Equispaced Design Research Articles

Optimal designs for some bivariate cokriging models

This article focuses on the estimation and design aspects of a bivariate collocated cokriging experiment. For a large class of covariance matrices, a linear dependency criterion is identified, which allows the best linear unbiased estimator of the primary variable in a bivariate collocated cokriging setup to reduce to a univariate kriging estimator. Exact optimal designs for efficient prediction for such simple and ordinary reduced cokriging models with one-dimensional inputs are determined. Designs are found by minimizing the maximum and the integrated prediction variance, where the primary variable is an Ornstein–Uhlenbeck process. For simple and ordinary cokriging models with known covariance parameters, the equispaced design is shown to be optimal for both criterion functions. The more realistic scenario of unknown covariance parameters is addressed by assuming prior distributions on the parameter vector, thus adopting a Bayesian approach to the design problem. The equispaced design is proved to be the Bayesian optimal design for both criteria. The work is motivated by designing an optimal water monitoring system for an Indian river.

Pointwise Wavelet Estimation of Density Function with Change-Points Based on NA and Biased Sample

This paper is concerned with the density estimation problem of negatively associated biased sample with the presence of multiple change-points. We use the peaks-over-threshold approach to estimate the number and locations of change-points and give an equispaced design estimation to evaluate the jump sizes for the underlying density function. Subsequently, we propose a nonlinear wavelet change-point estimation of the underlying density and show the convergence rate under poinwise risk over Besov space. It should be pointed out that the convergence rate of wavelet change-point estimation is near optimal (up to a logarithmic term) and remains the same as that of the usual wavelet density estimation without change-points.

Pointwise wavelet change-points estimation for dependent biased sample

This paper focuses on the multiple change-points detection for the biased regression function based on the dependent biased sample. A peaks-over-threshold method is utilized to detect the change-points and an equispaced design approach is provided to evaluate the jump sizes. On the basis of the change-points and jump sizes estimations, we propose a (Lp version of the) block thresholding estimator for the biased regression function. This block wavelet change-points estimator is adaptive and the optimal convergence rate of the proposed estimator is established on pointwise risk over Besov space with the presence of dependent bias. Additionally, the effectiveness of the proposed estimator is validated through some examples and numerical simulations.

Exact Bayesian designs for count time series

Exact D-optimal Bayesian designs for time series experiments are discussed in this article. This work is motivated by an RNA sequencing experiment and two disease surveillance studies, where the response is count type and has a correlated structure over time points. The conditional distribution of the count responses given a weakly stationary latent process is assumed to follow a log-linear model. The latent process allows for both overdispersion and autocorrelation in the responses. Linear predictor with the trend and seasonal components are studied. An estimating approach based on only the first two moments of the responses is used for parameter estimation. The D-optimality criterion based on minimization of the log determinant of the variance–covariance matrix of the parameter estimates is used for choosing the exact designs. To address the dependency of the design selection criterion on the unknown parameter values, prior distributions are assumed on the parameters. From the numerical results, it is noted that for linear predictors with only trend component, the optimal design is very close to the equispaced design for high correlation values. However, when the linear predictor has both the trend and seasonal components, the two designs are similar for smaller correlations.

Sharp oracle inequalities for Least Squares estimators in shape restricted regression

The performance of Least Squares (LS) estimators is studied in shape-constrained regression models under Gaussian and sub-Gaussian noise. General bounds on the performance of LS estimators over closed convex sets are provided. These results have the form of sharp oracle inequalities that account for the model misspecification error. In the presence of misspecification, these bounds imply that the LS estimator estimates the projection of the true parameter at the same rate as in the well-specified case. In isotonic and unimodal regression, the LS estimator achieves the nonparametric rate $n^{-2/3}$ as well as a parametric rate of order $k/n$ up to logarithmic factors, where $k$ is the number of constant pieces of the true parameter. In univariate convex regression, the LS estimator satisfies an adaptive risk bound of order $q/n$ up to logarithmic factors, where $q$ is the number of affine pieces of the true regression function. This adaptive risk bound holds for any collection of design points. While Guntuboyina and Sen [Probab. Theory Related Fields 163 (2015) 379–411] established that the nonparametric rate of convex regression is of order $n^{-4/5}$ for equispaced design points, we show that the nonparametric rate of convex regression can be as slow as $n^{-2/3}$ for some worst-case design points. This phenomenon can be explained as follows: Although convexity brings more structure than unimodality, for some worst-case design points this extra structure is uninformative and the nonparametric rates of unimodal regression and convex regression are both $n^{-2/3}$. Higher order cones, such as the cone of $\beta $-monotone sequences, are also studied.

Nonparametric regression estimation based on spatially inhomogeneous data: minimax global convergence rates and adaptivity

We consider the nonparametric regression estimation problem of recovering an unknown response function f on the basis of spatially inhomogeneous data when the design points follow a known density g with a finite number of well-separated zeros. In particular, we consider two different cases: when g has zeros of a polynomial order and when g has zeros of an exponential order. These two cases correspond to moderate and severe data losses, respectively. We obtain asymptotic (as the sample size increases) minimax lower bounds for the L 2 -risk when f is assumed to belong to a Besov ball, and construct adaptive wavelet thresholding estimators of f that are asymptotically optimal (in the minimax sense) or near-optimal within a logarithmic factor (in the case of a zero of a polynomial order), over a wide range of Besov balls. The spatially inhomogeneous ill-posed problem that we investigate is inherently more difficult than spatially homogeneous ill-posed problems like, e.g., deconvolution. In particular, due to spatial irregularity, assessment of asymptotic minimax global convergence rates is a much harder task than the derivation of asymptotic minimax local convergence rates studied recently in the literature. Furthermore, the resulting estimators exhibit very different behavior and asymptotic minimax global convergence rates in comparison with the solution of spatially homogeneous ill-posed problems. For example, unlike in the deconvolution problem, the asymptotic minimax global convergence rates are greatly influenced not only by the extent of data loss but also by the degree of spatial homogeneity of f .S pecif ically, even if 1/g is non-integrable, one can recover f as well as in the case of an equispaced design (in terms of asymptotic minimax global convergence rates) when it is homogeneous enough since the estimator is borrowing strength in the areas where f is adequately sampled.

Optimal designs for parameter estimation of the Ornstein–Uhlenbeck process

AbstractThis paper deals with optimal designs for Gaussian random fields with constant trend and exponential correlation structure, widely known as the Ornstein–Uhlenbeck process. Assuming the maximum likelihood approach, we study the optimal design problem for the estimation of the trend µ and the correlation parameter θ using a criterion based on the Fisher information matrix. For the problem of trend estimation, we give a new proof of the optimality of the equispaced design for any sample size (see Statist. Probab. Lett. 2008; 78:1388–1396). We also show that for the estimation of the correlation parameter, an optimal design does not exist. Furthermore, we show that the optimal strategy for µ conflicts with the one for θ, since the equispaced design is the worst solution for estimating the correlation. Hence, when the inferential purpose concerns both the unknown parameters we propose the geometric progression design, namely a flexible class of procedures that allow the experimenter to choose a suitable compromise regarding the estimation's precision of the two unknown parameters guaranteeing, at the same time, high efficiency for both. Copyright © 2008 John Wiley & Sons, Ltd.

Robust Optimum Designs for Transformation of the Responses in a Multivariate Chemical Kinetic Model

Transformation of both sides of a nonlinear model is often necessary to obtain tractable error distributions, but can have a dramatic effect on optimum designs for the parameters of the model. This article develops methods for the design of experiments when the value of the power transformation parameter is not precisely known. Optimum designs for particular transformations, together with the efficiency of these designs as the transformation varies, are studied for a multivariate model. It is shown that by a suitable choice of prior, Bayesian designs can be found that are robust to the correct transformation. A typical equispaced design is evaluated and the properties of designs are studied under various departures from the assumptions.

Wavelet estimation of a regression function with a sharp change point in a random design

In a random design nonparametric regression model, this paper deals with the detection of a sharp change point and the estimation of a regression function with a single jump point. A method based on design transformation and binning is used in order to convert a random design into an equispaced design whose number of points is a power of 2. Using the continuous wavelet transform of the data, we construct a sharp change point estimator and obtain its rate of convergence. Wavelet methods are well known for their good adaptivity around sudden local changes; however, in practice, the Gibbs phenomenon still exists. This difficulty is overcome by suitably adjusting the data with preliminary estimators for the location and the size of discontinuity. Global and local asymptotic results of the proposed method are obtained. The method is also tested on simulated examples and the results show that the proposed method alleviates the Gibbs phenomenon.

Smoothing Periodic Curves by a Method of Regularization

The present work develops an extension of the theory of polynomial smoothing splines for the statistical problem of estimating a periodic curve based on discrete and noisy observations of the curve itself or of a convolution functional of the curve. These estimates, called periodic $\alpha $-splines, are still of the method of regularization type, with a penalty involving Fourier coefficients. The questions of asymptotic rate of convergence of the integrated mean square error and the relationships with kernel estimates in the case of equispaced design points are addressed.