A geometrically motivated parametric model in manifold estimation

The general aim of manifold estimation is reconstructing, by statistical methods, an m-dimensional compact manifold S on ^d (with m≤d) or estimating some relevant quantities related to the geometric properties of S. Focussing on the cases d=2 and d=3, with m=d or m=d?1, we will assume that the data are given by the distances to S from points randomly chosen on a band surrounding S. The aim of this paper is to show that, if S belongs to a wide class of compact sets (which we call sets with polynomial volume), the proposed statistical model leads to a relatively simple parametric formulation. In this setup, standard methodologies (method of moments, maximum likelihood) can be used to estimate some interesting geometric parameters, including curvatures and Euler characteristic. We will particularly focus on the estimation of the (d?1)-dimensional boundary measure (in Minkowski's sense) of S. It turns out, however, that the estimation problem is not straightforward since the standard estimators show a remarkably pathological behaviour: while they are consistent and asymptotically normal, their expectations are infinite. The theoretical and practical consequences of this fact are discussed in some detail.     

Goodness‐of‐Fit Test for Monotone Functions

Abstract. In this article, we develop a test for the null hypothesis that a real‐valued function belongs to a given parametric set against the non‐parametric alternative that it is monotone, say decreasing. The method is described in a general model that covers the monotone density model, the monotone regression and the right‐censoring model with monotone hazard rate. The criterion for testing is an ‐distance between a Grenander‐type non‐parametric estimator and a parametric estimator computed under the null hypothesis. A normalized version of this distance is shown to have an asymptotic normal distribution under the null, whence a test can be developed. Moreover, a bootstrap procedure is shown to be consistent to calibrate the test.     

Non parametric regression estimations over Lp risk based on biased data

Using a wavelet basis, Chesneau and Shirazi study the estimation of one-dimensional regression functions in a biased non parametric model over L² risk (see Chesneau, C and Shirazi, E. Non parametric wavelet regression based on biased data, Communication in Statistics – Theory and Methods, 43: 2642–2658, 2014). This article considers d-dimensional regression function estimation over L^p?(1 ? p < ∞) risk. It turns out that our results reduce to the corresponding theorems of Chesneau and Shirazi’s theorems, when d = 1 and p = 2.     

Asymptotic results for empirical and partial-sum processes: A review

Two processes of importance in statistics and probability are the empirical and partial-sum processes. Based on d-dimensional data X₁, … X_a the empirical measure is defined for any A ⊂ R^d by the sample proportion of observations in A. When normalized, F_n yields the empirical process W_n: = n^1/2 (F_n - F), where F denotes the “true” probability measure. To define partial-sum processes, one needs data that are assigned to specified locations (in contrast to the above, where specified unit masses are assigned to random locations). A suitable context for many applications is that of data attached to points of a lattice, say {X_j:j ϵ J^d} where J = {1, 2,…}, for which the partial sums are defined for any A ⊂ R^d by Thus S(A) is the sum of the data contained in A. When normalized, S yields the partial-sum process. This paper provides an overview of asymptotic results for empirical and partial-sum processes, including strong laws and central limit theorems, together with some indications of their inferential implications.     

Semiparametric Mixtures of Symmetric Distributions

We consider in this paper the semiparametric mixture of two unknown distributions equal up to a location parameter. The model is said to be semiparametric in the sense that the mixed distribution is not supposed to belong to a parametric family. To insure the identifiability of the model, it is assumed that the mixed distribution is zero symmetric, the model being then defined by the mixing proportion, two location parameters and the probability density function of the mixed distribution. We propose a new class of M‐estimators of these parameters based on a Fourier approach and prove that they are ‐consistent under mild regularity conditions. Their finite sample properties are illustrated by a Monte Carlo study, and a benchmark real dataset is also studied with our method.     

Nonparametric Benchmark Dose Estimation with Continuous Dose‐Response Data

We propose a new method for risk‐analytic benchmark dose (BMD) estimation in a dose‐response setting when the responses are measured on a continuous scale. For each dose level d, the observation X(d) is assumed to follow a normal distribution: . No specific parametric form is imposed upon the mean μ(d), however. Instead, nonparametric maximum likelihood estimates of μ(d) and σ are obtained under a monotonicity constraint on μ(d). For purposes of quantitative risk assessment, a ‘hybrid’ form of risk function is defined for any dose d as R(d) = P[X(d) < c], where c > 0 is a constant independent of d. The BMD is then determined by inverting the additional risk functionR_A(d) = R(d) ? R(0) at some specified value of benchmark response. Asymptotic theory for the point estimators is derived, and a finite‐sample study is conducted, using both real and simulated data. When a large number of doses are available, we propose an adaptive grouping method for estimating the BMD, which is shown to have optimal mean integrated squared error under appropriate designs.     

An Independence Point Method of Confidence Band Construction for Multiple Linear Regression Models

This article addresses the problem of confidence band construction for a standard multiple linear regression model. An “independence point” method of construction is developed which generalizes the method of Gafarian (1964) for a simple linear regression model to a multiple linear regression model. Wynn (1984 Wynn , H. P. ( 1984 ). An exact confidence band for one-dimensional polynomial regression . Biometrika 71 : 375 – 9 .[Crossref], [Web of Science ®] , [Google Scholar]) pioneered the approach of basing confidence bands for a polynomial regression on a set of nodes where the function estimates are independent, and this approach is exploited in this article. This method requires only critical points from t-distributions so that the confidence bands are easy to construct. Both one-sided and two-sided confidence bands can be constructed using this method. An illustration of the new method is provided, and comparisons are made with other procedures.     

Central Limit Theorems for the Non‐Parametric Estimation of Time‐Changed Lévy Models

Abstract. Let {Z_t}_{t 0} be a Lévy process with Lévy measure ν and let be a random clock, where g is a non‐negative function and is an ergodic diffusion independent of Z. Time‐changed Lévy models of the form are known to incorporate several important stylized features of asset prices, such as leptokurtic distributions and volatility clustering. In this article, we prove central limit theorems for a type of estimators of the integral parameter β(?):=∫?(x)ν(dx), valid when both the sampling frequency and the observation time‐horizon of the process get larger. Our results combine the long‐run ergodic properties of the diffusion process with the short‐term ergodic properties of the Lévy process Z via central limit theorems for martingale differences. The performance of the estimators are illustrated numerically for Normal Inverse Gaussian process Z and a Cox–Ingersoll–Ross process .     

On the Validity of the Bootstrap in Non‐Parametric Functional Regression

Abstract. We consider the functional non‐parametric regression model Y= r( χ )+?, where the response Y is univariate, χ is a functional covariate (i.e. valued in some infinite‐dimensional space), and the error ? satisfies E(? | χ ) = 0. For this model, the pointwise asymptotic normality of a kernel estimator of r (·) has been proved in the literature. To use this result for building pointwise confidence intervals for r (·), the asymptotic variance and bias of need to be estimated. However, the functional covariate setting makes this task very hard. To circumvent the estimation of these quantities, we propose to use a bootstrap procedure to approximate the distribution of . Both a naive and a wild bootstrap procedure are studied, and their asymptotic validity is proved. The obtained consistency results are discussed from a practical point of view via a simulation study. Finally, the wild bootstrap procedure is applied to a food industry quality problem to compute pointwise confidence intervals.     

Coupling methods in approximations

Let X and Y be two arbitrary k-dimensional discrete random vectors, for k ≥ 1. We prove that there exists a coupling method which minimizes P( X ≠ Y ). This result is used to find the least upper bound for the metric d( X, Y ) = sup_A|P( X ∈ A ) ? P( Y ∈ A )| and to derive the inequality d(Σ X _i, Σ Y _i) ≤ Σd( X _i, Y _i). We thus obtain a unified method to measure the disparity between the distributions of sums of independent random vectors. Several examples are given.     

Interaction Terms in Distance-Based Regression

We propose a method of including polynomial and interaction terms in Distance-Based Regression (Cuadras and Arenas, 1990 Cuadras , C. M. , Arenas , C. ( 1990 ). A distance based regression model for prediction with mixed data . Commun. Statist. A Theor. Meth. 19 : 2261 – 2279 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), relying on properties of a semi-Hadamard or Khatri-Rao product of matrices. We demonstrate its application to real data examples.     

Empirical Likelihood Intervals for Conditional Value‐at‐Risk in Heteroscedastic Regression Models

Abstract. Non‐parametric regression models have been studied well including estimating the conditional mean function, the conditional variance function and the distribution function of errors. In addition, empirical likelihood methods have been proposed to construct confidence intervals for the conditional mean and variance. Motivated by applications in risk management, we propose an empirical likelihood method for constructing a confidence interval for the pth conditional value‐at‐risk based on the non‐parametric regression model. A simulation study shows the advantages of the proposed method.     

The inverse gaussian distribution: Some properties and characterizations

This article presents some structural properties of the inverse Gaussian distribution, together with several new characterizations based on constancy of regression of suitable functions on the sum of n independent identically distributed random variables. A decomposition of the statistic λσ (X^?1_i?X^?1) into n - 1 independent chi-squared random variables, each with one degree of freedom, is given when n is of the form 2^r.     

On multivariate variable-kernel density estimates for time series

Let X₁ be a strictly stationary multiple time series with values in R^d and with a common density f. Let X₁,.,.,X_n, be n consecutive observations of X₁. Let k = k_n, be a sequence of positive integers, and let H_ni be the distance from X_i to its kth nearest neighbour among X_j, j i. The multivariate variable-kernel estimate f_n, of f is defined by where K is a given density. The complete convergence of f_n, to f on compact sets is established for time series satisfying a dependence condition (referred to as the strong mixing condition in the locally transitive sense) weaker than the strong mixing condition. Appropriate choices of k are explicitly given. The results apply to autoregressive processes and bilinear time-series models.     

Uniform asymptotic linearity in a regression parameter of a process based on a rank statistic

For X₁, …, X_N a random sample from a distribution F, let the process S^δ_N(t) be defined as where K²_N = σ^N_i=1(c_i ? c?)² and R x_i, + Δd, is the rank of X_i + Δd_i, among X₁ + Δd₁, …, X_N + Δd_N. The purpose of this note is to prove that, under certain regularity conditions on F and on the constants c_i and d_i, S^Δ_N (t) is asymptotically approximately a linear function of Δ, uniformly in t and in Δ, |Δ| ≤ C. The special case of two samples is considered.     

On the Uniform Strong Consistency of Local Polynomial Regression Under Dependence Conditions

Abstract

In this article, nonparametric estimators of the regression function, and its derivatives, obtained by means of weighted local polynomial fitting are studied. Consider the fixed regression model where the error random variables are coming from a stationary stochastic process satisfying a mixing condition. Uniform strong consistency, along with rates, are established for these estimators. Furthermore, when the errors follow an AR(1) correlation structure, strong consistency properties are also derived for a modified version of the local polynomial estimators proposed by Vilar-Fernández and Francisco-Fernández (Vilar-Fernández, J. M., Francisco-Fernández, M. (2002 Vilar-Fernández, J. M. and Francisco-Fernández, M. 2002. Local polynomial regression smoothers with AR-error structure. TEST, 11(2): 439–464.  [Google Scholar]). Local polynomial regression smoothers with AR-error structure. TEST 11(2):439–464).     

Bayesian variable selection approach to a Bernstein polynomial regression model with stochastic constraints

This paper provides a Bayesian estimation procedure for monotone regression models incorporating the monotone trend constraint subject to uncertainty. For monotone regression modeling with stochastic restrictions, we propose a Bayesian Bernstein polynomial regression model using two-stage hierarchical prior distributions based on a family of rectangle-screened multivariate Gaussian distributions extended from the work of Gurtis and Ghosh [7 S.M. Curtis and S.K. Ghosh, A variable selection approach to monotonic regression with Bernstein polynomials, J. Appl. Stat. 38 (2011), pp. 961–976. doi: 10.1080/02664761003692423[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. This approach reflects the uncertainty about the prior constraint, and thus proposes a regression model subject to monotone restriction with uncertainty. Based on the proposed model, we derive the posterior distributions for unknown parameters and present numerical schemes to generate posterior samples. We show the empirical performance of the proposed model based on synthetic data and real data applications and compare the performance to the Bernstein polynomial regression model of Curtis and Ghosh [7 S.M. Curtis and S.K. Ghosh, A variable selection approach to monotonic regression with Bernstein polynomials, J. Appl. Stat. 38 (2011), pp. 961–976. doi: 10.1080/02664761003692423[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] for the shape restriction with certainty. We illustrate the effectiveness of our proposed method that incorporates the uncertainty of the monotone trend and automatically adapts the regression function to the monotonicity, through empirical analysis with synthetic data and real data applications.     

Goodness‐of‐fit Test for Directional Data

In this paper, we study the problem of testing the hypothesis on whether the density f of a random variable on a sphere belongs to a given parametric class of densities. We propose two test statistics based on the L² and L¹ distances between a non‐parametric density estimator adapted to circular data and a smoothed version of the specified density. The asymptotic distribution of the L² test statistic is provided under the null hypothesis and contiguous alternatives. We also consider a bootstrap method to approximate the distribution of both test statistics. Through a simulation study, we explore the moderate sample performance of the proposed tests under the null hypothesis and under different alternatives. Finally, the procedure is illustrated by analysing a real data set based on wind direction measurements.     

On Projection‐type Estimators of Multivariate Isotonic Functions

Abstract. Let M be an isotonic real‐valued function on a compact subset of and let be an unconstrained estimator of M. A feasible monotonizing technique is to take the largest (smallest) monotone function that lies below (above) the estimator or any convex combination of these two envelope estimators. When the process is asymptotically equicontinuous for some sequence r_n→∞, we show that these projection‐type estimators are r_n‐equivalent in probability to the original unrestricted estimator. Our first motivating application involves a monotone estimator of the conditional distribution function that has the distributional properties of the local linear regression estimator. Applications also include the estimation of econometric (probability‐weighted moment, quantile) and biometric (mean remaining lifetime) functions.