Andrey Feuerverger

On Some Fourier Methods for Inference

Abstract Common statistical procedures such as maximum likelihood and M-estimation admit generalized representations in the Fourier domain. The Fourier domain provides fertile ground for approaching a number of difficult problems in inference. In particular, the empirical characteristic function and its extension for stationary time series are shown to be fundamental tools which support numerically simple inference procedures having arbitrarily high asymptotic efficiency and certain robustness features as well. A numerical illustration involving the symmetric stable laws is given.

Sensitivity analysis and the “what if” problem in simulation analysis

We discuss some known and some new results on the score function (SF) approach for simulation analysis. We show that while simulating a single sample path from the underlying system or from an associated system and applying the Radon-Nikodym measure one can: estimate the performance sensitivities (gradient, Hessian etc.) of the underlying system with respect to some parameter (vector of parameters); extrapolate the performance measure for different values of the parameters; evaluate the performance measures of queuing models working in heavy traffic by simulating an associated (auxiliary) queuing model working in light (lighter) traffic; evaluate the performance measures of stochastic models while simulating random vectors (say, by the inverse transform method) from an auxiliary probability density function rather than from the original one (say by the acceptance-rejection method). Applications of the SF approach to a broad variety of stochastic models are given.

Statistical methods for assessing the dimensions of synaptic vesicles in nerve terminals

Chemical transmission between neurons occurs by the release of neurotransmitter packaged within vesicles of the presynaptic neuron onto a postsynaptic target. The amount of transmitter contained within a vesicle is in part regulated by the size of the vesicle. Thus, it is of general interest to quantify the dimension of vesicles in understanding the basic principles of chemical synaptic transmission. These vesicles can only be measured by electron microscopic techniques. Obtaining the true dimensions of synaptic structures is therefore complicated by stereological considerations. In this study, we suggest improved methods for determining the distributions (and mean sizes) for populations of vesicle diameters by mathematical processes involving (1) an implicit inversion of the empirical data distribution, (2) an explicit inversion approach, and (3) an approach based on substituting the empirical distribution into the inversion formula and then isotonizing using an iterated convex minorant algorithm. These procedures provide distributions that better represent the true population distributions (and means) for comparisons with other data sets of vesicle diameter measures.

Biases in the Shanks-Rényi prime number race

Rubinstein and Sarnak investigated systemsof inequalities of the form π(x; q, a1) > … > π(x; q, ar), where p(x; q, b) denotes the number of primes up to x that are congruent to b mod q. They showed, under standard hypotheses on the zeros of Dirichlet L-functions mod q, that the set of positive real numbers x for which these inequalities hold has positive (logarithmic) density δq;al, … .ar > 0. They also discovered the surprising fact that a certain distribution associated with these densities is not symmetric under permutations of the residue classes aj in general, even if the aj are all squaresor all nonsquares mod q (a condition necessary to avoid obvious biases of the type first observed by Chebyshev). This asymmetry suggests, contrary to prior expectations, that the densities δq;al , …,ar themselves vary under permutations of the aj. Here we derive (under the hypotheses used by Rubinstein and Sarnak) a general formula for the densities δq;al , …,ar, and We use this formula to calculate many of these densities when q ≤ 12 and r ≤ 4. For the special moduli q = 8 and q = 12, and for {al, a2,a3} a permutation of the nonsquares {3, 5, 7} mod 8 and {5, 7, 11} mod 12, respectively, we rigorously bound the error in our calculations, thus verifying that these densities are indeed asymmetric under permutation of the aj. We also determine several situations in which the densities δq;al , …, ar remain unchanged under certain permutations of the aj, and some situations in which they are provably different.

On Statistical Inference Based on Record Values

We develop methodology for conducting inference based on record values and record times derived from a sequence of independent and identically distributed random variables. The advantage of using information about record times as well as record values is stressed. This point is a subtle one, since if the sampling distribution F is continuous then there is no information at all about F in the record times alone; the joint distribution of any number of them does not depend on F. However, the record times and record values jointly contain considerably more information about F than do the record values alone. Indeed, in the case of a distribution with regularly varying tails, the rate of convergence of the exponent of regular variation is two orders of magnitude faster if information about record times is included. Optimal estimators and convergence rates are derived under simple, specific models, and shown to be surprisingly robust against significant departures from those models. However, even under our special models the estimators have irregular properties, including an undefined information matrix. To some extent these difficulties may be alleviated by conditioning and by considering the relationship between maximum likelihood and maximum probability estimators.

Positron Emission Tomography and Random Coefficients Regression

We further explore the relation between random coefficients regression (RCR) and computerized tomography. Recently, Beran et al. (1996, Ann. Statist., 24, 2569–2592) explored this connection to derive an estimation method for the non-parametric RCR problem which is closely related to image reconstruction methods in X-ray computerized tomography. In this paper we emphasize the close connection of the RCR problem with positron emission tomography (PET). Specifically, we show that the RCR problem can be viewed as an idealized (continuous) version of a PET experiment, by demonstrating that the nonparametric likelihood of the RCR problem is equivalent to that of a specific PET experiment. Consequently, methods independently developed for either of the two problems can be adapted from one problem to the other. To demonstrate the close relation between the two problems we use the estimation method of Beran, Feuerverger and Hall for image reconstruction in PET.

On Optimal Uniform Deconvolution

This paper concerns the nonstandard problem of uniform deconvolution for nonperiodic functions over the real line. New algorithms are developed for this nonstandard statistical problem and integrated mean squared error bounds are established. We show that the upper bound of the integrated mean squared error for our new procedure is the same as for the standard case; hence these new estimators attain the lower bound minimax, and hence optimal, rate of convergence. Our method has potential applications to such problems as the deblurring of optical images which have been subjected to uniform motion over a finite interval of time. We also treat the case when the support of the uniform is not given and must be estimated. The numerical properties of our algorithms are demonstrated and shown to be well behaved.

Methods for Density Estimation in Thick-Slice Versions of Wicksell's Problem

Abstract A new, implicit method is suggested for density estimation in inverse problems, where data are drawn not from the target distribution, but rather from its image under a transformation. The approach that we propose produces density estimators that are themselves densities, without the negativity problems known to plague more explicit inversion techniques. We also suggest a general empirical approach to selecting the smoothing parameter so as to optimize performance in the context of the target distribution, rather than its image after the transformation. We apply the new methods, and competing techniques, to a thick-section Wicksell-type problem, using data on the radii of nerve terminals from the electric organ of the electric ray Torpedo marmorata. It is shown that statistical properties of estimators in this problem are very different from those for the thin-slice, classical Wicksell problem, and so the two cases cannot be developed simply by analogy with one another.

Some aspects of probability forecasting

The problems of assessing, comparing and combining probability forecasts for a binary events sequence are considered. A Gaussian threshold model (analytically of closed form) is introduced which allows generation of different probability forecast sequences valid for the same events. Chi - squared type test statistics, and also a marginal-conditional method are proposed for the assessment problem, and an asymptotic normality result is given. A graphical method is developed for the comparison problem, based upon decomposing arbitrary proper scoring rules into certain elementary scoring functions. The special role of the logarithmic scoring rule is examined in the context of Neyman - Pearson theory.

Dating medieval English charters

Deeds, or charters, dealing with property rights, provide a continuous documentation which can be used by historians to study the evolution of social, economic and political changes. This study is concerned with charters (written in Latin) dating from the tenth through early fourteenth centuries in England. Of these, at least one million were left undated, largely due to administrative changes introduced by William the Conqueror in 1066. Correctly dating such charters is of vital importance in the study of English medieval history. This paper is concerned with computer-automated statistical methods for dating such document collections, with the goal of reducing the considerable efforts required to date them manually and of improving the accuracy of assigned dates. Proposed methods are based on such data as the variation over time of word and phrase usage, and on measures of distance between documents. The extensive (and dated) Documents of Early England Data Set (DEEDS) maintained at the University of Toronto was used for this purpose.