Estimation of the Sobol indices in a linear functional multidimensional model

We consider a functional linear model where the explicative variables are known stochastic processes taking values in a Hilbert space, the main example is given by Gaussian processes in L²([0,1])${\mathbb{L}}^2([0,1])$. We propose estimators of the Sobol indices in this functional linear model. Our estimators are based on U-statistics. We prove the asymptotic normality and the efficiency of our estimators and we compare them from a theoretical and practical point of view with classical estimators of Sobol indices.     

Minimum Hellinger distance estimation in a nonparametric mixture model

In this paper, we investigate the estimation problem of the mixture proportion λ$\lambda$ in a nonparametric mixture model of the form λF(x)+(1-λ)G(x)$\lambda F(x)+(1-\lambda )G(x)$ using the minimum Hellinger distance approach, where F and G are two unknown distributions. We assume that data from the distributions F and G   as well as from the mixture distribution λF+(1-λ)G$\lambda F+(1-\lambda )G$ are available. We construct a minimum Hellinger distance estimator of λ$\lambda$ and study its asymptotic properties. The proposed estimator is chosen to minimize the Hellinger distance between a parametric mixture model and a nonparametric density estimator. We also develop a maximum likelihood estimator of λ$\lambda$. Theoretical properties such as the existence, strong consistency, asymptotic normality and asymptotic efficiency of the proposed estimators are investigated. Robustness properties of the proposed estimator are studied using a Monte Carlo study. Two real data examples are also analyzed.     

Rates of convergence for the k-nearest neighbor estimators with smoother regression functions

Let (X, Y  ) be a R^d×R-valued${\mathbb{R}}^d\times \mathbb{R}\mathrm{-}\mathrm{valued}$ random vector. In regression analysis one wants to estimate the regression function m(x)?E(Y|X=x)$m(x)?\mathbf{E}(Y|X=x)$ from a data set. In this paper we consider the rate of convergence for the k-nearest neighbor estimators in case that X   is uniformly distributed on [0,1]^d$[\mathrm{0,1}{]}^d$, Var(Y|X=x)$\mathbf{Var}(Y|X=x)$ is bounded, and m is (p, C)-smooth. It is an open problem whether the optimal rate can be achieved by a k  -nearest neighbor estimator for 1<p≤1.5$1<p\le 1.5$. We solve the problem affirmatively. This is the main result of this paper. Throughout this paper, we assume that the data is independent and identically distributed and as an error criterion we use the expected L₂ error.     

Local asymptotic minimax theory for block-decreasing densities

In this paper, we study Lebesgue densities on (0,∞)^d$(0,\infty {)}^d$ that are non-increasing in each coordinate, while keeping all other coordinates fixed, from the perspective of local asymptotic minimax lower bound theory. In particular, we establish a local optimal rate of convergence of the order n^−1/(d+2)$n^{-1/(d+2)}$.     

Truncated linear estimation of a bounded multivariate normal mean

We consider the problem of estimating the mean θ$\theta$ of an _Np(θ,_Ip)${\mathrm{N}}_{\mathrm{p}}(\theta ,I_p)$ distribution with squared error loss ∥δ−θ∥²$\parallel \delta -\theta {\parallel }^2$ and under the constraint ∥θ∥≤m$\parallel \theta \parallel \le m$, for some constant m>0$m>0$. Using Stein's identity to obtain unbiased estimates of risk, Karlin's sign change arguments, and conditional risk analysis, we compare the risk performance of truncated linear estimators with that of the maximum likelihood estimator _δmle${\delta }_{\mathrm{mle}}$. We obtain for fixed (m,p)$(m,p)$ sufficient conditions for dominance. An asymptotic framework is developed, where we demonstrate that the truncated linear minimax estimator dominates _δmle${\delta }_{\mathrm{mle}}$, and where we obtain simple and accurate measures of relative improvement in risk. Numerical evaluations illustrate the effectiveness of the asymptotic framework for approximating the risks for moderate or large values of p.     

Efficiency in estimation of memory

We study the efficiency of semiparametric estimates of memory parameter. We propose a class of shift invariant tapers of order (p,q). For a fixed p, the variance inflation factor of the new tapers approaches 1 as q   goes to infinity. We show that for d∈(−1/2,p+1/2)$d\in (-1/2,p+1/2)$, the proposed tapered Gaussian semiparametric estimator has the same limiting distribution as the nontapered version for d∈(−1/2,1/2)$d\in (-1/2,1/2)$. The new estimator is mean and polynomial trend invariant, and is computationally advantageous in comparison to the recently proposed exact local Whittle estimator. The simulation study shows that our estimator has comparable or better mean squared error in finite samples for a variety of models.     

Skew Dyck paths,area, and superdiagonal bargraphs

Skew Dyck paths are a generalization of ordinary Dyck paths, defined as paths using up steps  U=(1,1)$U=(1,1)$, down steps  D=(1,-1)$D=(1,-1)$, and left steps  L=(−1,-1)$L=(-1,-1)$, starting and ending on the x-axis, never going below it, and so that up and left steps never overlap. In this paper we study the class of these paths according to their area, extending several results holding for Dyck paths. Then we study the class of superdiagonal bargraphs, which can be naturally defined starting from skew Dyck paths.     

Skew Dyck paths

In this paper we study the class S$\mathbb{S}$ of skew Dyck paths, i.e. of those lattice paths that are in the first quadrant, begin at the origin, end on the x-axis, consist of up steps  U=(1,1)$U=(1,1)$, down steps  D=(1,-1)$D=(1,-1)$, and left steps  L=(−1,-1)$L=(-1,-1)$, and such that up steps never overlap with left steps. In particular, we show that these paths are equinumerous with several other combinatorial objects, we describe some involutions on this class, and finally we consider several statistics on S$\mathbb{S}$.     

Asymptotic results in canonical discriminant analysis when the dimension is large compared to the sample size

This paper deals with the distributions of test statistics for the number of useful discriminant functions and the characteristic roots in canonical discriminant analysis. These asymptotic distributions have been extensively studied when the number p   of variables is fixed, the number q+1$q+1$ of groups is fixed, and the sample size N tends to infinity. However, these approximations become increasingly inaccurate as the value of p increases for a fixed value of N. On the other hand, we encounter to analyze high-dimensional data such that p is large compared to n. The purpose of the present paper is to derive asymptotic distributions of these statistics in a high-dimensional framework such that q   is fixed, p→∞$p\to \infty$, m=n-p+q→∞$m=n-p+q\to \infty$, and p/n→c∈(0,1)$p/n\to c\in (0,1)$, where n=N-q-1$n=N-q-1$. Numerical simulation revealed that our new asymptotic approximations are more accurate than the classical asymptotic approximations in a considerably wide range of (n,p,q)$(n,p,q)$.     

Prediction in moving average processes

For the stationary invertible moving average process of order one with unknown innovation distribution F, we construct root-n   consistent plug-in estimators of conditional expectations E(h(_Xn+1)|_X1,…,_Xn)$E(h(X_{n+1})|X_1,\dots ,X_n)$. More specifically, we give weak conditions under which such estimators admit Bahadur-type representations, assuming some smoothness of h or of F. For fixed h it suffices that h   is locally of bounded variation and locally Lipschitz in _L2(F)$L_2(F)$, and that the convolution of h and F   is continuously differentiable. A uniform representation for the plug-in estimator of the conditional distribution function P(_Xn+1?·|_X1,…,_Xn)$P(X_{n+1}?·|X_1,\dots ,X_n)$ holds if F has a uniformly continuous density. For a smoothed version of our estimator, the Bahadur representation holds uniformly over each class of functions h that have an appropriate envelope and whose shifts are F-Donsker, assuming some smoothness of F. The proofs use empirical process arguments.     

Sequential estimation of the sum of sinusoidal model parameters

Estimating the parameters of the sum of a sinusoidal model in presence of additive noise is a classical problem. It is well known to be a difficult problem when the two adjacent frequencies are not well separated or when the number of components is very large. In this paper we propose a simple sequential procedure to estimate the unknown frequencies and amplitudes of the sinusoidal signals. It is observed that if there are p components in the signal then at the k  th (k?p)$(k?p)$ stage our procedure produces strongly consistent estimators of the k   dominant sinusoids. For k>p$k>p$, the amplitude estimators converge to zero almost surely. Asymptotic distribution of the proposed estimators is also established and it is observed that it coincides with the asymptotic distribution of the least squares estimators. Numerical simulations are performed to observe the performance of the proposed estimators for different sample sizes and for different models. One ECG data and one synthesized data are analyzed for illustrative purpose.     

Mantel-Haenszel estimators for irregular sparse K2×J tables

This paper deals with sparse K2×J(J>2)$K2\times J(J>2)$ tables. Projection-method Mantel–Haenszel (MH) estimators of the common odds ratios have been proposed for K2×J$K2\times J$ tables, which include Greenland's generalized MH estimator as a special case. The method projects log-transformed MH estimators for all K2×2$K2\times 2$ subtables, which were called naive MH estimators, onto a linear space spanned by log odds ratios. However, for sparse tables it is often the case that naive MH estimators are unable to be computed. In this paper we introduce alternative naive MH estimators using a graph that represents K2×J$K2\times J$ tables, and apply the projection to these alternative estimators. The idea leads to infinitely many reasonable estimators and we propose a method to choose the optimal one by solving a quadratic optimization problem induced by the graph, where some graph-theoretic arguments play important roles to simplify the optimization problem. An illustration is given using data from a case–control study. A simulation study is also conducted, which indicates that the MH estimator tends to have a smaller mean squared error than the MH estimator previously suggested and the conditional maximum likelihood estimator for sparse tables.     

Constrained Bayes and empirical Bayes estimation under random effects normal ANOVA model with balanced loss function

The paper develops constrained Bayes and empirical Bayes estimators in the random effects ANOVA model under balanced loss functions. In the balanced normal–normal model, estimators of the Bayes risks of the constrained Bayes and constrained empirical Bayes estimators are provided which are correct asymptotically up to O(m^-1)$\mathrm{O}(m^{-1})$, that is the remainder term is o(m^-1)$\mathrm{o}(m^{-1})$, m$m$ denoting the number of strata.     

Tools for constructing optimal two-level factorial designs for a linear model containing main effects and one two-factor interaction

In Hedayat and Pesotan [1992, Two-level factorial designs for main effects and selected two-factor interactions. Statist. Sinica 2, 453–464.] the concepts of a g(n,e)$g(n,e)$-design and a g(n,e)$g(n,e)$-matrix are introduced to study designs of n$n$ factor two-level experiments which can unbiasedly estimate the mean, the n$n$ main effects and e$e$ specified two-factor interactions appearing in an orthogonal polynomial model and it is observed that the construction of a g-design is equivalent to the construction of a g  -matrix. This paper deals with the construction of D-optimal g(n,1)$g(n,1)$-matrices. A standard form for a g(n,1)$g(n,1)$-matrix is introduced and some lower and upper bounds on the absolute determinant value of a D-optimal g(n,1)$g(n,1)$-matrix in the class of all g(n,1)$g(n,1)$-matrices are obtained and an approach to construct D-optimal g(n,1)$g(n,1)$-matrices is given for 2?n?8$2?n?8$. For two specific subclasses, namely a certain class of g(n,1)$g(n,1)$-matrices within the class of g(n,1)$g(n,1)$-matrices of index one and the class C(H)$C(H)$ of g(8t+2,1)$g(8t+2,1)$-matrices constructed from a normalized Hadamard matrix H   of order 8t+4(t?1)$8t+4(t?1)$ two techniques for the construction of the restricted D-optimal matrices are given.     

Empirical Bayes regression analysis with many regressors but fewer observations

In this paper, we consider the prediction problem in multiple linear regression model in which the number of predictor variables, p, is extremely large compared to the number of available observations, n  . The least-squares predictor based on a generalized inverse is not efficient. We propose six empirical Bayes estimators of the regression parameters. Three of them are shown to have uniformly lower prediction error than the least-squares predictors when the vector of regressor variables are assumed to be random with mean vector zero and the covariance matrix (1/n)X^tX$(1/n){\mathit{X}}^{\mathrm{t}}\mathit{X}$ where X^t=(_x1,…,_xn)${\mathit{X}}^{\mathrm{t}}=({\mathit{x}}_1,\dots ,{\mathit{x}}_n)$ is the p×n$p\times n$ matrix of observations on the regressor vector centered from their sample means. For other estimators, we use simulation to show its superiority over the least-squares predictor.