Statistical inference for the optimal approximating model

In the setting of high-dimensional linear models with Gaussian noise, we investigate the possibility of confidence statements connected to model selection. Although there exist numerous procedures for adaptive (point) estimation, the construction of adaptive confidence regions is severely limited (cf. Li in Ann Stat 17:1001–1008, 1989). The present paper sheds new light on this gap. We develop exact and adaptive confidence regions for the best approximating model in terms of risk. One of our constructions is based on a multiscale procedure and a particular coupling argument. Utilizing exponential inequalities for noncentral χ ²-distributions, we show that the risk and quadratic loss of all models within our confidence region are uniformly bounded by the minimal risk times a factor close to one.     

Statistical inference for the index parameter in single-index models

In this paper, we are concerned with statistical inference for the index parameter in the single-index model . Based on the estimates obtained by the local linear method, we extend the generalized likelihood ratio test to the single-index model. We investigate the asymptotic behaviour of the proposed test and demonstrate that its limiting null distribution follows a χ²-distribution, with the scale constant and the number of degrees of freedom being independent of nuisance parameters or functions, which is called the Wilks phenomenon. A simulated example is used to illustrate the performance of the testing approach.     

Statistical inference for detrended point processes

We consider a multivariate point process with a parametric intensity process which splits into a stochastic factor b_t and a trend function a_t of a squared polynomial form with exponents larger than . Such a process occurs in a situation where an underlying process with intensity b_t can be observed on a transformed time scale only. On the basis of the maximum likelihood estimator for the unknown parameter a detrended (or residual) process is defined by transforming the occurrence times via integrated estimated trend function. It is shown that statistics (mean intensity, periodogram estimator) based on the detrended process exhibit the same asymptotic properties as they do in the case of the underlying process (without trend function). Thus trend removal in point processes turns out to be an appropriate method to reveal properties of the (unobservable) underlying process – a concept which is well established in time series. A numerical example of an earthquake aftershock sequence illustrates the performance of the method.     

Statistical inference in non-commutative probability

A theory of statistical inference, applicable to both classical and quantum systems, is first presented in an original version based on von Neumann algebras, then in aC ^*-algebraic generalization due to S. Gudder.     

Statistical inference in stochastic processes

(Probability: Pure and Applied, Series of Textbooks and Reference Books, vol. 6), edited by N. U. Prabhuand I. W. Basawa, Marcel Dekker, New York (1991), 288 pp. $89.75 (USA and Canda), $107.50 (All other countries). ISBN 0-8247-8417-0     

Statistical inference with minimumd-risk

Translated from Issledovaniya po Prikladnoi Matematike, No. 11, Part 2, pp. 25–39, 1984. Presented at the seminar of the Department of Probability Theory and Mathematical Statistics, NIIMM, May 21, 1981.     

Statistical inference in partially-varying-coefficient single-index model

Consider a varying-coefficient single-index model which consists of two parts: the linear part with varying coefficients and the nonlinear part with a single-index structure, and are hence termed as varying-coefficient single-index models. This model includes many important regression models such as single-index models, partially linear single-index models, varying-coefficient model and varying-coefficient partially linear models as special examples. In this paper, we mainly study estimating problems of the varying-coefficient vector, the nonparametric link function and the unknown parametric vector describing the single-index in the model. A stepwise approach is developed to obtain asymptotic normality estimators of the varying-coefficient vector and the parametric vector, and estimators of the nonparametric link function with a convergence rate. The consistent estimator of the structural error variance is also obtained. In addition, asymptotic pointwise confidence intervals and confidence regions are constructed for the varying coefficients and the parametric vector. The bandwidth selection problem is also considered. A simulation study is conducted to evaluate the proposed methods, and real data analysis is also used to illustrate our methods.     

Statistical inference in compound functional models

We consider a general nonparametric regression model called the compound model. It includes, as special cases, sparse additive regression and nonparametric (or linear) regression with many covariates but possibly a small number of relevant covariates. The compound model is characterized by three main parameters: the structure parameter describing the “macroscopic” form of the compound function, the “microscopic” sparsity parameter indicating the maximal number of relevant covariates in each component and the usual smoothness parameter corresponding to the complexity of the members of the compound. We find non-asymptotic minimax rate of convergence of estimators in such a model as a function of these three parameters. We also show that this rate can be attained in an adaptive way.     

Statistical inference in massive data sets

Analysis of massive data sets is challenging owing to limitations of computer primary memory. In this paper, we propose an approach to estimate population parameters from a massive data set. The proposed approach significantly reduces the required amount of primary memory, and the resulting estimate will be as efficient if the entire data set was analyzed simultaneously. Asymptotic properties of the resulting estimate are studied, and the asymptotic normality of the resulting estimator is established. The standard error formula for the resulting estimate is proposed and empirically tested; thus, statistical inference for parameters of interest can be performed. The effectiveness of the proposed approach is illustrated using simulation studies and an Internet traffic data example. Copyright © 2012 John Wiley & Sons, Ltd.     

Statistical inference in non-nested econometric models

The purpose of this paper is to discuss some procedures that are available for testing non-nested (or separate) hypotheses in the statistics and econometrics literature. Since many of these techniques may also be exploited in other disciplines, it is hoped that an elaboration of the principal theoretical findings may make them more readily accessible to researchers in other disciplines. Several simple examples are used to illustrate the concepts of nested and non-nested hypotheses and, within the latter category, “global” and “partial” non-nested hypotheses. Two alternative methods of testing non-nested hypotheses are discussed and contrasted: the first of these is Cox's modification of the likelihood-ratio statistic, and the second is Atkinson's comprehensive model approach. A major emphasis is placed on the role of the Cox principle of hypothesis testing, which enables a broad range of hypotheses to be tested within the same framework. The problem associated with the application of the comprehensive model approach to composite non-nested hypotheses is also highlighted; Roy's union-intersection principle is presented as a viable method of dealing with this problem. Simulation results concerning the finite-sample properties of various tests are discussed, together with an analysis of some attempts to correct the poor size of the Cox and related tests.     

Statistical inference for $$L^2$$-distances to uniformity

The paper deals with the asymptotic behaviour of estimators, statistical tests and confidence intervals for \(L^2\)-distances to uniformity based on the empirical distribution function, the integrated empirical distribution function and the integrated empirical survival function. Approximations of power functions, confidence intervals for the \(L^2\)-distances and statistical neighbourhood-of-uniformity validation tests are obtained as main applications. The finite sample behaviour of the procedures is illustrated by a simulation study.     

Statistical inference using higher-order information

This paper presents a class of minimum contrast estimators for stochastic processes with possible long-range dependence based on the information on higher-order spectral densities. The results on consistency and asymptotic normality of the proposed estimators are provided.     

STATISTICAL INFERENCE PROCEDURE FOR A BIVARIATE EXPONENTIAL DISTRIBUTION

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Statistical inference for DEA estimators of directional distances

In productivity and efficiency analysis, the technical efficiency of a production unit is measured through its distance to the efficient frontier of the production set. The most familiar non-parametric methods use Farrell–Debreu, Shephard, or hyperbolic radial measures. These approaches require that inputs and outputs be non-negative, which can be problematic when using financial data. Recently, Chambers et al. (1998) have introduced directional distance functions which can be viewed as additive (rather than multiplicative) measures efficiency. Directional distance functions are not restricted to non-negative input and output quantities; in addition, the traditional input and output-oriented measures are nested as special cases of directional distance functions. Consequently, directional distances provide greater flexibility. However, until now, only free disposal hull (FDH) estimators of directional distances (and their conditional and robust extensions) have known statistical properties (Simar and Vanhems, 2012). This paper develops the statistical properties of directional d estimators, which are especially useful when the production set is assumed convex. We first establish that the directional Data Envelopment Analysis (DEA) estimators share the known properties of the traditional radial DEA estimators. We then use these properties to develop consistent bootstrap procedures for statistical inference about directional distance, estimation of confidence intervals, and bias correction. The methods are illustrated in some empirical examples.     

Statistical inference on mixing proportion

In this paper, the interval estimation and hypothesis testing of the mixing proportion in mixture distributions are considered. A statistical inferential method is proposed which is inspired by the generalized p-values and generalized pivotal quantity. In some situations, the true levels of the tests given in the paper are equal to nominal levels, and the true coverage of the interval estimation or confidence bounds is also equal to nominal one. In other situations, under mild conditions, the tests are consistent and the coverage of the interval estimations or the confidence bounds is asymptotically equal to nominal coverage. Meanwhile, some simulations are performed which show that our method is satisfactory.     

Statistical inference for the Weitzman overlapping coefficient in a family of distributions

A usual problem in applied statistics is the one related to the estimation of the common area under two distributions, which is usually estimated by means of an overlapping coefficient. Particularly, for the Weitzman overlapping coefficient, we found that it is possible to provide a general expression that facilitates making inferences on this coefficient, for many distributions. This expression depends only on two parameters which are actually functions of the parameters of the selected models, among which we can mention the exponential, Weibull, Gumbel, Fréchet and some other distributions that arise under certain transformations of an exponential random variable. The simplicity of our unifying proposal is illustrated considering three well known distributions that have been individually analyzed in statistical literature. To illustrate the performance of the likelihood confidence intervals obtained for this overlapping coefficient, under our proposal, we carried out some simulation studies that yielded adequate coverage frequencies, and just for the sake of comparison we also computed Bootstrap confidence intervals. A real data set is analyzed to exemplify our proposal.     

线性常微分方程参数稳定性的统计推断

对具有随机误差的观测数据, 讨论了常系数线性常微分方程参数稳定性的统计推断问题. 通过残差项的Karhunen-Loeve 分解, 给出了变点检验步骤及其在原假设下的极限分布. 在对立假设下定义了变点的估计, 证明了检验以及估计的一致性. 对常系数二阶常微分方程进行了统计模拟, 结果表明原假设下的极限分布是对真实分布非常好的近似; 对立假设下, 即使输入函数的频率存在0.75% 的变化, 上述检验也能以大概率拒绝原假设. 最后利用上述方法研究了英国中部地区的气温数据, 揭示了数据一些新的特点.     

Statistical inference for Vasicek-type model driven by Hermite processes

Let $Z$ denote a Hermite process of order $q\ge 1$ and self-similarity parameter $H\in (\frac{1}{2},1)$. This process is $H$-self-similar, has stationary increments and exhibits long-range dependence. When $q=1$, it corresponds to the fractional Brownian motion, whereas it is not Gaussian as soon as $q?2$. In this paper, we deal with a Vasicek-type model driven by $Z$, of the form $d{X}_t=a(b?X_t)dt+d{Z}_t$. Here, $a>0$ and $b\in \mathbb{R}$ are considered as unknown drift parameters. We provide estimators for $a$ and $b$ based on continuous-time observations. For all possible values of $H$ and $q$, we prove strong consistency and we analyze the asymptotic fluctuations.     

Statistical inference for a certain class of bivariate distributions

The paper deals with statistical inference for a certain class of bivariate distributions. The class of marginal distributions is given and is shown to include distributions with only location and scale parameters. A normalizing transformation is applied to the marginal distributions and the parameters are estimated by maximum likelihood. For this class there is a great deal of simplification in the calculations for the asymptotic covariance matrix of the vector of parameter estimators. Statistics for tests of zero correlation are discussed. Also, the analysis is carried out for exponential marginal distributions.