Parametric statistics

About: Parametric statistics is a research topic. Over the lifetime, 39200 publications have been published within this topic receiving 765761 citations.

The psychometric function: II. Bootstrap-based confidence intervals and sampling.

Abstract: The psychometric function relates an observer's performance to an independent variable, usually a physical quantity of an experimental stimulus Even if a model is successfully fit to the data and its goodness of fit is acceptable, experimenters require an estimate of the variability of the parameters to assess whether differences across conditions are significant Accurate estimates of variability are difficult to obtain, however, given the typically small size of psychophysical data sets: Traditional statistical techniques are only asymptotically correct and can be shown to be unreliable in some common situations Here and in our companion paper (Wichmann & Hill, 2001), we suggest alternative statistical techniques based on Monte Carlo resampling methods The present paper's principal topic is the estimation of the variability of fitted parameters and derived quantities, such as thresholds and slopes First, we outline the basic bootstrap procedure and argue in favor of the parametric, as opposed to the nonparametric, bootstrap Second, we describe how the bootstrap bridging assumption, on which the validity of the procedure depends, can be tested Third, we show how one's choice of sampling scheme (the placement of sample points on the stimulus axis) strongly affects the reliability of bootstrap confidence intervals, and we make recommendations on how to sample the psychometric function efficiently Fourth, we show that, under certain circumstances, the (arbitrary) choice of the distribution function can exert an unwanted influence on the size of the bootstrap confidence intervals obtained, and we make recommendations on how to avoid this influence Finally, we introduce improved confidence intervals (bias corrected and accelerated) that improve on the parametric and percentile-based bootstrap confidence intervals previously used Software implementing our methods is available

Testing Continuous-Time Models of the Spot Interest Rate

Abstract: Different continuous-time models for interest rates coexist in the literature. We test parametric models by comparing their implied parametric density to the same density estimated nonparametrically. We do not replace the continuous-time model by discrete approximations, even though the data are recorded at discrete intervals. The principal source of rejection of existing models is the strong nonlinearity of the drift. Around its mean, where the drift is essentially zero, the spot rate behaves like a random walk. The drift then mean-reverts strongly when far away from the mean. The volatility is higher when away from the mean.

Nonparametric Econometrics: The np Package

Abstract: We describe the R np package via a series of applications that may be of interest to applied econometricians. The np package implements a variety of nonparametric and semiparametric kernel-based estimators that are popular among econometricians. There are also procedures for nonparametric tests of significance and consistent model specification tests for parametric mean regression models and parametric quantile regression models, among others. The np package focuses on kernel methods appropriate for the mix of continuous, discrete, and categorical data often found in applied settings. Data-driven methods of bandwidth selection are emphasized throughout, though we caution the user that data-driven bandwidth selection methods can be computationally demanding.

Nonparametric analysis of statistic images from functional mapping experiments

Abstract: The analysis of functional mapping experiments in positron emission tomography involves the formation of images displaying the values of a suitable statistic, summarising the evidence in the data for a particular effect at each voxel These statistic images must then be scrutinised to locate regions showing statistically significant effects The methods most commonly used are parametric, assuming a particular form of probability distribution for the voxel values in the statistic image Scientific hypotheses, formulated in terms of parameters describing these distributions, are then tested on the basis of the assumptions Images of statistics are usually considered as lattice representations of continuous random fields These are more amenable to statistical analysis There are various shortcomings associated with these methods of analysis The many assumptions and approximations involved may not be true The low numbers of subjects and scans, in typical experiments, lead to noisy statistic images with low degrees of freedom, which are not well approximated by continuous random fields Thus, the methods are only approximately valid at best and are most suspect in single-subject studies In contrast to the existing methods, we present a nonparametric approach to significance testing for statistic images from activation studies Formal assumptions are replaced by a computationally expensive approach In a simple rest-activation study, if there is really no activation effect, the labelling of the scans as “active” or “rest” is artificial, and a statistic image formed with some other labelling is as likely as the observed one Thus, considering all possible relabellings, a p value can be computed for any suitable statistic describing the statistic image Consideration of the maximal statistic leads to a simple nonparametric single-threshold test This randomisation test relies only on minimal assumptions about the design of the experiment, is (almost) exact, with Type I error (almost) exactly that specified, and hence is always valid The absence of distributional assumptions permits the consideration of a wide range of test statistics, for instance, “pseudo” t statistic images formed with smoothed variance images The approach presented extends easily to other paradigms, permitting nonparametric analysis of most functional mapping experiments When the assumptions of the parametric methods are true, these new nonparametric methods, at worst, provide for their validation When the assumptions of the parametric methods are dubious, the nonparametric methods provide the only analysis that can be guaranteed valid and exact

Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics

Abstract: Continuing the development of the structured singular value approach to robust control design, the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty. The problem is shown to be equivalent to a smooth constrained finite-dimensional optimization problem. In view of the fact that the functional to be maximized may have several local extrema, an upper bound on mu whose computation is numerically tractable is established; this leads to a sufficient condition of robust stability and performance. A historical perspective on the development of the mu theory is included. >