gene expression data sets we also survey past methods for extending lda to the high dimensional setting and explore their relationships with our proposal

several extensively tested multiple recursive random number generators abstract the multiple recursive generator mrg has been considered by many scholars as a very good random number rn generator this paper applies a sequential search to identify the mrgs of orders one two and three which are able to produce rns with good lattice structure in terms of the spectral value and beyer quotient to detect departures from local randomness and homogeneity extensive statistical tests including runs auto correlation chi square serial and the sparse occupancy tests have been conducted in approximately 19 3 billion candidates only four mrgs namely 1280550 45991 0 45991 1758790 885300443 0 1552858447 and 885300443 1546795921 598295599 have passed all the theoretical and empirical tests among which 0 45991 1758790 can be implemented efficiently by applying the approximate factoring method and is therefore most recommended

sparse discriminant analysis we consider the problem of performing interpretable classification in the high dimensional setting in which the number of features is very large and the number of observations is limited this setting has been studied extensively in the chemometrics literature and more recently has become commonplace in biological and medical applications in this setting a traditional approach involves performing feature selection before classification we propose sparse discriminant analysis a method for performing linear discriminant analysis with a sparseness criterion imposed such that classification and feature selection are performed simultaneously sparse discriminant analysis is based on the optimal scoring interpretation of linear discriminant analysis and can be extended to perform sparse discrimination via mixtures of gaussians if boundaries between classes are nonlinear or if subgroups are present within each class our proposal also provides low dimensional views of the discriminative directions

random number generation with the recursion x t x t 3p x t 3q abstract a generalized feedback shift register gfsr algorithm proposed by lewis and payne 1973 uses a primitive trinomial to generate a sequence of pseudorandom numbers we propose a similar algorithm which uses a primitive polynomial with many nonzero terms but generates a number as fast as the original gfsr algorithm our sequence is guaranteed to be equidistributed in higher dimensions and to have a good autocorrelation property extensive statistical tests have been performed on the sequences generated by our algorithm and the results were quite satisfactory

ideal approximation theory abstract let a e be an exact category and f ext a subfunctor a morphism in a is an f phantom if the pullback of an e conflation along is a conflation in f if the exact category a e has enough injective objects and projective morphisms it is proved that an ideal i of a is special precovering if and only if there is a subfunctor f ext with enough injective morphisms such that i is the ideal of f phantom morphisms a crucial step in the proof is a generalization of salce s lemma for ideal cotorsion pairs if i is a special precovering ideal then the ideal cotorsion pair i i generated by i in a e is complete this theorem is used to verify 1 that the ideal cotorsion pair cogenerated by the pure injective modules of r mod is complete 2 that the ideal cotorsion pair cogenerated by the contractible complexes in the category of complexes ch r mod is complete and using auslander and reiten s theory of almost split sequences 3 that the ideal cotorsion pair cogenerated by the jacobson radical jac mod of the category mod of finitely generated representations of an artin algebra is complete

computer implementation of random number generators abstract good theoretical work on algorithms is often defeated by a poor computer implementation this is particularly true of random number generators for two reasons the overwhelming concern for speed and the inherent ill conditioning of the problem the emphasis on speed has often resulted in shortcuts see carta 1990 these shortcuts often accept approximations which can be disastrous in random number generation other shortcuts are very machine specific and then often get used in the wrong environment the problem of random number generation using a deterministic recursion is ill conditioned in the sense that small deviations relative errors of the order of 10 10 can cause complete loss of accuracy this is why any approximation is disastrous in this paper we will concentrate on the coding considerations for a special case of a good algorithm namely a multiplicative congruential generator with properly chosen multiplier and modulus the discussion will use fortran constructs but it is applicable to any programming language we will also address implementation of recursive generators on vector processors