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Additive models for conditional quantile functions provide an attractive framework for nonparametric regression applications focused on features of the response beyond its central tendency. Total variation roughness penalities can be used to control the smoothness of the additive components much as squared Sobelev penalties are used for classical L2 smoothing splines. We describe a general approach to estimation and inference for additive models of this type. We focus attention primarily on selection of smoothing parameters and on the construction of confidence bands for the nonparametric components. Both pointwise and uniform confidence bands are introduced; the uniform bands are based on the Hotelling [Amer. J. Math.61 (1939) 440–460] tube approach. Some simulation evidence is presented to evaluate finite sample performance and the methods are also illustrated with an application to modeling childhood malnutrition in India.
A new class of geometric dispersion models associated with geometric sums is introduced by combining a geometric tilting operation with geometric compounding, in much the same way that exponential dispersion models combine exponential tilting and convolution. The construction is based on a geometric cumulant function which characterizes the geometric compounding operation additively. The so-called v-function is shown to be a useful characterization and convergence tool for geometric dispersion models, similar to the variance function for natural exponential families. A new proof of Rényi’s theorem on convergence of geometric sums to the exponential distribution is obtained, based on convergence of v-functions. It is shown that power v-functions correspond to a class of geometric Tweedie models that appear as limiting distributions in a convergence theorem for geometric dispersion models with power asymptotic v-functions. Geometric Tweedie models include geometric tiltings of Laplace, Mittag-Leffler and geometric extreme stable distributions, along with geometric versions of the gamma, Poisson and gamma compound Poisson distributions.
Several recent strands of work has led to the consideration of various types of continuous time stationary and infinitely divisible processes. A review of these types, with some new results, is presented.
KEYWORDS: Central approximation theorem, central limit theorem, curvature, empirical Fréchet mean, exponential map, Fréchet mean, gradient, Hessian, Kähler manifold, Lindeberg condition, Newton’s method, Riemannian centre of mass, Weak law of large numbers
We prove weak laws of large numbers and central limit theorems of Lindeberg type for empirical centres of mass (empirical Fréchet means) of independent nonidentically distributed random variables taking values in Riemannian manifolds. In order to prove these theorems we describe and prove a simple kind of Lindeberg–Feller central approximation theorem for vector-valued random variables, which may be of independent interest and is therefore the subject of a self-contained section. This vector-valued result allows us to clarify the number of conditions required for the central limit theorem for empirical Fréchet means, while extending its scope.
A prior for statistical inference can be one of three basic types: a mathematical prior originally proposed in Bayes [Philos. Trans. R. Soc. Lond.53 (1763) 370–418; 54 (1764) 269–325], a subjective prior presenting an opinion, or a truly objective prior based on an identified frequency reference. In this note we consider a method for deriving a mathematical prior based on approximate location models. This produces a mathematical posterior, and any practical interpretation of such a posterior is in terms of exact or approximate confidence under the postulated model. We describe how a proposed prior can be simply checked for consistency with confidence methods, using expansions about the maximum likelihood estimator.
The general theory of prediction-based estimating functions for stochastic process models is reviewed and extended. Particular attention is given to optimal estimation, asymptotic theory and Gaussian processes. Several examples of applications are presented. In particular, partial observation of a system of stochastic differential equations is discussed. This includes diffusions observed with measurement errors, integrated diffusions, stochastic volatility models, and hypoelliptic stochastic differential equations. The Pearson diffusions, for which explicit optimal prediction-based estimating functions can be found, are briefly presented.
With wireless sensor networks, preserving battery life is critical. For such sensors, data collection is relatively cheap while data transmission is relatively expensive. For such networks in ecological settings, certain processes are sufficiently predictable so that transmission of data at a particular time can be suppressed if it does not differ from what is expected at that time. That is, there will not be much loss of information with regard to inference. More precisely, there is a presumed model to explain the measurements collected at the sensors, which provides insight into what is expected at a given node, at a given time. Under the suppression, inference objectives include both estimation of the process parameters as well as reconstruction of the entire time series at each of the nodes.
In this paper, we build on the existing literature that has offered ways in which one can use suppression in wireless sensor networks to limit the number of transmissions. We introduce a new, computationally cheap, locally linear suppression scheme based upon process knowledge and compare it to the commonly used “constant” suppression scheme. Maintaining the same suppression threshold, we demonstrate decreased transmission rates under the new scheme while producing comparable posterior inference relative to constant suppression scheme. That is, the untransmitted readings are bounded to within an interval of the same length under both schemes, but the linear suppression scheme will transmit less data.
We implement this scheme for a synthetic dataset produced under the assumption of a diffusion model and show that even under high suppression rates, we are able to recover simulation parameters. We also implement linear suppression on data collected from a real wireless sensor network that measures the amount of light filtering through the forest canopy at a set of locations in the Duke Forest. We show that the in-sample predictive sum of squared errors from the suppressed data is only a bit larger than that from the full dataset.
Motivated by the need to smooth and to summarize multiple simultaneous time series arising from networks of environmental monitors, we propose a hierarchical wavelet model for which estimation of hyperparameters can be performed by marginal maximum likelihood. The result is an empirical Bayes thresholding procedure whose results improve on those of wavethresh in terms of mean square error. We apply the approach to data from the SensorScope environmental modelling system, and briefly discuss issues that arise concerning variance estimation in this context.
Motions of particles in fields characterized by real-valued potential functions, are considered. Three particular expressions for potential functions are studied. One, U, depends on the ith particle’s location, ri(t) at times ti. A second, V, depends on particle i’s vector distances from others, ri(t)−rj(t). This function introduces pairwise interactions. A third, W, depends on the Euclidian distances, ‖ri(t)−rj(t)‖ between particles at the same times, t. The functions are motivated by classical mechanics.
Taking the gradient of the potential function, and adding a Brownian term one, obtains the stochastic equation of motion
dri=−∇U(ri) dt−∑j≠i∇V(ri−rj) dt+σdBi
in the case that there are additive components U and V. The ∇ denotes the gradient operator. Under conditions the process will be Markov and a diffusion. By estimating U and V at the same time one could address the question of whether both components have an effect and, if yes, how, and in the case of a single particle, one can ask is the motion purely random?
An empirical example is presented based on data describing the motion of elk (Cervus elaphus) in a United States Forest Service reserve.
The standard estimator used in conjunction with importance sampling in Monte Carlo integration is unbiased but inefficient. An alternative estimator is discussed, based on the idea of a difference estimator, which is asymptotically optimal. The improved estimator uses the importance weight as a control variate, as previously studied by Hesterberg (Ph.D. Dissertation, Stanford University (1988); Technometrics37 (1995) 185–194; Statistics and Computing6 (1996) 147–157); it is routinely available and can deliver substantial additional variance reduction. Finite-sample performance is illustrated in a sequential testing example. Connections are made with methods from the survey-sampling literature.
KEYWORDS: CAN statistics, exponential family of densities, Hellinger-distance, LA(M)N, likelihood ratio, maximum likelihood estimator, nonparametrics, Rao’s score statistics, regular family of estimators, resampling plans
Wald-type test statistics based on asymptotically normally distributed estimators (not necessarily maximum likelihood estimation or best asymptotically normal) provides an easy access to have tests for statistical hypotheses, far beyond the parametric paradigms. The methodological perspectives rest on a basic consistent asymptotic normal (CAN) condition which is interrelated to the well-known local asymptotic normality (LAN) condition. Contiguity of probability measures facilitates the C(L)AN condition in a relatively easier way. For many regular families of distributions, when statistical hypotheses do not involve nonstandard constraints, verification of contiguity of probability measures is facilitated by the well-known LeCam’s First Lemma [see Hájek, Šidák and Sen Theory of Rank Tests (1999), Chapter 7]. For nonregular families, though contiguity may hold under different setups, CAN estimators are not fully exploitable in the Wald type testing theory. This simple feature is illustrated by a two-parameter exponential model. Guided by this simple example, mixture of distributions are appraised in the context of Wald-type tests and the so-called χ̅2- and E̅-test theory is thoroughly appraised. A general result on counter examples is presented in detail.