Nonparametric regression in exponential families

Most results in nonparametric regression theory are developed only for the case of additive noise. In such a setting many smoothing techniques including wavelet thresholding methods have been developed and shown to be highly adaptive. In this paper we consider nonparametric regression in exponential families with the main focus on the natural exponential families with a quadratic variance function, which include, for example, Poisson regression, binomial regression and gamma regression. We propose a unified approach of using a mean-matching variance stabilizing transformation to turn the relatively complicated problem of nonparametric regression in exponential families into a standard homoscedastic Gaussian regression problem. Then in principle any good nonparametric Gaussian regression procedure can be applied to the transformed data. To illustrate our general methodology, in this paper we use wavelet block thresholding to construct the final estimators of the regression function. The procedures are easily implementable. Both theoretical and numerical properties of the estimators are investigated. The estimators are shown to enjoy a high degree of adaptivity and spatial adaptivity with near-optimal asymptotic performance over a wide range of Besov spaces. The estimators also perform well numerically.Comment: Published in at http://dx.doi.org/10.1214/09-AOS762 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Robust nonparametric estimation via wavelet median regression

In this paper we develop a nonparametric regression method that is simultaneously adaptive over a wide range of function classes for the regression function and robust over a large collection of error distributions, including those that are heavy-tailed, and may not even possess variances or means. Our approach is to first use local medians to turn the problem of nonparametric regression with unknown noise distribution into a standard Gaussian regression problem and then apply a wavelet block thresholding procedure to construct an estimator of the regression function. It is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the Besov classes, without prior knowledge of the smoothness of the underlying functions or prior knowledge of the error distribution. The estimator also automatically adapts to the local smoothness of the underlying function, and attains the local adaptive minimax rate for estimating functions at a point. A key technical result in our development is a quantile coupling theorem which gives a tight bound for the quantile coupling between the sample medians and a normal variable. This median coupling inequality may be of independent interest.Comment: Published in at http://dx.doi.org/10.1214/07-AOS513 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Effect of mean on variance function estimation in nonparametric regression

Variance function estimation in nonparametric regression is considered and the minimax rate of convergence is derived. We are particularly interested in the effect of the unknown mean on the estimation of the variance function. Our results indicate that, contrary to the common practice, it is not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean when the mean function is not smooth. Instead it is more desirable to use estimators of the mean with minimal bias. On the other hand, when the mean function is very smooth, our numerical results show that the residual-based method performs better, but not substantial better than the first-order-difference-based estimator. In addition our asymptotic results also correct the optimal rate claimed in Hall and Carroll [J. Roy. Statist. Soc. Ser. B 51 (1989) 3--14].Comment: Published in at http://dx.doi.org/10.1214/009053607000000901 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Solution Structures of \u3cem\u3eMycobacterium tuberculosis\u3c/em\u3e Thioredoxin C and Models of Intact Thioredoxin System Suggest New Approaches to Inhibitor and Drug Design

Here, we report the NMR solution structures of Mycobacterium tuberculosis (M. tuberculosis) thioredoxin C in both oxidized and reduced states, with discussion of structural changes that occur in going between redox states. The NMR solution structure of the oxidized TrxC corresponds closely to that of the crystal structure, except in the C-terminal region. It appears that crystal packing effects have caused an artifactual shift in the α4 helix in the previously reported crystal structure, compared with the solution structure. On the basis of these TrxC structures, chemical shift mapping, a previously reported crystal structure of the M. tuberculosis thioredoxin reductase (not bound to a Trx) and structures for intermediates in the E. coli thioredoxin catalytic cycle, we have modeled the complete M. tuberculosis thioredoxin system for the various steps in the catalytic cycle. These structures and models reveal pockets at the TrxR/TrxC interface in various steps in the catalytic cycle, which can be targeted in the design of uncompetitive inhibitors as potential anti-mycobacterial agents, or as chemical genetic probes of function

Interval estimation for a binomial proportion

Abstract. We revisit the problem of interval estimation of a binomial proportion. The erratic behavior of the coverage probability of the standardWaldconfidence interval has previously been remarkedon in the literature (Blyth andStill, Agresti andCoull, Santner andothers). We begin by showing that the chaotic coverage properties of the Waldinterval are far more persistent than is appreciated. Furthermore, common textbook prescriptions regarding its safety are misleading and defective in several respects andcannot be trusted. This leads us to consideration of alternative intervals. A number of natural alternatives are presented, each with its motivation and context. Each interval is examinedfor its coverage probability andits length. Basedon this analysis, we recommendthe Wilson interval or the equaltailedJeffreys prior interval for small n andthe interval suggestedin Agresti andCoull for larger n. We also provide an additional frequentist justification for use of the Jeffreys interval. Key words and phrases: Bayes, binomial distribution, confidence intervals, coverage probability, Edgeworth expansion, expected length, Jeffreys prior, normal approximation, posterior

Comment: Fuzzy and Randomized Confidence Intervals and P-Values

Professor Geyer and Professor Meeden have given us an intriguing article with much material for thought and exploration, and they deserve our congratulations. Although the idea of randomized procedures has long existed, this paper has revitalized the discussion on randomized confidence intervals and randomized P -values. Interval estimation of a binomial proportion is a very basic but very important problem with an extensive literature. Brown, Cai and DasGupta (2001) revisited this problem and showed that the performance of the standard Wald interval, which is used extensively in textbooks and in practice, is far more erratic and inadequate than is appreciated. Several natural alternative confidence intervals for p were recommended in Brown, Cai and DasGupta (2001). See also Agresti and Coull (1998). These intervals are all what the authors call crisp intervals. The coverage probability of these crisp confidence intervals contains significant oscillation, which is intrinsic in all crisp intervals due to the lattice structure of the binomial distributions. In the present paper, Geyer and Meeden introduce the notion of fuzzy con- fidence intervals with the goal to eliminate oscillation and to have the exact coverage probability. The con- fidence intervals are obtained by inverting families of randomized tests. In addition, the authors introduce the notion of fuzzy P -values. The introduction of the critical function φ as a function of three variables x, α and θ provides a unified description of fuzzy decision, fuzzy confidence interval and fuzzy P -values. Our discussion here will focus on four issues: (1) What is new in this paper?; (2) exact versus approximate coverage; (3) expected length; (4) generalization of abstract randomized confidence intervals to simultaneous inferenc

Substrate Induced Structural and Dynamics Changes in Human Phosphomevalonate Iinase and Implications for Mechanism

Phosphomevalonate kinase (PMK) catalyzes an essential step in the mevalonate pathway, which is the only pathway for synthesis of isoprenoids and steroids in humans. PMK catalyzes transfer of the γ-phosphate of ATP to mevalonate 5-phosphate (M5P) to form mevalonate 5-diphosphate. Bringing these phosphate groups in proximity to react is especially challenging, given the high negative charge density on the four phosphate groups in the active site. As such, conformational and dynamics changes needed to form the Michaelis complex are of mechanistic interest. Herein, we report the characterization of substrate induced changes (Mg-ADP, M5P, and the ternary complex) in PMK using NMR-based dynamics and chemical shift perturbation measurements. Mg-ADP and M5P Kd\u27s were 6–60 μM in all complexes, consistent with there being little binding synergy. Binding of M5P causes the PMK structure to compress (τc = 13.5 nsec), whereas subsequent binding of Mg-ADP opens the structure up (τc = 15.6 nsec). The overall complex seems to stay very rigid on the psec-nsec timescale with an average NMR order parameter of S2 ∼0.88. Data are consistent with addition of M5P causing movement around a hinge region to permit domain closure, which would bring the M5P domain close to ATP to permit catalysis. Dynamics data identify potential hinge residues as H55 and R93, based on their low order parameters and their location in extended regions that connect the M5P and ATP domains in the PMK homology model. Likewise, D163 may be a hinge residue for the lid region that is homologous to the adenylate kinase lid, covering the “Walker-A” catalytic loop. Binding of ATP or ADP appears to cause similar conformational changes; however, these observations do not indicate an obvious role for γ-phosphate binding interactions. Indeed, the role of γ-phosphate interactions may be more subtle than suggested by ATP/ADP comparisons, because the conservative O to NH substitution in the β-γ bridge of ATP causes a dramatic decrease in affinity and induces few chemical shift perturbations. In terms of positioning of catalytic residues, binding of M5P induces a rigidification of Gly21 (adjacent to the catalytically important Lys22), although exchange broadening in the ternary complex suggests some motion on a slower timescale does still occur. Finally, the first nine residues of the N-terminus are highly disordered, suggesting that they may be part of a cleavable signal or regulatory peptide sequence. Proteins 2009. © 2008 Wiley-Liss, Inc

Molecular Docking and NMR Binding Studies to Identify Novel Inhibitors of Human Phosphomevalonate Kinase

Phosphomevalonate kinase (PMK) phosphorylates mevalonate-5-phosphate (M5P) in the mevalonate pathway, which is the sole source of isoprenoids and steroids in humans. We have identified new PMK inhibitors with virtual screening, using autodock. Promising hits were verified and their affinity measured using NMR-based 1H–15N heteronuclear single quantum coherence (HSQC) chemical shift perturbation and fluorescence titrations. Chemical shift changes were monitored, plotted, and fitted to obtain dissociation constants (Kd). Tight binding compounds with Kd’s ranging from 6–60 μM were identified. These compounds tended to have significant polarity and negative charge, similar to the natural substrates (M5P and ATP). HSQC cross peak changes suggest that binding induces a global conformational change, such as domain closure. Compounds identified in this study serve as chemical genetic probes of human PMK, to explore pharmacology of the mevalonate pathway, as well as starting points for further drug development