estimating the proportion of true null hypotheses with application to dna microarray data summary we consider the problem of estimating the proportion of true null hypotheses 0 i n a multiple hypothesis set up the tests are based on observed p values we first review published estimators based on the estimator that was suggested by schweder and spjotvoll then we derive new estimators based on nonparametric maximum likelihood estimation of the p value density restricting to decreasing and convex decreasing densities the estimators of 0 are all derived under the assumption of independent test statistics their performance under dependence is investigated in a simulation study we find that the estimators are relatively robust with respect to the assumption of independence and work well also for test statistics with moderate dependence

estimating the null and the proportion of nonnull effects in large scale multiple comparisons an important issue raised by efron in the context of large scale multiple comparisons is that in many applications the usual assumption that the null distribution is known is incorrect and seemingly negligible differences in the null may result in large differences in subsequent studies this suggests that a careful study of estimation of the null is indispensable in this article we consider the problem of estimating a null normal distribution and a closely related problem estimation of the proportion of nonnull effects we develop an approach based on the empirical characteristic function and fourier analysis the estimators are shown to be uniformly consistent over a wide class of parameters we investigate the numerical performance of the estimators using both simulated and real data in particular we apply our procedure to the analysis of breast cancer and human immunodeficiency virus microarray datasets the estimators perform favorably compared with existing methods

false discovery rate control with groups in the context of large scale multiple hypothesis testing the hypotheses often possess certain group structures based on additional information such as gene ontology in gene expression data and phenotypes in genome wide association studies it is hence desirable to incorporate such information when dealing with multiplicity problems to increase statistical power in this article we demonstrate the benefit of considering group structure by presenting a p value weighting procedure which utilizes the relative importance of each group while controlling the false discovery rate under weak conditions the procedure is easy to implement and shown to be more powerful than the classical benjamini hochberg procedure in both theoretical and simulation studies by estimating the proportion of true null hypotheses the data driven procedure controls the false discovery rate asymptotically our analysis on one breast cancer dataset confirms that the procedure performs favorably compared with the classical method

power enhanced multiple decision functions controlling family wise error and false discovery rates improved procedures in terms of smaller missed discovery rates mdr for performing multiple hypotheses testing with weak and strong control of the family wise error rate fwer or the false discovery rate fdr are developed and studied the improvement over existing procedures such as the sidak procedure for fwer control and the benjamini hochberg bh procedure for fdr control is achieved by exploiting possible differences in the powers of the individual tests results signal the need to take into account the powers of the individual tests and to have multiple hypotheses decision functions which are not limited to simply using the individual p values as is the case for example with the sidak bonferroni or bh procedures they also enhance understanding of the role of the powers of individual tests or more precisely the receiver operating characteristic roc functions of decision processes in the search for better multiple hypotheses testing procedures a decision theoretic framework is utilized and through auxiliary randomizers the procedures could be used with discrete or mixed type data or with rank based nonparametric tests this is in contrast to existing p value based procedures whose theoretical validity is contingent on each of these p value statistics being stochastically equal to or greater than a standard uniform variable under the null hypothesis proposed procedures are relevant in the analysis of high dimensional large m small n data sets arising in the natural physical medical economic and social sciences whose generation and creation is accelerated by advances in high throughput technology notably but not limited to microarray technology

simultaneous testing of grouped hypotheses finding needles in multiple haystacks in large scale multiple testing problems data are often collected from heterogeneous sources and hypotheses form into groups that exhibit different characteristics conventional approaches including the pooled and separate analyses fail to efficiently utilize the external grouping information we develop a compound decision theoretic framework for testing grouped hypotheses and introduce an oracle procedure that minimizes the false nondiscovery rate subject to a constraint on the false discovery rate it is shown that both the pooled and separate analyses can be uniformly improved by the oracle procedure we then propose a data driven procedure that is shown to be asymptotically optimal simulation studies show that our procedures enjoy superior performance and yield the most accurate results in comparison with both the pooled and separate procedures a real data example with grouped hypotheses is studied in detail using different methods both theoretical and numerical results demonstrate that exploit

exploring the information in p values for the analysis and planning of multiple test experiments a new methodology is proposed for estimating the proportion of true null hypotheses in a large collection of tests each test concerns a single parameter delta whose value is specified by the null hypothesis we combine a parametric model for the conditional cumulative distribution function cdf of the p value given delta with a nonparametric spline model for the density g delta of delta under the alternative hypothesis the proportion of true null hypotheses and the coefficients in the spline model are estimated by penalized least squares subject to constraints that guarantee that the spline is a density the estimator is computed efficiently using quadratic programming our methodology produces an estimate g delta of the density of delta when the null is false and can address such questions as when the null is false is the parameter usually close to the null or far away this leads us to define a falsely interesting discovery rate fidr a generalization of the false discovery rate we contrast the fidr approach to efron s 2004 journal of the american statistical association 99 96 104 empirical null hypothesis technique we discuss the use of g in sample size calculations based on the expected discovery rate edr our recommended estimator of the proportion of true nulls has less bias compared to estimators based upon the marginal density of the p values at 1 in a simulation study we compare our estimators to the convex decreasing estimator of langaas lindqvist and ferkingstad 2005 journal of the royal statistical society series b 67 555 572 the most biased of our estimators is very similar in performance to the convex decreasing estimator as an illustration we analyze differences in gene expression between resistant and susceptible strains of barley

optimal rates of convergence for estimating the null density and proportion of nonnull effects in large scale multiple testing an important estimation problem that is closely related to large scale multiple testing is that of estimating the null density and the proportion of nonnull effects a few estimators have been introduced in the literature however several important problems including the evaluation of the minimax rate of convergence and the construction of rate optimal estimators remain open in this paper we consider optimal estimation of the null density and the proportion of nonnull effects both minimax lower and upper bounds are derived the lower bound is established by a two point testing argument where at the core is the novel construction of two least favorable marginal densities f 1 and f 2 the density f 1 is heavy tailed both in the spatial and frequency domains and f 2 is a perturbation of f 1 such that the characteristic functions associated with f 1 and f 2 match each other in low frequencies the minimax upper bound is obtained by constructing estimators which rely on the empirical characteristic function and fourier analysis the estimator is shown to be minimax rate optimal compared to existing methods in the literature the proposed procedure not only provides more precise estimates of the null density and the proportion of the nonnull effects but also yields more accurate results when used inside some multiple testing procedures which aim at controlling the false discovery rate fdr the procedure is easy to implement and numerical results are given

a ridge parameter approach to deconvolution kernel methods for deconvolution have attractive features and prevail in the literature however they have disadvantages which include the fact that they are usually suitable only for cases where the error distribution is infinitely supported and its characteristic function does not ever vanish even in these settings optimal convergence rates are achieved by kernel estimators only when the kernel is chosen to adapt to the unknown smoothness of the target distribution in this paper we suggest alternative ridge methods not involving kernels in any way we show that ridge methods a do not require the assumption that the error distribution characteristic function is nonvanishing b adapt themselves remarkably well to the smoothness of the target density with the result that the degree of smoothness does not need to be directly estimated and c give optimal convergence rates in a broad range of settings

nonparametric regression estimation in the heteroscedastic errors in variables problem in the classical errors in variables problem the goal is to estimate a regression curve from data in which the explanatory variable is measured with error in this context nonparametric methods have been proposed that rely on the assumption that the measurement errors are identically distributed although there are many situations in which this assumption is too restrictive nonparametric estimators in the more realistic setting of heteroscedastic errors have not been studied in the literature we propose an estimator of the regression function in such a setting and show that it is optimal we give estimators in cases in which the error distributions are unknown and replicated observations are available practical methods including an adaptive bandwidth selector for the errors in variables regression problem are suggested and their finite sample performance is illustrated through simulated and real data examples

the influence of some fuzzy implication operators on the accuracy of a fuzzy model part ii abstract the influence of fuzzy implication operators and the connective also on the accuracy of a fuzzy model of a d c series motor is considered some typical fuzzy implication operators are applied to the construction of a fuzzy model of a d c series motor a root mean square error is used as the criterion of the fuzzy model s adequacy to the real system a number of mathematical operations necessary for the implementation of the fuzzy model are used as the criterion by which the fuzzy model s applicability if estimated from the point of view of computing techniques the best types of fuzzy relations representing fuzzy models of a real system are chosen in order to secure the least root mean square error with minimal number of mathematical operations necessary for computer implementation

sizer for exploration of structures in curves abstract in the use of smoothing methods in data analysis an important question is which observed features are really there as opposed to being spurious sampling artifacts an approach is described based on scale space ideas originally developed in the computer vision literature assessment of significant zero crossings of derivatives results in the sizer map a graphical device for display of significance of features with respect to both location and scale here scale means level of resolution that is bandwidth

multiple testing of composite null hypotheses in heteroscedastic models in large scale studies the true effect sizes often range continuously from zero to small to large and are observed with heteroscedastic errors in practical situations where the failure to reject small deviations from the null is inconsequential specifying an indifference region or forming composite null hypotheses can greatly reduce the number of unimportant discoveries in multiple testing the heteroscedasticity issue poses new challenges for multiple testing with composite nulls in particular the conventional framework in multiple testing which involves rescaling or standardization is likely to distort the scientific question we propose the concept of a composite null distribution for heteroscedastic models and develop an optimal testing procedure that minimizes the false nondiscovery rate subject to a constraint on the false discovery rate the proposed approach is different from conventional methods in that the effect size statistical significance and multiplicity issues are addressed in

operating characteristics and extensions of the false discovery rate procedure summary we investigate the operating characteristics of the benjamini hochberg false discovery rate procedure for multiple testing this is a distribution free method that controls the expected fraction of falsely rejected null hypotheses among those rejected the paper provides a framework for understanding more about this procedure we first study the asymptotic properties of the deciding point d that determines the critical p value from this we obtain explicit asymptotic expressions for a particular risk function we introduce the dual notion of false non rejections and we consider a risk function that combines the false discovery rate and false non rejections we also consider the optimal procedure with respect to a measure of conditional risk

empirical bayes analysis of a microarray experiment microarrays are a novel technology that facilitates the simultaneous measurement of thousands of gene expression levels a typical microarray experiment can produce millions of data points raising serious problems of data reduction and simultaneous inference we consider one such experiment in which oligonucleotide arrays were employed to assess the genetic effects of ionizing radiation on seven thousand human genes a simple nonparametric empirical bayes model is introduced which is used to guide the efficient reduction of the data to a single summary statistic per gene and also to make simultaneous inferences concerning which genes were affected by the radiation although our focus is on one specific experiment the proposed methods can be applied quite generally the empirical bayes inferences are closely related to the frequentist false discovery rate fdr criterion

on deconvolution with repeated measurements in a large class of statistical inverse problems it is necessary to suppose that the transformation that is inverted is known although in many applications it is unrealistic to make this assumption the problem is often insoluble without it however if additional data are available then it is possible to estimate consistently the unknown error density data are seldom available directly on the transformation but repeated or replicated measurements increasingly are becoming available such data consist of intrinsic values that are measured several times with errors that are generally independent working in this setting we treat the nonparametric deconvolution problems of density estimation with observation errors and regression with errors in variables we show that even if the number of repeated measurements is quite small it is possible for modified kernel estimators to achieve the same level of performance they would if the error distribution were known indeed density and regression estimators can be constructed from replicated data so that they have the same first order properties as conventional estimators in the known error case without any replication but with sample size equal to the sum of the numbers of replicates practical methods for constructing estimators with these properties are suggested involving empirical rules for smoothing parameter choice

proportion of non zero normal means universal oracle equivalences and uniformly consistent estimators d0 we propose a general approach to construct the universal oracle equivalence of the proportion the construction is based on the underlying characteristic function the oracle equivalence reduces the problem of estimating the proportion to the problem of estimating the oracle which is relatively easier to handle in fact the oracle equivalence naturally yields a family of estimators for the proportion which are consistent under mild conditions uniformly across a wide class of parameters the approach compares favourably with recent works by meinshausen and rice and genovese and wasserman in particular the consistency is proved for an unprecedentedly broad class of situations the class is almost the largest that can be hoped for without further constraints on the model we also discuss various extensions of the approach report results on simulation experiments and make connections between the approach and several recent procedures in large scale multiple testing including the false discovery rate approach and the local false discovery rate approach

kernel type estimators of jump points and values of a regression function in the fixed design nonparametric regression model kernel type estimators of the locations of jump points and the corresponding sizes of jump values of the regression function are proposed these kernel type estimators are analyzed with almost sure results and limiting distributions using the limiting distributions we are able to test the number of jump points and give asymptotic confidence intervals for the sizes of jump values of the regression function simulation studies demonstrate that the asymptotic results hold for reasonable sample sizes

multiple hypotheses testing and expected number of type i errors the performance of multiple test procedures with respect to error control is an old issue assuming that all hypotheses are true we investigate the behavior of the expected number of type i errors ene as a characteristic of certain multiple tests controlling the familywise error rate fwer or the false discovery rate fdr at a prespecified level we derive explicit formulas for the distribution of the number of false rejections as well as for the ene for single step step down and step up procedures based on independent p values moreover we determine the corresponding asymptotic distributions of the number of false rejections as well as explicit formulae for the ene if the number of hypotheses tends to infinity in case of fwer control we mostly obtain poisson distributions and in one case a geometric distribution as limiting distributions in case of fdr control we obtain limiting distributions which are apparently not named in the literature surprisingly the ene is bounded by a small number regardless of the number of hypotheses under consideration finally it turns out that in case of dependent test statistics the ene behaves completely differently compared to the case of independent test statistics

low order approximations in deconvolution and regression with errors in variables summary we suggest two new methods which are applicable to both deconvolution and regression with errors in explanatory variables for nonparametric inference the two approaches involve kernel or orthogonal series methods they are based on defining a low order approximation to the problem at hand and proceed by constructing relatively accurate estimators of that quantity rather than attempting to estimate the true target functions consistently of course both techniques could be employed to construct consistent estimators but in many contexts of importance e g those where the errors are gaussian consistency is from a practical viewpoint an unattainable goal we rephrase the problem in a form where an explicit interpretable low order approximation is available the information that we require about the error distribution the error in variables distribution in the case of regression is only in the form of low order moments and so is readily obtainable by a rudimentary analysis of indirect measurements of errors e g through repeated measurements in particular we do not need to estimate a function such as a characteristic function which expresses detailed properties of the error distribution this feature of our methods coupled with the fact that all our estimators are explicitly defined in terms of readily computable averages means that the methods are particularly economical in computing time

estimating the proportion of false null hypotheses among a large number of independently tested hypotheses we consider the problem of estimating the number of false null hypotheses among a very large number of independently tested hypotheses focusing on the situation in which the proportion of false null hypotheses is very small we propose a family of methods for establishing lower 100 1 a confidence bounds for this proportion based on the empirical distribution of the p values of the tests methods in this family are then compared in terms of ability to consistently estimate the proportion by letting 0 as the number of hypothesis tests increases and the proportion decreases this work is motivated by a signal detection problem that occurs in astronomy

false discovery rate adjusted multiple confidence intervals for selected parameters often in applied research confidence intervals cis are constructed or reported only for parameters selected after viewing the data we show that such selected intervals fail to provide the assumed coverage probability by generalizing the false discovery rate fdr approach from multiple testing to selected multiple cis we suggest the false coverage statement rate fcr as a measure of interval coverage following selection a general procedure is then introduced offering fcr control at level q under any selection rule the procedure constructs a marginal ci for each selected parameter but instead of the confidence level 1 q being used marginally q is divided by the number of parameters considered and multiplied by the number selected if we further use the fdr controlling testing procedure of benjamini and hochberg for selecting the parameters the newly suggested procedure offers cis that are dual to the testing procedure and are shown to be optimal in the independent case under the positive re

oracle and adaptive compound decision rules for false discovery rate control we develop a compound decision theory framework for multiple testing problems and derive an oracle rule based on the z values that minimizes the false nondiscovery rate fnr subject to a constraint on the false discovery rate fdr we show that many commonly used multiple testing procedures which are p value based are inefficient and propose an adaptive procedure based on the z values the z value based adaptive procedure asymptotically attains the performance of the z value oracle procedure and is more efficient than the conventional p value based methods we investigate the numerical performance of the adaptive procedure using both simulated and real data in particular we demonstrate our method in an analysis of the microarray data from a human immunodeficiency virus study that involves testing a large number of hypotheses simultaneously

asymptotic distribution of p values in composite null models abstract we investigate the compatibility of a null model h 0 with the data by calculating a p value that is the probability under h 0 that a given test statistic t exceeds its observed value when the null model consists of a single distribution the p value is readily obtained and it has a uniform distribution under h 0 on the other hand when the null model depends on an unknown nuisance parameter one must somehow get rid of e g by estimating it to calculate a p value various proposals have been suggested to remove each yielding a different candidate p value but unlike the simple case these p values typically are not uniformly distributed under the null model in this article we investigate their asymptotic distribution under h 0 we show that when the asymptotic mean of the test statistic t depends on the posterior predictive p value of guttman and rubin and the plug in p value are conservative i e their asymptotic distributions are more concentrated around 1 2 than a uni

some results on false discovery rate in stepwise multiple testing procedures the concept of false discovery rate fdr has been receiving increasing attention by researchers in multiple hypotheses testing this paper produces some theoretical results on the fdr in the context of stepwise multiple testing procedures with dependent test statistics it was recently shown by benjamini and yekutieli that the benjamini hochberg step up procedure controls the fdr when the test statistics are positively dependent in a certain sense this paper strengthens their work by showing that the critical values of that procedure can be used in a much more general stepwise procedure under similar positive dependency it is also shown that the fdr controlling benjamini liu step down procedure originally developed for independent test statistics works even when the test statistics are positively dependent in some sense an explicit expression for the fdr of a generalized stepwise procedure and an upper bound to the fdr of a step down procedure are obtained in terms of probability distributions of ordered components of dependent random variables before establishing the main results 1 introduction the main purpose of this paper is to present some formulas and inequalities providing theoretical insight into when and what stepwise procedures control the false discovery rate fdr at a prespecified level in multiple hypotheses testing the fdr first introduced in multiple testing by benjamini and hochberg 1995 is the expected proportion of erroneously rejected null hypotheses with v and r representing respectively the number of true null hypotheses rejected and the total number of null hypotheses rejected in a multiple testing procedure let q v r if r 0a nd 0i fr 0 then e q is the fdr of that procedure as they have argued it is an appropriate error rate to control in many practical situations particularly where a large number of null hypotheses are involved they also put forward an fdr controlling stepup procedure that is more powerful than comparable procedures controlling the traditional familywise error fwe rate when the underlying test statistics are independent benjamini and liu 1999 recently introduced a step down procedure with the same property the concept of fdr is now being used in a wide variety of applications abramovich and benjamini 1996 basford and tukey 1997 drigalenko

false discovery rates for spatial signals the problem of multiple testing for the presence of signal in spatial data can involve numerous locations traditionally each location is tested separately for signal presence but then the findings are reported in terms of clusters of nearby locations this is an indication that the units of interest for testing are clusters rather than individual locations the investigator may know a priori these more natural units or an approximation to them we suggest testing these cluster units rather than individual locations thus increasing the signal to noise ratio within the unit tested as well as reducing the number of hypothesis tests conducted because the signal may be absent from part of each cluster we define a cluster as containing a signal if the signal is present somewhere within the cluster we suggest controlling the false discovery rate fdr on clusters i e the expected proportion of clusters rejected erroneously out of all clusters rejected or its extension to general weights wfdr we int

a stochastic process approach to false discovery control this paper extends the theory of false discovery rates fdr pioneered by benjamini and hochberg j roy statist soc ser b 57 1995 289 300 we develop a framework in which the false discovery proportion fdp the number of false rejections divided by the number of rejections is treated as a stochastic process after obtaining the limiting distribution of the process we demonstrate the validity of a class of procedures for controlling the false discovery rate the expected fdp we construct a confidence envelope for the whole fdp process from these envelopes we derive confidence thresholds for controlling the quantiles of the distribution of the fdp as well as controlling the number of false discoveries we also investigate methods for estimating the p value distribution

estimation and confidence sets for sparse normal mixtures for high dimensional statistical models researchers have begun to focus on situations which can be described as having relatively few moderately large coefficients such situations lead to some very subtle statistical problems in particular ingster and donoho and jin have considered a sparse normal means testing problem in which they described the precise demarcation or detection boundary meinshausen and rice have shown that it is even possible to estimate consistently the fraction of nonzero coordinates on a subset of the detectable region but leave unanswered the question of exactly in which parts of the detectable region consistent estimation is possible in the present paper we develop a new approach for estimating the fraction of nonzero means for problems where the nonzero means are moderately large we show that the detection region described by ingster and donoho and jin turns out to be the region where it is possible to consistently estimate the expected fraction of nonzero coordinates this theory is developed further and minimax rates of convergence are derived a procedure is constructed which attains the optimal rate of convergence in this setting furthermore the procedure also provides an honest lower bound for confidence intervals while minimizing the expected length of such an interval simulations are used to enable comparison with the work of meinshausen and rice where a procedure is given but where rates of convergence have not been discussed extensions to more general gaussian mixture models are also given

the control of the false discovery rate in multiple testing under dependency benjamini and hochberg suggest that the false discovery rate may be the appropriate error rate to control in many applied multiple testing problems a simple procedure was given there as an fdr controlling procedure for independent test statistics and was shown to be much more powerful than comparable procedures which control the traditional familywise error rate we prove that this same procedure also controls the false discovery rate when the test statistics have positive regression dependency on each of the test statistics corresponding to the true null hypotheses this condition for positive dependency is general enough to cover many problems of practical interest including the comparisons of many treatments with a single control multivariate normal test statistics with positive correlation matrix and multivariate t furthermore the test statistics may be discrete and the tested hypotheses composite without posing special difficulties for all other forms of dependency a simple conservative modification of the procedure controls the false discovery rate thus the range of problems for which a procedure with proven fdr control can be offered is greatly increased 1 1 simultaneous hypotheses testing the control of the increased type i error when testing simultaneously a family of hypotheses is a central issue in the area of multiple comparisons rarely are we interested only in whether all hypotheses are jointly true or not which is the test of the intersection null hypothesis in most applications we infer about the individual hypotheses realizing that some of the tested hypotheses are usually true we hope not all and some are not we wish to decide which ones are not true indicating statistical discoveries an important such problem is that of multiple endpoints in a clinical trial a new treatment is compared with an existing one in terms of a large number of potential benefits endpoints

a direct approach to false discovery rates summary multiple hypothesis testing involves guarding against much more complicated errors than single hypothesis testing whereas we typically control the type i error rate for a single hypothesis test a compound error rate is controlled for multiple hypothesis tests for example controlling the false discovery rate fdr traditionally involves intricate sequential p value rejection methods based on the observed data whereas a sequential p value method fixes the error rate and estimates its corresponding rejection region we propose the opposite approach we fix the rejection region and then estimate its corresponding error rate this new approach offers increased applicability accuracy and power we apply the methodology to both the positive false discovery rate pfdr and fdr and provide evidence for its benefits it is shown that pfdr is probably the quantity of interest over fdr also discussed is the calculation of the q value the pfdr analogue of the p value which eliminates the need to set the error rate beforehand as is traditionally done some simple numerical examples are presented that show that this new approach can yield an increase of over eight times in power compared with the benjamini hochberg fdr method

succinct data structures for searchable partial sums with optimal worst case performance abstract the notion of succinct indexes can be dated back from the debut of jacobson s thesis 1988 14 and has triggered many results in the last decade in traditional indexing some given data are preprocessed so as to support online queries and updates on the data as efficiently as possible when succinctness is involved we are restricted to index the data using only an information theoretically minimum number of bits this paper concerns the succinct indexing schemes for a well studied problem called searchable partial sums sps in sps an array a of n non negative k bit integers is preprocessed so as to support online sum and search queries and possibly update operation of individual entry a succinct indexing scheme would allow only k n o k n bits to represent the array a the only known result is that when k 1 in this case it is known as the dynamic bit array problem we can support both queries in o log b n time and update in o b amortized time for any b with lg n lg lg n b n this paper shows that even for k o lg lg n we can index a succinctly such that both query and update operations can be supported using the same time complexities moreover the time for update becomes the worst case time furthermore the tradeoff between the query times and the update time is optimal as implied by pa trascu and demaine s lower bound result 2006 24 in general when k o lg u we show a lower bound of lg n lg lg n time for the search query irrespective of the update time this gives a tighter lower bound as compared to that of pa trascu and demaine s which is a consequence of the requirement of succinctness on the other hand we give a succinct index that can support sum in o log b n time search in o log b n time and update in o b time where min lg lg n lg lg u lg lg lg u lg n lg lg n the query times are optimal when b n this paper also extends the searchable partial sums with insert and delete operations and provides a succinct data structure for some cases

density estimation in the presence of heteroscedastic measurement error we consider density estimation when the variable of interest is subject to heteroscedastic measurement error the density is assumed to have a smooth but unknown functional form that we model with a penalized mixture of b splines we treat the situation in which multiple mismeasured observations of each variable of interest are observed for at least some of the subjects and the measurement error is assumed to be additive and normal the measurement error variance function is modeled with a second penalized mixture of b splines the article s main contributions are to address the effects of heteroscedastic measurement error effectively explain the biases caused by ignoring heteroscedasticity and present an equivalent kernel for a spline based density estimator derivation of the equivalent kernel may be of independent interest we use small sigma asymptotics to approximate the biases incurred by assuming that the measurement error is homoscedastic when it actually is heteroscedastic the biases incurred

on the optimal rates of convergence for nonparametric deconvolution problems deconvolution problems arise in a variety of situations in statistics an interesting problem is to estimate the density f of a random variable x based on n i i d observations from y x varepsilon where varepsilon is a measurement error with a known distribution in this paper the effect of errors in variables of nonparametric deconvolution is examined insights are gained by showing that the difficulty of deconvolution depends on the smoothness of error distributions the smoother the harder in fact there are two types of optimal rates of convergence according to whether the error distribution is ordinary smooth or supersmooth it is shown that optimal rates of convergence can be achieved by deconvolution kernel density estimators

the positive false discovery rate a bayesian interpretation and the q value multiple hypothesis testing is concerned with controlling the rate of false positives when testing several hypotheses simultaneously one multiple hypothesis testing error measure is the false discovery rate fdr which is loosely defined to be the expected proportion of false positives among all significant hypotheses the fdr is especially appropriate for exploratory analyses in which one is interested in finding several significant results among many tests in this work we introduce a modified version of the fdr called the positive false discovery rate pfdr we discuss the advantages and disadvantages of the pfdr and investigate its statistical properties when assuming the test statistics follow a mixture distribution we show that the pfdr can be written as a bayesian posterior probability and can be connected to classification theory these properties remain asymptotically true under fairly general conditions even under certain forms of dependence also a new quantity called the q value is introduced and investigated which is a natural bayesian posterior p value or rather the pfdr analogue of the p value 1 introduction when testing a single hypothesis one is usually concerned with controlling the false positive rate while maximizing the probability of detecting an effect when one really exists in statistical terms we maximize the power conditional on the type i error rate being at or below some level the field of multiple hypothesis testing tries to extend this basic paradigm to the situation where several hypotheses are tested simultaneously one must define an appropriate compound error measure according to the rate of false positives one is willing to encounter then a procedure is developed that allows one to control the error rate at a desired level while maintaining the power of each test as much as possible the most commonly controlled quantity when testing multiple hypotheses is the family wise error rate fwer which is the probability of yielding one or more false positives out of all hypotheses tested the most familiar example of this is the bonferroni method if there are m hypothesis tests each test is controlled so that the probability of a false positive is less than or equal to m for some chosen value of it then follows that the overall fwer is less than or equal to m any

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples