Bootstrap methods for stationary functional time series

Bootstrap methods for estimating the long-run covariance of stationary functional time series are considered. We introduce a versatile bootstrap method that relies on functional principal component analysis, where principal component scores can be bootstrapped by maximum entropy. Two other bootstrap methods resample error functions, after the dependence structure being modeled linearly by a sieve method or nonlinearly by a functional kernel regression. Through a series of Monte-Carlo simulation, we evaluate and compare the finite-sample performances of these three bootstrap methods for estimating the long-run covariance in a functional time series. Using the intraday particulate matter (\(\hbox {PM}_{10}\)) dataset in Graz, the proposed bootstrap methods provide a way of constructing the distribution of estimated long-run covariance for functional time series.     

Central quantile subspace

Quantile regression (QR) is becoming increasingly popular due to its relevance in many scientific investigations. There is a great amount of work about linear and nonlinear QR models. Specifically, nonparametric estimation of the conditional quantiles received particular attention, due to its model flexibility. However, nonparametric QR techniques are limited in the number of covariates. Dimension reduction offers a solution to this problem by considering low-dimensional smoothing without specifying any parametric or nonparametric regression relation. The existing dimension reduction techniques focus on the entire conditional distribution. We, on the other hand, turn our attention to dimension reduction techniques for conditional quantiles and introduce a new method for reducing the dimension of the predictor $$\mathbf {X}$$. The novelty of this paper is threefold. We start by considering a single index quantile regression model, which assumes that the conditional quantile depends on $$\mathbf {X}$$ through a single linear combination of the predictors, then extend to a multi-index quantile regression model, and finally, generalize the proposed methodology to any statistical functional of the conditional distribution. The performance of the methodology is demonstrated through simulation examples and real data applications. Our results suggest that this method has a good finite sample performance and often outperforms the existing methods.     

Nonparametric estimation of a multivariate multiple regression function

Let (X₁, X₂, Y₁, Y₂) be a four dimensional random variable having the joint probability density function f(x₁, x₂, y₁, y₂). In this paper we consider the problem of estimating the regression function \({{E[(_{Y_2 }^{Y_1 } )} \mathord{\left/ {\vphantom {{E[(_{Y_2 }^{Y_1 } )} {_{X_2 = X_2 }^{X_1 = X_1 } }}} \right. \kern-0em} {_{X_2 = X_2 }^{X_1 = X_1 } }}]\) on the basis of a random sample of size n. We have proved that under certain regularity conditions the kernel estimate of this regression function is uniformly strongly consistent. We have also shown that under certain conditions the estimate is asymptotically normally distributed.     

The type I distribution of the ratio of independent “Weibullized” generalized beta-prime variables

In this paper we introduce the distribution of , with c >  0, where X _i , i =  1, 2, are independent generalized beta-prime-distributed random variables, and establish a closed form expression of its density. This distribution has as its limiting case the generalized beta type I distribution recently introduced by Nadarajah and Kotz (2004). Due to the presence of several parameters the density can take a wide variety of shapes.      

Chain graph models: topological sorting of meta-arrows and efficient construction of $${\mathcal{B}}$$ -essential graphs

Essential graphs and largest chain graphs are well-established graphical representations of equivalence classes of directed acyclic graphs and chain graphs respectively, especially useful in the context of model selection. Recently, the notion of a labelled block ordering of vertices was introduced as a flexible tool for specifying subfamilies of chain graphs. In particular, both the family of directed acyclic graphs and the family of “unconstrained” chain graphs can be specified in this way, for the appropriate choice of . The family of chain graphs identified by a labelled block ordering of vertices is partitioned into equivalence classes each represented by means of a -essential graph. In this paper, we introduce a topological ordering of meta-arrows and use this concept to devise an efficient procedure for the construction of -essential graphs. In this way we also provide an efficient procedure for the construction of both largest chain graphs and essential graphs. The key feature of the proposed procedure is that every meta-arrow needs to be processed only once.     

Improvement of the Liu estimator in linear regression model

In the presence of stochastic prior information, in addition to the sample, Theil and Goldberger (1961) introduced a Mixed Estimator for the parameter vector β in the standard multiple linear regression model (T,Xβ,σ² I). Recently, the Liu estimator which is an alternative biased estimator for β has been proposed by Liu (1993). In this paper we introduce another new Liu type biased estimator called Stochastic restricted Liu estimator for β, and discuss its efficiency. The necessary and sufficient conditions for mean squared error matrix of the Stochastic restricted Liu estimator to exceed the mean squared error matrix of the mixed estimator will be derived for the two cases in which the parametric restrictions are correct and are not correct. In particular we show that this new biased estimator is superior in the mean squared error matrix sense to both the Mixed estimator and to the biased estimator introduced by Liu (1993).     

L models and multiple regressions designs

Given an orthogonal model

${{\bf \lambda}}=\sum_{i=1}^m{{{\bf X}}_i}{\boldsymbol{\alpha}}_i$

an L model

${{\bf y}}={\bf L}\left(\sum_{i=1}^m{{{\bf X}}_i}{\boldsymbol{\alpha}}_i\right)+{\bf e}$

is obtained, and the only restriction is the linear independency of the column vectors of matrix L. Special cases of the L models correspond to blockwise diagonal matrices L = D(L ₁, . . . , L _c ). In multiple regression designs this matrix will be of the form

${\bf L}={\bf D}(\check{{\bf X}}_1,\ldots,\check{{\bf X}}_{c})$

with \({\check{{\bf X}}_j, j=1,\ldots,c}\) the model matrices of the individual regressions, while the original model will have fixed effects. In this way, we overcome the usual restriction of requiring all regressions to have the same model matrix.

     

Estimation of a normal mean relative to balanced loss functions

LetX ₁,…,X _nbe a random sample from a normal distribution with mean θ and variance σ². The problem is to estimate θ with Zellner's (1994) balanced loss function, % MathType!End!2!1!, where 0<ω<1. It is shown that the sample mean % MathType!End!2!1!, is admissible. More generally, we investigate the admissibility of estimators of the form % MathType!End!2!1! under % MathType!End!2!1!. We also consider the weighted balanced loss function, % MathType!End!2!1!, whereq(θ) is any positive function of θ, and the class of admissible linear estimators is obtained under such loss withq(θ) =e ^θ .     

Comparing several exponential populations with more than one control

Suppose there are k ₁ (k ₁ ≥ 1) test treatments that we wish to compare with k ₂ (k ₂ ≥ 1) control treatments. Assume that the observations from the ith test treatment and the jth control treatment follow a two-parameter exponential distribution and , where θ is a common scale parameter and and are the location parameters of the ith test and the jth control treatment, respectively, i = 1, . . . ,k ₁; j = 1, . . . ,k ₂. In this paper, simultaneous one-sided and two-sided confidence intervals are proposed for all k ₁ k ₂ differences between the test treatment location and control treatment location parameters, namely , and the required critical points are provided. Discussions of multiple comparisons of all test treatments with the best control treatment and an optimal sample size allocation are given. Finally, it is shown that the critical points obtained can be used to construct simultaneous confidence intervals for Pareto distribution location parameters.     

A bound for the Euclidean distance between restricted and unrestricted estimators of parametric functions in the general linear model

Let [^(\varveck)]{\widehat{\varvec{\kappa}}} and [^(\varveck)]_r{\widehat{\varvec{\kappa}}_r} denote the best linear unbiased estimators of a given vector of parametric functions \varveck = \varvecKb{\varvec{\kappa} = \varvec{K\beta}} in the general linear models M = {\varvecy, \varvecX\varvecb, s²\varvecV}{{\mathcal M} = \{\varvec{y},\, \varvec{X\varvec{\beta}},\, \sigma^2\varvec{V}\}} and M_r = {\varvecy, \varvecX\varvecb | \varvecR \varvecb = \varvecr, s²\varvecV}{{\mathcal M}_r = \{\varvec{y},\, \varvec{X}\varvec{\beta} \mid \varvec{R} \varvec{\beta} = \varvec{r},\, \sigma^2\varvec{V}\}}, respectively. A bound for the Euclidean distance between [^(\varveck)]{\widehat{\varvec{\kappa}}} and [^(\varveck)]_r{\widehat{\varvec{\kappa}}_r} is expressed by the spectral distance between the dispersion matrices of the two estimators, and the difference between sums of squared errors evaluated in the model M{{\mathcal M}} and sub-restricted model M_r^*{{\mathcal M}_r^*} containing an essential part of the restrictions \varvecR\varvecb = \varvecr{\varvec{R}\varvec{\beta} = \varvec{r}} with respect to estimating \varveck{\varvec{\kappa}}.     

Conditional estimation of average on the basis of weighting data

LetF(x,y) be a distribution function of a two dimensional random variable (X,Y). We assume that a distribution functionF _x(x) of the random variableX is known. The variableX will be called an auxiliary variable. Our purpose is estimation of the expected valuem=E(Y) on the basis of two-dimensional simple sample denoted by:U=[(X ₁, Y₁)…(X_n, Y_n)]=[X Y]. LetX=[X ₁…X _n]andY=[Y ₁…Y _n].This sample is drawn from a distribution determined by the functionF(x,y). LetX _(k)be the k-th (k=1, …,n) order statistic determined on the basis of the sampleX. The sampleU is truncated by means of this order statistic into two sub-samples: % MathType!End!2!1! and % MathType!End!2!1!.Let % MathType!End!2!1! and % MathType!End!2!1! be the sample means from the sub-samplesU _k,1 andU _k,2, respectively. The linear combination % MathType!End!2!1! of these means is the conditional estimator of the expected valuem. The coefficients of this linear combination depend on the distribution function of auxiliary variable in the pointx _(k).We can show that this statistic is conditionally as well as unconditionally unbiased estimator of the averagem. The variance of this estimator is derived. The variance of the statistic % MathType!End!2!1! is compared with the variance of the order sample mean. The generalization of the conditional estimation of the mean is considered, too.     

Prevalent cohort studies and unobserved heterogeneity

Lifetime Data Analysis - Consider lifetimes originating at a series of calendar times $$ t_{1} ,t_{2} , \ldots $$ . At a certain time $$ t_{0} $$ a cross-sectional sample is taken, generating a...     

Hypotheses testing about the drift parameter in linear stochastic differential equation driven by stable processes

In this paper, we consider the problem of hypotheses testing about the drift parameter \(\theta \) in the process \(\text {d}Y^{\delta }_{t} = \theta \dot{f}(t)Y^{\delta }_{t}\text {d}t + b(t)\text {d}L^{\delta }_{t}\) driven by symmetric \(\delta \)-stable Lévy process \(L^{\delta }_{t}\) with \(\dot{f}(t)\) being the derivative of a known increasing function f(t) and b(t) being known as well. We consider the hypotheses testing \(H_{0}: \theta \le 0\) and \(K_{0}: \theta =0\) against the alternatives \(H_{1}: \theta >0\) and \(K_{1}: \theta \ne 0\), respectively. For these hypotheses, we propose inverse methods, which are motivated by sequential approach, based on the first hitting time of the observed process (or its absolute value) to a pre-specified boundary or two boundaries until some given time. The applicability of these methods is illustrated. For the case \(Y^{\delta }_{0}=0\), we are able to calculate the values of boundaries and finite observed times more directly. We are able to show the consistencies of proposed tests for \(Y^{\delta }_{0}\ge 0\) with \(\delta \in (1,2]\) and for \(Y^{\delta }_{0}=0\) with \(\delta \in (0,2]\) under quite mild conditions.     

Equalities between OLSE,BLUE and BLUP in the linear model

We consider equalities between the ordinary least squares estimator ( $\mathrm {OLSE} $ ), the best linear unbiased estimator ( $\mathrm {BLUE} $ ) and the best linear unbiased predictor ( $\mathrm {BLUP} $ ) in the general linear model $\{ \mathbf y , \mathbf X \varvec{\beta }, \mathbf V \}$ extended with the new unobservable future value $ \mathbf y _{*}$ of the response whose expectation is $ \mathbf X _{*}\varvec{\beta }$ . Our aim is to provide some new insight and new proofs for the equalities under consideration. We also collect together various expressions, without rank assumptions, for the $\mathrm {BLUP} $ and provide new results giving upper bounds for the Euclidean norm of the difference between the $\mathrm {BLUP} ( \mathbf y _{*})$ and $\mathrm {BLUE} ( \mathbf X _{*}\varvec{\beta })$ and between the $\mathrm {BLUP} ( \mathbf y _{*})$ and $\mathrm {OLSE} ( \mathbf X _{*}\varvec{\beta })$ . A remark is made on the application to small area estimation.     

From Optimality Robustness To Sufficiency and Completeness

Let (\Omega,{\cal A},{\cal P}) stand for a statistical experiment and {\cal B},{\cal C} for some sub- σ -algebras of {\cal A} with {\cal C}\subset {\cal B} . It is shown that for any {\cal B} -measurable d\in\bigcap_{P\in {\cal P}}\,{\cal L}_{2}(\Omega,{\cal A},P) there exists some d_{1}\in\bigcap_{P\in {\cal P}}{\cal L}_{2}(\Omega,{\cal A},P) being {\cal C} -measurable and a UMVU estimator in (\Omega,{\cal A},{\cal P}) and some conditional white noise d_{2}\in\bigcap_{P\in {\cal P}}\,{\cal L}_{2}(\Omega,{\cal A},P) , i.e. E_{P}(d_{2}\vert {\cal C})=0,P\in {\cal P} , satisfying d=d_{1}+d_{2} , where d_{j},j=1,2 , are uniquely determined up to P -zero sets, if and only if {\cal C} is sufficient and complete for {\cal P}\vert {\cal B} and {\cal B} is optimality robust for {\cal P} , i.e. any {\cal B} -measurable d\in\bigcap_{P\in {\cal P}}\,{\cal L}_{2}(\Omega,{\cal A},P) being some UMVU estimator in the restricted statistical experiment (\Omega,{\cal B},{\cal P}\vert {\cal B}) is already a UMVU estimator in the original statistical experiment (\Omega,{\cal A},{\cal P}) . In particular, the special case {\cal B}={\cal A} characterizes sufficiency and completeness of {\cal C} for {\cal P} and the special case {\cal B}={\cal C} optimality robustness and completeness of {\cal C} for {\cal P} from a decomposition theoretical point of view. As an application it is shown that a σ -algebra containing a sufficient and complete sub- σ -algebra is optimality robust without being itself in general neither sufficient nor complete.     

Estimation of general semi-parametric quantile regression

Quantile regression introduced by Koenker and Bassett (1978) produces a comprehensive picture of a response variable on predictors. In this paper, we propose a general semi-parametric model of which part of predictors are presented with a single-index, to model the relationship of conditional quantiles of the response on predictors. Special cases are single-index models, partially linear single-index models and varying coefficient single-index models. We propose the qOPG, a quantile regression version of outer-product gradient estimation method (OPG, Xia et al., 2002) to estimate the single-index. Large-sample properties, simulation results and a real-data analysis are provided to examine the performance of the qOPG.