The effect of the regularity of the error process on the performance of kernel regression estimators

This article considers estimation of regression function $f$ in the fixed design model $Y(x_i)=f(x_i)+ \epsilon (x_i), i=1,\ldots ,n$ , by use of the Gasser and Müller kernel estimator. The point set $\{ x_i\}_{i=1}^{n}\subset [0,1]$ constitutes the sampling design points, and $\epsilon (x_i)$ are correlated errors. The error process $\epsilon $ is assumed to satisfy certain regularity conditions, namely, it has exactly $k$ ( $=\!0, 1, 2, \ldots $ ) quadratic mean derivatives (q.m.d.). The quality of the estimation is measured by the mean squared error (MSE). Here the asymptotic results of the mean squared error are established. We found that the optimal bandwidth depends on the $(2k+1)$ th mixed partial derivatives of the autocovariance function along the diagonal of the unit square. Simulation results for the model of $k$ th order integrated Brownian motion error are given in order to assess the effect of the regularity of this error process on the performance of the kernel estimator.     

Generalized Bayes minimax estimators of location vectors for spherically symmetric distributions with residual vector

We consider estimation of the mean vector, $\theta $ , of a spherically symmetric distribution with known scale parameter under quadratic loss and when a residual vector is available. We show minimaxity of generalized Bayes estimators corresponding to superharmonic priors with a non decreasing Laplacian of the form $\pi (\Vert \theta \Vert ^{2})$ , under certain conditions on the generating function $f(\cdot )$ of the sampling distributions. The class of sampling distributions includes certain variance mixtures of normals.     

{\varvec{P}}-value model selection criteria for exponential families of increasing dimension

Let $\mathcal{M }_{\underline{i}}$ be an exponential family of densities on $[0,1]$ pertaining to a vector of orthonormal functions $b_{\underline{i}}=(b_{i_1}(x),\ldots ,b_{i_p}(x))^\mathbf{T}$ and consider a problem of estimating a density $f$ belonging to such family for unknown set ${\underline{i}}\subset \{1,2,\ldots ,m\}$ , based on a random sample $X_1,\ldots ,X_n$ . Pokarowski and Mielniczuk (2011) introduced model selection criteria in a general setting based on p-values of likelihood ratio statistic for $H_0: f\in \mathcal{M }_0$ versus $H_1: f\in \mathcal{M }_{\underline{i}}\setminus \mathcal{M }_0$ , where $\mathcal{M }_0$ is the minimal model. In the paper we study consistency of these model selection criteria when the number of the models is allowed to increase with a sample size and $f$ ultimately belongs to one of them. The results are then generalized to the case when the logarithm of $f$ has infinite expansion with respect to $(b_i(\cdot ))_1^\infty $ . Moreover, it is shown how the results can be applied to study convergence rates of ensuing post-model-selection estimators of the density with respect to Kullback–Leibler distance. We also present results of simulation study comparing small sample performance of the discussed selection criteria and the post-model-selection estimators with analogous entities based on Schwarz’s rule as well as their greedy counterparts.     

Estimation of the scale parameter of a log-logistic distribution

In this work the ranked set sampling technique has been applied to estimate the scale parameter $\alpha $ of a log-logistic distribution under a situation where the units in a sample can be ordered by judgement method without any error. We have evaluated the Fisher information contained in the order statistics arising from this distribution and observed that median of a random sample contains the maximum information about the parameter $\alpha $ . Accordingly we have used median ranked set sampling to estimate $\alpha $ . We have further carried out the multistage median ranked set sampling to estimate $\alpha $ with improved precision. Suppose it is not possible to rank the units in a sample according to judgement method without error but the units can be ordered based on an auxiliary variable $Z$ such that $(X, Z)$ has a Morgenstern type bivariate log-logistic distribution (MTBLLD). In such a situation we have derived the Fisher information contained in the concomitant of rth order statistic of a random sample of size $n$ from MTBLLD and identified those concomitants among others which possess largest amount of Fisher information and defined an unbalanced ranked set sampling utilizing those units in the sample and thereby proposed an estimator of $\alpha $ using the measurements made on those units in this ranked set sample.     

Random weighting estimation of stable exponent

This paper presents a new random weighting method to estimation of the stable exponent. Assume that $X_1, X_2, \ldots ,X_n$ is a sequence of independent and identically distributed random variables with $\alpha $ -stable distribution G, where $\alpha \in (0,2]$ is the stable exponent. Denote the empirical distribution function of G by $G_n$ and the random weighting estimation of $G_n$ by $H_n$ . An empirical distribution function $\widetilde{F}_n$ with U-statistic structure is defined based on the sum-preserving property of stable random variables. By minimizing the Cramer-von-Mises distance between $H_n$ and ${\widetilde{F}}_n$ , the random weighting estimation of $\alpha $ is constructed in the sense of the minimum distance. The strong consistency and asymptotic normality of the random weighting estimation are also rigorously proved. Experimental results demonstrate that the proposed random weighting method can effectively estimate the stable exponent, resulting in higher estimation accuracy than the Zolotarev, Press, Fan and maximum likelihood methods.     

D-optimal chemical balance weighing designs with autoregressive errors

In this paper, we consider the estimation problem of individual weights of three objects. For the estimation we use the chemical balance weighing design and the criterion of D-optimality. We assume that the error terms ${\varepsilon_{i},\ i=1,2,\dots,n,}$ are a first-order autoregressive process. This assumption implies that the covariance matrix of errors depends on the known parameter ρ. We present the chemical balance weighing design matrix ${\widetilde{\bf X}}$ and we prove that this design is D-optimal in certain classes of designs for ${\rho\in[0,1)}$ and it is also D-optimal in the class of designs with the design matrix ${{\bf X} \in M_{n\times 3}(\pm 1)}$ for some ρ ≥ 0. We prove also the necessary and sufficient conditions under which the design is D-optimal in the class of designs ${M_{n\times 3}(\pm 1)}$ , if ${\rho\in[0,1/(n-2))}$ . We present also the matrix of the D-optimal factorial design with 3 two-level factors.     

Hermite ranks and U-statistics

We focus on the asymptotic behavior of $U$ -statistics of the type $$\begin{aligned} \sum _{1\le i\ne j\le n} h(X_i,X_j)\\ \end{aligned}$$ in the long-range dependence setting, where $(X_i)_{i\ge 1}$ is a stationary mean-zero Gaussian process. Since $(X_i)_{i\ge 1}$ is Gaussian, $h$ can be decomposed in Hermite polynomials. The goal of this paper is to compare the different notions of Hermite rank and to provide conditions for the remainder term in the decomposition to be asymptotically negligeable.     

Underlying determinants driving agreement among coders

There are plenty of intercoder reliability indices, whereas the choice of them has been debated. With a Monte Carlo simulation, the determinants of the agreement indices were empirically tested. The chance agreement of Bennett’s S is found to be only affected by the number of categories. Consequently, S is a category based index. The chance agreements of Krippendorff’s $\alpha $ , Scott’s $\pi $ and Cohen’s $\kappa $ are affected by the marginal distribution, the level of difficulty and the interaction between them, and yet the difficulty level influences their chance agreements abnormally. The three indices are hence in general distribution based indices. Gwet’s $AC_1$ reversed the direction of the three aforementioned indices, but its chance agreement is additionally affected by the number of categories and the interaction between the number of categories and the marginal distribution. $AC_1$ can be classified into a class based on the number of categories, the marginal distribution and the level of difficulty. Both theoretical and practical implications were also discussed in the end.     

Stochastic orderings of convolution residuals

In this paper we study convolution residuals, that is, if $X_1,X_2,\ldots ,X_n$ are independent random variables, we study the distributions, and the properties, of the sums $\sum _{i=1}^lX_i-t$ given that $\sum _{i=1}^kX_i>t$ , where $t\in \mathbb R $ , and $1\le k\le l\le n$ . Various stochastic orders, among convolution residuals based on observations from either one or two samples, are derived. As a consequence computable bounds on the survival functions and on the expected values of convolution residuals are obtained. Some applications in reliability theory and queueing theory are described.     

An additive property of weak records from geometric distributions

Let $\{W_m\}{_{m\ge 1}}$ be the sequence of weak records from a discrete parent random variable, $X$ , supported on the non-negative integers. We obtain a new characterization of geometric distributions based on an additive property of weak records: $X$ follows a geometric distribution if and only if for certain integers, $n,\, s\ge 1, W_{n+s}\stackrel{d}{=}W_n+W^{\prime }_s$ , with $W^{\prime }_s$ independent of $W_n$ and $W^{\prime }_s\stackrel{d}{=} W_s$ .     

Estimation of the mean for critical branching process and its bootstrap approximation

This article focuses attention on the estimation of the mean of a sequence of branching process with immigration in the critical case. We get the limiting distribution of the pivot, and adopt a bootstrap procedure to bootstrap the least-square estimator with bootstrap sample size $m$ less than the size $n$ of the original sample. Under the assumptions that $m\rightarrow \infty $ and $m/n\rightarrow 0$ , the convergence in probability of the bootstrap distribution function is also established.     

Nonparametric density estimation in compound Poisson processes using convolution power estimators

Consider a compound Poisson process which is discretely observed with sampling interval $\Delta $ until exactly $n$ nonzero increments are obtained. The jump density and the intensity of the Poisson process are unknown. In this paper, we build and study parametric estimators of appropriate functions of the intensity, and an adaptive nonparametric estimator of the jump size density. The latter estimation method relies on nonparametric estimators of $m$ th convolution powers density. The $L^2$ -risk of the adaptive estimator achieves the optimal rate in the minimax sense over Sobolev balls. Numerical simulation results on various jump densities enlight the good performances of the proposed estimator.     

An extremal property of the generalized arcsine distribution

The main result of the paper is the following characterization of the generalized arcsine density p _γ (t) = t ^γ?1(1 ? t) ^γ?1/B(γ, γ)   with ${t \in (0, 1)}$ and ${\gamma \in(0,\frac12) \cup (\frac12,1)}$ : a r.v. ξ supported on [0, 1] has the generalized arcsine density p _γ (t) if and only if ${ {\mathbb E} |\xi- x|^{1-2 \gamma}}$ has the same value for almost all ${x \in (0,1)}$ . Moreover, the measure with density p _γ (t) is a unique minimizer (in the space of all probability measures μ supported on (0, 1)) of the double expectation ${ (\gamma-\frac12 ) {\mathbb E} |\xi-\xi^{\prime}|^{1-2 \gamma}}$ , where ξ and ξ′ are independent random variables distributed according to the measure μ. These results extend recent results characterizing the standard arcsine density (the case ${\gamma=\frac12}$ ).     

Asymptotic behaviour of near-maxima of Gaussian sequences

Let \((X_1,X_2,\ldots ,X_n)\) be a Gaussian random vector with a common correlation coefficient \(\rho _n,\,0\le \rho _n<1\) , and let \(M_n= \max (X_1,\ldots , X_n),\,n\ge 1\) . For any given \(a>0\) , define \(T_n(a)= \left\{ j,\,1\le j\le n,\,X_j\in (M_n-a,\,M_n]\right\} ,\,K_n(a)= \#T_n(a)\) and \(S_n(a)=\sum \nolimits _{j\in T_n(a)}X_j,\,n\ge 1\) . In this paper, we obtain the limit distributions of \((K_n(a))\) and \((S_n(a))\) , under the assumption that \(\rho _n\rightarrow \rho \) as \(n\rightarrow \infty ,\) for some \(\rho \in [0,1)\) .     

Some results on constructing general minimum lower order confounding 2^{n-m} designs for n\le 2^{n-m-2}

Zhang et al. (Stat Sinica 18:1689–1705, 2008) introduced an aliased effect-number pattern for two-level regular designs and proposed a general minimum lower-order confounding (GMC) criterion for choosing optimal designs. All the GMC \(2^{n-m}\) designs with \(N/4+1\le n\le N-1\) were constructed by Li et al. (Stat Sinica 21:1571–1589, 2011), Zhang and Cheng (J Stat Plan Inference 140:1719–1730, 2010) and Cheng and Zhang (J Stat Plan Inference 140:2384–2394, 2010), where \(N=2^{n-m}\) is run number and \(n\) is factor number. In this paper, we first study some further properties of GMC design, then we construct all the GMC \(2^{n-m}\) designs respectively with the three parameter cases of \(n\le N-1\) : (i) \(m\le 4\) , (ii) \(m\ge 5\) and \(n=(2^m-1)u+r\) for \(u>0\) and \(r=0,1,2\) , and (iii) \(m\ge 5\) and \(n=(2^m-1)u+r\) for \(u\ge 0\) and \(r=2^m-3,2^m-2\) .     

Nonparametric correlation integral–based tests for linear and nonlinear stochastic processes

The BDS test is the best-known correlation integral–based test, and it is now an important part of most standard econometric data analysis software packages. This test depends on the proximity ( $\varepsilon )$ and the embedding dimension ( $m)$ parameters both of which are chosen by the researcher. Although different studies (e.g., Kanzler in Very fast and correctly sized estimation of the BDS statistic. Department of Economics, Oxford University, Oxford, 1999) have been carried out to provide an adequate selection of the proximity parameter, no relevant research has yet been done on $m$ . In practice, researchers usually compute the BDS statistic for different values of $m$ , but sometimes these results are contradictory because some of them accept the null and others reject it. This paper aims to fill this gap. To that end, we propose a new simple, yet powerful, aggregate test for independence, based on BDS outputs from a given data set, that allows the consideration of all of the information contained in several embedding dimensions without the ambiguity of the well-known BDS tests.     

Variable selection for high-dimensional varying coefficient partially linear models via nonconcave penalty

In this paper, we consider the problem of simultaneous variable selection and estimation for varying-coefficient partially linear models in a “small $n$ , large $p$ ” setting, when the number of coefficients in the linear part diverges with sample size while the number of varying coefficients is fixed. Similar problem has been considered in Lam and Fan (Ann Stat 36(5):2232–2260, 2008) based on kernel estimates for the nonparametric part, in which no variable selection was investigated besides that $p$ was assume to be smaller than $n$ . Here we use polynomial spline to approximate the nonparametric coefficients which is more computationally expedient, demonstrate the convergence rates as well as asymptotic normality of the linear coefficients, and further present the oracle property of the SCAD-penalized estimator which works for $p$ almost as large as $\exp \{n^{1/2}\}$ under mild assumptions. Monte Carlo studies and real data analysis are presented to demonstrate the finite sample behavior of the proposed estimator. Our theoretical and empirical investigations are actually carried out for the generalized varying-coefficient partially linear models, including both Gaussian data and binary data as special cases.     

Asymptotic properties of M-estimators in linear and nonlinear multivariate regression models

We consider the (possibly nonlinear) regression model in \(\mathbb{R }^q\) with shift parameter \(\alpha \) in \(\mathbb{R }^q\) and other parameters \(\beta \) in \(\mathbb{R }^p\) . Residuals are assumed to be from an unknown distribution function (d.f.). Let \(\widehat{\phi }\) be a smooth \(M\) -estimator of \(\phi = {{\beta }\atopwithdelims (){\alpha }}\) and \(T(\phi )\) a smooth function. We obtain the asymptotic normality, covariance, bias and skewness of \(T(\widehat{\phi })\) and an estimator of \(T(\phi )\) with bias \(\sim n^{-2}\) requiring \(\sim n\) calculations. (In contrast, the jackknife and bootstrap estimators require \(\sim n^2\) calculations.) For a linear regression with random covariates of low skewness, if \(T(\phi ) = \nu \beta \) , then \(T(\widehat{\phi })\) has bias \(\sim n^{-2}\) (not \(n^{-1}\) ) and skewness \(\sim n^{-3}\) (not \(n^{-2}\) ), and the usual approximate one-sided confidence interval (CI) for \(T(\phi )\) has error \(\sim n^{-1}\) (not \(n^{-1/2}\) ). These results extend to random covariates.     

Bayesian prediction in doubly stochastic Poisson process

A stochastic marked point process model based on doubly stochastic Poisson process is considered in the problem of prediction for the total size of future marks in a given period, given the history of the process. The underlying marked point process \((T_{i},Y_{i})_{i\ge 1}\) , where \(T_{i}\) is the time of occurrence of the \(i\) th event and the mark \(Y_{i}\) is its characteristic (size), is supposed to be a non-homogeneous Poisson process on \(\mathbb {R}_{+}^{2}\) with intensity measure \(P\times \varTheta \) , where \(P\) is known, whereas \(\varTheta \) is treated as an unknown measure of the total size of future marks in a given period. In the problem of prediction considered, a Bayesian approach is used assuming that \(\varTheta \) is random with prior distribution presented by a gamma process. The best predictor with respect to this prior distribution is constructed under a precautionary loss function. A simulation study for comparing the behavior of the predictors under various criteria is provided.