Authors:Moosup Kim; Sangyeol Lee Pages: 945 - 968 Abstract: Abstract In this study, we consider the problem of estimating the tail exponent of multivariate regular variation. Since any convex combination of a random vector with a multivariate regularly varying tail has a univariate regularly varying tail with the same exponent under certain conditions, to estimate the tail exponent of the multivariate regular variation of a given random vector, we employ a weighted average of Hill’s estimators obtained for all of its convex combinations, designed to reduce the variability of estimation. We investigate the asymptotic properties and evaluate the finite sample performance of the weighted average of Hill’s estimators. A simulation study and real data analysis are provided for illustration. PubDate: 2017-10-01 DOI: 10.1007/s10463-016-0574-9 Issue No:Vol. 69, No. 5 (2017)
Authors:L. Baringhaus; B. Ebner; N. Henze Pages: 969 - 995 Abstract: Abstract We present a general result on the limit distribution of weighted one- and two-sample \(L^2\) -goodness-of-fit test statistics of some hypothesis \(H_0\) under fixed alternatives. Applications include an approximation of the power function of such tests, asymptotic confidence intervals of the distance of an underlying distribution with respect to the distributions under \(H_0\) , and an asymptotic equivalence test that is able to validate certain neighborhoods of \(H_0\) . PubDate: 2017-10-01 DOI: 10.1007/s10463-016-0567-8 Issue No:Vol. 69, No. 5 (2017)
Authors:Sunghoon Kwon; Jeongyoun Ahn; Woncheol Jang; Sangin Lee; Yongdai Kim Pages: 997 - 1025 Abstract: Abstract We propose a new penalty called the doubly sparse (DS) penalty for variable selection in high-dimensional linear regression models when the covariates are naturally grouped. An advantage of the DS penalty over other penalties is that it provides a clear way of controlling sparsity between and within groups, separately. We prove that there exists a unique global minimizer of the DS penalized sum of squares of residuals and show how the DS penalty selects groups and variables within selected groups, even when the number of groups exceeds the sample size. An efficient optimization algorithm is introduced also. Results from simulation studies and real data analysis show that the DS penalty outperforms other existing penalties with finite samples. PubDate: 2017-10-01 DOI: 10.1007/s10463-016-0571-z Issue No:Vol. 69, No. 5 (2017)
Authors:Omer Ozturk Pages: 1029 - 1057 Abstract: Abstract This article develops estimators for certain population characteristics using a judgment post stratified (JPS) sample. The paper first constructs a conditional JPS sample with a reduced set size K by conditioning on the ranks of the measured observations of the original JPS sample of set size \(H \ge K\) . The paper shows that the estimators of the population mean, median and distribution function based on this conditional JPS sample are consistent and have limiting normal distributions. It is shown that the proposed estimators, unlike the ratio and regression estimators, where they require a strong linearity assumption, only need a monotonic relationship between the response and auxiliary variable. For moderate sample sizes, the paper provides a bootstrap distribution to draw statistical inference. A small-scale simulation study shows that the proposed estimators based on a reduced set JPS sample perform better than the corresponding estimators based on a regular JPS sample. PubDate: 2017-10-01 DOI: 10.1007/s10463-016-0572-y Issue No:Vol. 69, No. 5 (2017)
Authors:Hanfang Yang; Yichuan Zhao Pages: 1059 - 1073 Abstract: Abstract In this paper, we propose a smoothed estimating equation for the difference of quantiles with two samples. Using the jackknife pseudo-sample technique for the estimating equation, we propose the jackknife empirical likelihood (JEL) ratio and establish the Wilk’s theorem. Due to avoiding estimating link variables, the simulation studies demonstrate that JEL method has computational efficiency compared with traditional normal approximation method. We carry out a simulation study in terms of coverage probability and average length of the proposed confidence intervals. A real data set is used to illustrate the JEL procedure. PubDate: 2017-10-01 DOI: 10.1007/s10463-016-0576-7 Issue No:Vol. 69, No. 5 (2017)
Authors:Nicolas Grosjean; Thierry Huillet Pages: 1075 - 1097 Abstract: Abstract We derive some additional results on the Bienyamé–Galton–Watson-branching process with \(\theta \) -linear fractional branching mechanism, as studied by Sagitov and Lindo (Branching Processes and Their Applications. Lecture Notes in Statistics—Proceedings, 2016). This includes the explicit expression of the limit laws in both the subcritical cases and the supercritical cases with finite mean, and the long-run behavior of the population size in the critical case, limits laws in the supercritical cases with infinite mean when the \(\theta \) process is either regular or explosive, and results regarding the time to absorption, an expression of the probability law of the \(\theta \) -branching mechanism involving Bell polynomials, and the explicit computation of the stochastic transition matrix of the \(\theta \) process, together with its powers. PubDate: 2017-10-01 DOI: 10.1007/s10463-016-0573-x Issue No:Vol. 69, No. 5 (2017)
Authors:Minggen Lu Pages: 1099 - 1127 Abstract: Abstract We consider a simple yet flexible spline estimation method for quasi-likelihood models. We approximate the unknown function by B-splines and apply the Fisher scoring algorithm to compute the estimates. The spline estimate of the nonparametric component achieves the optimal rate of convergence under the smooth condition, and the estimate of the parametric part is shown to be asymptotically normal even if the variance function is misspecified. The semiparametric efficiency of the model can be established if the variance function is correctly specified. A direct and consistent variance estimation method based on the least-squares estimation is proposed. A simulation study is performed to evaluate the numerical performance of the spline estimate. The methodology is illustrated on a crab study. PubDate: 2017-10-01 DOI: 10.1007/s10463-016-0575-8 Issue No:Vol. 69, No. 5 (2017)
Authors:James C. Fu; Wan-Chen Lee Pages: 1129 - 1139 Abstract: Abstract Suppose an urn contains m distinct coupons, labeled from 1 to m. A random sample of k coupons is drawn without replacement from the urn, numbers are recorded and the coupons are then returned to the urn. This procedure is done repeatedly and the sample sizes are independently identically distributed. Let W be the total number of random samples needed to see all coupons at least l times \((l \ge 1)\) . Recently, for \(l=1\) , the approximation for the first moment of the random variable W has been studied under random sample size sampling scheme by Sellke (Ann Appl Probab, 5:294–309, 1995). In this manuscript, we focus on studying the exact distributions of waiting times W for both fixed and random sample size sampling schemes given \(l \ge 1\) . The results are further extended to a combination of fixed and random sample size sampling procedures. PubDate: 2017-10-01 DOI: 10.1007/s10463-016-0578-5 Issue No:Vol. 69, No. 5 (2017)
Authors:Yusuke Shimizu Pages: 1141 - 1154 Abstract: Abstract In this paper, we study the uniform tail-probability estimates of a regularized least-squares estimator for the linear regression model. We make use of the polynomial type large deviation inequality for the associated statistical random fields, which may not be locally asymptotically quadratic. Our results enable us to verify various arguments requiring convergence of moments of estimator-dependent statistics, such as the mean squared prediction error and the bias correction for AIC-type information criterion. PubDate: 2017-10-01 DOI: 10.1007/s10463-016-0577-6 Issue No:Vol. 69, No. 5 (2017)
Authors:P. Vellaisamy Pages: 1155 - 1176 Abstract: Abstract Collapsibility deals with the conditions under which a conditional (on a covariate W) measure of association between two random variables Y and X equals the marginal measure of association. In this paper, we discuss the average collapsibility of certain well-known measures of association, and also with respect to a new measure of association. The concept of average collapsibility is more general than collapsibility, and requires that the conditional average of an association measure equals the corresponding marginal measure. Sufficient conditions for the average collapsibility of the association measures are obtained. Some interesting counterexamples are constructed and applications to linear, Poisson, logistic and negative binomial regression models are discussed. An extension to the case of multivariate covariate W is also analyzed. Finally, we discuss the collapsibility conditions of some dependence measures for survival models and illustrate them for the case of linear transformation models. PubDate: 2017-10-01 DOI: 10.1007/s10463-016-0580-y Issue No:Vol. 69, No. 5 (2017)
Authors:Darinka Dentcheva; Spiridon Penev; Andrzej Ruszczyński Pages: 737 - 760 Abstract: Abstract We address the statistical estimation of composite functionals which may be nonlinear in the probability measure. Our study is motivated by the need to estimate coherent measures of risk, which become increasingly popular in finance, insurance, and other areas associated with optimization under uncertainty and risk. We establish central limit theorems for composite risk functionals. Furthermore, we discuss the asymptotic behavior of optimization problems whose objectives are composite risk functionals and we establish a central limit formula of their optimal values when an estimator of the risk functional is used. While the mathematical structures accommodate commonly used coherent measures of risk, they have more general character, which may be of independent interest. PubDate: 2017-08-01 DOI: 10.1007/s10463-016-0559-8 Issue No:Vol. 69, No. 4 (2017)
Authors:Diane Donovan; Benjamin Haaland; David J. Nott Pages: 865 - 878 Abstract: Abstract Sliced Sudoku-based space-filling designs and, more generally, quasi-sliced orthogonal array-based space-filling designs are useful experimental designs in several contexts, including computer experiments with categorical in addition to quantitative inputs and cross-validation. Here, we provide a straightforward construction of doubly orthogonal quasi-Sudoku Latin squares which can be used to generate quasi-sliced orthogonal arrays and, in turn, sliced space-filling designs which achieve uniformity in one- and two-dimensional projections for the full design and uniformity in two-dimensional projections for each slice. These constructions are very practical to implement and yield a spectrum of design sizes and numbers of factors not currently broadly available. PubDate: 2017-08-01 DOI: 10.1007/s10463-016-0565-x Issue No:Vol. 69, No. 4 (2017)
Authors:Shaogao Lv; Xin He; Junhui Wang Pages: 897 - 923 Abstract: Abstract This paper focuses on the high-dimensional additive quantile model, allowing for both dimension and sparsity to increase with sample size. We propose a new sparsity-smoothness penalty over a reproducing kernel Hilbert space (RKHS), which includes linear function and spline-based nonlinear function as special cases. The combination of sparsity and smoothness is crucial for the asymptotic theory as well as the computational efficiency. Oracle inequalities on excess risk of the proposed method are established under weaker conditions than most existing results. Furthermore, we develop a majorize-minimization forward splitting iterative algorithm (MMFIA) for efficient computation and investigate its numerical convergence properties. Numerical experiments are conducted on the simulated and real data examples, which support the effectiveness of the proposed method. PubDate: 2017-08-01 DOI: 10.1007/s10463-016-0566-9 Issue No:Vol. 69, No. 4 (2017)
Authors:Huybrechts F. Bindele; Ash Abebe; Karlene N. Meyer Abstract: Abstract This study considers rank estimation of the regression coefficients of the single index regression model. Conditions needed for the consistency and asymptotic normality of the proposed estimator are established. Monte Carlo simulation experiments demonstrate the robustness and efficiency of the proposed estimator compared to the semiparametric least squares estimator. A real-life example illustrates that the rank regression procedure effectively corrects model nonlinearity even in the presence of outliers in the response space. PubDate: 2017-09-20 DOI: 10.1007/s10463-017-0618-9
Authors:Sam Efromovich; Jufen Chu Abstract: Abstract Left truncation and right censoring (LTRC) presents a unique challenge for nonparametric estimation of the hazard rate of a continuous lifetime because consistent estimation over the support of the lifetime is impossible. To understand the problem and make practical recommendations, the paper explores how the LTRC affects a minimal (called sharp) constant of a minimax MISE convergence over a fixed interval. The corresponding theory of sharp minimax estimation of the hazard rate is presented, and it shows how right censoring, left truncation and interval of estimation affect the MISE. Obtained results are also new for classical cases of censoring or truncation and some even for the case of direct observations of the lifetime of interest. The theory allows us to propose a relatively simple data-driven estimator for small samples as well as the methodology of choosing an interval of estimation. The estimation methodology is tested numerically and on real data. PubDate: 2017-09-20 DOI: 10.1007/s10463-017-0617-x
Authors:Jun Zhang; Zhenghui Feng; Xiaoguang Wang Abstract: Abstract Comparison of two-sample heteroscedastic single-index models, where both the scale and location functions are modeled as single-index models, is studied in this paper. We propose a test for checking the equality of single-index parameters when dimensions of covariates of the two samples are equal. Further, we propose two test statistics based on Kolmogorov–Smirnov and Cramér–von Mises type functionals. These statistics evaluate the difference of the empirical residual processes to test the equality of mean functions of two single-index models. Asymptotic distributions of estimators and test statistics are derived. The Kolmogorov–Smirnov and Cramér–von Mises test statistics can detect local alternatives that converge to the null hypothesis at a parametric convergence rate. To calculate the critical values of Kolmogorov–Smirnov and Cramér–von Mises test statistics, a bootstrap procedure is proposed. Simulation studies and an empirical study demonstrate the performance of the proposed procedures. PubDate: 2017-09-13 DOI: 10.1007/s10463-017-0616-y
Authors:David Kahle; Ruriko Yoshida; Luis Garcia-Puente Abstract: Abstract Exact conditional goodness-of-fit tests for discrete exponential family models can be conducted via Monte Carlo estimation of p values by sampling from the conditional distribution of multiway contingency tables. The two most popular methods for such sampling are Markov chain Monte Carlo (MCMC) and sequential importance sampling (SIS). In this work we consider various ways to hybridize the two schemes and propose one standout strategy as a good general purpose method for conducting inference. The proposed method runs many parallel chains initialized at SIS samples across the fiber. When a Markov basis is unavailable, the proposed scheme uses a lattice basis with intermittent SIS proposals to guarantee irreducibility and asymptotic unbiasedness. The scheme alleviates many of the challenges faced by the MCMC and SIS schemes individually while largely retaining their strengths. It also provides diagnostics that guide and lend credibility to the procedure. Simulations demonstrate the viability of the approach. PubDate: 2017-09-04 DOI: 10.1007/s10463-017-0615-z
Authors:Yoshihiko Maesono; Taku Moriyama; Mengxin Lu Abstract: Abstract We propose new smoothed sign and Wilcoxon’s signed rank tests that are based on kernel estimators of the underlying distribution function of the data. We discuss the approximations of the p-values and asymptotic properties of these tests. The new smoothed tests are equivalent to the ordinary sign and Wilcoxon’s tests in the sense of Pitman’s asymptotic relative efficiency, and the differences between the ordinary and new tests converge to zero in probability. Under the null hypothesis, the main terms of the asymptotic expectations and variances of the tests do not depend on the underlying distribution. Although the smoothed tests are not distribution-free, making use of the specific kernel enables us to obtain the Edgeworth expansions, being free of the underlying distribution. PubDate: 2017-08-23 DOI: 10.1007/s10463-017-0614-0
Authors:Graciela Boente; Daniela Rodriguez; Mariela Sued Abstract: Abstract In many situations, when dealing with several populations, equality of the covariance operators is assumed. An important issue is to study whether this assumption holds before making other inferences. In this paper, we develop a test for comparing covariance operators of several functional data samples. The proposed test is based on the Hilbert–Schmidt norm of the difference between estimated covariance operators. In particular, when dealing with two populations, the test statistic is just the squared norm of the difference between the two covariance operators estimators. The asymptotic behaviour of the test statistic under both the null hypothesis and local alternatives is obtained. The computation of the quantiles of the null asymptotic distribution is not feasible in practice. To overcome this problem, a bootstrap procedure is considered. The performance of the test statistic for small sample sizes is illustrated through a Monte Carlo study and on a real data set. PubDate: 2017-08-20 DOI: 10.1007/s10463-017-0613-1
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