71 research outputs found
Operating Characteristics and Extensions of the FDR Procedure
No full textWe investigate the operating characteristics of the Benjamini-Hochberg false discovery rate (FDR) procedure for multiple testing. This is a distribution free method that controls the expected fraction of falsely rejected null hypotheses among those rejected. This paper provides a framework for understanding how and why this procedure works. We start by studying the special case where the p-values under the alternative have a common distribution, where we are able to obtain many insights into this new procedure. We first obtain bounds on the ``deciding point'' D that determines the critical p-value. From this, we obtain explicit asymptotic expressions for a particular risk funciton. We introduce the dual notion of false non-rejections (FNR) and we consider a risk function that combines FDR and FNR. We also consider the optimal procedure with respect to a measure of conditional risk.</p
Frasian Inference
No full textDon Fraser has given an interesting account of the agreements and disagreements between Bayesian posterior probabilities and confidence levels. In this comment I discuss some cases where the lack of such agreement is extreme. I then discuss a few cases where it is possible to have Bayes procedures with frequentist validity. Such frequentist-Bayesian—or Frasian—methods deserve more attention.</p
Photochemical Products from Ergosterol in the Plasma Membrane of the Yeast Rhodotorula minuta cells illuminated by Near-UV
No full textWhen the plasma mumbrane of the yeast Rhodotorula minuta cells was exposed to near-UV at 0℃, three new compounds were formes photochemically with decrease in ergosterol content. When 0.4%-SDS solution containing ergosterol was illuminated with near-UV, the same three compounds as in the membrane were formed. As a result of instruments analysis, one of three photochemical products was identified as previtamine D2 and another two compounds were characterized chemically that they had the structure of ergosta-4, 7, 22-trein-3-on. In addition, these photochemical products did not affect the growth and carotenogenesis of the yeast Rhodotorulaminuta.赤色酵母R.minuta細胞膜に長波長紫外光を照射した時に生じるエルゴステロール由来の光化学反応生成物の単離・精製およびその同定を行った。結果、3種類の物質が単離・精製され、そのうち1つはPrevitamin D2であることが同定され、残りの2つについてはergosta-4,7,22-trien-3-on の何かの炭素にOH基が付いたものであることが分かった。単離した成分を培地に加え、菌体の増殖とカロテノイド生合成に対する影響を調べた結果、これらの物質はそのどちらにも影響を及ぼさなかった
Low-Noise Density Clustering
No full textWe study density-based clustering under low-noise conditions. Our framework allows for sharply defined clusters such as clusters on lower dimensional manifolds. We show that accurate clustering is possible even in high dimensions. We propose two data-based methods for choosing the bandwidth and we study the stability properties of density clusters. We show that a simple graph-based algorithm known as the ``friends-of-friends'' algorithm successfully approximates the high density clusters.</p
Exceedance Control of the False Discovery Proportion
No full textMultiple testing methods to control the False Discovery Rate (FDR), the expected proportion of falsely rejected null hypotheses among all rejections) have received much attention. It can be valuable instead to control not the mean of this false discovery proportion (FDP) but the probability that the FDP exceeds a specified bound. In this paper, we construct a general class of methods for exceedance control of FDP based on inverting tests of uniformity. The method also produces a confidence envelope for the FDP as a function of rejection threshold. We discuss how to select a procedure with good power.</p
Bayesian Frequentist Multiple Testing
No full textWe introduce a Bayesian approach to multiple testing. The method is an extension of the false discovery rate (FDR) method due to Benjamini and Hochberg (1995). We also examine the empirical Bayes approach to simultaneous inference proposed by Efron, Tibshirani, Storey and Tusher (2001). We show that, in contrast to the single hypothesis case - where Bayes and frequentist tests do not agree even asymptotically - in the multiple testing case we do have asymptotic agreement.</p
Adaptive Confidence Bands
No full textWe show that there do not exist adaptive confidence bands for curve estimation except under very restrictive assumptions. We propose instead to construct adaptive bands that cover a surrogate function f⋆ which is close to, but simpler than, f. The surrogate captures the significant features in f. We establish lower bounds on the width for any confidence band for f⋆ and construct a procedure that comes within a small constant factor of attaining the lower bound for finite-samples.</p
Symmetric, Coherent, Choquet Capacities
No full textChoquet capacities are a generalization of probability measures that
arise in robustness, decision theory and game theory. Many capacities
that arise in robustness are symmetric or can be transformed into symmetric
capacities. We characterize the extreme points of the set of upper
distribution functions corresponding to coherent, symmetric Choquet capacities
on [0,1]. We also show that the set of 2-alternating capacities is a
simplex and we give a Choquet representation of this set
Iterative Markov Chain Monte Carlo Computation of Reference Priors and Minimax Risk
No full textWe present an iterative Markov chain Monte Carlo algorithm for computing reference priors and minimax risk for general parametric families. Our approach uses MCMC techniques based on the Blahut-Arimoto algorithm for computing channel capacity in information theory. We give a statistical analysis of the algorithm, bounding the numbers of samples required for ties to chaotic algorithm to closely approximate the deterministic algorithm in each iteration. Simulations are presented for several examples from exponential families. Although we focus on applications to reference priors and minimax risk, the methods and analysis we develop are applicable to a much broader class of optimization problems and iterative algorithms.</p
Distribution Free Prediction Bands
No full textWe study distribution free, nonparametric prediction bands with a special focus on their finite sample behavior. First we investigate and develop different notions of finite sample coverage guarantees. Then we give a new prediction band estimator by combining the idea of "conformal prediction" (Vovk et al. 2009) with nonparametric conditional density estimation. The proposed estimator, called COPS (Conformal Optimized Prediction Set), always has finite sample guarantee in a stronger sense than the original conformal prediction estimator. Under regularity conditions the estimator converges to an oracle band at a minimax optimal rate. A fast approximation algorithm and a data driven method for selecting the bandwidth are developed. The method is illustrated first in simulated data. Then, an application shows that the proposed method gives desirable prediction intervals in an automatic way, as compared to the classical linear regression modeling.</p
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