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Alignment of curve data is an integral part of their statistical analysis, and can be achieved using modelor optimization-based approaches. The parameter space is usually the set of monotone, continuous warp maps of a domain. Infinite-dimensional nature of the parameter space encourages sampling based approaches, which require a distribution on the set of warp maps. Moreover, the distribution should also enable sampling in the presence of important landmark information on the curves which constrain the warp maps. For alignment of closed and open curves in [Formula: see text] , d = 1, 2, 3, possibly with landmark information, we provide a constructive, point-process based definition of a distribution on the set of warp maps of [0, 1] and the unit circle [Formula: see text] , that is, (1) simple to sample from, and (2) possesses the desiderata for decomposition of the alignment problem with landmark constraints into multiple unconstrained ones. For warp maps on [0, 1], the distribution is related to the Dirichlet process. We demonstrate its utility by using it as a prior distribution on warp maps in a Bayesian model for alignment of two univariate curves, and as a proposal distribution in a stochastic algorithm that optimizes a suitable alignment functional for higher-dimensional curves. Several examples from simulated and real datasets are provided.

We propose a new distribution for analyzing paleomagnetic directional data, that is, a novel transformation of the von Mises–Fisher distribution. The new distribution has ellipse-like symmetry, as ...

This article defines and studies a depth for multivariate functional data. By the multivariate nature and by including a weight function, it acknowledges important characteristics of functional data, namely differences in the amount of local amplitude, shape, and phase variation. We study both population and finite sample versions. The multivariate sample of curves may include warping functions, derivatives, and integrals of the original curves for a better overall representation of the functional data via the depth. We present a simulation study and data examples that confirm the good performance of this depth function. Supplementary materials for this article are available online.

In this article we investigate the problem of measuring deviations from stationarity in locally stationary time series. Our approach is based on a direct estimate of the L2-distance between the spectral density of the locally stationary process and its best approximation by a spectral density of a stationary process. An explicit expression of the minimal distance is derived, which depends only on integrals of the spectral density of the locally stationary process and its square. These integrals can be estimated directly without estimating the spectral density, and as a consequence, the estimation of the measure of stationarity does not require the specification of a smoothing bandwidth. We show weak convergence of an appropriately standardized version of the statistic to a standard normal distribution. The results are used to construct confidence intervals for the measure of stationarity and to develop a new test for the hypothesis of stationarity. Finally, we investigate the finite sample properties of t...

Consider a model parameterized by θ == (ψ, λ), where ψ is the parameter of interest. The problem of eliminating the nuisance parameter λ can be tackled by resorting to a pseudo-likelihood function L**(ψ) for ψ—namely, a function of ψ only and the data y with properties similar to those of a likelihood function. If one treats L**(ψ) as a true likelihood, the posterior distribution π**(ψ | y) ∝ π(ψ)L**(ψ) for ψ can be considered, where π(ψ) is a prior distribution on ψ. The goal of this article is to construct probability matching priors for a scalar parameter of interest only (i.e., priors for which Bayesian and frequentist inference agree to some order of approximation) to be used in π**(ψ || y ). When L**(ψ) is a marginal, a conditional, or a modification of the profile likelihood, we show that π(ψ) is simply proportional to the square root of the inverse of the asymptotic variance of the pseudo-maximum likelihood estimator. The proposed priors are compared with the reference or Jeffreys’ priors in four ...

This article is motivated by the estimation of the directions of the two rotation axes of the ankle. These axes carry information on individual ankles; they are useful in the construction of biomechanical models and the treatment of orthopedic problems. In biomechanics, the rotation axes of the ankle often are estimated using optimization techniques. This work investigates a statistical model for carrying out the estimation. The data set for analysis is a time-ordered sequence of 3×3 rotation matrixes giving the ankle's orientations as the foot moves with respect to the lower leg. These rotation matrixes are assumed to follow Fisher–von Mises distributions. The predicted values for the observed rotations feature four angles for the orientation of two rotation axes and two series of time-varying rotation angles about the two axes. Maximum likelihood estimators of the parameters are derived. Approximations to their sampling distributions are obtained when the errors are clustered around the identity matrix....

A new nonparametric method is described to estimate the distribution of mass within spherical galaxies. The problem of estimating the mass, M(r), within radius r is converted into a problem of estimating a regression function nonparametrically, subject to shape restrictions. We represent the restrictions by the interception of quadratic cones and use the second-order cone programming to estimate the unknown parameters. We establish asymptotic results that are used to construct confidence intervals on M(r). We apply the technique to new kinematic data for four dwarf galaxies. Results indicate that dark matter dominates the stellar kinematics of these systems at all radii.

The study of water waves has generated a wealth of sophisticated modeling developments in applied mathematics. Empirical observation capabilities have created a need for novel data analysis tools. This article is motivated by consideration of wind-generated wave field image data from a wave tank facility. A quantitative measure of wave field regularity is developed. The methodology is based on decomposition of the wave field into simple plane waves (an adaptation of projection pursuit). The percent variance explained as a function of the number of terms in the plane wave representation is used to define a variogram, and regularity of the wave field is assessed in terms of the weighted difference between the observed variogram and the expected variogram for a completely random field. The proposed regularity measure is illustrated by application to image data from wind-generated waves. The results suggest that the regularity measure is a function of parameters describing the generation of the wave field (wi...

We use the so-called “bent-cable” model to describe natural phenomena that exhibit a potentially sharp change in slope. The model comprises two linear segments, joined smoothly by a quadratic bend. The class of bent cables includes, as a limiting case, the popular piecewise-linear model (with a sharp kink), otherwise known as the broken stick. Associated with bent-cable regression is the estimation of the bend-width parameter, through which the abruptness of the underlying transition may be assessed. We present worked examples and simulations to demonstrate the regularity and irregularity of bent-cable regression encountered in finite-sample settings. We also extend existing bent-cable asymptotics that previously were limited to the basic model with known linear slopes of 0 and 1. Practical conditions on the design are given to ensure regularity of the full bent-cable estimation problem if the underlying bend segment has nonzero width. Under such conditions, the least-squares estimators are shown to be co...

We propose a new family of symmetric unimodal distributions on the circle that contains the uniform, von Mises, cardioid, and wrapped Cauchy distributions, among others, as special cases. The basic form of the densities of this family is very simple, although its normalization constant involves an associated Legendre function. The family of distributions can also be derived by conditioning and projecting certain bivariate spherically and elliptically symmetric distributions on to the circle. Trigonometric moments are available, and a measure of variation is discussed. Aspects of maximum likelihood estimation are considered, and likelihood is used to fit the family of distributions to an example set of data. Finally, extension to a family of rotationally symmetric distributions on the sphere is briefly made.

The use of Mahalanobis distances has a long history in statistics. Given a sample of size n and general location and scatter estimators, mn and Σn, we can define “generalized” radii as . If we wish to trim observations based on the estimators mn and Σn, then it is natural to first remove the most remote ones (i.e., those with the largest 's). With this in mind, we define a process that maps the trimming proportion, α in (0, 1], to the generalized radius of the observation that has just been removed by this level of trimming. We analyze the asymptotic behavior of this process for elliptically contoured distributions. We show that the limit law depends only on the elliptical family considered and how Σn serves to estimate the underlying “scale” factor through its determinant. We carry out Monte Carlo simulations for finite sample sizes, and outline an application for assessing fit to a fixed elliptical family and also for the case where a proportion of outlying observations is discarded.

This work considers a number of properties of space–time covariance functions and how these relate to the spatial-temporal interactions of the process. First, it examines how the smoothness away from the origin of a space–time covariance function affects, for example, temporal correlations of spatial differences. Models that are not smoother away from the origin than they are at the origin, such as separable models, have a kind of discontinuity to certain correlations that one might wish to avoid in some circumstances. Smoothness away from the origin of a covariance function is shown to follow from the corresponding spectral density having derivatives with finite moments. These results are used to obtain a parametric class of spectral densities whose corresponding space–time covariance functions are infinitely differentiable away from the origin and that allows for essentially arbitrary and possibly different degrees of smoothness for the process in space and time. Second, this work considers models that ...

Posterior means of positive functions can be expressed as the ratio of integrals which is called fully exponential form. When approximating the posterior means analytically, we usually use Laplace's method. Tierney and Kadane presented a second-order approximation by using the Laplace approximations in each of the numerator and denominator of the fully exponential form. However, Laplace's method requires the exact mode of the integrand. In this article we introduce the concept of asymptotic modes and present the Laplace method via an asymptotic mode under regularity conditions. Furthermore, we propose second-order approximations to the posterior means of positive functions without evaluating the third derivatives of a log-likelihood function and the exact modes of integrands. We also give an Edgeworth-like expansion for the random variable according to a posterior distribution using the Laplace method via an asymptotic mode.

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 θ one must somehow get rid of θ, (e.g., by estimating it) to calculate a p value. Various proposals have been suggested to “remove” θ, 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 θ, 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...

Motivated by interval/region prediction in nonlinear time series, we propose a minimum volume predictor (MV-predictor) for a strictly stationary process. The MV-predictor varies with respect to the current position in the state space and has the minimum Lebesgue measure among all regions with the nominal coverage probability. We have established consistency, convergence rates, and asymptotic normality for both coverage probability and Lebesgue measure of the estimated MV-predictor under the assumption that the observations are taken from a strong mixing process. Applications with both real and simulated data sets illustrate the proposed methods.

Multiple counts may occur in each cell of an a × b two-way layout (balanced or unbalanced) of two fixed factors A and B. Standard log-linear model analysis based on a Poisson distribution assumption of the cell counts is not applicable here, because of the unbalanced nature of the table or because the Poisson distribution assumption is not valid. We develop C(α) tests for interaction and main effects assuming data to be Poisson distributed and also assuming that data within the cells have extra (over/under) dispersion beyond that explained by a Poisson distribution. For this we consider an extended negative binominal distribution and a semiparametric model using the quasi-likelihood. We show that in all situations the C(α) tests for interaction are of very simple forms. For C(α) tests for the main effect in presence of no interaction, such simplification is possible only under certain conditions. A score test for detecting extra dispersion in presence of interaction is also obtained and is of sim...

In this article we explore the structure and usefulness of partial residual plots as tools for visualizing curvature as a function of selected predictors x 2 in a generalized linear model (GLM), where the vector of predictors x is partitioned as x T = (x T 1, x T 2). The GLM extension of ceres plots is discussed, but to a lesser extent. The usefulness of these plots for obtaining a good visual impression of curvature may be limited by the specified GLM, the link function, and the stochastic behavior of the predictors. Partial residual plots seem to work well when modeling is in a region where the conditional mean of the response given x stays well away from its extremes so that the link is essentially linear, and E(x 1 | x 2) is linear in x 2. ceres plots, however, require only the first condition. The behavior of these plots is contrasted with their behavior in additive-error models.

The recently developed theory of wavelets has a remarkable ability to “zoom in” on very short-lived frequency phenomena, such as transients in signals and singularities in functions, and hence provides an ideal tool to study localized changes. This article proposes a wavelet method for estimating jump and sharp cusp curves of a function in the plane. The method involves first computing wavelet transformation of data and then estimating jump and sharp cusp curves by wavelet transformation across fine scales. Asymptotic theory is established, and simulations are carried out to lend some credence to the asymptotic theory. The wavelet estimate is nearly optimal and can be computed by fast algorithms. The method is applied to a real image.

We introduce reprojection as a general purpose technique for characterizing the dynamic response of a partially observed nonlinear system to its observable history. Reprojection is the third step of a procedure wherein first data are summarized by projection onto a Hermite series representation of the unconstrained transition density for observables; second, system parameters are estimated by minimum chi-squared, where the chi-squared criterion is a quadratic form in the expected score of the projection; and third, the constraints on dynamics implied by the nonlinear system are imposed by projecting a long simulation of the estimated system onto a Hermite series representation of the constrained transition density for observables. The constrained transition density can be used to study the response of the system to its observable history. We utilize the technique to assess the dynamics of several diffusion models for the short-term interest rate that have been proposed and to compare them to a ne...

Two new approximations for multivariate normal probabilities for rectangular regions, based on conditional expectations and regression with binary variables, are proposed. One is a second-order approximation that is much more accurate but also more numerically time-consuming than the first-order approximation. A third approximation, based on the moment-generating function of a truncated multivariate normal distribution, is proposed for orthant probabilities only. Its accuracy is between the first- and second-order approximations when the dimension is less than seven and the correlations are not large. All of the approximations get worse as correlations get larger. These new approximations offer substantial improvements on previous approximations. They also compare favorably with the methods of Genz for numerical evaluation of the multivariate normal integral. The approximation methods should be especially useful within a quasi-Newton routine for parameter estimation in discrete models that involv...

A new family of bivariate densities with specified marginal densities is described. The bivariate dependence structure takes the form of a density weighting function constructed as follows. The space of continuous variates X and Y is mapped into a unit square through probability integral transformations. A polygonal partition of the unit square supports a bivariate covariance characteristic, which is structured as a functional of a univariate regression characteristic. This structure (a) exhibits all geometric features of positive (or negative) dependence implied by Frechet bounds, which it can attain, and (b) prescribes positive (or negative) mutual regression dependence between X and Y. To obtain a parametric family of bivariate densities, the regression characteristic is given a power form. The resultant two-parameter model has specified marginals and, within certain constraints, allows one to separately control the shape of the bivariate density and the degree of association between X and Y; ...

For two distribution functions, F and G, the shift function is defined by Δ(t) = G −1 · F(t) - t. The shift function is the distance from the 45° line and the quantity plotted in Q-Q plots. In the analysis of lifetime data, A represents the difference between two treatments. The shift function can also be used to find crossing points of two distribution functions. The large-sample distribution theory for estimates of Δ is studied for right-censored data. It turns out that the asymptotic covariance function depends on the unknown distribution functions F and G; hence simultaneous confidence bands cannot be directly constructed. A construction of simultaneous confidence bands for Δ is developed via the bootstrap. Construction and application of such bands are explored for the Q-Q plot.

This article gives asymptotic expansions for posterior expectations when the mode is on the boundary of the parameter space. The idea, based on the divergence theorem, is to reduce the high-dimensional integrals over the parameters space to surface integrals over the boundary of the parameter space and then apply the usual interior-mode Laplace method to the latter integrals. It is shown that these approximations have second-order accuracy. The method is illustrated with applications to a two-sample binomial problem and a random-eflects model.

Many finite populations which are sampled repeatedly change slowly over time. Then estimation of finite population characteristics for the current occasion, l, may be improved by the use of data from previous surveys. In this article we investigate the use of empirical Bayes procedures based on two superpopulation models. Each model has the same first stage: The values of the population units on the ith occasion are a random sample from the normal distribution with mean μ i and variance σ 2 i . At the second stage we assume that either (a) μ 1, …, μ l are a random sample from the normal distribution with mean θ and variance δ 2, or (b) given σ 2 i and τ, μ i has the normal distribution with mean θ and variance σ 2 i τ (independently for each i), whereas the σ 2 i are a random sample from the inverse gamma distribution with parameters η/2 and κ/2. In (a) the σ 2 i , θ, and σ 2 are assumed to be unknown, whereas in (b) θ, τ, and κ are unknown. We develop empirical Bayes point estimators and confide...

This article deals with the breakdown point ϵ* of simultaneous general M estimates () of the regression parameter vector θ and scale parameter σ of a linear model. A general expression for ϵ* is derived and its numerical values are computed for the case in which the carriers have a normal distribution. The numerical results show that ϵ* is in general much lower than the minimum of the breakdown points of with σ known and of with θ known. The values of ∈* for some real and simulated data sets are also given.

Prior knowledge on the space of possible images is given in the form of a function or template in some domain. The set of all possible true images is assumed to be formed by a composition of that function with continuous mappings of the domain into itself. A prior Gaussian distribution is given on the set of continuous mappings. The observed image is assumed to be a degradation of the true image with additive noise. Given the observed image, a posterior distribution is then obtained and has the form of a nonlinear perturbation of the Gaussian measure on the space of mappings. We present simulations of the posterior distribution that lead to structural reconstructions of the true image in the sense that it enables us to determine landmarks and other characteristic features of the image, as well as to spot pathologies in it. Moreover, we show that the reconstruction algorithm is relatively robust when the images are degraded by noise that is not necessarily additive.

A new transformation referred to as the G (m) n -transformation is introduced to establish a general methodology for finding functions that are easy to evaluate and give very good approximations to tail probabilities. The transformation is based only on a general class of differential equations that include the specified density function in the solution set, at least asymptotically. As a result this new method applies to a broad class of distributions. Not only is the method general, it is also accurate. Unlike many other approximation methods, the approximation functions produced maintain their high degree of accuracy even in the extreme tails, making them also suitable for extreme tail probability approximation. In this article the method is applied to the Pearson family members, the inverse Gaussian, and the ratio of χ 2 and lognormal that do not belong to the Pearson family. Furthermore, it is applicable in the case of all of the standard noncentral distributions. Tables are included that sho...

A comprehensive approach for the analysis of statistical univariate distributions is advanced. The method is based on recasting a distribution in a canonical set of first-order nonlinear differential equations called an S system, in which the derivative of each variable equals a difference of products of power-law functions. The recasting techniques are illustrated with some examples from central and noncentral probability distributions. The density and the cumulative, because of their close relationship, can be simultaneously evaluated within one S system. A simple inversion procedure permits the calculation of quantiles. When, in addition, a noncentral distribution and its central counterpart are represented in one S-system, the power function and the inverse power function are obtainable. Given a set of initial conditions, any differential equation solver can generate the desired solution. Particularly convenient and efficient is the program ESSYNS, designed specifically to solve and analyze s...

One of the main goals of modern crystallography is the determination of the detailed internal structure of crystalline matter, at the atomic level. Statistical analyses and, in particular, random-walk models play a central role in inferring structural information from crystallographic data. Such methods are routinely employed by crystallographers in the determination of crystal symmetry from the experimental data, and in the solution of the outstandingly important problem for this discipline, the phase problem. Three classes of approaches are discussed: (a) methods based entirely on the central limit theorem; (b) approximate expansions in terms of orthogonal polynomials that have the central-limit-theorem pdf as their weight function—that is, Gram–Charlier and Edgeworth expansions; and (c) pdf's that are exactly formulated and reduced to computable forms, represented as Fourier and Fourier–Bessel series. Both univariate and multivariate pdf's of crystallographic interest are derived and discussed...

(2003). Multivariate Dispersion, Central Regions and Depth: the Lift Zonoid Approach. Karl Mosler. Journal of the American Statistical Association: Vol. 98, No. 464, pp. 1082-1082.